OBSERVATIONS AND MODELS OF INFRARED DEBRIS DISK SIGNATURES AND THEIR EVOLUTION by

OBSERVATIONS AND MODELS OF INFRARED DEBRIS DISK SIGNATURES AND THEIR EVOLUTION by
OBSERVATIONS AND MODELS OF INFRARED DEBRIS DISK SIGNATURES
AND THEIR EVOLUTION
by
András Gáspár
A Dissertation Submitted to the Faculty of the
DEPARTMENT OF ASTRONOMY
In Partial Fulfillment of the Requirements
For the Degree of
DOCTOR OF PHILOSOPHY
In the Graduate College
THE UNIVERSITY OF ARIZONA
2011
2
T HE U NIVERSITY OF A RIZONA
G RADUATE C OLLEGE
As members of the Dissertation Committee, we certify that we have read the
dissertation prepared by András Gáspár entitled “Observations and models of
infrared debris disk signatures and their evolution” and recommend that it be
accepted as fulfilling the dissertation requirement for the Degree of Doctor of
Philosophy.
Date: 3 November 2011
Dr. George Rieke
Date: 3 November 2011
Dr. Dimitrios Psaltis
Date: 3 November 2011
Dr. Feryal Özel
Date: 3 November 2011
Dr. Kate Y. L. Su
Date: 3 November 2011
Dr. Dániel Apai
Final approval and acceptance of this dissertation is contingent upon the candidate’s submission of the final copies of the dissertation to the Graduate College.
I hereby certify that I have read this dissertation prepared under my direction
and recommend that it be accepted as fulfilling the dissertation requirement.
Date: 3 November 2011
Dissertation Director: Dr. George Rieke
3
S TATEMENT B Y A UTHOR
This dissertation has been submitted in partial fulfillment of requirements for
an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.
Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for
permission for extended quotation from or reproduction of this manuscript in
whole or in part may be granted by the head of the major department or the
Dean of the Graduate College when in his or her judgment the proposed use of
the material is in the interests of scholarship. In all other instances, however,
permission must be obtained from the author.
S IGNED: András Gáspár
4
A CKNOWLEDGMENTS
The path one walks down to reach the stage in life where they can actually
sit down to write an acknowledgment for their doctoral thesis is extremely long,
full of curves and obstacles. Along the way, I have met so many people that have
influenced, guided, and helped me that listing them all is mere impossible and
would over-tower the thesis itself.
The credit for teaching me how to write properly goes out to my Mother, who
put a lot of effort into teaching me about languages and literature. Obviously,
the biggest influence was that of my Father, who being a physical-chemist himself, always bestowed the logic of a scientist on me and allowed me to pursuit
and nurtured all of the interests I developed over the years. My elementary and
high school physics teachers, Mihály Kotormán and Dr. Ervin Szegedi, were also
invaluable in opening my eyes to the wonderful world of physics.
I have great deal of appreciation for all my previous advisors who took upon
the task of advising me. If it were not for Dr. László L. Kiss, I would definitely not
have started doing research as early as I have and maybe not even in astronomy.
He taught me all the initial tools an astronomer needs to know and that I still
use today. Dr. Zoltán Balog was not only a great advisor, but also a friend who I
could and still can count on. It was absolutely his influence that I started working
in infrared astronomy and that I applied to the University of Arizona later on. I
am also thankful for Dr. Melvin G. Hoare for giving me an opportunity to work
on the Science Verification phase of the UKIDSS project in Leeds, in the summer
of 2005. I also thank Kate Su for the help she gave with my first two papers at
Steward. I am grateful to Dr. Dimitrios Psaltis and Dr. Feryal Özel for teaching
me how to correctly and professionally approach a numerical coding project and
for devoting all the time they did to the final part of my thesis. Without their help
this thesis would have turned out differently.
I do have many people to thank and acknowledge. One special person however, my advisor Dr. George Rieke, does stands out from the crowd. He has,
next to his vast amount of duties, somehow always found time to advise me. He
always gave just the right amount of a push to get me going and always took
interest in my personal problems. He is someone who genuinely cares how his
students are doing. He always has and always will inspire me.
Last, but not least, I thank all my friends who have been part of my life and
supported me over the years. Zoltán Makai, Suresh Sivanandam, Dennis Just,
Alan Cooney and many others, thank you for all the fun times! And thank you,
my best friend, my dear wife, Bori, for everything, for your support, your understanding and all the joy I have gotten over the years. You make all this work and
invested time have a meaning. Thank you.
5
D EDICATION
I dedicate this thesis to my past, my present and my future.
My past being my grandparents, who have all taught me so many valuable
lessons in life and how to be a good person. They will always be an example to
follow.
My present being my mother, Irén, my father, Vilmos, my sister, Judit and
my wife, Bori. You have been with me throughout the struggles and the joyous
moments. You have encouraged me and given me goals to reach.
And my future being our little baby, whom I don’t even know yet, but cannot
wait to see in February. You and Mommy are the single reason this is all worth
doing.
6
TABLE
OF
C ONTENTS
L IST
OF
F IGURES
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L IST
OF
TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
A BSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
C HAPTER 1 I NTRODUCTION . . . . .
1.1 Motivation . . . . . . . . . . .
1.2 Extrasolar planets . . . . . . .
1.3 Star and planet formation . .
1.4 Circumstellar debris disks . .
1.5 The solar system’s debris disk
1.6 Outline of the thesis . . . . . .
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C HAPTER 2 FALSE SIGNS OF DEBRIS DISKS . . . . . . . . . . . . . . . . . . 29
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 Observations and Data Reduction . . . . . . . . . . . . . . . . . . . 32
2.3 The Bow Shock Model . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3.1 Physical description of the model . . . . . . . . . . . . . . . 38
2.3.2 Model Geometry and Parameters . . . . . . . . . . . . . . . . 40
2.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.4.1 Bow Shock Model Results . . . . . . . . . . . . . . . . . . . . 50
2.4.2 ISM Interactions . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.4.3 Implications for Diffusion/Accretion Model of λ Boötis Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
C HAPTER 3 D EBRIS DISK STUDY OF P RAESEPE . . . . . . . . . . . . . .
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Observations, data reduction, and photometry . . . . . . . . . .
3.3 Catalog surveys and the final sample . . . . . . . . . . . . . . . .
3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Color-color selection . . . . . . . . . . . . . . . . . . . . .
3.4.2 The SED fit selection . . . . . . . . . . . . . . . . . . . . .
3.4.3 Praesepe white dwarfs . . . . . . . . . . . . . . . . . . . .
3.4.4 Debris Disk Candidates . . . . . . . . . . . . . . . . . . .
3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Calulating errors on debris disk fractions . . . . . . . . .
3.5.2 The decay of the debris disk fraction in early-type stars .
3.5.3 The decay of the debris disk fraction for solar-type stars
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7
TABLE OF C ONTENTS — Continued
3.5.4 Evolutionary differences between the debris disks around
early- and solar-type stars . . . . . . . . . . . . . . . . . . . .
3.5.5 The results in context with the Late Heavy Bombardment .
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C HAPTER 4 M ODELING C OLLISIONAL C ASCADES I N D EBRIS D ISKS . .
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 The physical and numerical challenges of modeling debris disks
4.2.1 Collisional outcomes . . . . . . . . . . . . . . . . . . . . .
4.2.2 Incorporating the complete redistribution integral . . . .
4.2.3 The effect of radiation forces . . . . . . . . . . . . . . . .
4.3 The collisional model . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 The evolution equation . . . . . . . . . . . . . . . . . . .
4.3.2 The collisional term . . . . . . . . . . . . . . . . . . . . .
4.3.3 Collision outcomes . . . . . . . . . . . . . . . . . . . . . .
4.3.4 The initial distribution and fiducial parameters . . . . .
4.4 Simplified Dynamics . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Collisional velocities . . . . . . . . . . . . . . . . . . . . .
4.4.2 Reduced collisional probabilities of β critical particles . .
4.4.3 Reduced collisional probabilities of the largest particles .
4.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 Comparison to Thébault et al. (2003) . . . . . . . . . . . .
4.5.2 Comparison to Löhne et al. (2008) and Wyatt et al. (2011)
4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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C HAPTER 5 S TEEP D UST-S IZE D ISTRIBUTIONS . . . . . . . . . . . . . . . .
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Numerical modeling . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Evolution of the reference model . . . . . . . . . . . . . . . .
5.2.2 The dependence of the steady-state distribution function on
the collision parameters . . . . . . . . . . . . . . . . . . . . .
5.2.3 The dependence of the steady-state distribution function on
system variables . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.4 The dependence of the steady-state distribution function on
numerical parameters . . . . . . . . . . . . . . . . . . . . . .
5.2.5 The time to reach steady-state . . . . . . . . . . . . . . . . . .
5.2.6 The robustness of the solution . . . . . . . . . . . . . . . . .
5.3 Synthetic Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Relation between the particle mass distribution and the SED . . . .
5.5 Comparison to observations . . . . . . . . . . . . . . . . . . . . . . .
5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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8
TABLE
OF
C ONTENTS — Continued
C HAPTER 6 T HESIS C ONCLUSIONS . . . . . . . . . . . . . . . .
6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Evolution of the particle size distribution slope .
6.2.2 Warm debris disk models . . . . . . . . . . . . .
6.2.3 Stochastic events and debris disk evolution . . .
6.2.4 Debris disk haloes . . . . . . . . . . . . . . . . .
A PPENDIX A A GE
AND DISTANCE ESTIMATE OF
A PPENDIX B S TRENGTH
A PPENDIX C M ASS
CURVES
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P RAESEPE . . . . . . . . . 180
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CONSERVATION OF THE COLLISIONAL MODEL
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A PPENDIX D N UMERICAL EVALUATION OF THE COLLISIONAL MODEL AND
VERIFICATION TESTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
D.1 Taylor series expansion of TI . . . . . . . . . . . . . . . . . . . . . . . 193
D.2 Verification of the numerical precision of TI . . . . . . . . . . . . . . 197
D.3 Numerical evaluation of TII . . . . . . . . . . . . . . . . . . . . . . . 199
D.4 Convergence tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
D.5 The ODE solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
R EFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
9
L IST
1.1
OF
F IGURES
1.2
1.3
1.4
1.5
The distribution of known extrasolar planets in the mass/semimajor axis phase space . . . . . . . . . . . . . . . . . . . . . . . . . .
The early evolutionary stages of planets and planetary systems . .
The decay of planetary circumstellar primordial disks . . . . . . . .
The Fomalhaut debris disk . . . . . . . . . . . . . . . . . . . . . . . .
The β Pic debris disk . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
The 24 µm images of δ Velorum . . . . . . . . . . . . .
The 70 µm images of δ Velorum . . . . . . . . . . . . .
Nomenclature of angles . . . . . . . . . . . . . . . . .
Morphology of the bow shoch at δ Velorum. . . . . . .
Constrainment of the model parameters . . . . . . . .
The avoidance distance as a function of grain size . .
Contraining the dynamical parameters of interaction
Determining the density of the ISM . . . . . . . . . . .
Model images of δ Velorum . . . . . . . . . . . . . . .
Calculated temperature distribution of the bow shock
Model subtracted images . . . . . . . . . . . . . . . . .
Spectral energy distribution of the model . . . . . . .
λ Boötis spectral comparison . . . . . . . . . . . . . .
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3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
Praesepe Observed field . . . . . . . . . . . . . . .
The 24 µm photometry errors of Praespe . . . . . .
Spatial distribution and CMD position of sources .
Completeness limit of the 24 µm sample . . . . . .
Determining the KS -24 zero point . . . . . . . . . .
The color-color plot of Praesepe . . . . . . . . . . .
Images for the debris disk candidate stars . . . . .
SED fits of excess candidate sources . . . . . . . .
Decay of excess . . . . . . . . . . . . . . . . . . . .
Excess decay in binned data set . . . . . . . . . . .
Probability of LHB like event . . . . . . . . . . . .
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4.1
4.2
4.3
4.4
4.5
4.6
4.7
Illustration of the possible outcome scenarios of collisions . . . . . 108
Values for the radiation-force parameter β around stars . . . . . . . 115
Radiation force blowout vs. Poynting-Robertson drag timescales . . 117
The outcome possibilities as a function of colliding masses . . . . . 121
Collision probabilities as a function of particle mass and eccentricity 130
Distribution evolution compared to Thébault et al. (2003) . . . . . . 135
Mass evolution compared to Thébault et al. (2003) . . . . . . . . . . 136
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10
L IST OF F IGURES — Continued
4.8 Distribution evolution compared to Löhne et al. (2008) and Wyatt
et al. (2011) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.9 Mass evolution compared to Löhne et al. (2008) and Wyatt et al.
(2011) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.10 Binned mass evolution compared to Löhne et al. (2008) . . . . . . .
4.11 The effects of varying collisional weights in the Löhne et al. (2008)
runs on the distribution evolution . . . . . . . . . . . . . . . . . . .
4.12 The effects of varying collisional weights in the Löhne et al. (2008)
runs on the mass evolution . . . . . . . . . . . . . . . . . . . . . . .
4.13 Evolution of the particle size distribution compared to the Löhne
et al. (2008) and Wyatt et al. (2011) models. . . . . . . . . . . . . . .
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
Evolution of the reference model . . . . . . . . . . .
The model’s dependence on η0 and Mtot . . . . . . .
The model’s dependence on α and b . . . . . . . . .
The model’s dependence on Qsc , S and s . . . . . . .
The model’s dependence on R . . . . . . . . . . . . .
The model’s dependence on δ . . . . . . . . . . . . .
The compiled effects of the variables . . . . . . . . .
Silicate absorption efficiencies . . . . . . . . . . . . .
Synthetic SEDs . . . . . . . . . . . . . . . . . . . . . .
Observed SEDs in the submillimeter and millimeter
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138
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168
A.1 Isochrone fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
A.2 Isochrone fitting 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
A.3 Best fitting isochrone of Praesepe . . . . . . . . . . . . . . . . . . . . 187
The values of G(m, m′ ) as a function of the colliding masses . . . .
The largest X fragment produced by collisions . . . . . . . . . . . .
The error in the integration of TI . . . . . . . . . . . . . . . . . . . .
Iso-size contours for the produced Y fragments as a function of the
colliding body sizes and interaction velocities . . . . . . . . . . . . .
D.5 Description of the integration method used for TII . . . . . . . . . .
D.6 Convergence test results of our code . . . . . . . . . . . . . . . . . .
D.7 Numerical accuracy and speed of our ODE solver . . . . . . . . . .
D.1
D.2
D.3
D.4
194
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11
L IST
OF
TABLES
2.1
The parameters of δ Velorum . . . . . . . . . . . . . . . . . . . . . .
35
3.1
3.2
3.3
3.4
3.5
3.6
3.7
Photometry of Praesepe members in the [24] band . . . . . . . . . .
The probabilities of chance alignments for excess sources . . . . . .
The field star sample excess ratios for early-type stars . . . . . . . .
The cluster star sample excess ratios for early-type stars . . . . . . .
The field star sample excess ratios for solar-type stars . . . . . . . .
The cluster star sample excess ratios for solar-type stars . . . . . . .
The percent of debris disks in rebinned distributions, as a function
of spectral-types. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
79
87
88
90
91
94
4.1
4.2
The variables used in the model and their fiducial values . . . . . . 126
Parameters used for comparison models . . . . . . . . . . . . . . . . 132
5.1
5.2
The variables used in the reference model . . . . . . . . . . . . . . . 149
Observational data of debris disks . . . . . . . . . . . . . . . . . . . 166
A.1 The distance-modulus of Praesepe in the literature. . . . . . . . . . 181
A.2 The solutions for the fitting of isochrones . . . . . . . . . . . . . . . 184
12
A BSTRACT
In my thesis I investigate the occurrence of mid-infrared excess around stars
and their evolution. Since the launch of the first infrared satellite, IRAS, we have
known that a large fraction of stars exhibit significant levels of infrared emission above their predicted photospheric level. Resolved optical and infrared images have revealed the majority of these excesses to arise from circumstellar disk
structures, made up of distributions of planetesimals, rocks, and dust. These
structures are descriptively called debris disks.
The first part of my thesis analyzes the Spitzer Space Telescope Observations of δ
Velorum. The 24 µm Spitzer images revealed a bow shock structure in front of the
star. My analysis showed that this is a result of the star’s high speed interaction
with the surrounding interstellar medium. We place this observation and model
in context of debris disk detections and the origin of λ Boötis stars.
The second part of my thesis summarizes our observational results on the
open cluster Praesepe. Using 24 µm data, I investigated the fraction of stars with
mid-infrared excess, likely to have debris disks. I also assembled all results from
previous debris disk studies and followed the evolution of the fraction of stars
with debris disks.
The majority of debris disks systems are evolved, few hundred million or a
Gyr old. Since the dissipation timescale for the emitting dust particles is less
than the age of these systems, they have to be constantly replenished through
collisional grinding of the larger bodies. The last two chapters of my thesis is
a theoretical analysis of the collisional cascade in debris disks, the process that
produces the constant level of dust particles detected. I introduce a numerical
model that takes into account all types of destructive collisions in the systems
13
and solves the full scattering equation. I show results of comparisons between
my and other published models and extensive verification tests of my model. I
also analyze the evolution of the particle size distribution as a function of the
variables in my model and show that the model itself is quite robust against most
variations.
14
C HAPTER 1
I NTRODUCTION
1.1 Motivation
The genuine curiosity of mankind to understand the universe we live in has always been the driving force for our cognitive and technological evolution. We
want to understand how the world around us formed and how we in it came to
be. From the very early Greek philosophers, such as Epicurus (”There are infinite
worlds both like and unlike this world of ours”), to modern luminaries, such as Kant,
all intellectual thinkers have wondered about the possibility of life occurring elsewhere in this Universe. Our perspective of our place in the Universe has been
constantly reevaluated over the centuries, from Geocentric through Heliocentric
to a non-specifiable location, diminishing our special centrally located position.
We now understand that the single thing that makes Earth, the solar system and
the Milky Way special, is that we are in it. Our Sun is just one of ∼ 1 × 1023 other
stars in the world, making us absolutely insignificant and the possibility of other
life forms existing not insignificant. The quest to know whether we are alone
and to map the extent of our world is an ancient driving force that has led to the
discovery of the New World over five hundred years ago and to humankind’s
greatest technical achievement so far, the Moon landings.
We have proposed the existence of life even within the solar system already,
under the ice shelves of Europa, in the old dried out water flows of Mars, on
the surface of Titan, and have been conducting endless endeavors to find it. We
have also been increasing ambitious in searching for habitable planets around
other stars, so far with slim results, to keep the idea alive that other intelligent
life forms can also exist. We have found over 700 planets orbiting other stars (or
15
extrasolar planets), and their numbers keep rising. However, majority of them are
gas giants in close orbits, some in further orbits, and only a very few of them are
Super Earths. We are yet to detect an Earth mass planet orbiting in the habitable
zone of a late spectral type star, the types of stars that live long enough to support
the evolution of an intelligent life form on one of their planets. Human kind has
embarked on one of its likely endless and final quests, to find another habitable
planet and possibly intelligent life somewhere else in the Universe.
1.2 Extrasolar planets
When Galileo Galilei turned his spyglass to Jupiter, he was stunned to see four
moons orbiting the planet. Almost 400 years later, Marois et al. (2008) imaged
three planets orbiting HR 8799 and Kalas et al. (2008) imaged a planet orbiting
Fomalhaut. We are finally able to image large planets on distant orbits around
other stars. The very first extrasolar planet was discovered almost two decades
ago (Wolszczan & Frail, 1992), orbiting the billion year old 6.2 ms pulsar 1257+12,
by calculating the timing variations of the pulsar. The most successful technique,
however, in detecting extrasolar planets, has been the radial velocity variation
method. The first extrasolar planet around a main sequence star was detected
with this technique (Mayor & Queloz, 1995), as well as over 500 of the almost 700
planets found to date. By nature, this technique is selective to large planets on
close orbits.
Another successful technique, which is selective towards low inclination systems, is the transit method. Almost 200 planets have been detected (not all discovered) by this technique and many more are to be added to this list once the
Kepler Space Telescope’s detections are confirmed (Borucki et al., 2010). The transiting method is in theory not as selective towards close orbits as the radial ve-
16
100
Planetary mass (MJupiter)
10
1
0.1
Radial Velocity
Transit
Imaged
Microlensing
Pulsar timing
0.01
Earth mass
0.001
0.01
0.1
1
10
Semi-major axis (AU)
100
1000
Figure 1.1 The distribution of known extrasolar planets in the mass vs. semimajor axis phase space. Our detections are mostly limited to large mass and close
orbit systems (data collected from http://exoplanet.eu).
locity method is, but for it to detect large orbit systems long baselines in time
are required, which can render the observations impossible. As with the radial
velocity method, it is biased towards larger planets. In Figure 1.1, I show the distribution of known extrasolar planets in the planetary mass vs. semi-major axis
phase space. The plot clearly shows that the majority of planets discovered to
date have large masses and are located in close orbits, meaning their systems are
very different from ours. This is most likely all due to observational biases and is
no surprise.
The direct detection of Earth size planets with our current technology is im-
17
possible (Woolf & Angel, 1998; Perryman, 2000). The resolutions of our ground
based telescopes are limited by the atmosphere of our planet by the Fried parameter (the maximum size of an atmospheric isothermal patch that introduces one
radian of wavefront aberration) to ∼ λ/r0 , equaling roughly 1” in the optical. At
higher altitudes the Fried parameter increases, enabling better resolutions, down
to even 0.6” in the infrared wavelengths. Adaptive Optics (AO) systems can improve the resolution down close to the theoretical limit of the telescope. For an 8
m telescope in the near infrared this can be as good as 60 milliarcseconds. This
corresponds to a radial distance of 1.6 AU for a star at 10 pc. With AO systems
and the James Webb Space Telescope (JWST), we are just at the limit to be able to
resolve the Earth-Sun distance for a nearby star, where we hope to have terrestrial
planets located. However, the diffraction halo of the stars will still be significant,
especially when one considers that planets do not emit light. Any light we can
detect from any planet will be scattered or reprocessed. The light from the star
will still outshine the light from any possible planets.
Our observations have widened our imaginations as to what types of extrasolar planetary systems can exist, and given rise to new questions as to their
origin and the frequency of planetary systems similar to ours. They have also
demanded new methods to be explored to detect other types of systems. The
high frequency of planetary systems shows that planets may be a necessary byproduct of star formation. To detect planets, we must understand how they form,
for which we have to understand how stars form. Below, I give a short review
of our current understanding of star and planet formation, which will lead us to
understand our extrasolar planetary system detection method and model.
18
1.3 Star and planet formation
The formation of stars and planets cannot be separated from each other. The
formation of planets is a necessary by-product of the formation of stars, so we
must understand the basic steps that lead to the formation of a star. In Figure
1.2, I show the stages of star formation, given by Shu et al. (1987), which starts
off with a clump of gas and dust in a molecular cloud. In the left panels of the
Figure, I show the observable spectral energy distributions of the system, while in
the right panels I show simple sketches of the system itself as it goes through the
formation phases. While in the pre-stellar phase, the pre-stellar core only emits in
the submillimeter and mm wavelengths. The protostellar object is formed when
the pre-stellar core settles in hydrostatic equilibrium, which is considered to be
the time of birth for a star (Palla & Stahler, 1993). It is now called a Class 0 object.
The core is surrounded by the infalling envelope of dust and gas, which start to
form an accretion disk around the central object, as a result of the conservation of
angular momentum.
As the protostellar object starts fusing deuterium, it also starts emitting first in
the near-IR and later in the optical wavelengths, while heating the circumstellar
disk, which starts to thermally emit in the mid- and far-IR wavelengths. At these
stages, the disk still has a significant amount of gas in it and can be easily detected
from its emission lines and near-to-far infrared excess continuum emission. This
is also known as the T Tauri phase. The stellar wind, driven by convection, breaks
out along the rotational axis of the pre-main sequence star, resulting in collimated
jets from the poles, while the system is accreting matter from the disk. The accretion from the disk onto the star is terminated via viscous spreading in the inner
(Hartmann et al., 1998) and via photoevaporation in the outer regions of the disk
(Hollenbach et al., 1994; Gorti & Hollenbach, 2009), with stellar wind also speed-
19
Figure 1.2 The early evolutionary stages of planets and planetary systems
(fig. from Dauphas & Chaussidon, 2011).
20
ing the process (Matsuyama et al., 2009). The disk is continuously hollowed out
from the inside, while external sources sometime also play a role via photoevaporation (Balog et al., 2007). This phase, called the transition phase, is defined
when the disk itself does not produce significant excess emission in the near-IR
wavelengths. The pre-main sequence phase of the star ends when it starts fusing
hydrogen, however, leftover material in the disk will still contain both gas and
dust that are in the process of forming planets. The excess emission of the disk is
now strongest in the mid- and far-IR wavelengths, as its near-IR excess fades.
As the gas in the protoplanetary disk is accreted onto the planetesimals outside of the ice-line, and into the central star and also blown out of the system
and photoevaporated from the inner regions of the system, the excess emission
in the near-IR fades. In Figure 1.3, I show the decay in the measured fraction of
stars with circumstellar protoplanetary disks in stellar clusters detected by their
near-IR excess as a function of time (figure from Wyatt, 2008). This was originally
observed by Haisch et al. (2001). It is in this early stage of stellar evolution when
planetesimals and planetary cores start to form.
The processes that build planets are more complicated than those that build
the stars. From observations, we see that the gas rich disk disappears within 10
Myr (Figure 1.3). Since the gas giant planets have to acquire the elements to build
them, it is certain that they are formed in the first 3 Myr of a star’s life. By nature gas giants are formed somewhat differently from terrestrial planets. We also
know that they keep the majority of the angular momentum in the planetary systems, so they have great influence on the dynamical evolution of their planetary
systems.
There are two different theories for the formation of giant planets. The first,
core accretion, starts off with the same physics that is used to build terrestrial
21
Figure 1.3 The decay of circumstellar primordial disks (Haisch et al., 2001; Wyatt,
2008; Williams & Cieza, 2011).
planets and later builds the gas giant on the core (D’Angelo et al., 2010). As dust
and small grains coagulate into larger particles, they settle in the midplane of
the protoplanetary disk. Here, they aggregate and form the larger planetesimals.
The exact method they do this by is extremely complicated and not that well understood. The so called ”meter size barrier” makes it hard to build planetesimals
larger than a meter in size, as the collision of particles on this scale is always de-
22
structive and bodies of this size are also heavily affected by gas drag and quickly
brought inward to the central star. However, if a particle is able to grow larger,
then it is able to accumulate enough material to first form a planetary embryo
and then a protoplanet. Once this protoplanet is large enough that the escape velocity from its surface exceeds the thermal speed of the surrounding gas, it starts
to accrete gas onto the solid core. Since the planet’s surface and the gas is constantly heated by the infall of planetesimals, the amount of gas it is able to accrete
is limited. With the ever increasing core mass, however, after a certain point the
core and gas envelope contract, allowing more gas to be accreted. This phase,
known as runaway gas accretion, is only limited by the amount of the surrounding gas. The second, disk instability, assumes a similar core contraction to that of
a star via the gravitational instability (GI) in the protoplanetary disk (Boss, 1997).
While the first method is a bottom-up initially slow process, GI is a top down an
initially rapid process.
The formation of the terrestrial size planets is still not a completely resolved
problem. Radioactive isotope measurements suggest that the terrestrial bodies
in the solar system took less than 200 Myr to form. Adding this to their average
age of 4.5 Gyr, we match the age of the Sun, based on stellar evolution models. The surfaces of the non-eroding terrestrial bodies, such as that of the Moon,
Mercury, and asteroids show never ending fields of impact craters. The ages of
these craters suggest that impacts were much more common in the early solar
system. Traditionally, there are two scenarios that have been considered for the
formation of these large rocky planets. Either they form via pairwise collisions
between dust grains, or rapidly via gravitational instability in the midplane of
the protoplanetary disk. Unfortunately the effects of turbulence causes difficulties for the models. For the planetesimal theory, the large collisional velocities
23
make it difficult to build up objects larger than a meter in size. Due to turbulence, the collisional velocities between large, meter size objects is high enough
to produce destructive collisions. Small particles on the other hand are too hard
to merge with them. This is called the ”meter-size barrier.” For the GI model,
turbulence does not allow the larger dust and meter size particles to settle in the
disk midplane, thus the system cannot achieve gravitational instability. Two alternate models have recently been suggested to overcome these issues. Cuzzi
et al. (2001) proposed that small particles are able to stay for longer periods of
time between turbulent eddies. The probability of forming high density clumps
is small, but they can become gravitationally bound. As the sub-Keplerian orbital
velocity of the gas in the disk places ram pressure on the clumps, there is likely a
minimum mass needed for the survival of these clumps, which models place at
the mass of a solid planetesimal with a radius of 10-100 km. The second alternate
theory (Johansen et al., 2006) assumes that the planetesimal theory has already
produced a large number of meter sized objects. These meter size objects are effected by gas drag and move to pressure maxima points with short timescales.
Once a large number of the meter size planetesimals are gathered in the pressure
maxima, they start dragging the gas along and the radial drift is slowed. The
resulting streaming instability produces gravitationally bound clumps of meter
size objects, which then contract to form larger planetesimals/asteroids.
With either model, kilometer-size planetesimals are built within a few million
years, with the planets possibly reaching Earth-size in 10-100 Myr. A smaller,
Mars-size, planet can be built within 10 Myr. The final stage in planetary system
formation is the removal of the leftover material. As the asteroid belt and Kuiper
belt in the solar system show, this isn’t an absolute process. Asteroids are able
to remain in rings and haloes in the systems, and when undergoing gravitational
24
Figure 1.4 The Fomalhaut debris disk and its planet Fomalhaut b (Kalas et al.,
2008).
instabilities they can once again take part in massive collisions with each other.
1.4 Circumstellar debris disks
Almost three decades ago, the serendipitous discovery of IR excess around Vega
during routine calibration measurements came as a big surprise (Aumann et al.,
1984). Similar excess was found around other main sequence stars, such as β Pictoris and Fomalhaut. Later, optical images showed an extended disk structure
around β Pictoris (Smith & Terrile, 1984). The SEDs and the optical images were
all consistent with originating from a distribution of dust particles heated to temperatures between 80-100 K orbiting at significant distances (50-200 AU) from
25
Figure 1.5 The β Pic debris disk and its planet β Pic b (Lagrange et al., 2010).
the stars. Later analysis of IRAS data revealed over a 100 systems with similar
excesses (Mannings & Barlow, 1998) around stars of all ages and spectral types.
These circumstellar disks are the third type of disk class, after primordial planetary and transitional disks, called debris disks. With the Spitzer Space Telescope
we have detected many hundreds of them, both in the field (Rieke et al., 2005; Su
et al., 2006; Trilling et al., 2008; Carpenter et al., 2008, 2009; Moór et al., 2006) and
in stellar clusters, and up to many hundreds of parsecs.
What differentiates debris disks from the first two classes of circumstellar
disks, is that they are completely deficient in gas. Since they are found around
stars of all ages, and because the dissipation time of dust is significantly shorter
than the ages of the systems (Gillett, 1986), they are not primordial interstel-
26
lar grains left over from the formation of the system, but produced from massive collisional cascades between the planetesimals. Since collisional cascades
are likely initiated via dynamical instabilities in a disk due to planetary motions/migrations, the existence of these disks hint at the existence of planets in
these systems. Inner clearings, such as that at Fomalhaut, are even stronger evidence for the case. The later, successfully imaged planets at the inner edges of
the debris disks at Fomahaut (Figure 1.4; Kalas et al., 2008) and β Pic (Figure 1.5;
Lagrange et al., 2010) have confirmed this model.
As the Kuiper belt traces the solar system’s dynamical history, so do the resolved systems of debris disks. To deduce the evolution of a debris disk, which
is one of our main goals, the age of the system has to be known. This is not an
easy task (Moór et al., 2006). We also have to understand the outcomes of planetesimal/asteroid collisions. In Chapter 3, I introduce my Spitzer Space Telescope
observations on the open cluster Praesepe and deduced trends for the decay in
debris disk fractions as a function of the central stars’ spectral type. In Chapter
4, I detail my numerical model that solves the collisional evolution of the particle
size distribution in debris disks. Knowing the particle size distribution lets us
model the spectral energy distribution of the system and to analyze its evolution.
More introductory details on debris disks can be found in the introductions of
Chapter 3 and 4.
1.5 The solar system’s debris disk
Our interest in extrasolar debris disks is rooted in our own solar system’s debris
disk. Astronomers have known for a while that our solar system has a ring of
dust, seen as the zodiacal cloud at sunset and dawn, produced by these belts,
comets and also composed of interstellar grains, constantly brought inward to
27
the Sun. However, unlike extrasolar debris disks, ours is low density, not observable from other stars. As it is low density, the collision rates are low, resulting in
low production rates of small dust particles that could be blown out of the system. Instead, the dominant removal mechanism for the larger grains in our solar
system is the Poynting-Robertson drag, which brings the dust grains inward to
the the Sun. Over the 4.5 Gyr of our solar system’s evolution, the majority of
dust and planetesimals were removed via radiation pressure forces and due to
dynamical interactions. According to lunar crater counts, the inner parts of the
solar system underwent an intense period of planetesimal collisions roughly 3.8
Gyr ago (Tera et al., 1973, 1974). This could have been a result of such a dynamical instability, possibly caused by dynamical interactions of Jupiter and Saturn
(Gomes et al., 2005). The importance of dynamical effects can be also seen today, with the resonant trapping of the asteroids in the Main Asteroid Belt in the
Kirkwood zones, the warp in the zodiacal cloud (Dermott et al., 1999), and the
asymmetric component that co-orbits with Earth (Dermott et al., 1994).
1.6 Outline of the thesis
In the past thirty years our understanding of how extrasolar planetary systems
form and evolve has expanded by a great amount. We have imaged all stages
of it, detected many hundred of planets, and built up physical models to explain
the vast majority of observations. However, our knowledge is not yet complete,
with gray and white areas to be filled, mostly in the field of understanding terrestrial planets and our own solar system. Observations and consequent models
of debris disks will help us fill in many of these areas. Data from the Spitzer Space
Telescope and Herschel has and will reveal to us many new details. The goal of my
thesis is to understand certain aspects of debris disk evolution via observations
28
and models.
In Chapter 2, I present observations of a nearby star, δ Velorum. Spitzer images revealed a stunning bow shock structure in front of the star. Originally the
mid-IR excess around the star was contributed to a debris disk. Although the
topic of this chapter does not tie in directly to the wider theme of the thesis, it
is meant to present alternate ways of producing debris disk like excesses around
stars and to emphasize the existence of these systems. I model the bow shock as a
result of the star’s interaction with the Local Interstellar Cloud (LIC) in the Local
Bubble. The mere presence of the bow shock is surprising, as the density of the
Interstellar Medium is considerably lower in the Local Bubble, compared to other
nearby regions in the Milky Way. In Chapter 3, I present Spitzer observations of
the nearby open stellar cluster Praesepe. It is a good example of an older cluster,
but with rich membership. It is also relatively nearby (∼ 180 pc), meaning the
cluster could be observed to high completeness even for later spectral type stars.
The cluster’s age coincides with the age of the solar system, when it underwent
the Late Heavy Bombardment. I compile all available cluster and field star disk
fraction in the paper and analyze the decay of the [24] disk fraction. In Chapter
4, I present a detailed numerical model of the collisional cascades in debris disks.
My numerical model solves the full scattering equation and takes into account
both destructive collision types, erosive and catastrophic. I compare the model to
already published models in the literature and detail its strengths. In Chapter 5, I
use my numerical model to calculate the evolution of the mass distribution slope
in a collisional system. I show that the classic distribution function (Dohnanyi,
1969) produces a somewhat shallower slope than is yielded in our more accurate
numerical modeling. In the final chapter I conclude my results and give goals for
future work.
29
C HAPTER 2
FALSE
SIGNS OF DEBRIS DISKS
My advisor, Kate Su, discovered a bow shock shaped mid-infrared excess region
in front of δ Velorum using 24 µm observations obtained with the Multiband
Imaging Photometer for Spitzer (MIPS). The excess has been classified as a debris
disk from previous infrared observations. Although the bow shock morphology was only detected in the 24 µm observations, its excess was also resolved at
70 µm. I show that the stellar heating of an ambient interstellar medium (ISM)
cloud can produce the measured flux and morphology. Since δ Velorum was
classified as a debris disk star previously, our discovery may call into question
the same classification of other stars. I model the interaction of the star and ISM,
producing images that show the same geometry and surface brightness as is observed. The modeled ISM is ∼ 15 times overdense relative to the average Local
Bubble value, which is surprising considering the close proximity (24 pc) of δ
Velorum.
The abundance anomalies of λ Boötis stars have been previously explained as
arising from the same type of interaction of stars with the ISM. Low resolution
optical spectra of δ Velorum show that it does not belong to this stellar class. The
star therefore is an interesting testbed for the ISM accretion theory of the λ Boötis
phenomenon.
2.1 Introduction
Using IRAS data, more than a hundred main-sequence stars have been found to
have excess emission in the 12 - 100 µm spectral range (Backman & Paresce, 1993).
Many additional examples have been discovered with ISO and Spitzer. In most
30
cases the spectral energy distributions (SEDs) can be fitted by models of circumstellar debris systems of thermally radiating dust grains with temperatures of 50
to 200 K. Such grains have short lifetimes around stars: they either get ground
down into tiny dust particles that are then ejected by radiation pressure, or if
their number density is low they are brought into the star by Poynting-Robertson
drag. Since excesses are observed around stars that are much older than the time
scale for these clearing mechanisms, it is necessary that the dust be replenished
through collisions between planetesimals and the resulting collisional cascades of
the products of these events both with themselves and with other bodies. Thus,
planetary debris disks are a means to study processes occurring in hundreds of
neighboring planetary systems. Spitzer observations have revealed a general resemblance in evolutionary time scales and other properties to the events hypothesized to have occurred in the early Solar System.
Although the planetary debris disk hypothesis appears to account for a large
majority of the far infrared excesses around main-sequence stars, there are two
alternative possibilities. The first is that very hot gas around young, hot, and luminous stars can be responsible for free-free emission (e.g., Cote, 1987; Su et al.,
2006). The second possibility is that the excesses arise through heating of dust
grains in the interstellar medium around the star, but not in a bound structure
such as a debris disk. Kalas et al. (2002) noticed optical reflection nebulosities
around a number of stars with Vega-like excesses. These nebulosities show asymmetries that would not be typical of disks, they have complex, often striated
structures that are reminiscent of the Pleiades reflection nebulosities, and they
are much too large in extent to be gravitationally bound to the stars (see Gorlova
et al., 2006).
Dynamical rather than stationary interactions with the ISM are more interest-
31
ing (Charbonneau, 1991). Originally, it was proposed that ISM dust grains could
interact directly with material in debris disks (Lissauer & Griffith, 1989; Whitmire et al., 1992). However, it was soon realized that photon pressure from the
star would repel interstellar grains, resulting in grain-free zones with possible
bow-shock geometry around luminous stars (Artymowicz & Clampin, 1997).
This scenario has been proposed to account for the abundance anomalies associated with λ Boötis stars. These are late B to early F-type, Population I stars
with surface underabundances of most Fe-peak elements and solar abundances
of lighter elements, such as C, N, O and S. In the diffusion/accretion model (Venn
& Lambert, 1990; Kamp & Paunzen, 2002; Paunzen et al., 2003), it is suggested
that the abundance anomaly occurs when a star passes through a diffuse interstellar cloud. The radiation pressure repels the grains, and hence much of the
general ISM metals, while the gas is accreted onto the stellar surface. While the
star is within the cloud, a mid-infrared excess will result from the heating of the
interstellar dust; however, after the star has left the cloud the abundance anomalies may persist for ∼ 106 yr in its surface layers (Turcotte & Charbonneau, 1993)
without an accompanying infrared excess.
There have been few opportunities to test the predictions for dynamical interactions of main-sequence stars with the ambient interstellar medium. France et al.
(2007) have studied a bow shock generated by the O9.5 runaway star HD 34078.
Ueta et al. (2006) describe the bow shock between the mass loss wind of the AGB
star R Hya and the ISM. Noriega-Crespo et al. (1997) identified 58 runaway OB
stars with an observable bow shock structure using high resolution IRAS 60 µm
emission maps. Rebull et al. (2007) discovered that the young B5 star HD 281159
is interacting with the ISM, producing spherical shells of extended IR emission
centered on the star with a spike feature pointing from the star into the shells.
32
None of these cases correspond to the type of situation that might be mistaken
for a debris disk, nor which would be expected to produce a λ Boötis abundance
pattern.
δ Velorum is a nearby (∼ 24 pc) stellar system (at least five members)1 , with
modest excess in the IRAS data. It has been classified as an A-type star with a debris disk system (e.g., Aumann, 1985, 1988; Cote, 1987; Chen et al., 2006; Su et al.,
2006). Otero et al. (2000) observed a drop in the primary component’s brightness
(∼ 0.m 3) and showed that it is an eclipsing binary with probably two A spectral
type components. With the available data, Argyle et al. (2002) computed the system’s parameters. They suggested that the eclipsing binary (Aa) consists of two A
dwarfs with spectral types A1V and A5V and masses of 2.7 and 2.0 M⊙ and with
separation of 10 mas. The nearby B component is a G dwarf with mass around
1 M⊙ and separation of 0.′′6 from the main component. There is also another binary
(CD component) at 78′′ from the star.
In §2.2, I report measurements demonstrating that this star is producing a
bow shock as it moves through an interstellar cloud as hypothesized by Artymowicz & Clampin (1997). In §2.3, I model this behavior using simple dust grain
parameters and show satisfactory agreement with expectations for the ISM and
properties of the star. I discuss these results in §2.4, where I show that the star is
most likely not part of the λ Boötis stellar class. Thus, δ Velorum provides a test
of the diffusion/accretion hypothesis for λ Boötis behavior.
2.2 Observations and Data Reduction
I present observations of δ Velorum at 24 and 70 µm obtained with the Multiband
Imaging Photometer for Spitzer (MIPS) as part of three programs: PID 57 (2004
1
It is a complex multiple system: Otero et al. (2000); Hanbury Brown et al. (1974); Horch et al.
(2000); Argyle et al. (2002); Tango et al. (1979); Kellerer et al. (2007)
33
Feb 21), PID 20296 (2006 Feb 22, Apr 3) and PID 30566 (2006 June 12). For PID 57,
3 second exposures at four dither positions were taken, with a total integration
time of 193 seconds. The other observations at 24 µm (PID 20296) were done
in standard photometry mode with 4 cycles at 5 sub-pixel-offset cluster positions
and 3 sec integrations, resulting in a total integration of 902 sec on source for each
of the two epochs. The star HD 217382 was observed as a PSF standard (AOR ID
6627584) for PID 57, with the same observational parameters. The observation at
70 µm (PID 30566) was done in standard photometry default-scale mode with 10
sec integrations and 3 cycles, resulting in a total integration of 335 sec on source.
The binary component Aa was not in eclipsing phase according to the ephemeris equations by Otero et al. (2000) at either epoch. The period of the eclipse
is ∼ 45.16 days, and the system was ∼ 13 days before a primary minimum at the
first, ∼ 3 days before one at the second and ∼ 7.7 days before one at the third
epoch for the 24 µm observations. The 70 µm observation was 2.53 days before a
secondary minimum.
The data were processed using the MIPS instrument team Data Analysis Tool
(DAT, Gordon et al., 2005) as described by Engelbracht et al. (2007) and Gordon
et al. (2007). Care was taken to minimize instrumental artifacts.
Fitting the model described later demands flux measurements within a constant large external radius (see details in §2.3). Therefore, photometry for the
target was extracted using aperture photometry with a single aperture setting.
The center for the aperture photometry at both 24 and 70 µm was determined
by fitting and centroiding a 2-D Gaussian core. A radius of 56.′′025 was used for
both wavelengths, with sky annulus between 68.′′95 and 76.′′34. The aperture size
was chosen to be large enough to contain most of the flux from the bow shock,
but small enough to exclude the CD component to avoid contamination. The CD
34
component was bright at 24 µm at a distance of 78′′ from the AaB components,
but could not be detected at 70 µm. Aperture corrections were not applied because of the large size of the aperture. Conversion factors of 1.068 × 10−3 and
1.652 × 101 mJy arcsec−2 MIPS UNIT−1 were used to transfer measured instrumental units to physical units at 24 and 70 µm, respectively.
Faint extended asymmetric nebulosity offset from the central star is apparent
at 24 µm, with the dark Airy rings partially filled in. Using standard aperture and
point-spread-function (PSF) fitting photometry optimized for a point source, the
total flux is 1420 ± 42 mJy, ∼ 1.12 times the expected photospheric flux, which
was determined by fitting a Kurucz model (Castelli & Kurucz, 2003) to the optical
and near infrared photometry and extrapolating it to 24 and 70 µm. The large
aperture photometry value is greater by another factor of ∼ 1.1, which puts it
above the expected photospheric flux by a factor of ∼ 1.25. The final photometry
measurements (using the large aperture setting) are listed in Table 2.1. I also list
the modeled photospheric flux of the star and the modeled value of the IR excess.
Since the measured excess depends on the aperture used, to avoid confusion I
do not give a measured excess value, only the photospheric flux which can be
subtracted from any later measurements. The photospheric flux given in Table
2.1 does not include the contribution from the G dwarf (90 and 10 mJy at 24 and
70 µm, respectively). The top left panel in Figure 2.1 shows the summed image
from epochs 2 and 3, to demonstrate the asymmetry suggested even before PSF
subtraction.
For the first epoch 24 µm image, the reference star image was subtracted from
the image of δ Velorum, with a scale factor chosen as the maximum value that
would completely remove the image core without creating significant negative
flux residuals. The deeper exposures from the second and third epochs were
35
Table 2.1. The parameters of δ Velorum
F24 ∗
(mJy)
F70 ∗
(mJy)
1569 ± 42
237 ± 50
(10
ρISM
−24
−3
g cm
)
5.8 ± 0.4
vrel
(km s−1 )
36 ± 4
Fstar24 † Fstar70 † Fexcess24 ‡ Fexcess70 ‡
(mJy)
(mJy)
(mJy)
(mJy)
1277
∗
Observed fluxes with the large aperture
†
Photospheric values - not including G star component
‡
Modeled excesses at large aperture
147
174
141
designed to reveal faint structures far from the star, where the observed PSF is
difficult to extract accurately. Therefore, I used simulated PSFs (from STinyTim)
and the MIPS simulator 2 . Because bright structures nearly in the PSF contribute
to the residuals at large distances, I oversubtracted the PSF to compensate. The
first epoch PSF subtracted 24 µm image is shown in the bottom panels of Figure
2.1 and the composite from epochs 2 and 3 in the upper right.
The PSF subtracted images in Figure 2.1 show that the asymmetry is caused
by a bow shock. As shown in the lower left, the head of the bow shock points approximately toward the direction of the stellar proper motion. The bottom right
panel shows the excess flux contours and that it consists of incomplete spherical
shells centered on δ Velorum. Combined with the upper right image, there is also
a parabolic cavity, as expected for a bow shock. The stagnation points (where
photon pressure equals gravitational force) of the grains in the bow shock are
within ∼ 200 AU of the star, according to the sharp inner edges and the width
of the bow shock. A notable feature in the upper right is the wings of the bow
shock, which are detectable to ∼ 1500 AU.
2
Software designed to simulate MIPS data, including optical distortions, using the same observing templates used in flight.
36
Figure 2.1 The panels show 24 µm images of δ Velorum. All images are in logarithmic scaling, the FOV is ∼ 2.′74 × 2.′34. The scaling of the images are: −0.5
– 4 MJy sr−1 . Top-left panel: The original observed composite image from the 2nd
and 3rd epochs. Top-right panel: PSF oversubtracted image, which shows the bow
shock structure far from the star. Bottom-left panel: The intensity scaled PSF subtracted image (first epoch), which shows the bow shock structure close to the star.
This image shows the orientation of the images and the proper motion direction
of the star. The arrow bisecting the bow shock contour shows the calculated direction of the modeled relative velocity. Bottom-right panel: Same image as the
bottom-left panel, but with intensity contours plotted. The intensity contours
are at 0.25, 1.0, 1.75, 2.5 and 3.25 MJy sr−1 from the faintest to the brightest, respectively. The contours show that the extended emission consists of incomplete
spherical shells, centered on δ Velorum.
37
Figure 2.2 The panels show the 70 µm image of δ Velorum. All images are scaled
logarithmically from −0.5 – 3 MJy sr−1 . The FOV is ∼ 2.′46 × 3.′03. The orientation
of the images is the same as in Figure 2.1. First panel: the observed image. Middle
panel: the PSF subtracted image. The residual flux seems close to being concentric. Last panel: the intensity contours. They suggest that there is a faint concentric
70 µm excess further from the star that fades at the cavity region behind the star.
The 70 µm observation is shown in Figure 2.2. The PSF subtraction (scaled to
the point source flux of 125 mJy) does not reveal the bow shock structure at this
wavelength, only that there is extended excess. The total flux of the residual of the
PSF subtracted image is 119 mJy. The intensity contours (last panel) suggest that
the 70 µm excess fades at the cavity behind the star, but the effect is small. The
geometry and direction of the bow shock are discussed in more detail in §2.3.2.
2.3 The Bow Shock Model
Based on a previous suggestion by Venn & Lambert (1990), Kamp & Paunzen
(2002) proposed a physical model to explain the abundance pattern of λ Boötis
stars through star-ISM interaction and the diffusion/accretion hypothesis. Their
model is based on a luminous main-sequence star passing through a diffuse ISM
cloud. The star blows the interstellar dust grains away by its radiation pressure,
38
but accretes the interstellar gas onto its surface, thus establishing a thin surface
layer with abundance anomalies. So long as the star is inside the cloud, the dust
grains are heated to produce excess in the infrared above the photospheric radiation of the star. Martı́nez-Galarza et al. (2009) have developed a model of
this process and show that the global spectral energy distributions of a group of
λ Boötis type stars that have infrared excesses are consistent with the emission
from the hypothesized ISM cloud. Details of the model can be found in their paper. Here I adapt their model and improve its fidelity (e.g., with higher resolution
integrations), and also model the surface brightness distribution to describe the
observed bow shock seen around δ Velorum.
2.3.1 Physical description of the model
The phenomenon of star-ISM interactions generating bow shocks was first studied by Artymowicz & Clampin (1997). They showed that the radiative pressure
force on a sub-micron dust grain can be many times that of the gravitational force
as it approaches the star. The scattering surface will be a parabola with the star at
the focus point of the parabolic shaped dust cavity. Since the star heats the grains
outside of the cavity and close to the parabolic surface, an infrared-emitting bow
shock feature is expected.
The shape of the parabola (for each grain size) can be given in terms of the
distance between the star (focus) and the vertex. This so-called avoidance radius
(or the p/2 parameter of the scattering parabola) can be calculated from energy
conservation to be (Artymowicz & Clampin, 1997):
a
rav
=
2 (β a − 1) GM
,
2
vrel
(2.1)
where a is the radius of the particle, M is the mass of the star and vrel is the relative
velocity between the star and the dust grains.
39
β a is the ratio of photon pressure to gravitational force on a grain and it is
given by (Burns et al., 1979):
a
β =
0.57Qapr
L/L⊙
M/M⊙
a
µm
−1 δ
g cm−3
−1
,
(2.2)
where δ is the bulk density of the grain material and Qapr is the radiation pressure
efficiency averaged over the stellar spectrum. Qapr (λ) can be expressed in terms
of grain properties (absorption coefficient Qaab (λ), scattering coefficient Qasca (λ)
and the scattering asymmetry factor g = hcos α(λ)i, Burns et al., 1979; Henyey &
Greenstein, 1938):
Qapr (λ) = Qaab (λ) + Qasca (1 − g) ,
(2.3)
which gives
Qapr
=
R
Qapr (λ)B(T∗ , λ)dλ
R
,
B(T∗ , λ)dλ
(2.4)
where B(T∗ , λ) is the Planck function. I adopted astronomical silicates in our
model with δ = 3.3 g cm−3 from Draine & Lee (1984) and Laor & Draine (1993). I
considered a MRN (Mathis et al., 1977) grain size distribution in my model:
dn = Ca−γ da,
(2.5)
where C is a scaling constant and n is the number density of the cloud with γ =
3.5 and grain sizes ranging from 0.005 – 0.25 µm.
With these equations I was able to model the avoidance cavity for a grain that
encounters a star with known mass, luminosity and relative velocity. The model
describes a situation where the expelled grains are instantly removed from the
system rather than drifting away, but this only causes a minor discrepancy in
the wing and almost none in the apex of the parabola compared to the actual
case. In the actual scenario only those particles get scattered back upstream that
a
encounter the central star with small impact parameter (∼ rav
/2). This means
40
that most of the grains will get expelled toward the wings, where the grains go
further out and emit less infrared excess, thus their contribution to the total flux
will be small.
The model determines the number density of certain grain sizes and the position of their parabolic avoidance cavity. Outside of the cavity I assumed a constant number density distribution for each grain size. To calculate the surface
brightness of the system and its SED I assumed a thermal equilibrium condition,
with wavelength dependent absorption and an optically thin cloud.
2.3.2 Model Geometry and Parameters
The model described in §2.3.1 gives the distribution and temperature for each
grain size. This model was implemented in two ANSI C programs. The first
program fits the SED of the system to the observed photometry points, while
the second program calculates the surface brightness of the system. The fitted
photometry included uvby, UBV, HIPPARCOS V band, 2MASS, IRAS and MIPS
(24 and 70 µm) data. I subtracted the 24 and 70 µm flux contributed by the G star
(90 and 10 mJy, respectively) from the MIPS observations, because I wanted to
model the system consisting of the two A stars and the bow shock.
The input parameters are: stellar radius, mass-to-luminosity ratio (MLR), relative velocity of cloud and star, ISM dust density, cloud external radius and the
distance of the system. The stellar radius, MLR and the distance can be constrained easily. I determined the best-fit Kurucz model (Castelli & Kurucz, 2003)
by fitting the photometry points at wavelengths shorter than 10 µm. Since the
distance is known to high accuracy from HIPPARCOS I can determine the radius
and thus the luminosity of the star precisely. The mass was adopted from Argyle
et al. (2002). The G dwarf’s luminosity is only 1% of the system, so leaving the
star out does not cause any inconsistency. Its mass is only 17% of the total mass,
41
which can only cause minor changes in the determined final relative velocity, but
none in the final surface brightness or the computed ISM density. The model
then has three variable parameters: the density of the ISM grains (ρdust - does not
include gas), the relative velocity between the cloud and the star (vrel ) and the
external radius of the cloud (rext ). The model should describe the total flux from
exactly the area used for my photometry. The aperture radius of 56.′′025 (1366 AU
at the stellar distance of 24.45 pc) was used as rext . Both programs calculate the
a
a
Qapr , β a , rav
, n values and then the temperature at rav
for each grain size.
The SED modeling program decreases the temperature value from the one at
rav by 0.01 K steps and finds the radius for that corresponding grain temperature.
The program does not include geometrical parameters such as the inclination or
the rotation angle of the system, since these are irrelevant in calculating the total
flux. It calculates the contribution to the emitting flux for every grain size from
every shell to an external radius (rext ) and adds them up according to wavelength.
The program that calculates the surface brightness uses a similar algorithm as
the SED program, but it calculates the temperature at 1 AU distance steps from
a
rav
for every grain size and calculates the total flux in the line of sight in 1 AU2
resolution elements.
The total inclination ι of the bow shock was not included as a parameter, since
by eye the observed images seemed to show an inclination of ι ≈ 90◦ (a schematic
plot of the angle nomenclature is shown in Figure 2.3). This approximation is
strengthened by the radial velocity of the star, which is only ∼ 2 km s−1 compared
to the tangential velocity of ∼ 13 km s−1 . This assures that the motion of the
system is close to perpendicular to the line of sight. However, I have found that
the bow shock has similar appearance for a significant range of angles (±20◦ )
relative to ι = 90◦ . I illustrate this in Figure 2.4. If the relative velocity vector
42
would have a 70◦ (or 110◦ ) inclination it would only cause minor differences in the
modeled velocity (∆vrel ≈ 3 km s−1 ) and ISM density (∆ρISM ≈ 0.2×10−24 g cm−3 ).
At an inclination of 50◦ , the “wings” spread out and the bright rim at the apex
starts to become thin.
With interstellar FeII and MgII measurements Lallement et al. (1995) showed
that the Local Interstellar Cloud (LIC) has a heliocentric velocity of 26 km s−1
moving towards the galactic coordinate lII = 186 ± 3◦ , bII = −16 ± 3◦ . Since δ
Velorum is at lII ≈ 272◦ , bII ≈ −7◦ , the LIC is also moving perpendicular to our
line of sight at the star and in the direction needed to reach a high relative velocity
between the star and cloud. Crawford et al. (1998) showed a low velocity interstellar Ca K line component in the star’s spectrum with vhelio = 1.3 ± 0.4 km s−1 ,
which also proves that the ISM’s motion is perpendicular to our line of sight at δ
Velorum. The offset of the proper motion direction of the star from the head direction of the bow shock by a few degrees could be explained by the ISM velocity.
A simple vectorial summation of the star and the ISM velocities should give a net
motion in the direction of the bow shock.
2.3.3 Results
I first tried to find the best fitting SED to the photometry points corresponding to
wavelengths larger than 10 µm (MIPS, IRAS) with χ2 minimization in the vrel vs.
ρdust phase space. I defined χ2 as:
χ2 =
X (Fobs − Fcalc )2
2
σobs
(2.6)
The χ2 phase space with rext = 1366 AU showed no minimum (Figure 2.5, left
panel). The interpretation of the diagram is as follows: if the relative velocity is
small, then the avoidance radius will be large. Consequently the grains will be
at relatively low temperature and the amount of dust required to produce the
43
Figure 2.3 The nomenclature of the angles of the system. The heavy line is the
grain avoidance parabola. ϕ is the rotation angle of the system on the plane of
the sky (my initial guess was 4◦ N from the calculated direction of relative motion
shown in Figure 2.1), ι is the inclination and rav is an avoidance radius. The
observer is viewing from the axis pointing to the bottom left.
44
Figure 2.4 The panels show the 24 µm morphology of the bow shock viewed at
different inclinations, starting from 90◦ (left), 70◦ (middle) and 50◦ (right).
observed flux increases. On the other hand, if the relative velocity is large, then
the grains can approach closer to the star and heat up to higher temperatures. As
a result a smaller dust density is enough to produce the observed flux. Therefore,
the combination of the density of the cloud and the relative velocity can be well
constrained by the broad-band SED alone, but not each separately.
By using surface brightness values from the observations and the model calculations I was able to determine the vrel parameter and thus eliminate the degeneracy of the model. Since the bow shock is a parabolic feature it has only one
a
variable, the avoidance radius (rav
), which is the same as the p/2 parameter of the
parabola (with p being the distance between the focus point and the vertex). The
a
value of rav
does change as a function of grain size, but the head of the bow shock
will be near the value where the avoidance radius has its maximum as a function
of grain size. As can be seen in Figure 2.6, the avoidance radius has a maximum
at ∼ 0.06 µm grain size. The value of the avoidance radius on the other hand only
depends on the relative velocity between the ISM cloud and the star. This way I
can constrain the second parameter of the model (vrel ). The relative velocity has
to be set so that the avoidance radius of the ∼ 0.06 µm grain is around half the
45
χreduced2
0
100
200
300
400
500
600
35
30
15
80
25
40
20
20
10
χ2
ρdust [x10-26 g cm-3]
20
15
10
5
5
10
5
0
10 20 30 40 50 60 70 80 90
vrel [km s-1]
0
2
4
6
8
10
12
ρdust [x10-26 g cm-3]
Figure 2.5 Left panel: The χ2 phase space for ρdust vs. vrel with constrained rext =
1366 AU. Right panel: The ρ vs. vrel phase space (left panel) cut at vrel = 36 km s−1 .
parabola parameter value. This method gives a value that only approximates the
true one, but it can be used as an initial guess.
The vrel parameter was constrained by comparing the PSF subtracted image
“wings” with model images. Within a range of ±6 km s−1 of my initial guess
(vrel = 35 km s−1 ) with 1 km s−1 steps, I generated images of the surface brightness
distribution to a radius of 2500 AU. The computational time for a total 5000 ×
5000 × 5000 AU data cube was long, so I only calculated to a depth of 250 AU,
keeping the field of view (FOV) 5000 × 5000 AU. The fluxes of the generated
images were normalized (to ensure that the geometry was the main constraint
of the fit and not surface brightness variations) and rotated to angles ϕ = ±20◦
with 1◦ steps. After rotation, both the model images and the observed image
were masked with zeros where there was no detectable surface brightness in the
observed image.
46
800
700
solid line
δ = 3.3 g cm-3
dashed line δ = 2.2 g cm-3
rav(a) (AU)
600
500
400
300
200
100
0
0.01
0.1
Grain size (a) (µm)
Figure 2.6 The value of rav as a function of grain size. The solid lines are curves
for a silicate bulk density of 3.3 g cm−3 , while the dashed ones are for 2.2 g cm−3 .
The curves are for vrel values of 25, 30, 35, 40 and 45 km s−1 from top to bottom,
respectively.
47
The χ2 of the deviations of the model from the observed image were calculated. I was able to constrain the rotation angle of the model and the relative
velocity of the cloud to the star. The χ2 values in the ϕ vs. vrel phase space are
shown in Figure 2.7 (left panel). The small values at large rotation angles are artifacts due to the masking. The best-fit rotation angle is at ϕ = −4◦ to my initial
guess, which means that the direction of motion is 143◦ (CCW) of N. This is just
21◦ from the proper motion direction. The ISM velocity predicted from vectorial
velocity summation to fit this angle is 24 km s−1 , which is close to the ISM velocity value calculated by Lallement et al. (1995). The tangential velocity direction of
the ISM from the summation is ∼ 47◦ CW of N, which is pointing only 4.◦ 5 south
from the galactic plane.
χreduced2
65
20
70
75
80
85
90
91
90
10
5
89
χ2
Rotation angle [deg]
15
0
88
-5
-10
87
-15
86
-20
30
32
34
36
38
vrel [km s-1]
40
25
30
35
40
-1
vrel [km s ]
45
Figure 2.7 Left panel: χ2 in the phase space of ϕ vs. vrel . Right panel: The phase
space cut at ϕ = −4◦ , showing the best fit for vrel .
The vrel parameter and its error are calculated by fitting a Gaussian to the
phase space values at ϕ = −4◦ (Figure 2.7, right panel). One σ errors are given by
48
values at ∆χ2 = 1. The fits give vrel = 35.8±4.0 km s−1 . Figure 2.5 shows that if vrel
is constrained, then I can also determine the density of the cloud from simple SED
modeling. The vertical cut of Figure 2.5 (left panel) at vrel = 36 km s−1 is shown in
the right panel of the same Figure. ρdust = 6.43 × 10−26 g cm−3 is derived from this
fit, which gives an original ISM density of 6.43 × 10−24 g cm−3 assuming the usual
1:100 dust to gas mass ratio. This ρdust is an upper estimate of the actual value,
since the model computes what density would be needed to give the observed
brightness using a rext radius sphere. Since the line-of-sight distribution of the
dust is not cut off at rext , I used ρdust = 6.43 × 10−26 g cm−3 as an initial guess; a
range of density values was explored with model images.
Since the surface brightness scales with the density, only one image had to be
computed, which could be scaled afterwards with a constant factor. The resultant
χ2 distribution is shown in Figure 2.8. The calculated best fitting ISM density
is 5.8 ± 0.4 × 10−24 g cm−3 , assuming the average 1:100 dust to gas mass ratio.
The error was calculated at ∆χ2 = 1. This density (n ∼ 3.5 atoms cm−3 ) is only
moderately higher than the average galactic ISM density (∼ 1 atom cm−3 ). The
calculated surface brightness images for the three MIPS wavelengths are shown
in Figure 2.9. The closest stagnation point is for the 0.005 µm grains at 64 AU,
while the furthest is at 227 AU for 0.056 µm grains. The temperature coded image
in Figure 2.10 shows the surface brightness temperature of the bow shock (i.e. the
temperature of a black body, that would give the same surface brightness in the
MIPS wavelengths as observed). Table 2.1 shows good agreement between the
model and the measured values.
The original observed images at 24 µm and 70 µm were compared to the
model. I generated model images with high resolution that included the bow
shock and the central star with its photospheric brightness value at the central
49
100
χ2
80
60
40
20
0
0.2
0.4
0.6
0.8
1
ρfitted/ρcalculated
1.2
1.4
1.6
Figure 2.8 The final ISM density was determined from the best fitting surface
brightness image. This plot shows the χ2 of the fits of the model to the observed
image, where ρcalculated is the initial guess from Figure 2.5 right panel and ρfitted is
fitted density using surface brightness values.
pixel. I convolved these images with a 1.8 native pixel boxcar smoothed STinyTim PSF (see Engelbracht et al., 2007). These images were subtracted from the observed ones (Figure 2.11). The residuals are small and generally consistent with
the expected noise. Finally, the best fitting SED of the system (rext = 1366 AU) is
plotted in Figure 2.12. The total mass of the dust inside the rext = 1366 AU radius
is Mdust = 1.706 × 1024 g (0.023 MMoon ).
50
Figure 2.9 Top panels: Calculated high resolution surface brightnesses for 24, 70
and 160 µm, respectively. Bottom panels: The 24 and 70 µm image with MIPS
resolution, convolved with STinyTim PSFs. The images are not rotated to the
same angle as the observed bow shock.
2.4 Discussion
2.4.1 Bow Shock Model Results
My model gives a consistent explanation of the total infrared excess and the surface brightness distribution of the bow shock structure at δ Velorum. The question still remains how common this phenomenon is among the previously identified infrared-excess stars. Is it possible that many of the infrared excesses found
around early-type stars result from the emission of the ambient ISM cloud? The
51
Figure 2.10 Image of the bow shock generated by the model computations. The
image’s FOV is 2.′41 × 2.′41. The colorscale shows the integrated surface brightness
temperature of the bow shock (and not the radial temperature gradient of the
grains) in Kelvins.
majority of infrared excess stars are distant and cannot be resolved, so I cannot
answer for sure. However the excess at δ Velorum is relatively warm between 24
and 70 µm (F(24) ∼ 0.17 Jy, F(70) ∼ 0.14 Jy), and such behavior may provide an
indication of ISM emission. Another test would be to search for ISM spectral features. The ISM 9.7 µm silicate feature of the dust grains would have a total flux of
∼ 1 mJy for δ Velorum. Since the ∼ 1 mJy flux would originate from an extended
region and not a point source that could fit in the slit of IRS, it would be nearly
impossible to detect with Spitzer. Only a faint hint of the excess is visible in the
52
Figure 2.11 Top row: Left panel: Observed 24 µm image. Center panel: Model
24 µm image including both stellar photosphere and bow shock. Right panel:
Model image subtracted from observed. Bottom row: Left panel: Observed 70 µm
image. Center panel: Model 70 µm image including both stellar photosphere and
bow shock. Right panel: Model image subtracted from observed. The FOV is
∼ 2.′7 × 2.′1, N is up and E to the left.
8 µm IRAC images, consistent with the small output predicted by my model.
2.4.2 ISM Interactions
To produce a bow shock feature as seen around δ Velorum, the star needs to be
luminous, have a rather large relative velocity with respect to the interacting ISM,
and be passing through an ISM cloud. A relative velocity of ∼ 36 km s−1 is not
necessarily uncommon, since the ISM in the solar neighborhood has a space velocity of ∼ 26 km s−1 (Lallement et al., 1995) and stars typically move with similar
speeds. If the ISM encountering the star is not dense enough the resulting excess
will be too faint to be detected. The Sun and its close (∼ 100 pc) surrounding
are sitting in the Local Bubble (n(HI) < 0.24 cm−3 , T ≈ 7500 K, Lallement, 1998;
Jenkins, 2002). This cavity generally lacks cold and neutral gas up to ∼ 100 pc.
53
Flux [Jy]
1000
100
Flux [Jy]
10
1
0.3
30
1
50
λ [µm]
70
0.1
0.01
0.001
0.1
1
10
λ [µm]
100
1000
Figure 2.12 The best fit SED. The window in the upper-right corner is a magnified
part of the SED between 20 and 80 µm. The plotted fluxes are 24 and 70 µm MIPS
and 25 and 60 µm IRAS, with errorbars. The 9.7 µm silicate feature in the model
SED of the ISM cloud is very faint and on a bright continuum. The flux from the
G dwarf has been subtracted from the 24 and 70 µm MIPS observations.
The density I calculated at δ Velorum is ∼ 15 times higher than the average value
inside the Local Bubble. Observations over the past thirty years have shown
that this void is not completely deficient of material, but contains filaments and
cold clouds (Wennmacher et al., 1992; Herbstmeier & Wennmacher, 1998; Jenkins, 2002; Meyer et al., 2006). Talbot & Newman (1977) calculated that an average
galactic disk star of solar age has probably passed through about 135 clouds of
n(HI) ≥ 102 cm−3 and about 16 clouds with n(HI) ≥ 103 cm−3 . Thus the scenario
that I propose for δ Velorum is plausible.
54
2.4.3 Implications for Diffusion/Accretion Model of λ Boötis Phenomenon
Holweger et al. (1999) list δ Velorum as a simple A star, not a λ Boötis one. I downloaded spectra of the star from the Appalachian State University Nstars Spectra
Project (Gray et al., 2006). The spectra of δ Velorum, λ Boötis (prototype of its
group) and Vega (an MK A0 standard) are plotted in Figure 2.13. The metallic
lines are generally strong for δ Velorum. One of the most distinctive characteristics of λ Boötis stars is the absence or extreme weakness of the MgII lines at
4481 Å (Gray, 1988). Although the MgII line seems to be weaker than expected
for an A0 spectral type star, it still shows high abundance, which confirms that
δ Velorum is not a λ Boötis type star (Christopher J. Corbally, private communication). The overall metallicity ratio for δ Velorum is [M/H] = −0.33, while for
λ Boötis it is [M/H] = −1.86 (Gray et al., 2006). The G star’s contribution to the
total abundance in the spectrum is negligible, because of its relative faintness.
I used spectra from the NStars web site to synthesize a A1V/A5V binary composite spectrum and found only minor differences from the A1V spectrum alone.
Thus, the assigned metallicity should be valid.
These results show that at δ Velorum, where I do see the ISM interacting with
a star, there is no sign of the λ Boötis phenomenon or just a very mild effect. Turcotte & Charbonneau (1993) modeled that an accretion rate of ∼ 10−14 M⊙ yr−1
is necessary for a Teff = 8000 K main sequence star to show the spectroscopic
characteristics of the phenomenon. The λ Boötis abundance pattern starts to
show at 10−15 M⊙ yr−1 and ceases at 10−12 M⊙ yr−1 . To reach an ISM accretion
of 10−15 M⊙ yr−1 a collecting area of 2 AU radius would be needed with my modeled ISM density and velocity. For an accretion of 10−14 , 10−13 and 10−12 M⊙ yr−1
collecting areas of 6.5, 20 and 65 AU radii are needed, respectively.
With the accretion theory of Bondi & Hoyle (1944), I get an accretion rate of
55
1
norm. intensity
0.8
0.6
0.4
λ Boötis
Vega
δ Velorum
MgII
0.2
3800 3900 4000 4100 4200 4300 4400 4500 4600
λ [Å]
Figure 2.13 The spectra of δ Velorum (bottom line), Vega (middle line) and λ
Boötis (top line).
6.15×10−15 M⊙ yr−1 for δ Velorum. Thus, the accretion rate for this star is probably
not high enough to show a perfect λ Boötis spectrum, but should be high enough
for it to show some effects of accretion. This star is an exciting testbed for the
diffusion/accretion model of the λ Boötis phenomenon.
2.5 Summary
I observe a bow shock generated by photon pressure as δ Velorum moves through
an interstellar cloud. Although this star was thought to have a debris disk, its
infrared excess appears to arise at least in large part from this bow shock. I
present a physical model to explain the bow shock. My calculations reproduce
the observed surface brightness of the object and give the physical parameters
56
of the cloud. I determined the density of the surrounding ISM to be 5.8 ± 0.4 ×
10−24 g cm−3 . This corresponds to a number density of n ≈ 3.5 atoms cm−3 , which
means a ∼ 15 times overdensity relative to the average Local Bubble value. The
cloud and the star have a relative velocity of 35.8 ± 4.0 km s−1 . The velocity of the
ISM in the vicinity of δ Velorum I derived is consistent with LIC velocity measurements by Lallement et al. (1995). My best-fit parameters and measured fluxes are
summarized in Table 2.1.
Holweger et al. (1999) found that δ Velorum is not a λ Boötis star. The measurements from the Nstars Spectra Project also confirm this. Details regarding the
diffusion/accretion time scales for a complex stellar system remain to be elaborated. Nevertheless, the Spitzer observations of δ Velorum provide an interesting
testbed and challenge to the ISM diffusion/accretion theory for the λ Boötis phenomenon.
57
C HAPTER 3
D EBRIS
DISK STUDY OF
P RAESEPE
I present 24 µm photometry of the intermediate-age open cluster Praesepe. I
assemble a catalog of 193 probable cluster members that are detected in optical
databases, the Two Micron All Sky Survey (2MASS), and at 24 µm, within an area
of ∼ 2.47 square degrees. Mid-IR excesses indicating debris disks are found for
one early-type and for three solar-type stars. Corrections for sampling statistics
yield a 24 µm excess fraction (debris disk fraction) of 6.5 ± 4.1% for luminous and
1.9 ± 1.2% for solar-type stars. The incidence of excesses is in agreement with the
decay trend of debris disks as a function of age observed for other cluster and
field stars. The values also agree with those for older stars, indicating that debris
generation in the zones that emit at 24 µm falls to the older 1-10 Gyr field star
sample value by roughly 750 Myr.
I discuss the results in the context of previous observations of excess fractions
for early- and solar-type stars. I show that solar-type stars lose their debris disk
24 µm excesses on a shorter timescale than early-type stars. Simplistic Monte
Carlo models suggest that, during the first Gyr of their evolution, up to 15-30%
of solar-type stars might undergo an orbital realignment of giant planets such
as the one thought to have led to the Late Heavy Bombardment, if the length of
the bombardment episode is similar to the one thought to have happened in our
Solar System.
In Appendix A, I determine the cluster’s parameters via boostrap Monte Carlo
isochrone fitting, yielding an age of 757 Myr (± 36 Myr at 1σ confidence) and a
distance of 179 pc (± 2 pc at 1σ confidence), not allowing for systematic errors.
58
3.1 Introduction
Stars generally form with an accompanying circumstellar disk. Planets can grow
from this primordial disk over a few to a few tens of Myr. The Infrared Astronomy Satellite (IRAS) detected infrared excess emission from disks around stars
with ages much older than the clearing timescales of protoplanetary circumstellar disks (Aumann et al., 1984). These excesses arise from second-generation ”debris disks” that are the results of collisional cascades initiated by impacts between
planetesimals and of cometary activity (Backman & Paresce, 1993). The micronsized dust grains in debris disks are heated by the central star(s) and reradiate the
received energy at mid-infrared wavelengths. Studying this infrared emission
lets us probe the frequency of formation of planetary systems and to track their
evolution. For example, some of the relatively prominent disks may be analogs
to that in the Solar System at the epoch of Late Heavy Bombardment (LHB; e.g.
Gomes et al., 2005; Strom et al., 2005).
IRAS and Infrared Space Observatory (ISO) observations of debris disks suggest
that the excess rate steadily declines with stellar age, indicative of stars losing
these disks within a few hundred million years (Habing et al., 2001; Spangler
et al., 2001). A theoretical model that involved delayed stirring was developed
by Dominik & Decin (2003) to explain this phenomenon; however, a uniform evolutionary model could not be derived. There were a number of reasons. The sensitivity of these instruments was often inadequate for observations down to the
photospheric levels. The large beam sizes also occasionally confused the excesses
with background objects and/or the galactic cirrus. The Multiband Imaging Photometer for Spitzer (MIPS; Rieke et al., 2004) on the Spitzer Space Telescope has improved sensitivity and resolution in the mid-infrared and with it astronomers
have been able to carry out more detailed statistical studies of debris disks at a
59
wide range of stellar ages and spectral types.
Rieke et al. (2005) observed a large sample of nearby A-type field stars with
Spitzer, which combined with existing IRAS and ISO data definitively demonstrated that the frequency of debris disk excesses declines with age and that the
disk properties vary at all ages. Even by probing excesses down to 25% above the
photospheric level, Rieke et al. (2005) found that some stars at ages of only 10-20
Myr do not show any signs of excess. These results were confirmed by Su et al.
(2006). This behavior implies a very fast clearing mechanism for disks around
some of these stars, or perhaps that they form with only very low mass disks.
The models of Wyatt et al. (2007) provided a first-order explanation in terms of
a steady state evolution of the debris disks from a broad distribution of initial
masses.
An important question for habitable planet search/evolution is whether the
same processes occur for FGK-type stars. A number of surveys of solar-type
stars have been conducted with Spitzer. The MIPS Guaranteed Time Observers
(GTO) team has searched ∼ 200 field stars for excesses (Trilling et al., 2008), plus
many hundreds of open cluster members (e.g., Gorlova et al., 2006, 2007; Siegler
et al., 2007). The legacy survey by the Formation and Evolution of Planetary
Systems (FEPS) group has examined 328 stars (both field and open cluster members)(Mamajek et al., 2004; Meyer et al., 2004, 2008; Stauffer et al., 2005; Kim et al.,
2005; Silverstone et al., 2006).
Trilling et al. (2008) showed that solar-type stars of age older than 1 Gyr have
excess emission at 70 µm ∼ 16% of the time. Excesses at this wavelength are
expected to arise from Kuiper-Belt-like planetesimal regions, but with masses 10100 times greater. Meyer et al. (2008) find that 8.5-19% of solar-type stars at ages
< 300 Myr have debris disks detectable at 24 µm and that this number gradually
60
goes down to < 4% at older ages, augmenting work by Gorlova et al. (2006),
Siegler et al. (2007) and Trilling et al. (2008). Excesses at this wavelength around
solar-type stars probe the 1-40 AU range, the asteroidal and planetary region in
the Solar System.
The ideal laboratories to determine the stellar disk fractions with good number statistics are open clusters and associations. To investigate the fraction of
solar-type excess stars, the observations have to be able to detect the photospheres of the non-excess stars. The range of distances to suitable clusters compromises the uniformity of the results. The survey of h and χ Persei (Currie
et al., 2008) could only determine the early-type star excess fraction, while that
of NGC 2547 (Young et al., 2004; Gorlova et al., 2007) could only detect photospheres down to early G due to similar limits. The observations in M47 (Gorlova
et al., 2004) also yielded values to early G spectral type stars. The investigations
of IC 2391 (Siegler et al., 2007) and the Pleiades (Gorlova et al., 2006) gave insights
on debris disk evolution down as far as K spectral-type stars.
To study further the fraction of debris disks around solar-mass stars, I have
observed the nearby Praesepe (M44, NGC 2632, Beehive) open cluster. My observations, along with those of Cieza et al. (2008) on the Hyades cluster, fill the
gap in previous work on debris disk fractions in the age range of 600-800 Myr.
This range is of interest because it coincides with the LHB in the Solar System.
The close proximity of the cluster (∼ 180 pc) and its large number of members
ensured that good statistics would be achieved. Praesepe has been extensively
studied by many groups (Klein Wassink, 1927; Jones & Cudworth, 1983; Jones &
Stauffer, 1991; Hambly et al., 1995; Wang et al., 1995; Kraus & Hillenbrand, 2007),
providing a nearly full membership list to the completeness limit of [24] ∼ 9 mag
(the brightness of a G4 V spectral-type star at the distance of the cluster). The
61
member stars have high proper motions (∼ 39 mas yr−1 ), clearly distinguishing
them from field stars.
3.2 Observations, data reduction, and photometry
I used MIPS to observe Praesepe as part of the GTO program PID 30429 (2007
May 30). The center part of the cluster (8h 40m 21s , 19◦ 38′ 40′′ ) was imaged using three scan maps (with 12 legs in a single scan map overlapping with halfarray cross-scan). The map covers a field of ∼ 2.47 deg2 , as shown in Figure 3.1.
Medium scan mode was used, resulting in a total effective exposure time per
pixel of 80 s (at 24 µm). All data were processed using the MIPS instrument team
Data Analysis Tool (DAT, Gordon et al., 2005) as described by Engelbracht et al.
(2007).
Although MIPS in scan-mode provides simultaneous data from all three detectors (at 24, 70 and 160 µm), I base my study on the 24 µm channel data only.
The 70 and 160 µm detectors are insensitive to stellar photospheric emissions at
the distance of Praesepe. In retrospect, the rarity of excesses in the survey is consistent with the lack of detections at the longer wavelengths.
The initial coordinate list for the 24 µm photometry was assembled with the
daofind task under IRAF1 . I later expanded this list by visually examining the
images and manually adding all sources to the list that were missed by daofind.
The final list for photometry contained 1457 sources. To achieve high accuracy,
I performed point-spread function (PSF)–fitting photometry. The calibration star
HD 173398 was adopted as a PSF standard, with the final PSF constructed from 72
individual observations, kindly provided to us by C. Engelbracht. The standard
1
IRAF is distributed by the National Optical Astronomy Observatories, which is operated by
the Association of the Universities for Research in Astronomy, Inc. (AURA) under cooperative
agreement with the National Science Foundation.
62
20°30’
20°20’
20°10’
20°
δ2000
19°50’
19°40’
19°30’
19°20’
19°10’
19°
18°50’
m
44
m
43
m
42
m
41
m
40
α2000 8h
m
39
m
38
m
37
m
36
Figure 3.1 The observed field, showing the areas covered by three scanmaps and
the observed cluster member stars with sizes proportional to their brightness in
[24].
63
0.1
1
σ24 [mJy]
σ[24] [mag]
0.08
0.06
0.04
0.1
0.02
0
4
5
6
7
8
[24] [mag]
9
10
11
1
10
F24 [mJy]
100
Figure 3.2 The error of the 24 µm photometry is plotted as a function of brightness for cluster member sources. The left panel shows the flux and its error on
the magnitude scale, while the right panel shows them in mJy flux values. All
points have less than 0.1 magnitude error and nearly all stars brighter than 9th
magnitude have errors less than 0.04 magnitude.
IRAF tasks phot and allstar of the daophot package were used.
The observed field is free of nebulosity and stellar crowding, so I was able
to use a large PSF radius of 112′′, with fitting radius of 5.7′′ . The large PSF
radius ensured us that the aperture correction was negligible. The instrumental number counts were converted to flux densities with the conversion 1.068 ×
10−3 mJy arcsec−2 MIPS UNIT−1 (Engelbracht et al., 2007). I then translated these
values to 24 µm magnitudes taking 7.17 Jy for the [24] magnitude zero point,
which has error of ± 0.11 Jy (Rieke et al., 2008). I show the photometric error vs.
brightness plots of the measurements in Figure 3.2. Almost all sources brighter
than 9th magnitude (∼ 1.8 mJy) have errors less than 0.04 mag (∼ 0.07 mJy) and
all sources remain below errors of 0.1 mag; the average error is ∼ 5%. As a check,
I performed independent PSF photometry with StarFinder under IDL, obtaining photometry values within the errors of the IRAF photometry and with errors
similar to the ones given by daophot.
64
3.3 Catalog surveys and the final sample
I compiled a complete catalog for all sources in the field of view, including their
optical, near infrared, and 24 µm data. I expanded this catalog with all known
cluster members outside of the field of view (naturally without [24] data). This
enabled us to plot a full cluster optical color-magnitude diagram (CMD), which I
used to confirm the cluster’s age and distance (see Appendix A).
Optical data for the sources were obtained from the 5th data release of the
Sloan Digital Sky Survey (SDSS), while 2MASS provided J, H, and KS magnitudes. The SDSS photometry is generally unreliable for bright sources, the ones
mostly detected in the MIPS survey. To ensure I had good photometry for these
sources, I collected BV data (for 356 stars altogether) using the Webda database2 ,
providing an ensemble of data for high probability cluster members from various papers (Johnson, 1952; Anthony-Twarog, 1982; Dickens et al., 1968; Lutz &
Lutz, 1977; Upgren et al., 1979; Castelaz et al., 1991; Mermilliod et al., 1990; Weis,
1981; Stauffer, 1982; Andruk et al., 1995; Mendoza, 1967; Oja, 1985). The data
downloaded from the Webda database cover the brightest magnitude range of
the cluster, including stars avoided by modern CCD observations or where they
are saturated. I converted the BV magnitudes to SDSS r and g values by averaging the conversion slopes of Jester et al. (2005); Jordi et al. (2006); Zhao & Newberg
(2006) and Fukugita et al. (1996) and obtained
g = (0.607 ± 0.016)(B − V )
−(0.1153 ± 0.0095) + V
(3.1)
r = (−0.453 ± 0.028)(B − V )
+(0.1006 ± 0.0131) + V.
2
http://www.univie.ac.at/webda/
(3.2)
65
Where the calculated r or g brightnesses for the Webda catalog members differed
from the SDSS data by more than 0.5 magnitude, I replaced the SDSS data with
the calculated one.
Cluster membership was determined by compiling all accessible databases.
The largest membership lists are those of Wang et al. (1995) and Kraus & Hillenbrand (2007), which were supplemented by the Webda catalog search results.
Wang et al. (1995) give a list of 924 stars, out of which I chose only 198 that
are high probability members of the cluster according to the proper motion data
in the paper. The list of Kraus & Hillenbrand (2007) is much more robust with
1130 stars, all of which have membership probability > 50%; 1010 of them have
> 80% membership probability. The databases (SDSS, 2MASS, Webda, Wang
et al. (1995), Kraus & Hillenbrand (2007)) were cross-correlated with a maximum
matching radius of 3.6′′ . The closest member within this radius is matched as
a pair and all others are added to the catalog as new sources. The program excluded pairing members from the same catalog. The final cluster member list
contains 1281 candidates, of which 493 were in the observed field.
After plotting the color-magnitude diagram and doing an initial isochrone fit
on cluster members, I tested for bad photometry. I generated a list of all the
member stars that were further from the isochrone sequence than 0.3 magnitude,
examined all these stars for anomolies on SDSS images, and searched for BV magnitudes in Simbad. If the star was saturated or a calculated r, g magnitude differed from the SDSS r, g value by 0.5 magnitude or more (the same criteria as
used before), I used the calculated value.
In Figure 3.3, I show how the selection criteria narrow the CMD, and where
sources with different selection characteristics are distributed in the field. From
the 1457 sources identified in the 24 µm survey, 201 were cataloged as cluster
66
26
6
25
24
8
23
10
21
r [mag]
δ2000 [°]
22
20
19
12
14
18
16
17
18
16
15
20
9h
8h54m 8h48m 8h42m 8h36m 8h30m 8h24m 8h18m
α2000
0
0.5
1
g-r [mag]
1.5
2
Figure 3.3 The spatial and CMD position of the selected sources. Red dots: The
combined list for all sources in the observed field; Green dots: All cluster members outside the observed field; Blue dots: All cluster members in the observed
field that could not be identified in the 24 µm survey; Magenta dots: All cluster
members that were identified in the 24 µm survey.
members by previous work. Of these, 193 also have data in the optical and near
infrared. The survey’s completeness limit compared to 2MASS is at J = 10 mag
([24] ∼ 9 mag), as is shown in Figure 3.4. This limit corresponds to a G4 V star at
the distance of Praesepe. The completeness limit for the cluster member sources
is also shown in Figure 3.4. Between 10th and 11th magnitude in J I achieve 75%
completeness for cluster members.
For the [24] magnitude values to be comparable to the 2MASS KS photometry,
I fitted a Gaussian to the binned number distribution of the KS -[24] values of
all member sources with r-KS < 0.8 (∼ A stars). I derived a general correction
factor of -0.032±0.002 magnitude (∼ 3%) for the [24] values. The Gaussian fits
are shown in Figure 3.5. This same method has been used by Rieke et al. (2008)
to obtain the average ratio of KS to 24 µm flux densities. By optimizing the fit of
80
100% 100% 100% 100% 100% 98% 75% 47% 1.5%
100
70
Number of sources
Completeness percentage [%]
67
80
60
40
20
60
50
40
30
20
10
0
0
4
6
8
10
12
J2MASS [mag]
14
16
18
4
6
8
10
12
J2MASS [mag]
14
16
18
Figure 3.4 Left Panel: The completeness limit of the survey is shown as a function
of 2MASS J magnitude. I detect almost all sources brighter than 10th magnitude,
corresponding to a ∼ G4 V star. For sources fainter than 14th magnitude random
associations begin to occur. Right Panel: The total number of cluster members
within the field of view (light gray) and the number of members detected (dark
gray) at [24] are shown as a function of J magnitude.
the [24] data to the 2MASS KS data, I eliminated any absolute calibration offsets.
The average variance of the fitted gaussians is σ = 0.047 mag, consistent with the
average [24] error value of ∼ 0.05 mag.
I summarize the [24] photometry results for the 193 cluster members that were
identified in all wavelength regions in Table 3.1. The first column of the table
gives the designated number, while the coordinates are that of the 24 µm flux
source. As a source/coordinate comparison I also list the 2MASS source associated with the 24 µm emission. The table contains the KS adjusted [24] magnitude,
the original flux values (in mJy) and the ”best” r and g photometry value. Cluster
membership probability is shown by either the proper motion of the source or by
the Wang et al. (1995) catalog number of the source. Sources that are missing both
values were listed as cluster members either in the Webda database or in Kraus
& Hillenbrand (2007).
68
14
bin=0.04
<Ks - [24]> = -0.0301
bin=0.05
<Ks - [24]> = -0.0310
bin=0.02
<Ks - [24]> = -0.0342
bin=0.03
<Ks - [24]> = -0.0334
12
# stars in bin
10
8
6
4
2
14
12
# stars in bin
10
8
6
4
2
-0.2 -0.1
0
0.1 0.2
K-[24] (mag)
0.3
-0.2 -0.1
0
0.1 0.2
K-[24] (mag)
0.3
Figure 3.5 The panels show the Gaussian fits to the number distribution of A type
stars, within certain KS -[24] bins.
Table 3.1. Photometry of Praesepe members in the [24] band
α2000
[h:m:s]
δ2000
[◦ :′ :′′ ]
g∗
[mag]
r∗
[mag]
K2MASS
[mag]
[24]†
[mag]
F24 †
[mJy]
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
8:36:29.83
8:36:48.95
8:37:02.04
8:37:16.35
8:37:18.29
8:37:26.51
8:37:27.58
8:37:27.95
8:37:28.22
8:37:33.84
8:37:36.33
8:37:37.00
8:37:40.71
8:37:42.36
8:37:46.35
8:37:46.64
8:37:46.77
8:37:47.30
8:37:49.99
8:37:52.08
8:37:57.06
8:38:07.63
8:38:08.08
8:38:14.11
8:38:14.28
8:38:23.16
8:38:24.31
8:38:29.70
8:38:32.18
8:38:34.27
8:38:37.43
8:38:37.78
8:38:37.88
8:38:46.97
8:38:50.05
8:38:53.57
18:57:56.52
19:15:26.06
19:36:17.42
19:29:11.58
19:41:56.33
19:29:13.06
19:37:03.29
19:33:45.25
19:09:44.32
20:00:49.39
19:15:53.96
19:43:58.69
19:31:06.38
19:08:01.57
19:35:57.26
19:26:18.10
19:16:02.03
19:06:24.01
19:53:28.75
19:59:13.85
19:14:09.67
19:59:16.40
20:26:20.83
19:47:23.82
19:21:55.37
20:12:26.60
20:06:21.92
19:51:45.83
19:27:55.04
19:51:36.90
19:01:14.81
19:38:47.69
19:59:23.14
19:30:03.53
20:04:03.29
19:34:17.90
9.57
11.54∗
9.34
14.67
11.75
14.48
11.97
9.91∗
9.65
8.76∗
14.05∗
7.77∗
8.29
10.05∗
12.75
10.85∗
6.75∗
12.71
11.78∗
11.54
12.23
12.48
12.08∗
15.56
11.20
8.01∗
10.80
14.67∗
10.46∗
9.68
14.45
10.73∗
8.16∗
9.08
11.01
14.82∗
9.33∗
10.92∗
9.06∗
13.38
11.23
13.33
11.40
9.64
9.37∗
8.66∗
13.05∗
7.79∗
8.20∗
9.75∗
12.04
10.50
6.76∗
11.96∗
11.13
11.07
11.59
11.82
11.47∗
14.17
10.31
7.71∗
10.40
13.53
9.65
8.87
13.27
10.43
8.16∗
8.95
10.64
13.53∗
8.30±0.01
9.69±0.02
8.06±0.01
10.47±0.02
9.80±0.02
10.83±0.02
9.81±0.02
8.46±0.02
8.40±0.01
7.95±0.03
10.76±0.02
7.29±0.01
7.66±0.01
8.58±0.02
10.24±0.02
9.28±0.02
6.17±0.01
10.20±0.02
9.33±0.02
9.69±0.02
10.04±0.02
9.90±0.02
9.93±0.02
10.91±0.04
9.19±0.02
6.65±0.01
9.18±0.02
10.93±0.02
7.54±0.01
6.69±0.01
10.61±0.01
9.22±0.02
7.61±0.01
8.22±0.02
9.37±0.02
10.96±0.02
8.23±0.05
9.35±0.06
8.01±0.02
10.06±0.07
9.69±0.05
10.77±0.15
9.75±0.06
8.48±0.02
8.39±0.02
7.95±0.02
10.34±0.09
7.30±0.01
7.63±0.01
8.58±0.02
10.14±0.06
9.32±0.04
6.12±0.01
10.07±0.07
9.10±0.04
9.55±0.04
10.11±0.09
10.02±0.07
10.03±0.17
10.00±0.13
9.12±0.03
6.64±0.01
9.23±0.03
10.56±0.14
7.46±0.01
6.66±0.01
10.16±0.18
9.24±0.05
7.64±0.02
8.08±0.02
9.28±0.03
10.92±0.17
3.54± 0.15
1.26± 0.07
4.32± 0.09
0.65± 0.04
0.93± 0.04
0.34± 0.05
0.87± 0.04
2.80± 0.06
3.06± 0.05
4.57± 0.10
0.51± 0.04
8.31± 0.09
6.14± 0.08
2.57± 0.05
0.61± 0.03
1.29± 0.04
24.79± 0.26
0.65± 0.04
1.59± 0.05
1.05± 0.04
0.63± 0.05
0.68± 0.05
0.68± 0.11
0.69± 0.08
1.56± 0.05
15.37± 0.13
1.41± 0.04
0.41± 0.06
7.18± 0.08
15.03± 0.18
0.60± 0.10
1.40± 0.06
6.08± 0.08
4.06± 0.06
1.35± 0.03
0.30± 0.05
µα
[mas yr−1 ]
µδ
[mas yr−1 ]
-34.60
-36.30
-34.30
-34.70
-37.20
-42.70
-34.10
-36.60
-36.20
-35.70
-35.30
N/A
-34.80
-36.60
-37.80
-36.10
N/A
-35.60
-31.80
-38.80
-35.40
-38.10
-36.40
N/A
-35.00
N/A
-36.30
-40.10
N/A
N/A
-37.70
N/A
-37.40
-34.80
-36.60
-40.90
-12.60
-12.80
-13.00
-15.40
-15.20
-14.40
-12.60
-13.20
-13.40
-13.10
-11.20
N/A
-12.50
-13.50
-9.40
-13.40
N/A
-15.10
-19.20
-14.60
-13.70
-13.50
-14.40
N/A
-13.70
N/A
-13.10
-13.20
N/A
N/A
-6.80
N/A
-13.60
-12.60
-15.40
-21.30
W#‡
2MASS
267
268
274
277
2
3
5
288
6
10
12
295
13
14
299
304
310
325
346
347
358
21
24
27
394
35
37
423
432
08362985+1857570
08364896+1915265
08370203+1936171
08371635+1929103
08371829+1941564
08372638+1929128
08372755+1937033
08372793+1933451
08372819+1909443
08373381+2000492
08373624+1915542
08373699+1943585
08374070+1931063
08374235+1908015
08374640+1935575
08374660+1926181
08374675+1916020
08374739+1906247
08374998+1953287
08375208+1959138
08375703+1914103
08380758+1959163
08380808+2026223
08381421+1947234
08381427+1921552
08382311+2012263
08382429+2006217
08382963+1951450
08383216+1927548
08383425+1951369
08383723+1901161
08383776+1938480
08383786+1959231
08384695+1930033
08385001+2004035
08385354+1934170
69
#
Table 3.1—Continued
α2000
[h:m:s]
δ2000
[◦ :′ :′′ ]
g∗
[mag]
r∗
[mag]
K2MASS
[mag]
[24]†
[mag]
F24 †
[mJy]
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
8:38:55.07
8:39:01.89
8:39:02.27
8:39:02.84
8:39:03.24
8:39:03.60
8:39:04.09
8:39:05.25
8:39:06.12
8:39:06.55
8:39:09.11
8:39:10.15
8:39:12.20
8:39:15.05
8:39:19.77
8:39:21.88
8:39:24.99
8:39:28.63
8:39:29.42
8:39:30.44
8:39:33.44
8:39:36.35
8:39:38.29
8:39:42.66
8:39:42.81
8:39:43.35
8:39:44.68
8:39:45.78
8:39:50.74
8:39:50.86
8:39:52.35
8:39:55.08
8:39:56.51
8:39:57.78
8:39:58.09
8:39:58.40
19:11:54.02
20:00:19.62
19:19:35.36
19:43:28.99
20:02:35.12
19:59:59.24
19:31:23.20
20:07:01.92
19:40:36.59
19:00:36.68
19:35:32.68
19:40:42.56
19:06:56.45
20:12:39.35
20:03:10.91
19:51:40.86
19:27:33.70
19:28:25.00
19:47:11.51
20:04:08.69
20:10:10.52
19:15:39.67
19:26:26.02
19:46:42.49
20:05:10.46
19:25:10.52
19:16:30.94
19:22:01.06
19:32:26.92
19:33:02.23
19:18:45.61
20:03:54.47
19:33:10.91
19:32:29.26
19:12:05.98
20:09:29.99
10.84
12.48∗
12.83
9.48
15.18
8.33∗
14.40
9.51
7.48∗
13.83
8.54∗
9.55∗
10.86
11.61
9.78
12.97∗
10.75
12.07∗
13.09∗
10.68∗
9.79
15.10∗
13.13
6.69∗
7.75∗
12.27
7.68∗
10.93
7.06∗
12.15∗
10.68
10.37
7.32∗
7.58∗
9.71
8.71
9.87
11.45∗
12.06
9.19∗
13.86
8.32∗
13.28
9.31∗
7.43∗
13.09
8.49∗
9.32∗
10.41
11.13
8.97
12.20
10.01
11.32∗
12.29
10.11
8.67
13.87
12.28
6.65∗
7.73∗
10.86
7.69∗
10.50
6.26∗
11.56∗
10.07
10.02
7.33∗
7.53∗
9.38
8.86
8.04±0.01
9.07±0.02
10.26±0.02
8.12±0.01
11.05±0.02
7.77±0.01
10.86±0.01
8.41±0.02
6.71±0.01
11.30±0.02
7.88±0.02
8.41±0.01
9.26±0.02
9.65±0.02
7.08±0.01
10.37±0.02
9.00±0.01
9.53±0.02
10.06±0.01
8.81±0.01
6.13±0.01
11.01±0.02
10.29±0.02
6.00±0.01
7.16±0.01
7.90±0.02
7.09±10.00
9.26±0.02
4.39±0.04
10.00±0.02
9.01±0.02
8.96±0.02
6.79±0.01
7.01±0.02
8.48±0.02
8.10±0.01
7.90±0.02
8.96±0.03
10.20±0.08
8.08±0.02
10.88±0.13
7.71±0.02
10.46±0.13
8.38±0.03
6.73±0.01
11.16±0.30
7.87±0.03
8.40±0.02
9.18±0.05
9.63±0.06
7.02±0.01
10.07±0.10
8.89±0.03
9.40±0.05
9.94±0.22
8.74±0.02
6.00±0.02
10.49±0.12
10.22±0.09
5.98±0.01
7.16±0.03
7.76±0.01
7.15±0.01
9.39±0.04
4.32±0.01
9.77±0.06
8.90±0.03
8.86±0.04
6.82±0.01
7.01±0.01
8.36±0.02
8.03±0.07
4.81± 0.08
1.80± 0.05
0.57± 0.04
4.06± 0.06
0.31± 0.04
5.69± 0.09
0.45± 0.05
3.09± 0.07
14.03± 0.17
0.24± 0.07
4.93± 0.12
3.03± 0.04
1.47± 0.07
0.97± 0.05
10.80± 0.14
0.65± 0.06
1.92± 0.04
1.21± 0.06
0.73± 0.14
2.22± 0.03
27.56± 0.38
0.44± 0.05
0.57± 0.04
28.02± 0.37
9.47± 0.23
5.44± 0.06
9.53± 0.10
1.21± 0.05
129.25± 1.24
0.86± 0.05
1.91± 0.05
1.99± 0.07
12.99± 0.15
10.84± 0.09
3.15± 0.07
4.26± 0.28
µα
[mas yr−1 ]
µδ
[mas yr−1 ]
N/A
N/A
-36.60
-35.80
-40.70
-34.20
-37.00
-35.70
N/A
N/A
-35.30
-36.10
-37.00
-35.20
N/A
-36.40
-37.00
-36.10
-38.90
-35.80
N/A
-33.30
-33.00
N/A
N/A
N/A
N/A
-35.40
N/A
-36.10
-34.80
-37.50
N/A
N/A
-37.40
-36.00
N/A
N/A
-10.50
-11.20
-14.30
-13.30
-14.60
-12.10
N/A
N/A
-12.00
-13.70
-13.40
-14.70
N/A
-8.80
-14.90
-10.80
-9.00
-13.40
N/A
-24.60
-9.60
N/A
N/A
N/A
N/A
-12.80
N/A
-13.90
-14.30
-13.90
N/A
N/A
-12.50
-13.80
W#‡
2MASS
448
46
47
48
450
49
50
457
52
54
57
477
65
66
506
69
70
523
77
79
80
82
83
86
87
89
93
94
96
97
98
08385506+1911539
08390185+2000194
08390228+1919343
08390283+1943289
08390321+2002376
08390359+1959591
08390411+1931216
08390523+2007018
08390612+1940364
08390649+1900360
08390909+1935327
08391014+1940423
08391217+1906561
08391499+2012388
08391972+2003107
08392185+1951402
08392498+1927336
08392858+1928251
08392940+1947118
08393042+2004087
08393342+2010102
08393643+1915378
08393836+1926272
08394265+1946425
08394279+2005103
08394333+1925121
08394466+1916308
08394575+1922011
08395072+1932269
08395084+1933020
08395234+1918455
08395506+2003541
08395649+1933107
08395777+1932293
08395807+1912058
08395838+2009298
70
#
Table 3.1—Continued
α2000
[h:m:s]
δ2000
[◦ :′ :′′ ]
g∗
[mag]
r∗
[mag]
K2MASS
[mag]
[24]†
[mag]
F24 †
[mJy]
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
8:39:59.10
8:39:59.19
8:39:59.58
8:39:59.84
8:40:00.01
8:40:00.64
8:40:01.32
8:40:01.72
8:40:04.20
8:40:04.92
8:40:05.70
8:40:06.28
8:40:06.37
8:40:06.44
8:40:09.74
8:40:11.46
8:40:12.32
8:40:13.45
8:40:15.36
8:40:15.59
8:40:15.72
8:40:17.63
8:40:18.10
8:40:18.97
8:40:20.16
8:40:20.75
8:40:22.09
8:40:22.33
8:40:22.73
8:40:23.29
8:40:23.48
8:40:25.55
8:40:26.14
8:40:26.30
8:40:26.76
8:40:27.03
20:01:53.15
19:40:08.58
18:56:35.30
19:34:00.55
19:34:39.86
19:48:23.44
20:08:08.38
18:59:59.17
19:47:04.24
19:43:45.48
19:01:30.18
19:27:14.80
19:18:26.46
20:00:28.12
19:37:17.83
19:58:16.21
19:38:22.78
19:46:45.08
19:59:39.66
19:27:29.84
19:54:54.07
19:47:15.14
19:31:55.13
20:11:31.16
19:20:56.44
19:41:12.23
19:40:11.82
20:06:24.88
19:27:53.46
19:40:23.95
19:50:06.04
19:28:32.92
19:41:11.33
19:13:11.06
20:10:55.34
19:32:41.42
9.35
9.86
10.08∗
12.41∗
13.35∗
10.49∗
9.82∗
10.47
12.11
9.95
13.20
10.55
11.58∗
6.88∗
12.54∗
6.78∗
10.07∗
13.75∗
8.88∗
14.61
13.15
10.20
7.52∗
13.59
6.83∗
7.68∗
6.95∗
10.26
10.94∗
10.61∗
8.09∗
9.75
9.50∗
13.40
8.27∗
6.27∗
9.15
9.69
9.95
11.57∗
12.51∗
10.17∗
9.57∗
9.93
11.54
9.67
12.31
10.10
10.76
6.06∗
11.71
6.71
9.79∗
12.79∗
8.77∗
13.47
12.29
9.73∗
7.57∗
12.38∗
6.77∗
7.69∗
6.08∗
9.97
10.53∗
10.20∗
8.04
9.37
9.27∗
12.50
8.17∗
6.31∗
8.21±0.02
8.78±0.02
9.30±0.01
9.48±0.02
10.55±0.02
9.08±0.02
8.62±0.01
8.70±0.02
10.00±0.02
8.65±0.02
10.01±0.02
8.87±0.02
9.23±0.02
4.20±0.02
10.13±0.02
6.53±0.02
8.67±0.02
10.64±0.02
8.04±9.99
10.69±0.02
10.01±0.02
8.58±0.02
7.16±0.01
10.04±0.01
6.04±0.01
7.28±0.02
4.18±0.03
8.85±0.01
9.34±0.02
9.01±0.02
7.59±0.01
8.76±0.02
8.37±0.02
10.46±0.02
7.43±0.01
5.88±0.01
8.22±0.01
8.73±0.03
9.32±0.05
9.31±0.04
10.20±0.06
8.97±0.03
8.53±0.02
8.48±0.03
10.05±0.10
8.54±0.03
9.90±0.07
8.80±0.03
9.23±0.03
4.13±0.01
10.04±0.08
6.56±0.01
8.56±0.03
10.75±0.13
8.03±0.02
10.52±0.12
9.92±0.06
8.60±0.03
7.18±0.01
9.71±0.06
6.01±0.01
7.30±0.01
4.07±0.01
8.64±0.03
9.25±0.04
9.01±0.03
7.55±0.02
8.71±0.03
8.28±0.02
10.56±0.13
7.38±0.01
5.92±0.01
3.56± 0.05
2.24± 0.06
1.29± 0.06
1.31± 0.05
0.57± 0.03
1.80± 0.05
2.67± 0.05
2.81± 0.07
0.66± 0.06
2.65± 0.06
0.76± 0.05
2.10± 0.05
1.40± 0.04
153.97± 1.35
0.67± 0.05
16.42± 0.17
2.62± 0.07
0.35± 0.04
4.27± 0.06
0.43± 0.05
0.75± 0.04
2.52± 0.07
9.34± 0.08
0.90± 0.05
27.23± 0.31
8.36± 0.11
162.87± 1.71
2.43± 0.06
1.39± 0.05
1.73± 0.04
6.65± 0.10
2.28± 0.06
3.37± 0.05
0.42± 0.05
7.73± 0.09
29.83± 0.29
µα
[mas yr−1 ]
µδ
[mas yr−1 ]
-36.40
N/A
N/A
-33.80
-39.40
-36.30
-36.00
-36.50
-33.00
-36.10
-35.70
N/A
-34.30
N/A
-33.90
N/A
-36.90
-31.50
-35.80
-36.30
-38.00
-35.50
N/A
-37.40
N/A
N/A
N/A
-36.60
-37.80
-37.00
N/A
-36.80
-37.20
-38.40
N/A
N/A
-16.20
N/A
N/A
-12.20
-4.20
-13.10
-14.50
-11.70
-13.70
-12.50
-12.20
N/A
-14.70
N/A
-10.50
N/A
-14.50
-14.40
-12.30
-8.50
-13.20
-13.60
N/A
-14.00
N/A
N/A
N/A
-12.20
-13.30
-11.80
N/A
-13.30
-11.90
-7.00
N/A
N/A
W#‡
2MASS
99
565
100
101
102
103
576
106
578
582
111
114
115
116
117
119
601
120
122
123
607
125
127
128
129
131
132
133
134
135
624
136
137
08395908+2001532
08395915+1940083
08395957+1856357
08395983+1934003
08395998+1934405
08400062+1948235
08400130+2008082
08400171+1859595
08400416+1947039
08400491+1943452
08400571+1901307
08400627+1927148
08400635+1918264
08400643+2000280
08400968+1937170
08401145+1958161
08401231+1938222
08401345+1946436
08401535+1959394
08401549+1927310
08401571+1954542
08401762+1947152
08401810+1931552
08401893+2011307
08402013+1920564
08402075+1941120
08402209+1940116
08402231+2006243
08402271+1927531
08402327+1940236
08402347+1950059
08402554+1928328
08402614+1941111
08402624+1913099
08402675+2010552
08402702+1932415
71
#
Table 3.1—Continued
α2000
[h:m:s]
δ2000
[◦ :′ :′′ ]
g∗
[mag]
r∗
[mag]
K2MASS
[mag]
[24]†
[mag]
F24 †
[mJy]
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
8:40:27.46
8:40:27.52
8:40:28.68
8:40:31.72
8:40:31.85
8:40:32.97
8:40:33.48
8:40:39.25
8:40:39.94
8:40:41.91
8:40:42.51
8:40:43.22
8:40:46.09
8:40:47.23
8:40:48.01
8:40:48.32
8:40:52.52
8:40:52.53
8:40:54.93
8:40:56.29
8:40:56.76
8:40:56.95
8:41:04.79
8:41:07.34
8:41:07.39
8:41:09.61
8:41:09.82
8:41:10.02
8:41:10.32
8:41:10.70
8:41:13.04
8:41:13.80
8:41:15.43
8:41:16.04
8:41:18.42
8:41:19.96
19:16:40.87
19:39:20.05
20:18:44.86
19:51:01.84
20:12:5.98
19:11:39.59
19:38:00.42
19:13:41.88
19:40:09.37
19:13:25.68
19:33:57.85
19:43:09.62
19:18:34.67
19:32:37.64
19:39:31.57
19:55:19.02
20:15:59.87
19:28:59.77
19:56:06.25
19:34:49.26
19:44:05.50
19:56:05.57
19:31:22.94
19:26:48.08
19:04:16.43
19:51:18.32
19:56:07.04
19:30:32.18
19:49:07.10
19:49:46.38
19:32:34.26
19:55:19.24
20:02:15.04
19:44:54.13
19:15:39.38
19:38:04.20
11.45
13.77∗
12.04∗
11.98∗
11.85∗
8.72
12.63∗
7.82∗
11.44∗
10.86∗
11.66∗
7.07∗
9.79
10.87
11.56
11.29
8.52
10.55
12.50∗
6.76∗
12.64∗
8.79∗
11.35
13.01
10.56
11.01∗
14.26∗
10.35∗
11.84∗
9.06
15.06
8.35∗
14.99∗
14.06
7.95∗
14.20
10.96∗
12.83
11.35
11.38∗
11.28
8.55∗
11.91∗
7.80∗
10.66
10.43∗
11.11∗
6.85
9.45
10.08
10.79
10.86
8.47∗
10.15
11.80∗
6.79∗
11.94∗
8.67∗
10.60
12.12
10.04
10.50∗
13.19
9.98
11.29∗
8.86
13.73
8.35∗
13.78∗
13.44
7.90∗
13.09
9.65±0.02
10.69±0.02
9.46±0.02
9.91±0.02
9.83±0.01
7.96±0.00
10.17±0.02
7.23±0.01
9.19±0.02
9.06±0.02
9.71±0.02
6.33±0.01
8.53±0.02
8.20±0.01
9.25±0.02
9.51±0.02
7.80±0.01
9.05±0.02
10.13±0.02
6.28±0.01
10.21±0.02
8.05±0.02
8.75±0.02
10.29±0.02
8.64±0.02
8.94±0.02
10.76±0.02
8.91±0.02
9.75±0.02
8.19±0.01
10.35±0.02
7.77±0.01
11.02±0.02
11.71±0.02
7.29±0.02
10.76±0.02
9.58±0.04
10.84±0.12
9.42±0.07
9.72±0.09
9.81±0.06
7.82±0.02
10.05±0.08
7.25±0.01
9.18±0.03
9.04±0.03
9.69±0.04
6.33±0.01
8.49±0.02
8.10±0.01
9.25±0.03
9.50±0.04
7.78±0.02
8.97±0.03
10.16±0.09
6.28±0.01
10.13±0.11
7.95±0.01
8.68±0.03
10.30±0.20
8.53±0.03
8.61±0.03
10.67±0.10
8.83±0.13
9.62±0.05
8.19±0.02
10.22±0.08
7.69±0.02
10.79±0.11
10.50±0.09
7.04±0.01
10.22±0.17
1.02± 0.04
0.32± 0.04
1.18± 0.07
0.90± 0.07
0.83± 0.04
5.16± 0.10
0.66± 0.05
8.73± 0.10
1.47± 0.05
1.68± 0.04
0.92± 0.04
20.45± 0.14
2.79± 0.05
3.99± 0.05
1.39± 0.04
1.10± 0.04
5.36± 0.08
1.79± 0.05
0.60± 0.05
21.23± 0.28
0.61± 0.06
4.57± 0.06
2.33± 0.06
0.53± 0.09
2.68± 0.06
2.48± 0.06
0.37± 0.04
2.04± 0.23
0.98± 0.04
3.66± 0.06
0.56± 0.05
5.82± 0.09
0.34± 0.03
0.44± 0.03
10.61± 0.12
0.56± 0.09
µα
[mas yr−1 ]
µδ
[mas yr−1 ]
-33.30
-33.40
-37.40
-35.60
-36.40
-37.40
-38.90
N/A
-35.50
-35.90
-36.70
N/A
-37.10
N/A
-37.60
-35.50
-34.60
-37.00
-37.20
N/A
-36.10
-36.10
N/A
-43.70
-39.90
-36.70
-42.40
-36.90
-36.50
-37.80
-37.60
-36.90
-37.30
N/A
-37.40
-36.30
-12.10
-12.20
-15.90
-12.90
-13.90
-14.20
-10.60
N/A
-11.30
-13.00
-13.80
N/A
-13.20
N/A
-14.50
-13.00
-12.70
-13.20
-14.90
N/A
-11.10
-15.40
N/A
-8.10
-14.10
-13.90
-15.10
-12.00
-13.20
-13.70
-9.70
-12.60
-11.90
N/A
-12.90
-12.00
W#‡
2MASS
628
138
631
640
641
141
142
150
151
153
154
156
158
161
162
166
167
677
170
171
172
176
177
179
180
182
183
184
709
188
189
190
192
721
08402743+1916409
08402751+1939197
08402863+2018449
08403169+1951010
08403184+2012060
08403296+1911395
08403347+1938009
08403924+1913418
08403992+1940092
08404189+1913255
08404248+1933576
08404321+1943095
08404608+1918346
08404720+1932373
08404798+1939321
08404832+1955189
08405247+2015594
08405252+1928595
08405487+1956067
08405630+1934492
08405669+1944052
08405693+1956055
08410478+1931225
08410725+1926489
08410737+1904164
08410961+1951186
08410979+1956072
08411002+1930322
08411031+1949071
08411067+1949465
08411319+1932349
08411377+1955191
08411541+2002160
08411602+1944514
08411840+1915394
08411992+1938047
72
#
Table 3.1—Continued
α2000
[h:m:s]
δ2000
[◦ :′ :′′ ]
g∗
[mag]
r∗
[mag]
K2MASS
[mag]
[24]†
[mag]
F24 †
[mJy]
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
8:41:22.48
8:41:23.93
8:41:25.89
8:41:26.98
8:41:28.65
8:41:33.89
8:41:35.09
8:41:35.90
8:41:36.20
8:41:37.43
8:41:42.31
8:41:43.68
8:41:43.85
8:41:45.49
8:41:47.74
8:41:48.24
8:41:49.34
8:41:50.09
8:41:51.98
8:41:53.16
8:41:54.37
8:41:55.90
8:41:57.84
8:41:58.86
8:42:05.50
8:42:06.51
8:42:10.79
8:42:11.50
8:42:12.34
8:42:12.85
8:42:15.50
8:42:18.85
8:42:20.16
8:42:21.62
8:42:24.74
8:42:32.27
18:56:00.17
20:14:57.30
19:56:36.85
19:32:32.71
19:44:49.13
19:58:08.83
19:39:45.04
19:06:25.16
19:08:33.58
19:31:13.08
19:39:37.98
19:57:43.85
20:13:37.06
19:16:02.17
19:24:43.88
19:27:30.49
19:11:47.51
19:52:27.19
20:10:01.99
20:09:34.16
19:15:27.14
19:41:22.96
18:54:42.08
20:06:26.82
19:35:57.95
19:24:40.72
18:56:03.62
19:16:36.37
19:12:48.20
19:16:03.79
19:41:15.47
20:24:36.22
20:02:11.72
20:10:53.72
19:35:17.27
19:23:46.25
13.49
15.46∗
10.92∗
10.05
11.43
12.07
9.17∗
14.84
9.57
14.19
9.72∗
12.73
10.69
10.35
11.66∗
14.28∗
15.24
7.37∗
12.44∗
8.61∗
11.65
11.39
9.66∗
13.77∗
11.07
7.96∗
7.92∗
12.57∗
14.32
14.10∗
10.12
12.59
9.91
9.32
11.21
11.38
12.60
14.11∗
10.55
9.73∗
10.76
11.47∗
8.12
13.59
9.23
13.09
9.50∗
12.05
10.34
9.98∗
11.28
13.24
13.90
6.56∗
11.76∗
8.51∗
11.03
10.86
9.35
12.83
10.33∗
7.97∗
7.93∗
11.85∗
13.20
13.07
9.78
11.91
9.56
9.13
10.83∗
10.84
10.54±0.02
10.78±0.02
9.33±0.02
8.72±0.02
9.47±0.02
9.93±0.01
5.96±0.02
10.98±0.02
8.35±0.02
10.73±0.01
8.48±0.02
10.26±0.02
9.14±0.01
8.93±0.02
10.10±0.02
10.73±0.01
10.83±0.01
4.68±0.00
10.09±0.02
7.79±0.01
9.64±0.02
9.54±0.01
8.43±0.02
10.60±0.01
8.38±0.03
7.43±0.02
7.35±0.01
10.17±0.01
10.83±0.01
10.48±0.01
8.77±0.01
10.19±0.02
8.41±0.02
8.28±0.01
9.48±0.01
9.46±0.01
10.20±0.15
10.11±0.13
9.37±0.04
8.79±0.03
9.44±0.13
9.94±0.07
5.86±0.01
10.55±0.11
8.34±0.02
10.50±0.14
8.38±0.03
10.15±0.10
8.99±0.04
8.92±0.03
9.96±0.07
10.62±0.10
10.55±0.11
4.63±0.01
9.78±0.08
7.71±0.01
9.56±0.05
9.59±0.07
8.43±0.03
10.32±0.16
8.30±0.02
7.39±0.01
7.35±0.02
9.99±0.06
10.56±0.10
10.11±0.06
8.72±0.02
10.05±0.12
8.40±0.02
8.13±0.02
9.34±0.04
9.50±0.06
0.58± 0.08
0.63± 0.08
1.24± 0.05
2.12± 0.06
1.16± 0.14
0.73± 0.05
31.26± 0.30
0.42± 0.04
3.21± 0.05
0.44± 0.06
3.09± 0.08
0.60± 0.10
1.76± 0.07
1.88± 0.05
0.72± 0.05
0.39± 0.04
0.42± 0.04
97.51± 1.02
0.85± 0.07
5.71± 0.07
1.04± 0.05
1.02± 0.07
2.94± 0.07
0.52± 0.08
3.32± 0.05
7.64± 0.06
7.95± 0.11
0.70± 0.04
0.41± 0.04
0.63± 0.04
2.24± 0.05
0.66± 0.08
3.03± 0.06
3.89± 0.06
1.27± 0.05
1.10± 0.06
µα
[mas yr−1 ]
µδ
[mas yr−1 ]
-34.00
N/A
-36.30
-37.30
-39.00
-39.40
N/A
-31.10
-36.00
-40.90
-37.30
-40.50
-37.40
-38.10
-30.30
-40.80
-33.90
N/A
-40.60
-38.20
-34.80
-37.60
-34.30
-40.30
N/A
-38.40
-34.10
-37.60
-33.50
-33.00
-37.60
N/A
-35.70
-36.80
N/A
-36.50
-9.90
N/A
-13.70
-12.40
-13.50
-14.40
N/A
-9.60
-14.30
-12.00
-13.80
-13.10
-15.70
-13.20
-9.50
-9.80
-10.80
N/A
-15.70
-13.70
-13.20
-12.10
-11.10
-11.80
N/A
-12.10
-12.10
-10.00
-9.60
-11.70
-15.00
N/A
-15.60
-14.40
N/A
-12.50
W#‡
2MASS
726
194
195
196
198
201
751
204
758
206
769
771
208
774
776
212
213
214
215
217
218
792
223
224
817
822
824
226
839
228
229
232
08412258+1856020
08412390+2014572
08412584+1956369
08412698+1932329
08412869+1944481
08413384+1958087
08413506+1939449
08413599+1906255
08413620+1908335
08413741+1931140
08414229+1939379
08414368+1957437
08414382+2013368
08414549+1916023
08414776+1924439
08414818+1927312
08414934+1911471
08415008+1952270
08415199+2010013
08415314+2009340
08415437+1915266
08415587+1941229
08415782+1854422
08415884+2006272
08420547+1935585
08420650+1924405
08421080+1856037
08421149+1916373
08421233+1912488
08421285+1916040
08421549+1941156
08421883+2024350
08422012+2002117
08422162+2010539
08422471+1935175
08423225+1923463
73
#
74
3.4 Results
In this section I present the results on the debris disk fraction I observed in Praesepe and place it in context with previous results on the evolution of debris disks.
There are two basic methods to detect 24 µm excess. The first is to use a colorcolor diagram, with one of the colors determining the stars’ spectral type and
the other being KS -[24]. The r-KS color is ideal to differentiate spectral types,
while the KS -[24] color depends only weakly on the spectral type of the star since
both wavelengths fall on the Rayleigh-Jeans part of the spectral energy distribution (SED) for all sources hotter than early M type (Teff > 3200 K) (Gautier et al.,
2007). For non-excess stars the KS -[24] color should stay close to zero. Any excess
measured in KS -[24] is most likely caused by circumstellar material.
The second method is to fit the observed optical and near-infrared photometry
with theoretical SEDs based on stellar photosphere models. Excesses are revealed
if the 24 µm flux density is significantly greater than the predicted flux.
3.4.1 Color-color selection
I used the color-color diagram shown in Figure 3.6 as the primary method to identify sources as excess candidates. I plot all cluster members that have magnitude
values in r, KS , and [24], 193 sources altogether.
Gautier et al. (2007) show the trend of KS -[24] photospheric color with spectral type for stars of low effective temperature. The empirical locus of stars on
the color-color plot in Figure 3.6 was derived by fitting a curve to a sample of
field stars (from Gautier et al. (2007) and Trilling et al. (2008)). I then converted
the fitted V-KS colors to r-KS colors through conversion tables in Cox (2000) and
Kraus & Hillenbrand (2007). The final color-color curve for r-KS vs. KS -[24] for
Table 3.1—Continued
#
α2000
[h:m:s]
δ2000
[◦ :′ :′′ ]
g∗
[mag]
r∗
[mag]
K2MASS
[mag]
[24]†
[mag]
F24 †
[mJy]
181
182
183
184
185
186
187
188
189
190
191
192
193
8:42:40.19
8:42:40.73
8:42:42.51
8:42:43.72
8:42:44.44
8:43:00.59
8:43:05.96
8:43:08.24
8:43:10.82
8:43:20.20
8:43:32.42
8:43:35.56
8:44:07.37
19:07:58.87
19:32:35.34
19:05:59.78
19:37:23.52
19:34:48.11
20:20:15.79
19:26:15.36
19:42:47.59
19:31:33.64
19:46:08.58
19:44:38.00
20:11:22.63
20:04:36.23
12.55∗
10.03
12.04∗
12.76
10.09
11.76
10.25
13.92
12.20
11.05
12.92
10.29
10.30
11.86∗
9.66
11.38
11.76
9.53
11.24
9.65
12.89
11.58
10.62
12.01
9.99
10.05
10.19±0.01
8.72±0.02
9.88±0.02
9.80±0.02
8.63±0.02
9.77±0.02
8.46±0.02
10.67±0.01
10.01±0.02
9.36±0.02
10.22±0.02
8.92±0.02
9.06±0.01
9.83±0.06
8.67±0.03
9.88±0.06
9.64±0.05
8.48±0.02
9.57±0.06
8.40±0.01
10.15±0.09
10.49±0.22
9.26±0.04
9.94±0.08
8.95±0.03
8.95±0.03
0.81± 0.04
2.36± 0.07
0.78± 0.04
0.97± 0.04
2.82± 0.05
1.03± 0.06
3.02± 0.04
0.60± 0.05
0.44± 0.09
1.37± 0.05
0.73± 0.05
1.82± 0.05
1.82± 0.04
µα
[mas yr−1 ]
µδ
[mas yr−1 ]
-35.30
-38.40
-37.40
-36.40
-38.20
-37.30
-36.60
-33.70
-38.50
-39.40
-40.10
-39.30
N/A
-10.90
-12.70
-13.50
-14.20
-13.50
-16.00
-13.80
-11.60
-17.50
-13.40
-16.50
-14.70
N/A
W#‡
2MASS
863
235
236
868
238
887
248
899
902
255
919
257
-
08424021+1907590
08424071+1932354
08424250+1905589
08424372+1937234
08424441+1934479
08430055+2020161
08430593+1926152
08430822+1942475
08431076+1931346
08432019+1946086
08433239+1944378
08433553+2011225
08440734+2004369
∗ The r and/or g magnitudes marked with a star were calculated from B and V magnitudes as described in §3.3, while the rest are the original SDSS
values.
† The [24] magnitudes are the ones that were calibrated to the 2MASS K magnitudes, while the mJy values in the F
24 column are the original flux
S
values.
‡ The
numbers in this column represent the numbering of Wang et al. (1995).
75
76
3.5
146
24
139
3
M0
58
31
152
2.5
K5
43
11 144
188
110 90
145
106
2
77
52
191
181
163
134
r-Ks
22
189
1.5
64
K0
G5
G0
2
100
F5
1
F0
143
0.5
A5
0
A0
-0.4
0
0.4
0.8
Ks-[24]
Figure 3.6 The color-color plot for the cluster members with photometric measurements in r, KS , and [24]. The 1σ measurement error in [24] is plotted for stars
that are outside of the trend curve. The nomenclature is from Table 3.1.
77
main-sequence (MS) stars is:
KS − [24] = 3.01 × 10−5 (r − KS ) + 0.0233(r − KS )2
+0.0072(r − KS )3 − 0.0015(r − KS )4 .
(3.3)
In Figure 3.6, I plot this curve and the 3σ average confidence level for the photometry in [24] (∼ 0.15 mag) (the errors of the curve itself are minor compared to
the photometric errors). The majority of the stars (> 86%) lie within this band.
The errors plotted for the stars outside of the MS fitted curve are the 1σ errors
in the [24] photometry. To use the KS -[24] color as an excess diagnostic tool, one
must make sure that the KS magnitude is truly photospheric. I examined the J, H,
and KS fits to theoretical SEDs (Castelli & Kurucz, 2003) and concluded that all
KS magnitudes are truly photospheric; the largest difference (from the debris disk
candidate sample introduced later) is in the case of star 143, where the measured
value is above the predicted SED value by 5.6%.
As Figure 3.6 shows, I have 7 stars in the ”blue” region of the color-color plot,
which I used to establish a selection rule to clean the excess region of the plot of
possible spurious detections. I only accepted stars as true excess stars that: 1)
lie at least 3σ24 (their own σ and not the average [24] error) from the trend line;
2) have [24] data at least 3σ24 from the best fitting SED solution also; and 3) are
point sources on the images and have no noise anomalies. All stars in the ”blue”
region failed these criteria. From the 19 stars that lie in the excess (”red”) region
of the color-color diagram, fifteen were eliminated as debris disk candidates for
the following reasons. Only 8 were 3σ24 from the trend line: 181, 143, 100, 77, 134,
188, 24, and 2 (in the nomenclature of Table 3.1 and Figure 3.6). Star 188 turned
out to be contaminated by a minor planet, which was identified by comparing
scanlegs separately. Stars 24 and 2 are resolved doubles on the higher resolution
2MASS and SDSS images, so I excluded them from the list.
78
Star 100 is contaminated by a faint background galaxy, which was visible as a
faint nebulosity next to the star. The probability for other sources of a 24 µm excess arising through a chance alignment with distant galaxies can be determined
from galaxy counts (Papovich et al., 2004). The ∼ 0.15 mag [24] excess criterion
results in different flux values identified as excesses as a function of source brightness. I estimated the probability of chance alignments by dividing the sample into
1 magnitude bins and running a Monte Carlo code with the number of sources in
the bin and the number of extragalactic sources corresponding to 0.15 magnitude
excess value for the specific bin. The matching radius for the chance alignment
was chosen to be r = 3.6′′ and the code was ran to 10000 simulations per magnitude bin. I summarize the simulation numbers with the probabilities of at least
n number of chance alignments in each bin in Table 3.2. The probability that star
143 with [24] = 7.04 mag is a chance alignment with a background galaxy is very
low (< 3%), so it is very likely to be a true debris disk star. The probability that
at least two sources (star 100 and 134) are contaminated by a background galaxy
in the 8-9 magnitude bin is also very low (< 4%), and since star 100 is already
contaminated, I classify star 134 as a real debris disk star also. The likelihood that
stars 77 and 181 are contaminated within 3.6′′ is high (∼ 90%). However, there is
no indication of any positional offset between KS and [24], even at the 1′′ level, so
this likelihood is probably overestimated.
I determine stars 143 and 134 to be definite debris disk stars in Praesepe and
list stars 77 and 181 as possible debris disk stars. I show these sources in Figure
3.7 and detail their properties in §3.4.4. Figure 3.7 shows that the fields are clean
and that the sources are point-like. The PSFs were centered on the 24 µm sources
with IRAF’s centroid algorithm. As Table 3.1 shows, the coordinate center of the
excesses is closer than 1′′ to the 2MASS coordinates for the debris disk candidates.
Table 3.2. The probabilities of chance alignments for the sources with
background galaxies as a function of [24] brightness.
[24] bin
[mag]
N∗
[#]
Flux
[mJy]
Excess
[mJy]
Ngalaxies
[sr−1 ]
4-5
5-6
6-7
7-8
8-9
9-10
10-11
11-12
4
2
11
26
48
53
48
1
125.900
50.122
19.954
7.944
3.162
1.259
0.501
0.199
16.245
6.467
2.575
1.025
0.408
0.162
0.065
0.026
2×104
7×104
4×105
1×106
7×106
4×107
8×107
1×108
[0]∗
P of at least n chance alignments
[1]†
[2]†
[3]†
[4]†
99.99%
99.98%
99.54%
97.46%
72.35%
12.60%
2.47%
91.20%
0.01%
0.02%
0.46%
2.54%
27.65%
87.40%
97.53%
8.80%
∼0%
∼0%
∼0%
0.04%
3.80%
60.26%
87.34%
-
∼0%
∼0%
∼0%
0.01%
31.93%
69.30%
-
∼0%
∼0%
∼0%
∼0%
13.14%
47.06%
-
∗
The probability that none of the cluster member sources are chance aligned with a background galaxy in the appropriate magnitude range.
†
The probability that at least 1, 2, 3 or 4 cluster member sources are chance aligned with a
background galaxy in the appropriate magnitude range.
79
80
Figure 3.7 SDSS, 2MASS, 24 micron, and 24 micron PSF subtracted images for
stars 77, 134, 143, and 181. The fields of view (FOV) for the images are 69.9′′ ×
37.97′′ and they have linear flux scaling.
81
3.4.2 The SED fit selection
The 2MASS data are only useful in selecting debris disk candidates if the KS magnitude is photospheric. Since the threshold of the KS -[24] color above which a star
is selected to be a debris disk candidate depends on the spectral type (the determination of which depends on correct r band photometry), I also fit the photometric
data for all stars within the trend curves with model spectra to look for excess
candidates. To be considered a debris disk candidate in this region required even
stronger selection criteria then in the case of the ”excess region stars.” Stars were
selected to be candidates from this region if their [24] photometry was at least
3σ24 from the fitted SED and if the star was 3σ24 + 10% (0.1 mag) from the trend
line in the color-color plot. The 10% is an allowance for systematic errors. None
of the stars within the trend curves passed these criteria.
3.4.3 Praesepe white dwarfs
I also checked whether any of the known eleven Praesepe white dwarfs (Dobbie
et al., 2006) were detected, indicating a possible white dwarf debris disk. WD
0837+199 showed a strong signal in [24]. The UKIRT Infrared Deep Sky Survey (UKIDSS) survey team (Sarah Casewell, private communication 2008) have
found that this signal originates from a background galaxy a few arcseconds
north of the WD.
3.4.4 Debris Disk Candidates
I discuss the four debris disk candidate stars in this section. None of these stars
show extended emission (resolved disk), implying that the excess is confined to
the radius of the MIPS beam of 6′′ (Rieke et al., 2004), which is ∼ 1000 AU at the
distance of Praesepe. This is consistent with the sizes of already resolved mid-IR
debris disks (Stapelfeldt et al., 2004; Su et al., 2005, 2008; Backman et al., 2009).
82
Flux [Jy]
10
0
10
-1
10
-2
10
-3
Flux [Jy]
10-1
Star 143
Teff=7500 K
log(g)=5.0
Star 134
Teff=5500 K
log(g)=4.5
Star 77
Teff=5000 K
log(g)=4.5
Star 181
Teff=5250 K
log(g)=4.5
10-2
10-3
1
10
Wavelength [µm]
1
10
Wavelength [µm]
Figure 3.8 The best fitting SEDs of the debris disk candidate stars with available
optical, 2MASS, and [24] photometry. The [24] photometry is plotted with 1 and
3σ errors.
The best fitting SEDs of the debris disk candidate stars are plotted in Figure 3.8.
3.4.4.1 Star #77
This star was identified on three separate scanlegs, with no contamination by
minor planets. It is rather faint with mV = 12.88 mag. Its optical and NIR photometry was best fitted by the Teff = 5000 K and log g = 4.5 (K3 V) Kurucz model
(Castelli & Kurucz, 2003). Franciosini et al. (2003) used XMM-Newton to detect
X-ray emission from it with a flux of LX = 1.67×1028 erg s−1 in the ROSAT 0.1-2.4
keV band. They point out that the flux measured by ROSAT (Randich & Schmitt,
1995) is a magnitude higher than theirs. The star is a cluster member cataloged in
83
many papers (Wang et al., 1995; Klein Wassink, 1927; Jones & Cudworth, 1983).
3.4.4.2 Star #134
Star #134 (WJJP 179, KW 367) is a bright cluster member, with mV = 10.71 mag. It
was imaged on two scanlegs with high S/N. Its optical and NIR photometry was
best fitted by the Teff = 5500 K and log g = 4.5 (G8 V) Kurucz model (Castelli &
Kurucz, 2003) and I detect no extent to the stellar PSF core (Figure 3.7). North of
it by 6′′ , a fainter extended source is visible on both scanlegs. It has been found to
be a triple system by Mermilliod et al. (1994) and the mass of the components was
estimated by Halbwachs et al. (2003) using CORAVEL radial velocity measurements. The system consists of a wide pair, one of which is a spectroscopic binary
with a period of 3.057 days. It is also a definite cluster member (Wang et al., 1995;
Klein Wassink, 1927; Jones & Cudworth, 1983).
3.4.4.3 Star #143
This is the brightest of all debris disk stars I observed, with mV = 8.04 mag. Its
optical and NIR photometry was best fitted with a Teff = 7500 K, log g = 5.0 (A7
V) Kurucz model (Castelli & Kurucz, 2003). The PSF subtraction was very clean,
with no hint of any extended emission (Figure 3.7). The star was discovered to
be a δ Scuti type of pulsating variable by Paparo & Kollath (1990) (HI Cnc, HD
73890, BD+19 2078). It has been cataloged as a definite cluster member in many
papers (Wang et al., 1995; Klein Wassink, 1927; Jones & Cudworth, 1983). With
high-resolution imaging surveys, Mason et al. (1993) found it to be a single star.
3.4.4.4 Star #181
The star was identified on three separate scanlegs. The best fit to its photometry
points was with a Teff = 5250 K, log g = 4.5 (K0 V) Kurucz model (Castelli &
Kurucz, 2003). It has been identified as a cluster member in many catalogs (Wang
84
et al., 1995; Hambly et al., 1995; Klein Wassink, 1927; Jones & Stauffer, 1991). It
was not identified as a close binary star in the surveys of Bouvier et al. (2001) and
Mermilliod & Mayor (1999). No extended emission is seen in the PSF subtracted
image (Figure 3.7).
3.5 Discussion
I have found 4 sources out of 193 in the spectral range from A0 to K3 showing excess at 24 µm. One of the sources (star 143) is an A7 type star (out of 29 early-type
stars), while the remaining three are G8, K0, and K3 (out of 164 solar-type stars),
based on their photometric colors and fitted SEDs. Although the probability of
chance alignments with faint background galaxies within 3.6′′ are rather high for
the K0 and K3 spectral-type sources, since the peaks of their emission are well
within 1′′ of the 2MASS coordinates they are likely excess sources. However, the
statistics are incomplete to their spectral limit. In the field of view there are 106
stars within F0 and G8 spectral-type, of which I detected 98, meaning I have an
almost complete sample of sources within this spectral band. I use the excess
fraction of 1/106 for the solar-type star sample.
The excesses found around early type stars (B8-A9) are usually dealt with
separately in the literature from the ones found around solar-type stars (F0-K4),
because the dominant grain removal processes in the debris disks may not be the
same and the 24 µm excesses probe significantly different distances from the stars.
These populations are also separated observationally, by the natural detection
limits.
In the following sections I analyze the results in the context of previous debris
disk fractions observed around early- and solar-type stars. The errors on the
debris disk fractions are given by Bayesian statistics detailed in the following
85
§3.5.1. I contrast the results for early- and solar-type stars in §3.5.2 and §3.5.3 and
discuss the implications for debris disk decay time scales in §3.5.4. In §3.5.5, I
compare these results with a simple model for the incidence of episodes like the
LHB around other stars.
3.5.1 Calulating errors on debris disk fractions
Due to the small number of observations, I estimated the debris disk fractions
and associated uncertainties using a Bayesian approach, which I outline in this
section.
If the fraction of objects with disks is fdisk , derived from the observed number
of disks (n) from a sample size of N, then the posterior probability that fdisk has a
certain value will be
P (fdisk |n, N) ∝ P (fdisk)P (n|fdisk, N).
(3.4)
Here, P (fdisk|n, N) if the probability distribution for fdisk , given that n and N are
known. P (fdisk) is the prior distribution of fdisk and P (n|fdisk , N) is the probability of observing that n of N sources have a disk, assuming a certain value
of fdisk . P (fdisk|n, N) will be the posterior probability distribution for fdisk and
P (n|fdisk, N) is the likelihood function. If no prior assumption is made on the
value of fdisk , then the prior will be uniform, i.e. P (fdisk) = 1. This will be assumed, so that all information on fdisk originates from the data itself. The likelihood function, P (n|fdisk, N), is a binomial distribution, therefore
P (fdisk|n, N) ∝ fdiskn (1 − fdisk )N −n ,
(3.5)
where the binomial coefficient has been dropped because of its non-dependence
on fdisk , making it irrevelant in the posterior distribution.
This equation is equivalent to a Beta (B) distribution with parameters α = n+1
and β = N − n + 1. The expectation value (posterior mean) of the B distribution
86
is simply
E(fdisk ) =
n+1
α
=
,
α+β
N +2
(3.6)
while its mode gives the regular ratio of n/N (if n > 1 and N > 2). The 1σ confidence region can be found by integrating the central region that contains 68.3%
of the probability for the B distribution. This was done here by Monte Carlotype calculations. I simulated 107 random variables from a B distribution and
searched for the bottom and upper limits at the 15.85% and 84.15% percentiles.
I give the results with the expectation values and the upper and lower errors
from the 1σ limits. I decided to use expectation values (posterior mean) over
mode averages based on that the fractions are usually low making the distributions skewed. In such cases, they are better described by their mean. For example,
this will give an expected debris disk fraction of
E(fdisk ) =
1+1
= 1.85%
106 + 2
(3.7)
for the solar-type stars.
3.5.2 The decay of the debris disk fraction in early-type stars
A-type stars are well suited to search for excess emission originating from debris
disks. The extended surveys of Rieke et al. (2005) and Su et al. (2006), probed
the excess fraction for A-type stars in the field and in associations between the
ages of 5 and 850 Myr. Numerous observations have also determined the excess
fraction for early-type stars in open clusters and associations (e.g. Young et al.,
2004; Gorlova et al., 2004, 2006; Siegler et al., 2007; Cieza et al., 2008).
I compared the early spectral type excess fraction to the ones in the literature.
I combined the data of Rieke et al. (2005) and Su et al. (2006), removing cluster
and association members. Sources that were listed in both catalogs were adopted
from Su et al. (2006), due to the improved reduction methods and photospheric
87
Table 3.3. The field star sample excess ratios at 24 µm at certain age bins for
early type stars (Rieke et al., 2005; Su et al., 2006).
Age
[Myr]
3.16 - 10 . . . . . . .
10 - 31.6 . . . . .
31.6 - 100 . . . . . .
100 - 316 . . . . . .
316 - 1000 . . . . .
Excess fraction
[#]
[%]
6/10
3/4
4/10
13/39
3/31
58.3 ± 14.2
66.7+18.7
−19.1
41.7 ± 14.2
34.2 ± 7.4
12.1 ± 5.5
model fits in the latter paper. Sources were counted as excess sources if their
relative excess exceeded 15%. IRAS and ISO sources from the Rieke et al. (2005)
sample were removed, due to their higher – 25% – excess thresholds. The final
age bins from the combined catalogs are listed in Table 3.3.
I also compared the results to those from open cluster (and OB association)
surveys by other groups. I list these clusters, their excess fraction, age and the
references for these parameters in Table 3.4. The majority of these clusters are
from MIPS group papers, that used the same 15% excess level threshold as I did
in my study of Praesepe. The few others used similar thresholds, or as in the case
of the β Pic MG study (Rebull et al., 2008), all excess sources that were identified
exceeded their 20% threshold with no sources between 15 and 20%. I plot the
excess fractions from all surveys with the field star samples in the top left panel of
Figure 3.9.
The fifteen open clusters and associations follow the same trend as the field
star sample, with the exception of IC 2602 (Su et al., 2006) and IC 2391 (Siegler
et al., 2007). Possible explanations for this deviation are explored in Siegler et al.
(2007) and they conclude that the most likely cause is the lack of a statistically
88
Table 3.4. The excess fraction at 24 µm for early type stars in
clusters/associations.
Name
Age
[Myr]
Upper Sco . . . . . .
Orion OB1b . . . .
Orion OB1a . . . .
β Pic MG . . . . . . .
Upper Cen . . . . .
NGC 2547 . . . . . .
IC 2602 . . . . . . . . .
IC 2391 . . . . . . . . .
α Per . . . . . . . . . . .
Pleiades . . . . . . . .
Pleiades . . . . . . . .
NGC 2516 . . . . . .
Ursa M . . . . . . . . .
Coma Berenices
Hyades . . . . . . . . .
Hyades . . . . . . . . .
Praesepe . . . . . . .
Praesepe . . . . . . .
5±1
5±1
8.5±1.5
12+8
−4
25±5
30±5
30±5
50±5
65±15
115±10
115±10
145±5
400±100
500±50
625±50
625±50
757±114
757±114
∗
Excess fraction
[#]
[%]
0/3
6/22
8/21
3/5
7/17
8/18
1/8
1/10
2/5
2/7
5/20
13/51
1/7
0/5
1/12
2/11
1/29
0/5
36.9∗
29.2±9.2
39.1±10.2
57.1+18.6
−18.7
42.1±11.3
45.0±11.1
20.0+12.3
−12.0
16.7+10.4
−10.1
42.9+18.7
−18.6
33.3+15.6
−15.5
27.3+9.5
−9.4
26.4±6.0
22.2+13.5
−13.2
26.4∗
14.3+9.0
−8.8
23.1+11.5
−11.4
6.5±4.1
26.4∗
Excess Age
Reference
1,2
3
3
4
1,2
5,6
1
7
1,2
1
6
2
1
1,2
1
9
10
1
11
12
12
13
14
5,6
15
16
17,18
18,19,20
18,19,20
19,21
22,23,24
25
26
26
10
10
Upper limit
References. — (1) Su et al. (2006); (2) Rieke et al. (2005); (3) Hernández
et al. (2006); (4) Rebull et al. (2008); (5) Young et al. (2004); (6) Gorlova
et al. (2007); (7) Siegler et al. (2007); (8) Gorlova et al. (2006); (9) Cieza
et al. (2008); (10) This work; (11) Preibisch et al. (2002); (12) Briceño et al.
(2005); (13) Ortega et al. (2002); (14) Fuchs et al. (2006); (15) Stauffer et al.
(1997); (16) Barrado y Navascués et al. (2004); (17) Song et al. (2001); (18)
Martı́n et al. (2001); (19) Meynet et al. (1993); (20) Stauffer et al. (1998);
(21) Jeffries et al. (2001); (22) Soderblom & Mayor (1993); (23) Castellani
et al. (2002); (24) King et al. (2003); (25) Odenkirchen et al. (1998); (26)
Perryman et al. (1998)
89
large sample. The peak near ∼ 12 Myr observed by Currie et al. (2008) is suggested. Thereafter, the excess fraction shows a steady decline to the age of Praesepe (∼ 750 Myr). Although the single A7 debris disk star I observed is not a statistically high number, the sample of 29 stars it was drawn from is high enough
to indicate a real lack of debris disks around early-type stars at ∼ 750 Myr.
3.5.3 The decay of the debris disk fraction for solar-type stars
Detailed studies of the frequency of debris disks as a function of system age
are useful tools to characterize belts of planetesimals and their collisions around
solar-type stars. They provide important proxies for comparisons between the
Solar System and exoplanetary systems in terms of planetary system formation
and evolution. For example, observations at 70 µm show that Kuiper-belt-like
planetesimal systems around solar-type stars can be rather common (∼ 16%;
Trilling et al., 2008)(∼ 14%; Hillenbrand et al., 2008), but are not necessarily accompanied by 24 µm excess, which would be indicative of terrestrial planet formation.
To provide a large sample, I merged the 24 µm data of Trilling et al. (2008),
Beichman et al. (2006) and that of the FEPS group (Carpenter et al., 2008, 2009;
Meyer et al., 2008) resulting in a database of 425 solar-type field stars with age
estimates in the range from 3.16 Myr to 10 Gyr. The tables in Trilling et al. (2008)
include the results of Bryden et al. (2006) and Beichman et al. (2005) with their
photometry data reevaluated with the same procedures as the newer Trilling et al.
(2008) sample. I divided this database into the same logarithmic age bins as I
did for the early-type field star sample and calculated the debris disk fraction in
these bins using the 15% threshold in excess emission at 24 µm. The debris disk
fractions are summarized in Table 3.5.
I also compiled results at 24 µm from the literature on debris disk fractions
90
Table 3.5. The field star sample excess ratios at 24 µm for solar-type stars at
certain age bins from the compiled sample of Trilling et al. (2008), Beichman
et al. (2006) and the FEPS collaboration (Carpenter et al., 2008, 2009; Meyer et al.,
2008).
Age
3.16 - 10
10 - 31.6
31.6 - 100
100 - 316
316 - 1000
1 - 3.16
3.16 - 10
Excess fraction
[#]
[%]
Myr .
Myr .
Myr .
Myr .
Myr .
Gyr .
Gyr .
2/12
2/8
2/38
7/48
7/58
2/94
6/167
21.4+10.8
−10.6
30.0+14.4
−14.2
7.5±4.0
16.0±5.1
13.3±4.3
3.1±1.7
4.1±1.5
around solar-type stars in open clusters and associations. They are summarized
in Table 3.6. The excess fractions for the combined sample of solar-type stars
are plotted in the top right panel of Figure 3.9. The plots show a significantly
larger scatter in the excess fractions for solar-type than for early-type stars. A
second interesting feature is a possible environmental effect on the fraction of
debris disks around solar-type stars. Although not pronounced – and possibly
strongly effected by sampling biases – there seems to be higher fraction of debris
disk stars in clusters/associations than in the field.
In Praesepe, the few debris disk candidate stars (from a statistically large sample of 106 stars) implies that the planetary systems in the 1-40 AU zones around
solar-type stars have generally reached a quiescent phase. This behavior can be
compared with that of the field star sample, which levels off at a few percent at
ages > 1 Gyr. This result may seem surprising given the LHB period of the Solar System, but it is actually consistent with the models of Gomes et al. (2005)
and Thommes et al. (2008). The LHB was modeled in these papers to be a result
91
Table 3.6. The excess fraction in [24] for solar-type stars in
clusters/associations.
Name
Age
[Myr]
Orion OB1b . . . .
Upper Sco . . . . . .
Upper Sco . . . . . .
η Cha . . . . . . . . . . .
Orion OB1a . . . .
β Pic MG . . . . . . .
Lower Cen C . . .
Lower Cen C . . .
Upper Cen L . . .
Upper Cen L . . .
NGC 2547 . . . . . .
Tuc-Hor . . . . . . . .
IC 2602 . . . . . . . . .
IC 2391 . . . . . . . . .
α Per . . . . . . . . . . .
Pleiades . . . . . . . .
Pleiades . . . . . . . .
Hyades . . . . . . . . .
Hyades . . . . . . . . .
Praesepe . . . . . . .
5±1
5±1
5±1
8+7
−4
9±2
12+8
−4
16±1
16±1
17±1
17±1
30±5
30±5
30±5
50±5
65±15
115±10
115±10
625±50
625±50
757±114
∗
Excess fraction
[#]
[%]
7/12
5/16
2/5
8/13
4/5
5/25
11/24
5/14
3/11
1/23
8/20
1/7
1/5
5/16
2/13
5/53
5/20
0/67
0/22
1/106
57.1±13.2
33.3+11.1
−11.0
42.9+18.7
−18.6
60±12.6
71.4+16.4
−16.9
22.2±7.9
46.2±9.8
37.5+12.1
−12.0
30.8+12.7
−12.6
8.0+5.1
−5.0
40.9±10.5
22.2+13.5
−13.2
28.6+16.9
−16.4
33.3+11.1
−11.0
20.0+10.1
−10.0
10.9±4.1
27.3±9.4
2.7∗
7.7∗
1.9±1.2
Excess Age
Reference
1
2
3
4
1
5
3
2
3
2
6
5
2
7
2
8
2
9
2
10
11
12
12
13,14
11
15
16
16
12
12
6
17
18
19
20,21
21,22,23
21,22,23
24
24
10
Upper limit
References. — (1) Hernández et al. (2006); (2) Carpenter et al. (2008);
(3) Chen et al. (2005); (4) Gautier et al. (2008); (5) Rebull et al. (2008); (6)
Gorlova et al. (2007); (7) Siegler et al. (2007); (8) Gorlova et al. (2006); (9)
Cieza et al. (2008); (10) This work; (11) Briceño et al. (2005); (12) Preibisch
et al. (2002); (13) Mamajek et al. (1999); (14) Lyo et al. (2004); (15) Ortega
et al. (2002); (16) Mamajek et al. (2002); (17) Rebull et al. (2008), with arbitrary errors adopted from similar age clusters; (18) Stauffer et al. (1997);
(19) Barrado y Navascués et al. (2004); (20) Song et al. (2001); (21) Martı́n
et al. (2001) (22) Meynet et al. (1993); (23) Stauffer et al. (1998); (24) Perryman et al. (1998)
92
of instability in the planetary system, caused by either strong interaction at the
mean motion resonances of Jupiter and Saturn or that of Uranus and Neptune. In
both cases the outer planetary disk is destabilized, causing planetesimals to migrate inward and initiate a collisional cascade. The models of Gomes et al. (2005)
show a wide range of ages (192 Myr – 1.1 Gyr) when the LHB can occur, but they
are more likely to be initiated at the earlier ages. The timing of the cascade depends on a few initial conditions that can be set to realistic parameters to give
any of the solutions. The paper by Strom et al. (2005) also agrees that the LHB
was a catastrophic event, lasting between 10 and 150 Myr, however they argue
that the characteristics of the craters found on the inner planets originating from
that epoch are more likely to be from main belt asteroids.3 The collisional cascade
or ”terminal cataclysm” model is also supported by recent studies of Hadean-era
zircons on Earth (Trail et al., 2007).
3.5.4 Evolutionary differences between the debris disks around early- and solartype stars
To illustrate the differences between the evolution of debris disks around earlyand solar-type stars, I combined the top panel plots in Figure 3.9 in the bottom
panel of the same figure. There appears to be an upper envelope to the excess
fraction as a function of age, as if there were a theoretical maximum number of
debris disks possible at any age. There is substantial scatter below this envelope.
Figure 3.9 shows that there is a subtle difference between the evolution of
debris disks around early- and solar-type stars. To reduce the effects of observational biases (such as detection thresholds) and sampling differences (number
3
Hartmann et al. (2000) and Morbidelli et al. (2001) argued that the LHB was the tail end of
a monotonically decreasing impactor population. This theory was questioned by Bottke et al.
(2007), who computed the probability of the cratering records being created by it, and could rule
it out at a 99.7% (3σ) confidence level.
93
of stars in clusters), I rebinned all the data to a more homogeneous sampling. I
used the same logarithmic age bins as I did for the field star samples: 3.16-10,
10-31.6, 31.6-100, 100-316 Myr, and 0.316-1, 1-3.16, and 3.16-10 Gyr. The result is
shown in Figure 3.10, along with a second plot that shows the decay trends for A,
F and G spectral-type stars separately. The data for all rebinned decay trends are
summarized in Table 3.7. The ”rise-and-fall” characteristics for early-type stars
is confirmed (Currie et al., 2008), but with a quick drop-off at later ages. The
solar-type stars show a monotonic decaying trend that reaches a constant of a
few percent at later ages. The most important feature though is that the trends
have different timescales.
The fraction of infrared excesses at a given age range is set by the interplay
of the occurrence rate of the collisional cascades for each system, the longevity
of the dust produced in these cascades, and my ability to detect the debris at the
distance of the given cluster. Detailed modeling of these processes is required to
interpret the different rate of decline in the debris disk fraction between earlyand solar-type stars. Although such modeling is beyond the scope of this work,
three possible explanations can be invoked to explain qualitatively the faster decline of excess fraction around solar-type stars. First, the dust must be in the 24
µm emitting regions and solar-type stars have about 50× smaller disk surface
area in which a collisional cascade can produce warm enough dust. Second, the
orbital velocity of planetesimals in the 24 µm emitting zone will be higher around
solar-type than the early-type stars, possibly accelerating the evolution of their
debris disks. Third, the dust size distributions and lifetimes are different for the
two groups of stars.
Table 3.7. The percent of debris disks in a rebinned distribution, as a function
of stellar spectral-type.
Age
[yr]
3.16 - 10 M
10 - 31.6 M
31.6 - 100 M
100 - 316 M
316 - 1000 M
1 - 3.16 G
3.16 - 10 G
Early-type stars
Excess fraction
[#]
[%]
20/56
22/52
7/25
33/117
8/100
-
36.2±6.3
42.6±6.7
29.6+8.8
−8.7
28.6±4.1
8.8±2.8
-
Solar-type stars
Excess fraction
[#]
[%]
28/63
37/137
9/67
17/121
8/253
2/94
6/167
44.6±6.2
27.3±3.8
14.5±4.2
14.6±3.2
3.5±1.1
3.1±1.7
4.1±1.5
A-type stars
Excess fraction
[#]
[%]
19/35
11/20
4/6
13/44
5/62
-
54.05±8.2
54.55±10.6
62.50+17.1
−16.9
30.43±6.8
9.38±3.6
-
F-type stars
Excess fraction
[#]
[%]
12/20
19/42
2/8
2/11
4/30
2/52
2/57
59.09±10.5
45.45±7.5
30.00+14.2
−14.4
23.08+11.4
−11.5
15.62±6.3
5.56±3.0
5.08±2.7
G-type stars
Excess fraction
[#]
[%]
3/15
7/39
2/29
10/42
3/70
0/37
0/85
23.5+10.1
−10.2
19.5±6.1
9.7±5.1
25.0±6.5
5.6±2.6
2.6+2.1
−2.2
1.2±1.0
94
95
3.5.5 The results in context with the Late Heavy Bombardment
The cratering record of all non geologically active rocky planets and moons in
the inner Solar System reveal a period of very intense past bombardment. Geochronology of the lunar cratering record shows that this bombardment ended
abruptly at ∼ 700 Myr (see e.g. Tera et al., 1973, 1974; Chapman et al., 2007), but
the scarcity of the lunar rock record prior to this event hinders accurate assessment of the temporal evolution of the impact rates or the length of the bombardment period. Dynamical simulations of different possible impactor populations
show that an unrealistically massive impactor population would be required to
maintain the impact rate measured at the end of the bombardment for a prolonged period, thus convincingly arguing for the bombardment being a shortduration spike in the impact rate (Bottke et al., 2007). A possible explanation for
this is that a dynamical instability initiated by the migration of the giant planets caused minor planetary bodies to migrate inwards from the outer region of
the Solar System, bombarding the inner planets. Modeling shows that this scenario can occur over a wide range of ages (Gomes et al., 2005). Strom et al. (2005)
show that it is possible instead that main belt asteroids bombarded the planetary
system.
I performed a Monte Carlo simulation to evaluate the observed debris disk
fraction in the context of the evidence from the LHB. The goal was to constrain
the fraction of the solar-type stars that undergo LHB (or fine dust generation)
and the duration of these events. I presumed in my models that all LHB events
could be detected in the existing debris disk surveys and that they had an equal
probability of occurring once from 100 Myr to 1 Gyr. Both of these are strong
assumptions. There is significant uncertainty on how much dust was generated
and under what time scales during the LHB, making it difficult to relate the LHB
96
unambiguously to debris disks. However, given that Spitzer measurements of
24 µm excess emission are typically sensitive to a collisional cascade involving
mass on the order of a few lunar masses, and that such an episode has clearing
time scale >
∼ 2 Myr (Grogan et al., 2001), it seems plausible that the destruction
of a few large asteroids can be detected in most observed systems. In my code
I modeled clusters with 135 (106+29) members in 20000 simulations. I varied
the overall percentage of stars that will ever generate a debris disk from 0 to 100
% and the duration of their bombardment episodes from 0 to 500 Myr. If the
number of disks at 750 Myr were within my measured excess fraction of 1-3%,
the simulation was tagged as being consistent with the measurements, else it was
tagged inconsistent. The overall probability of a given parameter pair is given by
dividing the number of consistent simulations at a certain total disk fraction and
duration timescale by the number of simulations (20000).
The calculated probability map is shown in Figure 3.11. The plot shows that
the results are degenerate in the parameter space of dt and pd , with dt being
the duration of a bombardment episode and pd the percentage of stars to ever
undergo such an event. Between the extremes of a very large percentage of the
stars undergoing debris disk generation, but with a very short lifetime (∼ 5-10
Myr) and a very small percentage (< 5 %), with a long (> 300 Myr) lifetime there
is a continuous set of solutions.
My simple model allows the quantitative assessment of the probability of different types of LHB-like episodes. For example, I can exclude at a 3 % significance
level that 60% of the stars undergo major orbital rearrangements, if this leads to
debris production over 100 Myr. Similarly, very short debris producing events
are unlikely, because they would not produce observable disks, inconsistent with
the results.
97
If we seek to evaluate the probability of strictly LHB-like debris producing
episodes, we can fix the length of the episode to 75 Myr, consistent with the duration estimated for the inner Solar System and the other timescales discussed in
§3.5.3. In this case my results show that up to 15-30 % of the stars should undergo such a major orbital reorientation during the first Gyr of their evolution to
be consistent with the modeling.
3.6 Summary
I conducted a 24 µm photometric survey for debris disks in the nearby (∼ 180 pc)
relatively old (750 Myr) Praesepe open cluster. The combined sample of SDSS,
Webda, and 2MASS gave us a robust highly probable cluster member list. With
simultaneous fitting of cluster distance and age I derived a series of solutions for
both parameters as a function of metallicity (see Appendix A). The derived age
for Praesepe is 757 Myr (± 114 Myr at 3σ confidence) and a distance of 179 pc (±
6 pc at 3σ confidence)
Out of the 193 cluster members that I detected at all wavelengths in the combined catalog, 29 were early (B5-A9) and 164 later (F0-M0) spectral types. I found
one star in the early and three in the later spectral type groups that show excess emission. Up to near the completeness limit, with one debris disk star, there
are 106 sources in the later spectral-type sample. This result shows that only
6.5 ± 4.1% of early- and 1.9 ± 1.2% of solar-type stars are likely to possess debris
disks in the 1-40 AU zones. These values are similar to that found for old (> 1
Gyr) field stars.
I place my results in context with the Late Heavy Bombardment theory of the
Solar System. With simple Monte Carlo modeling I show that the observations
are consistent with 15-30% of the stars undergoing a major re-arrangement of the
98
planetary orbits and a subsequent LHB-like episode once in their lifetime, with a
duration period of 50-100 Myr.
I also summarize the results in the literature on the decay timescales of debris
disks around early- and solar-type stars. I find that the decay timescale for solartype stars is shorter than for earlier-type stars.
99
90
Field stars
Clusters/Associations
80
[24] excess fraction (%)
[24] excess fraction (%)
90
70
60
50
40
30
20
10
0 B5-A9 stars
1 Myr
10 Myr
100 Myr
1 Gyr
10 Gyr
80
70
60
50
40
30
20
10
F-K stars
0
1 Myr
10 Myr
Age (Myr)
90
Field Stars
Clusters/Associations
100 Myr
1 Gyr
10 Gyr
Age
Field Stars (early-type)
Field Stars (solar-type)
Clusters/Associations (early-type)
Clusters/Associations (solar-type)
80
[24] excess fraction (%)
70
60
50
40
30
20
10
Full sample
0
1 Myr
10 Myr
100 Myr
Age
1 Gyr
10 Gyr
Figure 3.9 Top Left Panel: The decay of the debris disk fraction for early type stars.
Top Right Panel: The decay of the debris disk fraction for solar-type stars. Bottom
Panel: The combined plot of all excess fractions. The errors in excess fraction are
the 1σ errors from the beta distribution calculations (§3.5.1) while the age errors
are from the literature. The age ”errors” for the field star sample show the age
bins.
100
60
early-type stars
solar-type stars
[24] excess fraction (%)
50
40
30
20
10
0
1 Myr
10 Myr
100 Myr
Age
1 Gyr
80
10 Gyr
A stars
F stars
G stars
70
[24] excess fraction (%)
60
50
40
30
20
10
0
1 Myr
10 Myr
100 Myr
Age
1 Gyr
10 Gyr
Figure 3.10 Top panel: The difference between the decaying trend for early- and
solar-type stars, in a binned data plot. Bottom panel: The difference between the
decaying trend for A, F and G spectral-type stars, in a binned data plot. The
errors in the excess fraction are from beta distribution calculations (§3.5.1), while
the ”error bars” in the ages show the age bins. The numerical values for the data
points are summarized in Table 3.7.
101
Probability of detecting 1-3% Debris Disk Fraction at 750 Myr
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Percentage of Stars with a Bombardment Episode [%]
100
80
60
40
20
0
0
100
200
300
400
Duration of the Bombardment [Myr]
500
Figure 3.11 The probability of detecting the 1-3% (2/(106+29)) debris disk fraction
observed at Praesepe, as a function of the percentage of stars that undergo LHB
type debris disk generation and the duration of the events. The contour lines are
at 20, 40 and 60% probability.
102
C HAPTER 4
M ODELING C OLLISIONAL C ASCADES I N D EBRIS D ISKS
I develop a new numerical algorithm to model collisional cascades in debris
disks. Because of the large dynamical range in particle masses, I solve the integrodifferential equations describing erosive and catastrophic collisions in a particlein-a-box approach, while treating the orbital dynamics of the particles in an approximate fashion. I employ a new scheme for describing erosive (cratering) collisions that yields a continuous set of outcomes as a function of colliding masses.
I demonstrate the stability and convergence characteristics of my algorithm and
compare it with other treatments. I show that incorporating the effects of erosive
collisions results in a decay of the particle distribution that is significantly faster
than with purely catastrophic collisions.
4.1 Introduction
More than 500 extrasolar planets have been identified to date in over 400 planetary systems.1 Most of these planets were discovered via radial velocity measurements. As a result, only a handful of them are less than 10 Earth masses; the
vast majority are gas giants resembling Jupiter. They are also in extremely close
orbits to their host stars, making these systems dramatically different from ours.
A large number of additional candidate systems have been found recently with
the Kepler mission (Borucki et al., 2010).
In contrast to the great majority of known exoplanet systems, our own solar
system has a complex configuration with gas giants at significant distances from
their central star and rocky planets/asteroids within the giant planet zone. The
1
http://exoplanet.eu
103
detection of rocky planets and planetesimals around other stars is difficult. One
of the most productive approaches is indirectly via the thermal emission of their
planetary debris dust belts. Ever since the discoveries with IRAS, we know that
extrasolar systems harbor disks of dust/debris that are generated by planetesimal
collisions and are similar to our Kuiper belt but much more massive (Aumann
et al., 1984; Backman & Paresce, 1993). The dust reprocesses the stellar light and
emits it as thermal radiation in the infrared, submillimeter and radio wavelength
regime. A prototypical example of such a system is Fomalhaut, where a planet
is shepherding the star’s debris disk resolved in both scattered light (Kalas et al.,
2008) and in infrared emission (Holland et al., 2003; Stapelfeldt et al., 2004; Marsh
et al., 2005). Debris disks highlight the constituents of planetary systems that are
many to hundreds of AU away from their stars.
With the launch of the Spitzer Space Telescope, many observations have been
obtained to detect and possibly to resolve debris disks in the infrared regime.
Debris disks have been probed around all types of stars, both in stellar clusters
and in the field. These observations showed that even though debris disks are
common around stars of all spectral types, they are more likely to be detected
in the earlier stages of stellar evolution (Wyatt, 2008). We have also learned that
debris disks may be located close to or far from their central stars (Morales et al.,
2011), that there are systems with multiple debris rings (Hillenbrand et al., 2008,
such as our solar system), and that there can be wide varieties of mineralogical
compositions within the disks (Weinberger et al., 2003; Currie et al., 2011). Debris disk studies are now a major component of the Herschel observing program
(Matthews et al., 2010; Eiroa et al., 2010), which will provide substantial advances
in our understanding of their outer zones.
Interpreting these results demands theoretical insights in a variety of areas.
104
For example, attempts have been made to understand the evolution of debris
disks as a function of stellar type by studying them in stellar clusters of different
ages. As concluded in Gáspár et al. (2009), solar-type stars in the field (Beichman et al., 2006; Trilling et al., 2008; Carpenter et al., 2008, 2009) and in clusters
(Gorlova et al., 2006, 2007; Siegler et al., 2007) may show a faster decay trend
compared to that observed for earlier-type stars (Rieke et al., 2005; Su et al., 2006),
although the difference is subtle and needs confirmation. The decay trends of the
fractional luminosity (fd = Lexc /L∗ ) show a large range in values. Spangler et al.
(2001) find a decay ∝ t−1.76 when fitting ISO/IRAS data, while Greaves & Wyatt
(2003) get a much shallower decay ∝ t−0.5 . The majority of surveys however find
a decay ∝ t−1 (Liu et al., 2004; Moór et al., 2006; Rieke et al., 2005). A better theoretical understanding is needed to sort out these results and to provide testable
hypotheses that can be compared with the observations.
Only a handful of debris disks have been resolved; for the majority, we only
know the integrated infrared excess emission. Finding the underlying spatial
distribution of the debris in these disks is not straightforward, as any spectral
energy distribution (SED) can be modeled with a degenerate set of debris rings
at different distances. Although much of the uncertainty is associated with the
optical constants of the grains, another under-appreciated issue is the grain size
distribution. Collisional models can reduce the number of free parameters in the
SED models by determining the stable size distribution of particles in the disks.
Observations of resolved debris disks also have raised questions that can best
be addressed by theoretical models. For example, Spitzer MIPS images have
shown a significant extended halo of dust around Vega (Su et al., 2005), both
at 24 and 70 µm. Initial calculations hypothesized the halo around Vega to be a
result of a high outflow of dust due to radiation pressure from a recent high-mass
105
collisional event (Su et al., 2005), while Müller et al. (2010) model it as a result of
weakly bound particles on highly eccentric orbits. Further modeling and deep
observations of additional systems will help distinguish these two possibilities.
In this chapter, I describe a new algorithm for modeling debris disks, in which
I refine the physics and numerical methods used in collisional cascade models. In
§4.2, I briefly outline previous models and introduce the basics of my algorithm.
In §4.3, I detail my numerical methods, followed in §4.4 by my approach for including simplified dynamics. In the last section, I compare my numerical algorithm to previous ones and discuss in detail the differences between the codes
and the effects those differences have on the outcome of the collisional cascades.
I also supplement the work with an extended appendix that covers the numerical
methods and the verification tests of the code.
4.2 The physical and numerical challenges of modeling debris disks
Collisional cascades have been studied both analytically and using collisional
integro-differential numerical models. The classic analytic models of Dohnanyi
(1969), Hellyer (1970), and Bandermann (1972) took into account both erosive
and catastrophic collisional outcomes, assumed a material strength that was independent of the particle mass, a particle mass distribution with no cut-offs, and
a constant interaction velocity. They yielded steady state power-law mass distribution indices of -11/6. This result was in general agreement with the measured
size distribution of asteroids in the solar system. More recently, analytic models
by Dominik & Decin (2003) and Wyatt et al. (2007) showed that the fractional infrared luminosity in a collision-dominated steady-state system decays following
a t−1 power-law, in agreement with most observations. Wyatt et al. (2007) also
derived a maximum mass and fractional luminosity as a function of age and dis-
106
tance from the central star, which they then used to classify systems with possible
recent transient events.
However, numerical models are needed to expand on these results. In the
particular case of our Solar System, sophisticated numerical models were developed to track the evolution of the largest asteroids (Greenberg et al., 1978). They
have been further improved to reproduce the observed wavy structure in the size
distribution at the very highest masses (e.g., O’Brien & Greenberg, 2005; Bottke
et al., 2005). These models yield power-law distributions that deviate from the
classic solution of Dohnanyi (1969), with certain regions steeper than it and others shallower. Using a steeper or shallower distribution and extrapolating it to
dust sizes can result in substantial offsets in the number of particles and thus in
the infrared emission originating from them for a given planetesimal mass. Conversely, the particle size distribution affects the underlying disk mass calculated
from the observed infrared emission.
A complete numerical model of collisional cascades would follow outcomes
from all types of collisions, include a kinematic description of the system, incorporate coagulation below certain thresholds, and do all this with high numerical
fidelity. Although such a model has not yet been built because of its complexity,
there are a number of approaches in the literature that model collisional cascades
down to particles of micron size, each with distinctive strengths and weaknesses.
Dullemond & Dominik (2005) modeled the coagulation of dust particles to
study the formation of planetesimals in protoplanetary systems. They show that
fragmentation is important even at early ages. A purely coagulating system loses
all of its dust in less than a million years, which is inconsistent with observations.
Their models suggest that protoplanetary disks reach an equilibrium between
grain growth and fragmentation, which maintains their infrared signatures for a
107
few million years.
The collisional code ACE has been used in many studies (Krivov et al., 2000,
2005, 2006, 2008; Löhne et al., 2008; Müller et al., 2010). It follows the evolution
of the particle size distributions as well as the spatial distribution of the dust in
debris disks. The code initially only accounted for collisions resulting in catastrophic outcomes, while the latest version (Krivov et al., 2008; Müller et al., 2010)
includes erosive (cratering) events as well. The collisional outcome prescriptions
are based on the Dohnanyi (1969) particle-in-a-box model, but with a more elaborate description of material strengths in collision outcomes as well as the radiation force blowout. The strength of the code is that it calculates the dynamical
evolution of the systems, as well. Since following the dynamical evolution of a
system makes large demands on computer memory space and CPU speed, the
code can only model the size distribution with a low number of mass grid points;
it originally used a first order Euler Ordinary Differential Equation (ODE) solving algorithm, but has been modified to include a more precise one (A. Krivov,
priv. comm.). Krivov et al. (2006) and Müller et al. (2010) applied this algorithm
to debris disks in general and to the specific example of the Vega system. They
followed the orbital paths of fragments and placed special emphasis on radiation
effects. Löhne et al. (2008) modeled debris disk evolution around solar-type stars
and found, both with analytic and numerical analysis, that the majority of physical quantities, such as the mass and the infrared luminosity, decrease with time
as t−0.3 to t−0.4 . This is in contrast with the observed t−1 decay found by some
obervations (Liu et al., 2004; Moór et al., 2006; Rieke et al., 2005). However, the
population synthesis verification tests in Löhne et al. (2008) yield good agreement
with the latest Spitzer observations.
Thébault et al. (2003, 2007) study the evolution of extended debris disks with a
log10[n(m)]
108
Lightly Erosive
40
Erosive/Catastrophic
Super-Catastrophic
20
Y
X
YX
0
YX
-20
-20
-10
0
10
log10(m) (kg)
20
-20
-10
0
10
log10(m) (kg)
20
-20
-10
0
10
log10(m) (kg)
20
Figure 4.1 Illustration of the possible outcome scenarios of collisions. In all collisions, a single largest X fragment is created as well as a power-law distribution
of fragments with a largest mass of Y .
particle-in-a-box algorithm. They include both catastrophic and erosive collisions
and employ resolution and numerical methods similar to the ones implemented
in the ACE code. They model the extended disk structure by dividing the disk
into separate, but interacting rings. Their model does not include dynamics.
Campo Bagatin et al. (1994) showed that a series of wave patterns is produced
in the mass distribution of particles when a low-mass cutoff is enforced, such
as in the case of a radiation force blowout limit. This signature is produced by
the other debris disk numerical models as well (Thébault et al., 2003; Thébault &
Augereau, 2007; Krivov et al., 2006; Löhne et al., 2008; Wyatt et al., 2011). However, the conditions under which these waves are produced have not been completely analyzed. Wyatt et al. (2011) do show that the amplitude and wavelength
of the waves is collisional velocity dependent. Such strong features in the particle
size distribution are not observed in the dust collected within the solar system.
The interplanetary dust flux model of Grun et al. (1985), which used in situ satellite measurements of the micro-meteroid flux in the solar system, and the terrestrial particle flux measurements of the LDEF satellite (Love & Brownlee, 1993)
109
only show a single peak at ∼ 100µm in the dust distribution. However, these
measurements detected particles that were brought inward from the outer parts
of the solar system via Poynting-Robertson drag and particles removed from the
inner parts of the solar system via radiation force blowout. Their results are reflections of more than a single parent distribution and of multiple physical effects.
The dynamical code of Kuchner & Stark (2010) models the evolution and 3D
structure of the Kuiper belt, with a Monte Carlo algorithm and a simple treatment of particle collisions. Their models predict that grain-grain collisions are
important even in a low density debris ring such as our Kuiper belt.
Kuchner & Stark (2010) and Dullemond & Dominik (2005) both emphasize the
strong effects of fragmentation in their models. Müller et al. (2010) point out that
including erosive (cratering) events is necessary for their models to reproduce the
observed surface brightness profiles of Vega. Thébault & Augereau (2007) also
show that a complete collisional treatment will result in significant deviations
from the classic power-law solution.
My goal is to set up a numerical model that places special emphasis on investigating these issues. My new empirical description of collisional outcomes avoids
discontinuities between erosive and catastrophic collisions and thus enables a
more stable and accurate calculation. I also solve the full scattering integral, thus
ensuring mass conservation and the propagation of the largest remnants of collision outcomes. Finally, I use second order integration and fourth order ODE
solving methods to improve the numerical accuracy. Below, I outline the physical
and numerical techniques I will employ. In the Appendix, I present verification
tests of these treatments.
110
4.2.1 Collisional outcomes
In collision theory, two types of outcomes are generally distinguished: catastrophic and erosive (the latter also known as cratering). In the case of a catastrophic collision (CC), both colliding bodies are completely destroyed and their
masses are redistributed in a power-law distribution. In the case of an erosive
collision (EC), the projectile is much smaller than the target, resulting in one big
fragment whose mass is close to the original target mass. The cratered mass plus
the original projectile mass is redistributed also in a power-law distribution. The
boundary between these two classifications is drawn by convention at the point
where the largest fragment mass is half of the original target mass. I illustrate
these outcomes in Figure 4.1.
In the first panel of Figure 4.1, I plot the outcome distribution of an erosive
collision, where there is a single large X fragment and a distribution of dust at
much lower masses. The X fragment is over half the mass of the original target
mass M. The redistributed mass is equal to the cratered mass plus the projectile
mass. The largest fragment in the distribution, Y , is arbitrarily set to be 20% of
the cratered mass. In §4.3.3 I elaborate on the validity of this arbitrary value.
In the second panel of Figure 4.1, I plot the outcome at the boundary case
between catastrophic and erosive collisions, where the single largest fragment X
is exactly half of the original target mass M. The redistributed mass is equal to
the other half of the target mass M plus the projectile mass. The largest fragment
in the distribution, Y , is arbitrarily set to be 20% of the cratered mass here as well
(10% of the target mass).
Finally, in the third panel of Figure 4.1, I plot the outcome of a super-catastrophic collision, where the target and projectile masses are equal. The mass of
the single largest fragment X is given by the relation of Fujiwara et al. (1977).
111
The redistributed mass is equal to M − X plus the projectile mass. The largest
fragment in the distribution, Y , is arbitrarily set to be at 0.5X.
In reality, there is no strict boundary between catastrophic and erosive collisions (Holsapple et al., 2002). The outcomes between these two extreme scenarios
should be continuous. In laboratory experiments, however, it is easier to test the
extreme outcomes. In my model, I use the laboratory experiments to describe the
extreme solutions and connect them with simple interpolations throughout the
parameter space. I revise the currently used models to include an X fragment
for both erosive and catastrophic collisions as a separate new gain term, thus
avoiding a discontinuous mathematical assumption. In this treatment, the placement of the X fragments is grid size independent, further improving precision
and guaranteeing the accurate downward propagation of these fragments. I am
able to express the loss term in a much simpler form, including collisions from
both regimes. Previous models only included a full loss term for catastrophic
collisions and removed fractions of particles for erosive collisions.
The slope of the power-law particle redistribution has been studied extensively. Dohnanyi (1969) used a single power-law value from the largest mass
to the smallest. Later experiments have shown that a double (or even a triple)
power-law distribution is a more likely outcome (see, e.g., Davis & Ryan, 1990).
This has led to the widespread use of a double power-law for the redistribution
in numerous collision models. I conducted numerical tests that demonstrated
that there is negligible difference in using a wide range of slopes with a single
power-law. Therefore, I have used the simplest method of redistributing the
fragmented particles with a single power-law slope from the second largest Y
fragment downwards, scaled to conserve mass. As a nominal value, I will use
the classic -11/6 redistribution slope.
112
4.2.2 Incorporating the complete redistribution integral
The classic solution to the collisional evolution of an asteroid system involves
solving the Smoluchowski (1916) integro-differential equation. This was first
done by Dohnanyi (1969). Because erosive collisions remove only a small part
of the target mass in a collision, Dohnanyi (1969) expressed the erosive removal
term in a differential form. This is not appropriate for my case. My system has
well defined boundaries; thus a continuity equation cannot be used. The locality
of the collisional outcomes in phase space is not certain either.
Therefore, to solve the Smoluchowski equation for the problem at hand, I need
to solve the full scattering integral. This is complicated numerically, as the integrations must extend over the entire dynamical range of ∼ 40 orders of magnitude in mass. To be able to perform accurate integrations over such a large
interval, I need to use a large number of grid points and sophisticated numerical
methods. To achieve this in a reasonable time, I drop the radial dependence of
the various quantities and to solve the equations under a “particle-in-a-box” approximation. With this approach, I lose radial and velocity information but gain
accuracy.
4.2.3 The effect of radiation forces
Poynting-Robertson drag can be an effective form of removing particles from the
disk, so I include it in my model. However, the strongest and most dominating
radiation effect is the removal of particles via radial radiation forces. These act on
orbital timescales and can remove or place particles on extremely eccentric orbits.
This gives rise to the challenge of incorporating a radial dependent removal term
into a particle-in-a-box model that does not carry radial information. In §4.3.1.1
and §4.3.1.2, I discuss my approach for incorporating these radiation effects.
Stellar wind drag is an important dust removal effect for very late-type stars,
113
such as in the case of AU Mic (Augereau & Beust, 2006; Strubbe & Chiang, 2006),
which is an M1 spectral-type star. I concentrate on modeling debris disks in earlyand solar-type systems, so I chose to neglect the effects of stellar wind drag. I do
not take into account the Yarkovsky effect either, as it is small in high-density
debris disks compared to the other radiation effects.
4.3 The collisional model
I now discuss my collisional code (CODE-M - COllisional Disk Evolution Model),
which solves a system of integro-differential equations that describe the evolution of the number densities of particles of different masses. The code includes
outcomes from erosive (cratering) collisions and catastrophic collisions and qualitatively follows the effects originating from radiation forces and Poynting-Robertson drag.
The system-dependent parameters are: the spectral-type of the central star
(which defines the stellar mass M∗ and the magnitude of the radiation effects it
will have on the particles), the minimum and maximum particle masses (mmin
and mmax , respectively), the radius, width, and height of the debris ring (R, ∆R,
and h, respectively), the total mass within the ring (MTOT ), and the slope of the
initial size distribution of the particles (η). I estimate the total volume of the
narrow ring, V, as
V = 2πhR∆R,
which together with MTOT defines a mass density.
(4.1)
114
4.3.1 The evolution equation
In general, the change in the differential number density n(m, t) at any given time
for a particle of mass m is given by (Smoluchowski, 1916)
d
n(m, t) = TPRD + Tcoll ,
dt
(4.2)
where TPRD is the Poynting-Robertson drag (PRD) term and Tcoll is the sum of the
collisional terms. I define the differential number density of particles such that
N(t) =
Z
n(m, t)dm
(4.3)
is the time-dependent total number density of particles within the ring.
Effects such as radiation force blowout and Poynting-Robertson drag are able
to deplete the low-mass end of the distribution, which in turn alters the evolution
of the disk and more importantly, its infrared signature. Because I do not follow
the radial profile of the various debris disk quantities in my algorithm, I can only
capture the effects of radiation forces in a simplified way.
4.3.1.1 Poynting-Robertson drag term
Poynting-Robertson drag has an important effect on the orbits of particles around
stars. A complete analysis is given by Burns et al. (1979), who correct many errors
made in previous work.
Poynting-Robertson drag arises from the fact that particles re-radiate the energy they absorb from the central star preferentially in their direction of motion.
This eventually causes the particles to slow in their orbit and follow an inward
spiral. Burns et al. (1979) show that the change in the orbital distance due to
Poynting-Robertson drag can be written as
dR(m)
2GM∗ β(m)
=−
,
dt
cR
(4.4)
β(m)
115
10
2
10
1
10
0
10
-1
10
-2
Particle size (µm)
0.1
1
0.01
10
100
A0
F0
G0
G5
K0
10-3
10-4
10-20
10-18
10-16 10-14 10-12
Particle mass (kg)
10-10
10-8
Figure 4.2 Calculated values for the radiation-force parameter β around stars of
spectral-type A0, F0, G0, G5 and K0. The thin double-dashed black line is at
the critical value β=0.5, above which radiation forces are able to remove particles
from circular orbits.
where G is the gravitational constant, c is the speed of light, and β(m) is a parameter for a particle of mass m that measures the ratio of radiation to gravitational
force the particle experiences.
I calculate the β(m) values as a function of the particle masses, optical constants, and the spectral type of the central star following Gáspár et al. (2008). For
the calculations I assume a silicate composition for the particles and a bulk density of 2.7 g cm−3 . I show the calculated β(m) values for a few different spectral
type stars in Figure 4.2.
I use equation (4.4) to derive an approximate term that captures the effect of
116
Poynting-Robertson drag as (see eq. [4.2])
TPRD = −
n(m, t)
,
τPRD (m)
(4.5)
where
τPRD (m) =
c
R∆R .
2GM∗ β(m)
(4.6)
The mass dependence of the timescale comes from the mass dependence of the
parameter β. In principle, once a particle is removed from the collisional system
it still radiates in the IR; it just does not take part in the collisional cascade. I keep
track of the removal rate of these particles, but do not follow the total amount
removed or their infrared emission.
4.3.1.2 Radiation force blowout
The effects of the radiation force blowout are incorporated in my code with the
simplified dynamics treatment introduced in §4.4, and not by the inclusion of a
separate term in the differential equation as are the effects of Poynting-Robertson
drag.
Removing a particle from the collisional system via radiation force blowout
requires roughly an orbital timescale
τRF B = 2π
s
R3
.
GM∗
(4.7)
As I will show in §4.4, under my assumptions a newly created particle of mass
m gets removed via radiation force blowout if β(m) ≥ 0.5 and is unaffected by
radiation forces when β(m) < 0.5.
Although the radiation force blowout timescale is not used in my code, in
Figure 4.3 I compare it to the Poynting-Robertson drag timescale around an A0
spectral-type star, assuming a disk width-to-radius ratio of 0.1. The plot shows
that within reasonable disk radii estimates, radiation force blowout will always
R (AU)
117
10
1
10
0
10
Particle size (µm)
0.1
1
0.01
10
1
-1
0.1
10
0.01
-2
10-3
10-4
0.001
τPRD=τRFB
τPRD (yr)
τRFB (yr)
10-20
10-18
0.0001
(dR/R=0.1, sp=A0)
10-16
10-14
Particle mass (kg)
10-12
Figure 4.3 Comparison of the radiation force blowout (RFB) to the PoyntingRobertson drag (PRD) timescales for an A0 spectral-type star, as a function of
particle mass and distance from the star, with a disk width of dR/R = 0.1. The
dashed lines give the orbital distances as a function of particle size, where the
Poynting-Robertson drag and blowout timescales are 0.1, 0.01, 0.001 and 0.0001
years. The solid red line gives the distance where the timescale for PoyntingRobertson drag is equal to the radiation force blowout timescale. Above the solid
red line, radiation force blowout dominates, while below it Poynting-Robertson
drag does. The plot shows that within reasonable disk radii estimates, radiation
force blowout will always dominate in the β(m) > 0.5 domain, while outside of
it Poynting-Robertson drag will be the stronger effect.
dominate in the β(m) > 0.5 domain, while outside of it Poynting-Robertson drag
will be the stronger effect. Whether Poynting-Robertson drag is an effective form
of removal in the β(m) < 0.5 domain depends on the number density of particles
118
in the ring, i.e., the collisional timescale of the system (Wyatt et al., 2005). The
outcomes are similar for all spectral type stars and for realistic ∆R/R values.
4.3.2 The collisional term
The probability of a collision between two particles is a function of their number
densities and their collisional cross section. I express the collisional cross section
for particles of mass m and m′ as
2
σ (m, m′ ) = π [r (m) + r (m′ )] ,
(4.8)
where r(m) is the radius of each particle. I express the differential rate of collisions between the two masses as
P (m, m′ ; t) = n(m, t)n(m′ , t)V σ (m, m′ )
= n(m, t)n(m′ , t)V π [r (m) + r (m′ )]
1
1 2
= κn(m, t)n(m′ , t)V π m 3 + m′ 3
2
(4.9)
where V is their characteristic collisional velocity, n(m, t) and n(m′ , t) are the differential number densities for particles of mass m and m′ ,
κ≡
3
4πρ
23
,
(4.10)
and ρ is the bulk mass density of the particles. The number densities in the problem are naturally all time dependent. However, for brevity, hereafter I drop the
time dependence from my notation.
The decrease or increase in number density at a certain mass will be determined by three separate events: the removal of particles caused by their interaction with all other particles, the addition of the X particles from the interactions
of other particles (see §4.2.1 and Figure 4.1), and the addition of particles from the
redistribution of smaller fragments originating from collisions of other particles.
119
I express the first event, which describes the removal of particles, as
m
Zmax
d
dm′ P (m, m′ ) .
n(m) = −
dt
rem
(4.11)
mmin
I completely remove all particles from all grid points if they take part in a collision, even if they are the target objects in erosive collisions.
The second event to be described is the addition of the large X fragments. To
this end, I need to calculate the mass M that will produce a particle of mass m =
X when interacting with a particle of mass m′ . I achieve this with a root finding
algorithm from the collisional equations presented in the following sections and
calculate it only once in the beginning of each run. In equation form,
µZ
X (m)
d
dm′ P (M, m′ ) .
=
n(m)
dt
′
m≡X(m ,M )
(4.12)
mmin
The lower limit of the integration is the minimum mass in the distribution. I
denote the largest mass m′ that can create a particle of mass m as the X fragment
as µX (m). Its value can also be calculated via root finding algorithms and has to
be calculated for each value of m once in the beginning of each run (see Figure
D.2 in the Appendix).
These first two integrals may catastrophically cancel, meaning that the difference between the two terms may be significantly smaller than the absolute value
of each, causing the former to be artificially set to zero when evaluated numerically. It is therefore useful to combine these terms into a single integral in a way
that will lessen the probability of catastrophic cancellation:
( µZ
X (m)
1
1
1 2
1 2
′
′
′3
′3
3
3
TI (m) =
−V πκ
dm n(m ) n(m) m + m
− n(M) M + m
mmin
+
m
Zmax
µX (m)
1
1
′3
dm′ n(m′ )n(m) m 3 + m
2
)
,
(4.13)
120
Unfortunately TI can still suffer from catastrophic cancellation (when m′ is much
smaller than m, and by definition only for the first integral). I overcome this issue
by employing a Taylor-series expanded form of TI , as given in the Appendix.
The third event in the collisional term is the addition of the power-law fragments back to the distribution. The description of this process is quite simple;
however, its precise calculation is not. I write in general
TII (m) =
Z
mmax
mmin
dµ
Z
mmax
dMP (µ, M)A(µ, M) × H [Y (µ, M) − m] m−γ , (4.14)
µ
where µ is the projectile mass, M is the target mass, A is the scaling of the powerlaw distribution, and H is the Heaviside function. The total redistributed mass
is
Mredist. (µ, M) =
Z
Y (µ,M )
A(µ, M)m−γ+1 dm ,
(4.15)
0
where Y is the largest fragment within the redistribution (i.e., the second largest
fragment in the collision, after X; see §4.2.1). This gives the scaling factor
A(µ, M) =
(2 − γ) Mredist. (µ, M)
.
Y 2−γ (µ, M)
(4.16)
The precision of this integration depends strongly on the resolution of the grid
points, due to the integration limits set by the Heaviside function. I discuss in
detail the integration methods I used in the Appendix.
4.3.3 Collision outcomes
The collisional equations can be integrated if the values of X(µ, M), Y (µ, M), and
Mredist (µ, M) are known as a function of the colliding masses. Their values are
strongly dependent on the outcome of the collision they originate from, which is
determined by the energies of the colliding parent bodies. I show the domains
of erosive, interpolated erosive (explained later in the section), and catastrophic
collisions as a function of the colliding body masses in Figure 4.4 for collisional
121
In
te
rp
ol
at
io
ed
ns
1 km
Erosive Collisions
1 km
1m
tro
C
at
as
a(M)
C
ol
ic
C
ol
C
at
as
tro
ph
ic
C
ol
ic
tro
as
C
at
0
a(µ)
1m
lis
Erosive Collisions
ph
5
1 mm
In
te
rp
ol
at
io
ed
ns
1 km
lis
10
a(µ)
1m
lis
Erosive Collisions
15
log10(M) (kg)
1 mm
In
te
rp
ol
at
io
ed
ns
20
1 km
ph
a(µ)
1m
1 mm
1 mm
-5
-1
-1
V = 0.5 km s
-10
-10
-5
0
5 10
log10(µ) (kg)
15
20
-1
V = 1.0 km s
-10
-5
0
5 10
log10(µ) (kg)
15
20
V = 1.5 km s
-10
-5
0
5 10
log10(µ) (kg)
15
20
Figure 4.4 The outcome possibilities as a function of colliding masses plotted for
collisional velocities of 0.5, 1 and 1.5 km s−1 . These collisional velocities roughly
correspond to debris ring radii of 100, 25 and 10 AU around an A spectral type
star, respectively (see §4.4). I note that higher collisional velocities can occur in
some systems.
velocities of V = 0.5, 1.0 and 1.5 km s−1 . I introduce the method for calculating
collisional velocities from orbital velocities in §4.4, but it is a good general approximation that the collisional velocity is roughly an order of magnitude smaller than
the orbital velocity. These collisional velocities will then correspond to debris ring
radii of 100, 25 and 10 AU around an A spectral type star, respectively.
A collision is considered to be catastrophic if
Q(µ, M)impact ≡
µV2
≥ Q∗ (M) ,
2M
(4.17)
where Q∗ (M) is the dispersion strength parameter of the target mass M, µ is the
projectile mass, and V is the relative velocity of the projectile compared to the
parent ring (§4.3.1). I use the dispersion strength description of Benz & Asphaug
(1999) and discuss my choice in the Appendix. Note that, in a more accurate
treatment, I would redistribute the relative kinetic energy to both masses and
122
not just to the target mass (i.e., divide by µ + M instead of M). I am, however,
using the original definition of Qimpact (as opposed to using the relative kinetic
energy) because the Q∗ (M) values that I will be comparing it to were defined
the same way (Benz & Asphaug, 1999) and this definition makes the problem
more tractable numerically. I note that some work has indicated that the tensile
strength curve itself may be collision velocity dependent (Holsapple et al., 2002;
Stewart & Leinhardt, 2009), which I currently do not take into account.
In catastrophic collisions both particles are completely destroyed. Based on
experimental evidence (Fujiwara et al., 1977; Matsui et al., 1984; Takagi et al.,
1984; Holsapple et al., 2002), I will assume that apart from the largest fragment
X(µ, M), the total mass is redistributed as a power-law distribution of particles
up to a mass that I denote as Y (µ, M). I calculate the largest single mass created
using the relation (Fujiwara et al., 1977)
−βX
µV2
1
.
X(µ, M) = M
2 2MQ∗ (M)
(4.18)
At the separatrix between catastrophic and erosive collisions, Q(µ, M)impact =
Q∗D (M), and X(µ, M) = M/2, which is exactly what I expect. The βX factor is
measured to be 1.24 by Fujiwara et al. (1977) and this is the fiducial value that
I use. Some experiments have shown that the shape and material of the target
have an effect on the exact value of βX (Matsui et al., 1984; Takagi et al., 1984).
The second largest fragment, Y , is always a fraction (0 < fY < 1.0) of the
cratered mass, Mcr , in the erosive collision domain up to the erosive/catastrophic
collision boundary. In the catastrophic collision domain, Y is a fraction (fX ) of
the X fragment. I interpolate fX from its value defined by fY at the separatrix
(where fY = fX , as X = Mcr ) to a specified value fXmax at the super catastrophic
123
collision case of µ = M as
i
h

µ
max ln


2Q∗ (M )M V −2
f
i .
h
fX = exp ln (fY ) + ln X


M
fY
ln
(4.19)
2Q∗ (M )M V −2
My fiducial values for these fractions are fY = 0.2 and fXmax = 0.5. I express the
remaining mass in catastrophic collisions as
Mredist (µ, M) = µ + M − X(µ, M) ,
(4.20)
which is redistributed as a large number of smaller particles.
I can check the plausibility of the fY = 0.2 value by taking the asteroid family
Eunomia as an example. Its largest member is said to be 70% of the original parent body’s (meaning that the collision generating the family was erosive). The asteroid family’s largest member is 264 km in diameter, while the second is 116 km
(Leliwa-Kopystyński et al., 2009). Assuming equal densities, the second ranked
body therefore accounts for 20% of the mass removed from the parent body. Since
this is a single case, I cannot be sure that this is a good description in every case.
Erosive collisions are more complicated and less well understood. A collision
will be erosive if
Q(µ, M)impact ≡
µV2
< Q∗D (M) .
2M
(4.21)
As described in §4.3, an erosive collision will result in a single large fragment,
which will be a remnant of the target body, and a distribution of smaller particles.
I use the formula of Koschny & Grün (2001a,b), i.e.,
µV 2
Mcr = α
2
b
,
(4.22)
to calculate the total mass cratered from the target, where α and b are constants,
with fiducial values of α = 2.7 × 10−6 and b = 1.23. This formula is only valid
for small cratered masses; it can lead to artificially high values for the cratered
124
masses (much larger than the target mass) even in the erosive collision domain.
When the cratered mass given by this formula is larger than an arbitrarily set
fraction fM of the target mass, I use the following interpolation formula


ln µV 2 /Q 

l
2M
0.5
,
Mcr = M × exp ln(fM ) + ln

fM ln [Q∗D (M)/Ql ] 
(4.23)
(4.24)
where
Ql =
fM 1−b
M
α
1/b
.
I choose an arbitrary fiducial value for fM of 10−4 .
In erosive collisions, the single large fragment is expressed as
X = M − Mcr .
(4.25)
As defined before, the largest fragment of the redistributed mass is a fraction fY
of the cratered mass
Y (µ, M) = fY Mcr (µ, M) ,
(4.26)
while the redistributed mass is
Mredist (µ, M) = µ + Mcr .
(4.27)
Thus, the X(µ, M), Y (µ, M), and Mredist (µ, M) parameters can be summarized as

h
i−βX


µV2
M 1
in CC
∗
2 2M Q (M )
X(µ, M) =
(4.28)


M − Mcr (µ, M)
in EC



fX (µ, M)X(µ, M)
in CC
Y (µ, M) =
(4.29)


fY Mcr (µ, M)
in EC



µ + M − X(µ, M) in CC
Mredist (µ, M) =
(4.30)


µ + Mcr (µ, M)
in EC
125
The Mredist is redistributed as a large number of smaller particles in both collision types, with a slope of γ = 11/6 and a scaling given by equation (4.16). I
choose a redistribution slope of γ = 11/6, which is a value close to that given by
experimental results (Davis & Ryan, 1990) and is the same as used by Dohnanyi
(1969).
I give a list of the variable collisional parameters of my model and their fiducial values in Table 4.1.
4.3.4 The initial distribution and fiducial parameters
I use the Dohnanyi (1969) steady-state solution of η = 11/6 as the initial distribution, where η is the slope of the initial distribution and yields an initial number
density of n(m) = Cm−η , where C is an appropriate scaling constant for the distribution. The exact value of this slope is unknown for all real systems. Fortunately,
the convergent solutions and the timescales of reaching a convergent solution are
fairly insensitive to this value.
4.4 Simplified Dynamics
For the smallest particles, which I am particularly interested in modeling, radiation forces lead to effects such as reduced collisional probabilities in thin ring
disks and increased collisional velocities in extended disks. In this section, I describe my approximate treatment of these effects.
The radiation originating from the central star effectively modifies the mass
of the star seen by the particles; the orbits themselves remain conic sections. I
calculate the particle orbits using
mR̈ = Frad (m) + Fgrav (m) = −
[1 − β(m)] GM∗ m R
·
R2
R
(4.31)
If β(m) < 1, then the net force is still attractive, so all conic sections are possible
Table 4.1. The numerical, collisional and system parameters used in my model
and their fiducial values
Variable
δ
ρ
mmin
mmax
Mtot
η
R
∆R
h
Sp
γ
βX
α
b
fM
fY
max
fX
Θ
p
Qsc
S
G
s
g
Description
Numerical variable
Neighboring grid point mass ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
System variables
Bulk density of particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mass of the smallest particles in the system . . . . . . . . . . . . . . . . . . . . . . . . .
Mass of the largest particles in the system . . . . . . . . . . . . . . . . . . . . . . . . . .
The total mass within the debris ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Initial power-law distribution of particle masses . . . . . . . . . . . . . . . . . . . .
The distance of the debris ring from the star . . . . . . . . . . . . . . . . . . . . . . . .
The width of the debris ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The height of the debris ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The spectral-type of the star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Collisional variables
Redistribution power-law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Power exponent in X particle equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Scaling constant in Mcr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Power-law exponent in Mcr equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Interpolation boundary for erosive collisions . . . . . . . . . . . . . . . . . . . . . . .
Fraction of Y /Mcr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Largest fraction of Y /X at super catastrophic collision boundary . . .
Constant in smoothing weight for large-mass collisional probability .
Exponent in smoothing weight for large-mass collisional probability
The total scaling of the Q∗ strength curve . . . . . . . . . . . . . . . . . . . . . . . . . .
The scaling of the strength regime of the Q∗ strength curve . . . . . . . . .
The scaling of the gravity regime of the Q∗ strength curve . . . . . . . . . .
The power exponent of the strength regime of the Q∗ strength curve
The power exponent of the gravity regime of the Q∗ strength curve .
Fiducial value
1.1
Notes
§B
Eq. (4.10)
§4.3, Eq. (4.13)
§4.3, Eq. (4.13)
§4.3
§4.3
Eqs. (4.1, 4.6, 4.7, 4.42, 4.46)
Eqs. (4.1, 4.6, 4.42, 4.46)
Eqs. (4.1, 4.43)
§4.4
11/6
1.24
2.7 × 10−6
1.23
10−4
0.2
0.5
106 mmax
16
1
3.5 × 107 erg/g
0.3 erg cm3 /g2
-0.38
1.36
Eqs. (4.14, 4.15, 4.16)
Eqs. (4.18, 4.28)
Eqs. (4.22, 4.24)
Eqs. (4.22, 4.24)
Eqs. (4.23, 4.24)
Eq. (4.19, 4.26, 4.29)
Eq. (4.19)
Eq. (4.48)
Eq. (4.48)
Eq. (B.1)
Eq. (B.1)
Eq. (B.1)
Eq. (B.1)
Eq. (B.1)
126
127
as orbital paths. Reaching a hyperbolic orbit is possible as long as the specific
orbital energy of a particle,
GM∗ m
1 2
−
E = mvorb
2
R
(4.32)
is positive, where vorb is orbital velocity. According to equation (4.32), the velocity
needed to obtain a positive specific orbital energy is
2
vorb
≥ [1 − β(m)]
2GM∗
.
R
(4.33)
while the velocity the particle inherits from its parent body is approximately
2
vorb
≈
GM∗
,
R
(4.34)
assuming that both particles that collided were in circular orbits and they collided
with a relative velocity significantly smaller than their orbital velocities. From the
last two equations we see, that
β(m) ≥
1
2
(4.35)
is required for a newly created particle to be put on a hyperbolic orbit, which I
take as the requirement for radiation force blowout to occur. A more detailed
analysis of the energetics can be found in Kresak (1976), Burns et al. (1979), and
references therein.
The effects of dynamical evolution on the collisional cascade can be traced to
the eccentricity of the orbits. To follow the orbital path of a dust grain that has
been created in a collision, I assume that the parent bodies were on circular orbits
at radius R and had β(m) ≈ 0. I also assume that the produced grain will be
created with very small relative velocity with respect to the parent bodies. The
total energy per unit mass of the grain is
E(m)
M∗ G M∗ Gβ(m)
=−
+
.
m
2R
R
(4.36)
128
The first term of the RHS of the equation gives the total energy per unit mass
before the collision (which I assume not to be affected by the radiation force) and
the second part gives the decrease in potential energy per unit mass after the
break-up. I can also write the total energy per unit mass as
M∗ G [1 − β(m)]
E(m)
=−
,
m
2a(m)
(4.37)
where a(m) is the semi-major axis of the orbit the particle of mass m will acquire.
Equating these two expressions I get
a(m) =
1 − β(m)
R.
1 − 2β(m)
(4.38)
As expected, at β(m) = 0.5 the semi-major axis becomes infinite while at
β(m) = 0 it is equal to the semi-major axis of the colliding particles’ original
orbit. The eccentricity (eβ ) of the orbit can be determined from the fact that the
periapsis will equal the original orbital distance
a(m) [1 − eβ (m)] = R ,
(4.39)
yielding
eβ (m) =



β(m)/ [1 − β(m)] if β ≤ 0.5


> 1
.
(4.40)
if β > 0.5
At β(m) = 0.5, the eccentricity equals 1, and at β(m) = 0, it equals zero, consistent with my expectations. Similar derivations can be found elsewhere (e.g.,
Harwit, 1963; Kresak, 1976). A particle on an eccentric orbit will have a modified
probability of interaction with other particles in the parent ring, which I address
in §4.4.2.
4.4.1 Collisional velocities
Lissauer & Stewart (1993) give the velocity of a planetesimal relative to the other
129
planetesimals in the swarm (i.e., the collisional velocity), averaged over an epicycle and over a vertical oscillation as
s
V = vorb
5 e 2
+
4 2
2
i
,
2
(4.41)
where e is the maximum eccentricity and i is the maximum inclination in the
system. This equation is valid for a swarm of particles in Rayleigh distributed
equilibrium. This condition is true for a system in quasi-collisional equilibrium.
I use this equation to estimate the collisional velocity of all particles, setting
e=
∆R
2R
(4.42)
i=
h
.
2R
(4.43)
and
The smallest particles that are in highly eccentric orbits will have varying velocities along their trajectories. However, when at their periapsis, they will have
their original orbital velocities, as by definition they are on eccentric orbits due to
their original periapsis velocity. Because of this, in the simplified dynamical treatment I only use a single collisional velocity for all particles, which is described by
equation (4.41).
4.4.2 Reduced collisional probabilities of β critical particles
Particles with β(m) less than 0.5, but which are still non-zero, called β critical particles, are thought to produce halos around debris disks via the highly eccentric
orbits radiation forces place them on (Thébault & Wu, 2008; Müller et al., 2010).
For a particle to go into an eccentric orbit, it must acquire a radial velocity
component that is different than zero. In collisions, fragments will be ejected in
all directions with a certain velocity distribution. Since the smallest fragments
will tend to escape with the highest velocities (e.g., Jutzi et al., 2010), it is a fair
130
a(m) (µm)
1
0.8
0.6
0.4
0.2
0
-14
10
100
w(m)
w(m)
1
-13
-12
-11
-10 -9
-8
log10(m) (kg)
-7
-6
-5
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
e(m)
Figure 4.5 Left Panel: The weighted collisional probabilities of particles as a function of their mass m (and size). Right Panel: The weighted collisional probabilities
of particles as a function of their eccentricity e. The particles are assumed to be
within a narrow (dR/R = 0.1) ring at 25 AU from an A0 spectral-type star.
question to ask whether thermalization of velocity vectors and their high values
is a stronger effect in placing dust particles on higher eccentricity orbits compared
to radiation effects that reduce the effective stellar mass.
To answer this question, I need to examine the origin of the particles that
contribute to the increase of the differential density in each mass grid point. I
calculate TII only integrating in µ space, thus calculating the rate of increase of
the differential number density of particles with mass m that originate from collisions with a target of mass M. My calculations show that there is a pronounced
peak from M masses roughly on the same scale (at most one order of magnitude higher) as m itself. Most particles originate from targets ∼ 3 − 5× larger
in size than the particle itself. The results of Jutzi et al. (2010) clearly show that
the velocities acquired by collision fragments at 1/3 sizes are more than an order
of magnitude lower than the collisional velocities, meaning that, in the most extreme case, a fragment will receive up to a few tenths of a km s−1 radial velocity
compared to its 10-30 km s−1 orbital velocity. I can thus safely say that particles
131
that are created with β(m) values similar to 0.5 tend to be placed on eccentric orbits by the radiation forces rather than being dispersed. These orbits will extend
out from the initial debris disk ring, shielding the particles from being destroyed
and from them creating other particles.
My approach to calculating a weighting factor for each particle mass, determined by the fraction of its orbital period it spends in the parent ring is similar to
that of Thébault & Wu (2008). The orbital time of a particle in an elliptical orbit
as a function of its distance from the center of mass is (Taff, 1985)
s
r
2
a−l
GM
a
−
l
−1
− eβ 1 −
cos
,
= (t − t0 )
aeβ
aeβ
a3
(4.44)
where t0 = 0 is the initial time at periapsis, l is its distance at time t from the
center of mass, and I omit the m dependences of eβ and a for clarity. I estimate
the semi-major axis as
a=
R − ∆R/2
,
1 − eβ
(4.45)
and I calculate the time ∆t needed for a particle to reach the outer edge of the disk
at l = R + dR/2. Dividing ∆t by the half of the orbital period gives the weighting
factor for each mass m as i
s
)
(
1
∆R
(e
−
1)
(∆R
−
2Re
)
∆R
(2
−
e
)
−
2Re
β
β
β
β
w=
cos−1
−2
π
eβ (∆R − 2R)
(∆R − 2R)2
(4.46)
I plot these weighting factors as a function of the particle mass and orbital eccentricity in Figure 4.5. When analyzing the particle distributions, I only plot the
number of particles within the parent ring, which I calculate as
nring (m) = n(m)w(m) .
(4.47)
4.4.3 Reduced collisional probabilities of the largest particles
The very last grid point in the domain of solution will only reduce its number
density, as it cannot gain from larger masses. The lack of a positive term for the
132
Table 4.2. Parameters used for comparison models
Variable
ρ (kg m−3 )
mmin (kg)
mmax (kg)
Mtot (M⊕ )
η
R (AU)
∆R (AU)
h (AU)
Sp
PRD
Comparison to
Thébault (2003)
Comparison to
Löhne (2008)
2700
1.42×10−21
1.78×1018
0.0030221
11/6
5
1
0.5
A0
off
2500
1.42×10−21
4.20×1018
1.0
1.87
11.25
7.5
3.4
G5
off
last grid point causes its evolution time to become very small compared to all
others and leads to a numerical instability. In order to avoid this, I multiply the
collisional rates with a weight that smooths to zero for the largest particles
"
#p
−m
1 − exp − mmax
Θ ,
(4.48)
σw (m) =
1 − exp − mmax
Θ
for both the projectile and target particle. I chose Θ to be a number a few orders of
magnitude larger than mmax and use an arbitrary p = 16. The modified collisional
rates, therefore, read
1
1 2
′3
3
P (m, m ) = V πκn(m)n(m )w(m)w(m ) × σw (m)σw (m ) m + m
.
′
′
′
′
(4.49)
I discuss the implications of the choice of the weight function and of its parameters in §4.5.2.
4.5 Results
As I discussed in §4.2, collisional cascades in debris disks have been studied extensively in the past decades, with many different analytic and numerical solu-
133
tions to the problem. To demonstrate the similarities and differences between my
model and some earlier ones, I show in the following subsection the results of a
few comparison tests. The system variables used by my code for these runs are
summarized in Table 4.2.
I compare my numerical model to three previous well known algorithms, the
particle-in-a-box code of Thébault et al. (2003), the dynamical code ACE (Krivov
et al., 2000, 2005; Löhne et al., 2008; Müller et al., 2010), and the 1D steady-state
solver code of Wyatt et al. (2011). Although I do make an effort to model their systems as accurately as possible, a true benchmark between the codes is impossible.
This is due in part to the fact that all models have somewhat different collisional
and dynamical prescriptions.
4.5.1 Comparison to Thébault et al. (2003)
A relatively straightforward comparison can be made between CODE-M and the
Thébault et al. (2003) model. Although the initial Thébault et al. (2003) model has
been subsequently improved in Thébault & Augereau (2007), I chose to compare
my results with the former, as they are both particle-in-a-box approaches to the
collisional cascade, with some dynamical effects included in a simplified manner.
Thébault et al. (2003) aimed to model the inner 10 AU region of the β Pictoris
disk, with their reference model being a dense debris ring at 5 AU, with a width
of 1 AU and height of 0.5 AU. I adopted their largest particle size of 54 km for
the comparison run. However, I adopted a smaller minimum particle size than
they did (in my case well below the blowout size), to be able to follow the removed particles more completely. This treatment does not have effects on the actual distribution within the ring. I also had Poynting-Robertson drag turned off.
Although both of our models include erosive (cratering) collisions, the Thébault
et al. (2003) prescription uses hardness constants (α) of much softer material than
134
that of my nominal case and a linear relationship between cratered mass and
impact energy (the prescription for erosive collisions has been changed in their
later paper Thébault & Augereau 2007). For a better agreement, I also model a
modified cratered prescription case, where I set b = 1, α = 10−4 and fM = 0.01.
With these adjustments our cratering prescriptions agree better; however my interpolation formula is offset compared to Thébault et al. (2003). While ours has a
continuous prescription at the CC/EC boundary (i.e., the cratered mass is 0.5M),
the Thébault et al. (2003) interpolation does not (i.e., the cratered mass is 0.1M).
Figure 4.6 compares the evolution of the distribution of particles between the
Thébault et al. (2003) nominal case and my runs. In the vertical axes I plot n(m) ×
m2 , which is similar to the “mass/bin” value used by Thébault et al. (2003). To
make them exact, I divide the Thébault et al. (2003) values by (δ-1), which places
them on the same scale. A few similarities and a few significant differences can
be noted. Generally both models show wavy structure - which is a well studied
phenomenon (see e.g., Campo Bagatin et al., 1994; Wyatt et al., 2011) - but the
exact structure of the waves differs.
My modified erosive (cratering) prescription model gives a much better agreement with the Thébault et al. (2003) results than my fiducial prescription, in the
sense that it yields a deeper first wave in the distribution with a larger wavelength. The offset between the locations of the first dip and the subsequent peak
in the two models could likely result from the higher collisional velocities that
Thébault et al. (2003) calculate for the smallest particles. My modified erosion
prescription gives a good agreement with the Thébault et al. (2003) results for
particles larger than a km in size, which is a surprise as the Thébault et al. (2003)
erosive constants are for much softer materials than my nominal values are for.
Just above the blow-out regime my model becomes abundant in dust particles,
135
1021
Thebault (2003)
CODE-M
CODE-M w/ mod Mcr
Thebault (2003)
CODE-M
CODE-M w/ mod Mcr
n(m) x m2 (kg)
1020
1019
1018
1017
1016
1015
1021
t = 2500 yr
Thebault (2003)
CODE-M
CODE-M w/ mod Mcr
t = 25000 yr
Thebault (2003)
CODE-M
CODE-M w/ mod Mcr
n(m) x m2 (kg)
1020
1019
1018
1017
1016
1015
t = 0.5 Myr
1 µm
1 mm
1m
Particle size
1 km
t = 10 Myr
1 µm
1 mm
1m
Particle size
1 km
Figure 4.6 Comparison of the evolution of the dust distribution around the β
Pictoris disk modeled by Thébault et al. (2003) and the model presented in this
work. The thin solid line is the initial distribution. The modified cratered mass
(Mcr ) model uses an erosive collision prescription that is a closer analog to the
original Thébault et al. (2003) soft material one. See text for more details.
as more and more dust is placed on highly eccentric orbits. Although some
smoothing is expected in reality, I do expect the number of dust particles near
the blowout limit to increase.
While both my distributions show the typical double power-law feature of
quasi steady-state collisional cascades (see e.g., Wyatt et al., 2011) above and below the change in the strength curve, it is masked in the Thébault et al. (2003)
136
10-2
Mdust/Mplanetesimals
1
M(t)/M0
0.9
0.8
0.7
0.6
Thebault (2003)
CODE-M
CODE-M w/ mod Mcr
103
104
105
time (yr)
106
107
10-3
10-4
Thebault (2003)
CODE-M
CODE-M w/ mod Mcr
10-5 3
10
104
105
time (yr)
106
107
Figure 4.7 Comparison of the evolution of the total disk mass and the dust-toplanetesimal mass ratio around the β Pictoris disk modeled by Thébault et al.
(2003) and the models presented in this work. The modified cratered mass (Mcr )
model uses an erosive collision prescription that is a closer analog to the original
Thébault et al. (2003) one. See text for more details.
model, due to the high amplitude wavy structures.
In Figure 4.7, I show the differences in the evolution of the disk mass between
the nominal case of Thébault et al. (2003) and my models. The left panel shows
the evolution of the total disk mass within the debris ring, while the right panel
shows the evolution of the dust-to-planetesimal mass ratio. These figures are
equivalent to Thébault et al. (2003) figures 2 and 3 (except that these are in plotted
in logarithmic scales). My nominal model predicts a faster decay of the total disk
mass, reaching 25% mass loss, while the modified erosive prescription agrees
with the Thébault et al. (2003) model and loses ∼ 12% of its initial mass. The
evolution of the dust-to-planetesimal disk mass differs while the quasi steadystate is being reached, after which all models decay with the same slope. My
nominal case model has an order of magnitude larger dust-to-planetesimal mass
ratio at all times compared to the Thébault et al. (2003) model, while my modified
erosive collision prescription case is close to it.
137
There are some easily identified differences between our models. Thébault
et al. (2003) use the same Benz & Asphaug (1999) dispersive strength curve as I
do, although they do average it to account for impact angle variations. This is an
unnecessary step, as the Benz & Asphaug (1999) strength curve is already impact
angle averaged, and is corrected in Thébault & Augereau (2007). However, I find
that this scaling offset does not have a significant effect on the outcome of the distribution evolution. Thébault et al. (2003) use a double power-law for fragment
redistribution, while I use only a single power-law. In Chapter 5, I will show that
varying the slope of the single power-law does not have a significant effect on
the evolution of the distribution either, so it appears that this difference is also
likely not a significant contributing factor to our discrepancies. A noteworthy
difference between our models is that Thébault et al. (2003) calculate fragment
re-accumulation, while I do not. This is a possible explanation for our discrepancies at high masses, and the offsets I have in the total mass decay.
The most significant difference between the models is that ours uses a single
interaction velocity, while Thébault et al. (2003) model the interaction velocity
between the β critical elliptical orbit smallest particles and the parent ring. This
is likely to account for some of the additional offsets for the smallest particles, as
higher interaction velocities have been shown to initiate higher amplitude waves
(Campo Bagatin et al., 1994; Wyatt et al., 2011). Thébault et al. (2003) also take into
account the constant presence of particles smaller than the blowout limit within
the parent ring, which I do not. Within denser disks this step may smooth out
any features near the blowout limit.
4.5.2 Comparison to Löhne et al. (2008) and Wyatt et al. (2011)
As introduced in §4.2, the numerical code ACE (Krivov et al., 2000, 2005, 2006,
2008; Löhne et al., 2008; Müller et al., 2010) solves the dynamical evolution of the
138
10-1
n(m) x m2 (MEarth)
10-2
Lohne (2008)
Wyatt (2011)
CODE-M only CC
CODE-M
Lohne (2008)
Wyatt (2011)
CODE-M only CC
CODE-M
10-3
10-4
10-5
10-6
10-7
10-8
t = 0.5 Myr
1 µm
1 mm
1m
Particle radius
t = 50 Myr
1 km
1 µm
1 mm
1m
Particle radius
1 km
Figure 4.8 Comparison of the evolution of the particle distribution within a debris
disk around a solar-type star modeled by Löhne et al. (2008), Wyatt et al. (2011)
and the code presented in this work. The ”only CC” CODE-M model uses only
catastrophic collisions. The thin solid line is the initial distribution. See text for
more details.
1
103
Disk mass (MEarth)
CL (MEarth-1 Gyr-1)
104
102
101
100
Lohne (2008)
CODE-M full
CODE-M CC only
10-1 4
10
105
106
107
time (yr)
108
109
0.8
0.6
0.4
Lohne (2008)
CODE-M full
CODE-M CC only
0.2
104
105
106
107
time (yr)
108
109
Figure 4.9 Comparison of the evolution of the dust mass within a debris disk
around a solar-type star modeled by Löhne et al. (2008) and the model presented
in this work. See text for details.
collisional system as well as the collisional fragmentation, thus a straight comparison to CODE-M cannot be performed. I use their ii-0.3 model (Löhne et al.,
139
"Bin mass" (MEarth)
10-1
10-2
10-3
10-4
10-5
10-6
10-7 0
10 101 102 103 104 105 106 107 108 109
time (yr)
Figure 4.10 Evolution of the total mass within consecutive mass regions from the
smallest to the largest particles in the system for the full collisional system, using
the ii-0.3 parameters of Löhne et al. (2008). The plot can be compared to the
top panel of Figure 4. of Löhne et al. (2008).
2008) for comparison, which is for a relatively wide (7.5-15 AU), extremely dense
(1 M⊕ total mass with a largest planetesimal size of 74 km) debris ring. I turned
off the effects of Poynting-Robertson drag for this comparison model. The initial parameters I assumed are summarized in Table 4.2. This same system was
modeled by Wyatt et al. (2011), whose results I also use for comparison.
In Figure 4.8, I show the evolution of the dust distribution of the system given
by CODE-M, ACE, and Wyatt et al. (2011). As the version of ACE used in Löhne
et al. (2008) only modeled catastrophic collisions and the Wyatt et al. (2011) model
uses catastrophic collision rates, I also include a CODE-M run in the plot that only
models the outcomes of catastrophic collisions. Since the Löhne et al. (2008) values are already downscaled by (δ − 1), no additional scaling was required of their
140
6
-1
n(m) x m2 (MEarth)
10
10-2
6
θ=10
θ=1024
θ=10
θ=108
θ=1010
θ=10
θ=1024
θ=10
θ=108
θ=1010
10-3
10-4
10-5
10-6
t = 0.5 Myr
t = 50 Myr
-7
10
n(m) x m2 (MEarth)
10-1
10-2
p=16
p=4
p=8
p=12
p=20
p=24
p=16
p=4
p=8
p=12
p=20
p=24
10-3
10-4
10-5
10-6
10-7
t = 0.5 Myr
1 µm
1 mm
1m
Particle radius
1 km
t = 50 Myr
1 µm
1 mm
1m
Particle radius
1 km
Figure 4.11 Comparison of the evolution of my model distribution using the
Löhne et al. (2008) ii-0.3 parameters, when varying the parameters of the
weights in the collision cross sections for large particles.
data. The Wyatt et al. (2011) data points are divided by δ − 1 = 0.0626 to convert
them to differential number densities. Qualitatively, all distributions agree much
better than in the Thébault et al. (2003) comparison (Figure 4.6); however, there is
some scaling offset between the full CODE-M and the other models, especially at
large masses.
The wavelengths of the waves roughly agree between CODE-M and ACE, with
the single difference being the absence of the strong offset of the first crest in
my model; the agreement is also good between CODE-M and Wyatt et al. (2011).
141
1
103
Disk mass (MEarth)
CL (MEarth-1 Gyr-1)
104
102
101
100
10-1
θ=1062
θ=106
θ=108
θ=1010
θ=10
0.8
0.6
0.4
0.2
θ=1062
θ=106
θ=108
θ=1010
θ=10
Disk mass (MEarth)
CL (MEarth-1 Gyr-1)
1
103
102
101
100
p=16
p=4
p=8
p=12
p=20
p=24
10-1 4
10
105
106
107
time (yr)
108
109
0.8
0.6
0.4
p=16
p=4
p=8
p=12
p=20
p=24
0.2
104
105
106
107
time (yr)
108
109
Figure 4.12 The change in the dust mass evolution when varying the parameters
of the weights in the collision cross sections for large particles.
The double power-law distribution due to the change from strength to gravity
dominated thresholds in the strength curve (Benz & Asphaug, 1999) can be distinguished in all three models, with roughly the same slopes. The ACE and the
Wyatt et al. (2011) models maintain their initial −1.87 number density distribution slope, while CODE-M becomes somewhat steeper for the smallest particles.2
My catastrophic-collision-only model has a smaller amplitude and wavelength
wave structure than my full model or the other models. The most significant difference between the models is the scaling offset of the full CODE-M model, which
I analyze below.
In Figure 4.9, I show a comparison to figures 1 and 2 of Löhne et al. (2008). In
the left panel I show the evolution of CL (CLöhne ), which is introduced in Equation
2
In Figure 4.8, I plot in the y-axis the product n(m)m2 , so that a steeper number density slope
will show up as a flatter distribution.
Mdust/Mplanetesimal
1
CODE-M full model
CODE-M CC only
Lohne (2008)
Wyatt (2011)
3.90
3.80
3.70
0.1
3.60
3.50
η(a) with same mass ratio
142
3.40
0.01
102
103
104
105 106
time (yr)
107
Figure 4.13 The evolution of the dust-to-planetesimal mass ratio of the CODE-M
models and the values for the Löhne et al. (2008) and Wyatt et al. (2011) models at 0.5 and 50 Myr. I calculate the dust mass from 0.1 mm to 10 cm and the
planetesimal mass from 10 cm to 100 m. On the right vertical axis I give the
value of the slope of the power-law distribution that would give the same dustto-planetesimal mass ratio.
(11) of Löhne et al. (2008) as
CL = −
Ṁdisk
,
2
Mdisk
(4.50)
This quantity is inversely proportional to the characteristic timescale of the system. As expected, since my system evolves faster, its characteristic timescale is
shorter, so the CL factor for my models is larger. This can be seen in the right
panel as well, where I plot the decay of the total mass in my system and that
given in figure 2 of Löhne et al. (2008). I adopted the exact strength curve of
Löhne et al. (2008) in this run of my model, with the corrections given by Wyatt
et al. (2011).
143
In Figure 4.10, I show a comparison to the top panel of figure 4 in Löhne
et al. (2008), which shows the evolution of the total mass within each of their
mass bins. In this plot, I show the evolution of the full collisional system, which
includes erosive and catastrophic collisions. Since I do not use mass bins, but
rather a differential number density griding, I integrate the distribution between
14 grid points for each mass value, which roughly corresponds to a single mass
bin of Löhne et al. (2008). Up to roughly a few hundred meters in size (where the
strength curve has its minimum) all mass “bins” decay in close parallel slopes
to each other after reaching their quasi steady-state around 10,000 yr. This is in
contrast to the Löhne et al. (2008) results and agrees more with figure 2 of Wyatt
et al. (2011), who model the same system. The intermediate size planetesimals
(∼ km) show a steeper decay than that modeled by either Löhne et al. (2008) or
Wyatt et al. (2011).
The obvious difference between ACE and CODE-M, is that ACE also evolves the
dynamics in the system and takes into account the varying collisional velocities
in the system from particles that are in elliptical orbits within the parent ring.
This could easily explain the offset of the first wave given by ACE in Figure 4.8.
The increasing offset between the full CODE-M run and the other two models is likely due to the absence of erosive collisions by Löhne et al. (2008) and
to using catastrophic collision rates in Wyatt et al. (2011). Kobayashi & Tanaka
(2010) have shown earlier erosive (cratering) collisions to be the dominant effect
for mass loss in collisionally evolving systems. This effect is demonstrated by
the CODE-M model I run with only catastrophic collisions included, which scales
exactly with the ACE and Wyatt et al. (2011) models. Since CODE-M does not include aggregation, the collisions of the smallest particles with the largest bodies
is not modeled perfectly. I assume the realistic distribution decay to lie between
144
the two models given by CODE-M.
As introduced in §4.4.3, I artificially reduce the collision cross section of the
largest particles in the system to zero in order to avoid numerical instabilities.
However, as this is a completely arbitrarily defined numerical necessity, I investigate its effects on the total mass decay, where I expect it to be the strongest.
I reproduce Figures 4.8 & 4.12, but with varying the values of Θ and p in the
smoothing function, in Figures 4.11 & 4.12. As can be seen in these plots, the
variable Θ does not affect either the evolution of the distribution or the total mass
decay, as long as it is larger than one. Varying the values of p does have an effect
on both the evolution of the distribution and the total mass decay. The effect is
only on the largest bodies in the system; below a size of one hundred meters, the
shapes of the distributions remain unchanged, with the only differences being
scaling offsets.
Visual examination of the distributions in Figure 4.9 hint at a slightly steeper
distribution slope for the CODE-M models than the Löhne et al. (2008) and Wyatt
et al. (2011) ones. Since the distributions have wavy structures in them, this is
difficult to show with a slope fit. Therefore, I calculate the dust-to-planetesimal
mass ratios of the distributions. I define the dust sizes to be from 0.1 mm to 10
cm and the planetesimal sizes to be from 10 cm to 100 m. In Figure 4.13, I plot
the evolution of the mass ratios of my models and the mass ratios for the Löhne
et al. (2008) and Wyatt et al. (2011) models at 0.5 and 50 Myr. My models have
significantly higher mass ratios than theirs. This is likely a result of the differences
between our collisional equations.
145
4.6 Conclusions
In this chapter, I present a numerical model of the evolution of the distribution
of dust in dense debris disks. I calculate my model with a new numerical code,
CODE-M, which I extensively verify and test in the Appendix. I compare my code
to the previously published numerical models of collisional cascades in debris
disks, showing general agreement. Unlike previous codes, which include features such as a detailed treatment of particle dynamics and extended debris disks,
CODE-M only models rings but with improved fidelity in this situation, because
I solve the full scattering integral and use solvers that achieve high numerical
accuracy.
My model shows faster decays than previously published ones (Thébault et al.,
2003; Löhne et al., 2008; Wyatt et al., 2011) and also yields slightly higher dustto-planetesimal mass ratios. I attribute these characteristics to be a result of the
accurate treatment of collisional cascades.
146
C HAPTER 5
S TEEP D UST-S IZE D ISTRIBUTIONS
In this chapter, I explore the evolution of the mass distribution of dust in collisiondominated debris disks, using the collisional code introduced in the previous
chapter. I analyze the equilibrium distribution and its dependence on model parameters by evolving over 100 models to 10 Gyr. With numerical models, I show
that systems reach collisional equilibrium with a mass distribution that is steeper
than the ones given by earlier analytic or current numerical methods. My model
yields a steady state slope of n(m) ∼ m−1.88 [n(a) ∼ a−3.65 ]. This steeper solution
has observable effects in the submillimeter and millimeter wavelength regimes
of the electromagnetic spectrum. I assemble data for nine debris disks that have
been observed at these wavelengths and, using a simplified absorption efficiency
model, show that the predicted slope of the particle mass distribution generates
SEDs that are in agreement with the observed ones.
5.1 Introduction
The total mass within debris disks as well as the infrared excess emission produced by their dust are generally calculated assuming the analytic estimate of
the distribution of masses in the asteroid belt by Dohnanyi (1969). This solution
predicts that the sizes follow a power-law, with their numbers increasing with
decreasing size a as n(a) ∼ a−3.5 . However, a number of recent efforts to model
observations of debris disks have found it necessary to adopt steeper slopes (Krist
et al., 2010; Golimowski et al., 2011).
Durda & Dermott (1997) showed that a steep tensile strength curve, i.e., the
function that gives the minimum energy required to disrupt a body catastrophi-
147
cally (see, e.g., Holsapple et al., 2002; Benz & Asphaug, 1999; Gáspár et al., 2011),
results in a steeper steady-state distribution than the traditional solution. Collisional models of the dust in circumstellar disks (Thébault et al., 2003; Krivov
et al., 2005; Thébault & Augereau, 2007; Löhne et al., 2008; Müller et al., 2010;
Wyatt et al., 2011) have also shown that the dust particles will settle with a distribution n(a) ∼ a−3.61 , on top of which additional structures appear. This steeper
distribution has readily observable effects at the far-IR and submm wavelengths.
It also results in higher total dust mass and lower planetesimal mass estimates
for the systems.
In this chapter, I investigate the slope of the mass distribution and the physical parameters that influence it with the numerical code, introduced in Chapter 4.
My code has been developed to calculate the evolution of the particle mass distribution in collisional systems, taking into account both erosive and catastrophic
collisions. In §5.2, I introduce models for the numerical analysis of the collisional
cascades and give my findings. In §5.3, I generate a set of synthetic spectra in order to analyze the effects certain distribution parameters have on different parts
of the SEDs. In §5.4, I introduce a simple relation between the Rayleigh-Jeans
part of the spectral energy distributions and the particle size distribution. In §5.5,
I compare the results to the observed far-IR and sub millimeter data for nine
sources.
5.2 Numerical modeling
In this section, I analyze the steady-state dust distribution with the full numerical
code. I run a set of numerical models to study the evolution of the slope of the
steady-state distribution function and its dependence on the model parameters.
I investigate the time required for the distribution to settle into its steady-state
148
and, with a wide coverage of the parameter space, I also examine the robustness
of the solution.
5.2.1 Evolution of the reference model
I set up a reference debris disk as a basis for comparison to all other model
runs. This model consists of a moderately dense debris disk situated at 25 AU
around an A0 spectral-type star, with a width and height of 2.5 AU. This radial
distance ensures a moderate evolution speed, but with a peak emission in the
mid-infrared. It also guarantees a Rayleigh-Jeans tail in the far-infrared regime,
which is the primary imaging window for the Herschel Space Telescope. The total
mass in the debris disk is 1 M⊕ , distributed within minimum and maximum particle masses that correspond to radii of 5 nm and 1000 km, when assuming a bulk
density of 2.7 g cm−3 . I summarize the disk parameters of the reference model in
Table 5.1. I evolve the reference model for 10 Gyr.
In Figure 5.1, I show the evolution of the particle distribution, plotting it at
six different points in time up to 10 Gyr. I plot log10 [n (m) × m2 ] on the vertical axis, which can be related to the “mass/bin” that is frequently used in other
simulations. Even though the number densities decrease with increasing particle
masses, the mass distribution increases towards the larger masses in this representation, as long as the mass distribution slope is smaller than 2.
The smallest particles reach collisional equilibrium first, roughly at 1 Myr,
followed by larger particle sizes as the system evolves. After 50-100 Myr of
evolution, the upper, gravity dominated part of the distribution (m > 1013 kg)
also reaches equilibrium. The distribution maintains its slope for masses below
1010 kg, which roughly corresponds to a planetesimal radius of 100 m. The kink in
the distribution at the upper end is due to the change in the strength curve slope
(O’Brien & Greenberg, 2005; Bottke et al., 2005).
149
Table 5.1. Numerical, Collisional, and System parameters of the reference
model
Variable
ρ
mmin
mmax
Mtot
η0
R
∆R
h
Sp
γ
βX
α
b
fM
fY
max
fX
Qsc
S
G
s
g
δ
Θ
P
Description
System variables
Bulk density of particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mass of the smallest particles in the system . . . . . . . . . . . . . . . . . . . . . . . .
Mass of the largest particles in the system . . . . . . . . . . . . . . . . . . . . . . . . . .
Total mass within the debris ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Initial power-law distribution of particle masses . . . . . . . . . . . . . . . . . . .
Distance of the debris ring from the star . . . . . . . . . . . . . . . . . . . . . . . . . . .
Width of the debris ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Height of the debris ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Spectral-type of the star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Collisional variables
Redistribution power-law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Power exponent in X particle equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Scaling constant in Mcr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Power-law exponent in Mcr equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Interpolation boundary for erosive collisions . . . . . . . . . . . . . . . . . . . . . . .
Fraction of Y /Mcr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Largest fraction of Y /X at super catastrophic collision boundary . . .
Total scaling of the Q∗ strength curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Scaling of the strength regime of the Q∗ strength curve . . . . . . . . . . . . .
Scaling of the gravity regime of the Q∗ strength curve . . . . . . . . . . . . . .
Power exponent of the strength regime of the Q∗ strength curve . . .
Power exponent of the gravity regime of the Q∗ strength curve . . . .
Numerical parameters
Neighboring grid point mass ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Constant in smoothing weight for large-mass collisional probability
Exponent in smoothing weight for large-mass collisional probability
Fiducial value
2.7 g cm−3
1.42×10−21 kg
1.13×1022 kg
1 M⊕
1.87
25 AU
2.5 AU
2.5 AU
A0
11/6
1.24
2.7 × 10−6
1.23
10−4
0.2
0.5
1
3.5 × 107 erg/g
0.3 erg cm3 /g2
-0.38
1.36
1.104
106 mmax
16
150
1 mm
1m
1 km
-12
log10[n(m) x m2/(kg m-3)]
-13
-14
-15
-16
-17
0 yr
0.07 Myr
1.76 Myr
50 Myr
1 Gyr
10 Gyr
-18
-19
-10
-5
0
5
log10(m/kg)
10
15
20
Figure 5.1 Particle mass distribution of the reference model plotted at various
points in time.
Structures in the distribution slope, such as waves, may in principle occur at
the low-mass end when assuming softer material properties or higher collision
velocities. The distribution may also acquire some curvature (see Chapter 4).
Because of these effects, I evaluate the average slope of the distribution by fitting
a power-law over a large mass range, but one that remains below the kink in the
distribution. Specifically, I fit the distribution between masses 10−8 and 104 kg,
which roughly correspond to sizes of 0.1 mm and 1 m.
I examine the dependence of the steady-state distribution slope on the ad hoc
initial conditions parametrized by the initial mass-distribution slope η0 and the
-12
-16
1.9
-18
-20
-10
-12
-14
1.8
1.5
1.66
1.87
1.90
1.99
1.7
1.6
η0
1.5
1.89
0.01 M⊕
0.1 M⊕
1 M⊕
10 M⊕
100 M⊕
Mtot
1.88
Slope
2
η0
Slope
2
-14
-3
log10[n(m) x m /(kg m )]
2
1.5
1.66
1.87
1.90
1.99
-3
log10[n(m) x m /(kg m )]
151
-16
0.01 M⊕
0.1 M⊕
1 M⊕
10 M⊕
100 M⊕
1.86
-18
-20
-20
1.87
Mtot
1.85
-10
0
10
log10(m/kg)
20
0
3
6
log10(t/yr)
9
Figure 5.2 (Left panels) Particle mass distribution at 10 Gyr, when varying the initial mass distribution slope (top) and the total mass of the system(bottom). (Right
panels) Evolution of the dust-mass distribution slopes when varying the initial
mass distribution slope (top) and the total mass of the system (bottom). The
steady-state distribution slope is practically independent of these initial conditions.
initial total mass in the disk Mtot . In Figure 5.2, I show the evolution of the particle
mass distribution and its slope as a function of these parameters. The left panels
show the dust distribution after 10 Gyr of evolution for different values of the
two parameters, while the right panels show the evolution of the dust-mass distribution slope. The top right panel shows the evolution of the dust distribution
152
slope for varying initial mass-distribution slopes. Variations in η0 do not affect
the final dust distributions, although the high mass end evolves differently or
reaches equilibrium at different timescales. A distribution with less dust initially
(η0 < 1.87) also takes more time to reach equilibrium. A shallow distribution
with an initial slope of η0 = 1.5 takes as much as ∼ 1 Gyr to reach equilibrium,
although such initial distribution slopes are unlikely. As shown by Löhne et al.
(2008), the evolution of the particle mass distribution is scalable by the total mass
(or number densities of particles), which is what we see in the bottom two panels
of Figure 5.2. All systems with different initial masses reach the same equilibrium
mass distribution, but on different timescales. More massive systems evolve on
shorter timescales, thus reaching their equilibrium more quickly, while less massive systems evolve more slowly.
5.2.2 The dependence of the steady-state distribution function on the collision
parameters
The parameters that describe the outcomes of collisions, in principle, should be
roughly the same for all collisional systems. These are the fragmentation constants and the parameters of the strength curve (Benz & Asphaug, 1999). In order
to investigate their effects on the evolution of the particle mass distribution, I
vary their nominal values and evolve the models to the same 10 Gyr, as I did for
the reference model.
I give here a detailed analysis of the effects of varying only five of the twelve
parameters (α, b, Qsc , s, S), as the remaining seven (γ, βX , fY , fXmax , fM , g, G) have
no significant effects (see Table 5.1 for the description of these parameters).
In Figure 5.3, I plot the resulting mass distributions when varying the cratered
mass parameters α and b (Koschny & Grün, 2001a,b). The parameter α is the total
scaling and b is the exponent of the projectile’s kinetic energy in the equation of
-3
2.4x10-5
2.4x10-6
2.4x10-7
2.4x10
-16
-18
α
-20
-20
-10
-12
1.16
1.23
1.26
1.34
-14
2
-14
-4
log10[n(m) x m /(kg m )]
-12
2
-3
log10[n(m) x m /(kg m )]
153
0
10
log10(m/kg)
20
-16
-18
b
-20
-20
-10
0
10
log10(m/kg)
20
Figure 5.3 Particle mass distribution at 10 Gyr, when varying the parameters α
(left) and b (right) of the cratered mass (equation 5.1).
the cratered mass
Mcr = α
µV 2
2
b
.
(5.1)
As can be seen in Figure 5.3, the resulting mass distributions depend on the values of α and b only in the gravity dominated regime. At these larger masses,
the model is incomplete, because I do not include aggregation. When increasing
α, i.e., basically softening the materials or increasing the effects of erosions, the
number of eroded particles in the gravity-dominated regime increases rapidly. A
similar effect can be observed when increasing the value of b. However, within
reasonable values of α and b, the variation of the equilibrium particle mass distribution slope in the dust mass regime is negligible.
In Figure 5.4, I plot the resulting distributions and the evolution of the dust
distribution slope when I vary the parameters Qsc , S, and s. These are all variables in the tensile strength curve, which is given as (Benz & Asphaug, 1999)
h a s
a g i J g
Q∗ (a) = 10−4 Qsc S
.
+ Gρ
cm
cm
erg kg
(5.2)
The variable Qsc is a global scaling factor, S is the scaling of the strength-domina-
-12
-16
-0.01
-0.2
-0.38
-0.6
-1.0
1.92
Slope
-14
1.94
-0.01
-0.2
-0.38
-0.6
-1.0
-18
1.9
1.88
1.86
s
s
1.84
-20
-12
3.5x105
3.5x10710
3.5x1015
3.5x10
1.92
Slope
-14
1.94
3.5x105
3.5x10710
3.5x1015
3.5x10
-16
-18
1.9
1.88
1.86
S
S
1.84
-20
-12
1.94
3
-14
101
10
1
10-2
-16
-18
1.9
1.88
1.86
Qsc
-20
-20
3
101
10
1
10-2
1.92
Slope
log10[n(m) x m2/(kg m-3)]
log10[n(m) x m2/(kg m-3)]
log10[n(m) x m2/(kg m-3)]
154
-10
0
10
log10(m/kg)
20
Qsc
1.84
0
3
6
log10(t/yr)
9
Figure 5.4 (Left panels) Particle mass distribution at 10 Gyr, when varying the values of the tensile strength curve parameters Qsc , S and s. (Right panels) Evolution
of the dust-mass distribution slopes when varying the values of the same parameters.
155
ted regime, s is the power dependence on particle size of the strength-dominated
regime, G is the scaling of the gravity-dominated regime, and g is the power dependence on particle size of the gravity-dominated regime. The tensile strength
curve has been extensively studied for decades. However, as it is dependent
on various material properties and the collisional velocity (Stewart & Leinhardt,
2009; Leinhardt & Stewart, 2011), its parameters do not have universally applicable values. Determining the tensile strength curve at large and small sizes is also
extremely difficult experimentally. However, because variations in the gravitydominated regime of the curve (G and g) do not have significant effects on the
equilibrium dust-mass distribution, I do not consider these parameters further in
the following discussion.
The slope s of the strength curve in the strength-dominated regime depends
on the Weibull flaw-size distribution. Its measured values range anywhere between -0.7 and -0.3 (Holsapple et al., 2002). Steeper values of s make smaller
materials harder to disrupt, which results in a steeper dust distribution slope. At
s = 1.0, the smallest particles are hard enough to resist catastrophic disruption
even when the projectile mass equals the target mass. This results in a mass distribution with a slope equal to the redistribution slope γ = 1.83. At s = 0.6, the
smallest particles are still able to destroy each other and generate a dust distribution slope that is close to 1.91.
The scaling constants of the tensile strength curve are the dominant parameters in the evolution toward the steady state distribution. When reducing the
complete tensile strength curve scaling Qsc , wave structures form more easily, as
a particle becomes capable of affecting the evolution of particles much larger than
itself (see Chapter 4). When upscaling the tensile strength curve, the steady-state
distribution slope starts to resemble the redistribution slope, as it is the particle
156
redistributions that lead the evolution of the particle mass distribution. When
varying the scaling of only the strength side of the curve S, similar effects can be
seen.
I conclude that, for all reasonable values of the collisional parameters, the
steady-state dust-mass distribution slope is larger or equal to 1.88.
5.2.3 The dependence of the steady-state distribution function on system variables
There are a number of parameters that can change from one collisional system to
another: the material density ρ, the maximum and the minimum particle mass in
the system mmin and mmax , the radial distance R, height h, and width ∆R of the
disk, and the spectral type of the central star. All these parameters affect three
properties of the collisional model: the blow-out mass, the collisional velocity,
and the number density of particles. Varying these parameters will change the
timescale of the evolution and affect the steady-state distribution slope. In this
subsection, I analyze the effects of varying the radial distance, as it modifies the
equilibrium mass distribution by setting the collisional velocity. Modifying either
disk parameters ∆R and h or the spectral-type of the star would have similar
effects. I do not discuss the variations in the timescales.
In the left panel of Figure 5.5, I show the effects of varying the radial distance, R, on the mass distribution evolved to 10 Gyr. Variations in the radial
distance affect both the number density of particles and the collisional velocity,
naturally changing the timescale of the evolution and also the outcome of the
collisions. Decreasing the radial distance will increase the collisional velocity, resulting in the appearance of waves at the small-mass end of the distribution. It
also generates a much more pronounced kink at the high mass end. On the other
hand, when the velocity is decreased at large radii, the low mass end of the distri-
-12
-14
-16
1.9
5 AU
10 AU
25 AU
50 AU
100 AU
R
1.88
Slope
log10[n(m) x m2/(kg m-3)]
157
5 AU
10 AU
25 AU
50 AU
100 AU
1.86
-18
R
-20
-20
-10
0
10
log10(m/kg)
1.84
20
0
3
6
log10(t/yr)
9
Figure 5.5 (Left panel) Particle mass distribution at 10 Gyr, when varying the value
of the radial distance of the disk. (Right panel) Evolution of the dust-mass distribution slopes when varying the same parameter.
bution starts resembling the redistribution function, as smaller particles are not
destroyed due to the lower energy collisions. Moreover, no kink is produced at
the high mass end. In the right panel of Figure 5.5, I show the evolution of the
particle-mass distribution slope as a function of the collisional velocity. For high
velocity collisions, the waves render the fitting of a single mass distribution slope
ill constructed but the underlying slope of the wavy mass distribution is slightly
steeper than for the smaller collisional velocity case.
5.2.4 The dependence of the steady-state distribution function on numerical parameters
I discuss here the effects of three non-physical variables that appear in the numerical algorithm. They are: the mass ratio δ between neighboring grid points and
the parameters of the large particle collisional cross-section smoothing formula,
Θ and p (see Chapter 4). In Figure 5.6, I plot the distribution of the model with
varying values of δ at 10 Gyr (left panel) and the evolution in the slope of the dust
-12
-14
1.9
2.685 (N=101)
1.280 (N=401)
1.131 (N=801)
1.104 (N=1001)
1.064 (N=1601)
-16
δ
-18
-20
-10
0
10
log10(m/kg)
2.685 (N=101)
1.280 (N=401)
1.131 (N=801)
1.104 (N=1001)
1.064 (N=1601)
1.89
Slope
log10[n(m) x m2/(kg m-3)]
158
δ
1.88
1.87
20
0
3
6
log10(t/yr)
9
Figure 5.6 (Left panel) Particle mass distribution at 10 Gyr, when varying the resolution of the numerical model. (Right panel) Evolution of the dust-mass distribution slopes when varying the same parameter.
distribution (right panel). The evolution of the dust distribution is affected by the
number of grid points used, converging at δ = 1.13; this corresponds to 801 grid
points between my mmin and mmax mass range. Using a lower number of grid
points leads to errors in the numerical integration for the redistribution, leading
to an offset larger than 7% for the smallest particles in the system, when only using half as many grid points. I find that the dust distribution slope is practically
independent of the smoothing variables Θ and p.
5.2.5 The time to reach steady-state
In the model calculations, the dust distributions in the vast majority of cases reach
steady state by 10-20 Myr and only in a few cases do they take somewhat longer.
The characteristic time is less than 100 Myr for all realistic cases. This shows that,
apart from second generation debris disks, the majority of debris disks around
stars of ages over 100 Myr are most likely to be in collisional steady-state, at least
for the smallest particles (< 1 cm). However, young and extended systems, such
159
3.7
Effects of 50% variation in parameters
1.88
3.6
1.86
1.84
Dohnanyi (1969)
3.5
Size-distribution slope
Mass-distribution slope
1.9
1.82
lo
)
(Θ
g 10
η 0 (P)
g 10
lo (M)
g 10 )
(G
g 10
lo
lo
g
ρ
)
f Y (f M
g 10
ax
lo
m
f X (α)
g 10
lo
βX
γ (R)
g 10
lo (S)
g 10 sc)
(Q
g 10
lo
lo
h
b
∆R
s
Figure 5.7 Effect on the equilibrium particle mass and size distributions of varying each collisional and system variable by 50%.
as β Pic (Smith & Terrile, 1984; Vandenbussche et al., 2010), might not be in complete steady-state at the outer parts of their disks, where interaction timescales
are longer.
5.2.6 The robustness of the solution
One of the most surprising results from the wide range of numerical models computed is the robustness of the steady state distribution. Varying the values of the
model parameters does not result in significant changes in the slope of the distributions. In Figure 5.7, I show the effect of varying each model parameter by
50% on the equilibrium slope of the dust-mass distribution function. I order the
parameters on the horizontal axis as a function of decreasing magnitude of their
effects. Note that I varied the parameters that typically span many orders of magnitude by 50% in log. The plot shows that the dominant parameter, by far, is the
slope of the strength curve in the strength-dominated regime. This is followed by
variables that affect the collisional velocity (∆R and h) and the power b of the erosive cratered mass formula (equation [5.1]). The plot also shows that neither of
160
the arbitrarily chosen collision prescription constants have any significant effects
on the outcome of the collisional cascade. The model runs predict an equilibrium
dust mass distribution slope of η = 1.88 ± 0.02 (ηa = 3.65 ± 0.05), taking the maximum offsets originating from the 50% variation in the model parameters test as
the error.
5.3 Synthetic Spectra
In the following sections, I compare the emission that results from the predicted
particle-mass distributions to observations. As a first step, I generate an array of
synthetic spectra using realistic astronomical silicate emission properties. I then
analyze how the spectra are influenced by the particle mass distribution function.
The flux emitted by a distribution of particle masses at a certain frequency is
equal to
vπ
Fν = 2
D
x
Z
da n(a)a2 Qabs (a, ν) Bν (T ) ,
(5.3)
n
where Qabs is the absorption efficiency coefficient, Bν (T ) is the blackbody function, and v is the total volume of the emitting region. Since in infrared astronomy
it is customary to express the flux density as a function of wavelength, I rewrite
this also as
vπλ2
Fν = 2
D c
Z
x
da n(a)a2 Qabs (a, λ) Bλ (T ) .
(5.4)
n
The exact function of the absorption efficiencies of particles in the interstellar
medium or in circumstellar disks is largely unknown. The most commonly used
particle types for SED calculations are artificial astronomical silicate (the properties of which are adjusted to reproduce the typical 10 µm silicate feature and
measured laboratory dielectric functions) and graphite (Draine & Lee, 1984). In
Figure 5.8, I plot the absorption efficiency as a function of wavelength for a few
astronomical silicate particle sizes. Particles larger than 10 µm have nearly con-
161
101
100
Qabs(λ,a)
10-1
10-2
10-3
0.1
1.0
10.0
20.0
40.0
80.0
10-4
10-5
1
10
100
Wavelength (µm)
1000
Figure 5.8 Absorption efficiency of astronomical silicates as a function of wavelength for a range of particle sizes between 0.1 and 80 µm. Solid lines are the
∼ λ−β approximations to the long-wavelength regimes of the curves that I employ in this work.
stant absorption efficiency curves at shorter wavelengths (λ < 2πa, where a is
the particle radius) with Qabs = 1, which is followed by a power-law cut off. The
slope of this power-law becomes constant for wavelengths larger than ∼ 8πa, and
is commonly denoted by the variable β. Astronomical silicates of all sizes have a
typical value of β = 2 (Draine & Lee, 1984).
In Figure 5.9, I show synthetic SEDs, all scaled to the same flux level at 1000 µm.
The top panel shows spectra calculated around an A0 spectral-type star, with debris rings placed at various distances between 2 and 292 AU. The minimum particle size cut-off was set at ∼ 5 µm, in accordance with the model (see Chapter 4).
162
2 AU
100
5 AU
Radial distance
11 AU
25 AU
Fν (Jy)
10
58 AU
130 AU
1
292 AU
0.1
0.01
10
100
1000
100
Minimum cut-off
1 µm
Fν (Jy)
10
30 µm
1
0.1
0.01
10
100
100
1.99
1000
Particle distribution slope
Fν (Jy)
10
1.80
1
0.1
0.01
10
100
λ (µm)
1000
Figure 5.9 Synthetic SEDs for
an array of model powerlaw particle mass distributions, with varied parameters.
The fluxes are scaled to match
at 1000 µm. In the top panel, I
show the synthetic SEDs generated for a variety of radial
distances; in the middle panel,
I show synthetic SEDs generated for a variety of minimum cut-offs (1, 2, 4, 8, 15
and 30 µm); and in the bottom panel, I show synthetic
SEDs generated for a variety of particle-mass distribution slopes (1.80, 1.84, 1.88,
1.92, 1.96 and 1.99). Note: In
the middle-panel I vary the
minimum cut-off of the particle mass distributions, even
though it is a parameter inherently set by equations in the
collisional model. Since in reality, the placement of the cutoff is set by the optical properties and structural build of
the micron size particles, it
is generally treated as a variable in SED models. In the
plot, I show that such variations in the placement of the
minimum cut-off do not affect the long-wavelength part
of the SEDs.
163
All disks with radial distances below ∼ 130 AU have a common slope for wavelengths larger than 250 µm, and the furthest disk at 292 AU joins this common
slope around ∼ 350 µm.
The blow-out size in a system depends on grain structure (porosity) and the
exact value of the optical constants for small grains (which is largely unknown
and is a function of grain material). For this reason, I also calculated synthetic
SEDs for a debris ring at 25 AU around an A0 spectral-type star, with the minimum particle size of the distribution artificially cut off at sizes between 5 and
30 µm. (Note that I normally calculate the blow-out mass self-consistently as described in Chapter 4.) I plot these SEDs in the middle panel of Figure 5.9. The
offsets between the SEDs become apparent for wavelengths shorter than 200 µm,
while for longer wavelengths, the emission profiles agree and have a common
Rayleigh-Jeans slope.
Finally, I explore the dependence of the SED on the slope of the steady-state
particle-mass distribution. The bottom panel of Figure 5.9 shows synthetic SEDs
generated for a debris ring at 25 AU around an A0 spectral type star, with a
minimum particle cut-off size at 5 µm, but with particle mass distribution slopes
between 1.81 and 1.99. These plots show that the slope of the Rayleigh-Jeans part
of the emission is greatly influenced by the particle size distribution slope. In
fact, they depend almost solely on it, with the temperature of the grains having
mild effects at large orbital distances.
5.4 Relation between the particle mass distribution and the SED
The absorption efficiency curves can be simplified and described as



1
λ < 4πa
Qabs (λ, a) ∝
β


 xa
λ > 4πa
λ
164
where x is a scaling constant for the power-law part of the function. Fitting the
silicate absorption efficiency functions, I find
x = 12
a
µm
−0.5
.
(5.5)
Using this simplified absorption efficiency model and assuming that all particles
contribute to the Rayleigh-Jeans tail of the SED with their own Rayleigh-Jeans
emission, I estimate the emitted flux density at long wavelengths as
"
#
β Z λ
Z ∞
4π
β
12
2vπkb T Cdisk
10−3
da a2−ηa .
Fν =
da a2+ 2 −ηa +
λ
D 2 λ2
λ
0
4π
(5.6)
Here I assumed a β parameter that is independent of the particle size. The variable Cdisk is the number density scaling (see Chapter 4), kb is the Boltzmann constant, T is the temperature of the dust grains (which I also assume to be particle
size independent), and D is the distance of the system from the observer. The
quantity ηa is the steady state particle size distribution slope, and can be calculated from the mass distribution slope as ηa = 3η − 2. Integrating these functions,
I get
β
Fν (λ) = C1 λ1− 2 −ηa + C2 λ1−ηa ,
(5.7)
where
2vπkb T Cdisk 10−3 12β 22ηa −5−β π ηa −3−β/2
×
D2
6 + β − 2ηa
ηa −3
2vπkb T Cdisk (4π)
×
.
=
D2
ηa − 3
C1 =
(5.8)
C2
(5.9)
Assuming β = 2, which is appropriate for astronomical silicates, I find that the
slope of the SED is equal to −ηa for the short wavelength part of the RayleighJeans tail of the SED and 1 − ηa for the long wavelength regime. Similar results
have been found by Wyatt & Dent (2002).
165
The models yield a steady-state distribution slope of ηa ≈ 3.65, meaning that
the Rayleigh-Jeans tail end of the SEDs should be proportional to
Fν ∝ λ−2.65 ,
(5.10)
as long as the particles are in collisional steady state.
5.5 Comparison to observations
To compare the computed spectra of steady-state collisional disks to data, I assembled the available data for debris disks with far-IR and submillimeter observations. As a result of the analysis in §5.3, where I determined the wavelength
range that is least sensitive to parameters, I use only data at wavelengths larger
than 250 µm. To fit a power-law to the Rayleigh-Jeans regime of the SEDs, I
need a minimum of three data points above the wavelength cut-off. I found
a total of only nine sources that fulfill these requirements. I present the farIR/submillimeter fluxes for these sources in Table 5.2. Occasionally, published
submillimeter measurements do not account for systematic errors. In these cases,
I applied a total of 30% error to all ground based measurements at 350 and 450 µm
and 15% for all Herschel data and measurements above 850 µm. I also made sure
that the data included all the flux from each source and applied an aperture correction estimate otherwise. All corrections are listed as notes in Table 5.2.
I perform individual power-law fits to the data of each source as well as a fit to
all sources simultaneously with a common Rayleigh-Jeans slope. In Figure 5.10,
I present the photosphere subtracted excess emissions for each source in the left
panels and plot the best-fit power-law spectrum of the form
−l
λ
,
Fν = A ×
200 µm
(5.11)
obtained from individual fits. I show in the right panels of Figure 5.10 the error
166
Table 5.2. Observational data of debris disks
Star
β Pic
ǫ Eri
Fomalhaut
HD 8907
HD 104860
HD 107146
HR 8799
Vega
HD 207129
λ (µm) Flux (mJy) Error (mJy) Excess (mJy)
250
350
500
800
850
850
1200
1200
1300
350
450
850
1300
350
450
850
1300
450
850
1200
350
450
850
1200
3000
350
450
850
1300
3000
350
850
1200
250
350
500
850
250
350
500
870
1,900.0
720.0
380.0
115.0
85.2
104.3
24.3
35.9
24.9
366.0
250.0
37.0
24.2
1,180.0
595.0
97.0
21.0
22.0
4.8
3.2
50.1
47.0
6.8
4.4
1.35
319.0
130.0
20.0
10.4
1.42
89.0
15.0
4.0
1,680.0
610.0
210.0
45.7
113.0
44.3
25.9
5.1
285.0
108.0
57.0
30.0
13.0
16.0
4.0
5.0
4.0
109.8
75.0
5.55
4.0
354.0
200.0
14.55
3.5
11.0
1.2
0.9
15.0
14.0
1.2
1.1
0.67
90.0
39.0
3.2
3.0
0.3
26.0
3.0
2.7
260.0
100.0
40.0
7.0
18.0
9.0
8.0
2.7
1,897.5
718.7
379.4
114.8
85.0
103.8
24.2
35.8
24.8
359.45
246.06
35.92
23.74
1,168.3
587.8
95.1
20.17
21.87
4.76
3.18
50.0
46.9
6.78
4.39
1.35
318.8
129.9
19.96
10.39
1.41
88.8
14.96
3.98
1,617.6
578.5
194.8
40.5
111.78
43.68
25.60
5.00
Reference
Vandenbussche et al. (2010)
Vandenbussche et al. (2010)
Vandenbussche et al. (2010)
Zuckerman & Becklin (1993)
Holland et al. (1998)
Holland et al. (1998)
Liseau et al. (2003)
Liseau et al. (2003)
Chini et al. (1991)
Backman et al. (2009)
Greaves et al. (2005)
Greaves et al. (2005)
Chini et al. (1991)
Marsh et al. (2005)
Holland et al. (2003)
Holland et al. (2003)
Chini et al. (1991)
Najita & Williams (2005)
Najita & Williams (2005)
Roccatagliata et al. (2009)
Roccatagliata et al. (2009)
Najita & Williams (2005)
Najita & Williams (2005)
Roccatagliata et al. (2009)
Carpenter et al. (2005)
Roccatagliata et al. (2009)
Najita & Williams (2005)
Najita & Williams (2005)
Najita & Williams (2005)
Carpenter et al. (2005)
Patience et al. (2011)
Williams & Andrews (2006)
Bockelée-Morvan et al. (1994)
Sibthorpe et al. (2010)
Sibthorpe et al. (2010)
Sibthorpe et al. (2010)
Holland et al. (1998)
Marshall et al. (2011)
Marshall et al. (2011)
Marshall et al. (2011)
Nilsson et al. (2010)
167
contours of the slope and normalization of the power-law at the 1, 2, and 3σ
levels. The plots also indicate the χ2min /d.o.f. (the minimum of the reduced χ2 ) of
each fit. The solid red line represents the Rayleigh-Jeans slope calculated from
the Dohnanyi (1969) analytic solution, the green band represents the best slope
given by the reference model calculation (including errors from 50% variations
in the slope of the strength curve, see Figure 5.7), and the blue band yields the
global fit solution of
l = 2.60 ± 0.06 .
(5.12)
The global fit and the reference model agree within the errors of the prediction.
5.6 Conclusions
In this Chapter, I used the numerical model introduced in Chapter 4 to follow
the evolution of a distribution of particle masses. I varied all twelve collisional,
all six system, and all three numerical variables of my model and examined the
effects of these variations on the evolution of the particle mass distribution. My
numerical model has been built to ensure mass conservation and that the resulting distribution of particles is not artificially offset due to numerical errors, as the
integrations of the model span over 40 orders of magnitude in mass. In §5.2.4
of this Chapter, I demonstrate that lower precision integrations can lead to shallower particle distributions.
The steady-state particle distribution of the collisional system is extremely
robust against variations in its variables, with the strongest effects occurring from
changes to the tensile strength curve (Holsapple et al., 2002; Benz & Asphaug,
1999). Even these variations have mild effects on the slope of the particle mass
distribution, modifying it only between the values of 1.84 and 1.94 (3.52 and 3.82
in mass space, respectively). I find the dust distribution of the reference model to
168
100
3.6
3.2
10-1
Slope
Fν (Jy)
3.4
10-2
10-3
10
3
2.8
2.6
2.4
HD 207129 Excess
10
100
1000
0.1
3.6
3.2
10-1
Slope
Fν (Jy)
3.4
10-2
10
HR 8799
2.2
10
100
1000
0.1
3.6
3.2
Slope
Fν (Jy)
0.5
2.8
10-2
0.25
0.5
0.75
1
χ2min = 0.61
l = 2.00+0.56
-0.00
3
2.8
2.6
2.4
HD 8907 Excess
HD 8907
2.2
10
100
1000
0.1
0
3.6
3.4
3.2
10-1
Slope
Fν (Jy)
0.4
χ2min = 1.03
l = 2.21+0.33
-0.21
2.4
HR 8799 Excess
10-1
10-2
10-3
0.3
3
3.4
10
0.2
2.6
0
10-3
HD 207129
2.2
0
10-3
χ2min = 0.36
l = 2.42+0.65
-0.42
1
χ2min = 1.50
l = 2.34+0.42
-0.34
3
2.8
2.6
HD 107146 Excess
2.4
HD 107146
2.2
10
100
1000
Wavelength (µm)
0.5
1
Normalization
2
3
Figure 5.10 Observed SEDs of debris disks with submillimeter and millimeter
data. The left panels are the photosphere-subtracted fluxes of the excess emissions with the best fitting slopes, while the right panels are the 68%, 95% and
99% confidence contours of the individual fits. The error contours also show the
slope given by the Dohnanyi (1969) mass distribution function in red, the value
predicted by the numerical code in the green band, and the best global fit with
errors in the blue band.
169
10-1
Slope
Fν (Jy)
100
10-2
10-3
HD 104860 Excess
10
100
1000
Slope
Fν (Jy)
101
100
10-1
10-2
Fomalhaut Excess
10
100
100
Slope
Fν (Jy)
101
10-2
10-4
100
Slope
10
2
10
1
ε Eri Excess
10
100
10-1
10-2
10-3
β Pic Excess
10
100
1000
Wavelength (µm)
Figure 5.10. (Cont.)
0.4
0.5
3
4
5
10
15
χ2min = 0.01
l = 3.01+0.30
-0.28
Vega
3.6
3.4
3.2
3
2.8
2.6
2.4
2.2
2
2
3
4
5
6
χ2min = 2.86
l = 2.14+0.49
-0.14
ε Eri
1000
100
2
1
10-1
0.3
Fomalhaut
3.6
3.4
3.2
3
2.8
2.6
2.4
2.2
2
1000
100
10-2
Fν (Jy)
10
1
1
Slope
Fν (Jy)
10
Vega Excess
0.2
χ2min = 0.58
l = 3.11+0.31
-0.34
1
102
10-3
HD 104860
3.6
3.4
3.2
3
2.8
2.6
2.4
2.2
2
1000
10-1
χ2min = 1.32
l = 2.07+0.57
-0.07
0.1
102
10-3
3.6
3.4
3.2
3
2.8
2.6
2.4
2.2
2
3.6
3.4
3.2
3
2.8
2.6
2.4
2.2
2
2
3
χ2min = 1.36
l = 2.63+0.24
-0.26
β Pic
1
10
Normalization
170
be 1.88 (3.65 in size space). I find that waves occur when the collisional velocities
are high or when particle strengths are low at the mass distribution cut-off, where
the radiation force blowout dominates the dynamics.
The Rayleigh-Jeans tail of the debris disk SEDs is dominated by the medium
sized particles, whose mass distribution is less affected by possible wavy structures. I derive a simple formula that gives the slope of the measured flux density
in the Rayleigh-Jeans part of the SEDs as
Fν ∝ λ1−ηa .
This implies that the mass distribution slope itself could, in principle, be measured from long-wavelength observations. I assemble a list of nine debris disks
that have been measured at the far-IR, submillimeter, and millimeter wavelengths
and examine the Rayleigh-Jeans slope of their emissions. My predictions of a
slope of l = 2.65 ± 0.05 agrees well with the observations, which have a global
slope fit of l = 2.60 ± 0.06.
171
C HAPTER 6
T HESIS C ONCLUSIONS
This thesis gives an analysis of the nature of debris disk infrared signatures, using
observational data obtained with the Spitzer Space Telescope and theoretical models. Below I give a brief summary of the results presented in the thesis followed
by an outlook for future work.
6.1 Summary
In Chapter 2, I present an observational example for a false debris disk signature resulting from the thermal heating of the interstellar dust repelled and accumulated in front of the A spectral-type star δ Velorum. The star was originally
assumed to have a debris disk based on its mid-IR excess emission. This was
later revisited when precision PSF subtraction from 24 µm Spiter MIPS images
revealed an impressive bow shock structure in front of the star. Assuming that
this structure is produced by a dynamical interaction between the star and the
surrounding interstellar medium, I introduce a model that calculates the geometry of the bow shock that expects the dynamics of the system to be dominated
by the radiation and gravitational forces the interstellar dust particles experience
from δ Velorum. I also calculate the total surface brightness emitted by the system, when assuming the particles to be in thermal equilibrium. My model gives
a consistent explanation of the total infrared excess and the surface brightness
distribution of the bow shock structure at δ Velorum. I determined the density of
the surrounding ISM to be 5.8 ± 0.4 × 10−24 g cm−3 . This corresponds to a number
density of n ≈ 3.5 atoms cm−3 , which means a ∼ 15 times overdensity relative to
the average Local Bubble value. The cloud and the star have a relative velocity of
172
35.8 ± 4.0 km s−1 . The velocity of the ISM in the vicinity of δ Velorum I derived is
consistent with LIC velocity measurements by Lallement et al. (1995). The question still remains how common this phenomenon is among the previously identified infrared-excess stars. Is it possible that many of the infrared excesses found
around early-type stars result from the emission of the ambient ISM cloud? The
majority of infrared excess stars are distant and cannot be resolved, so we cannot
answer for sure. However the excess at δ Velorum is relatively warm between 24
and 70 µm (F(24) ∼ 0.17 Jy, F(70) ∼ 0.14 Jy), and such behavior may provide an
indication of ISM emission.
In Chapter 3, I show the results of the Spitzer 24 µm survey of the nearby
(∼ 180 pc), relatively old (750 Myr) galactic open cluster Praesepe. I compiled a
cluster member list using data from SDSS, Webda, and 2MASS. Out of the 193
cluster members detected at all wavelengths in the combined catalog, 29 were
early (B5-A9) and 164 later (F0-M0) spectral types. I found one star in the early
and three in the later spectral type groups that show excess emission. Up to near
the completeness limit, with one debris disk star, there are 106 sources in the
later spectral-type sample. This result shows that only 6.5 ± 4.1% of early- and
1.9 ± 1.2% of solar-type stars are likely to possess debris disks in the 1-40 AU
zones. These values are similar to that found for old (> 1 Gyr) field stars. This
result can be placed in context with the Late Heavy Bombardment theory of the
Solar System. With simple Monte Carlo modeling, I show that the observations
are consistent with 15-30% of the stars undergoing a major re-arrangement of the
planetary orbits and a subsequent LHB-like episode once in their lifetime, with a
duration period of 50-100 Myr. I also summarize the results in the literature on
the decay timescales of debris disks around early- and solar-type stars, and find
that the decay timescale for solar-type stars is shorter than for earlier-type stars.
173
In Chapter 4, I present the main theoretical work of my thesis, which is the numerical modeling of the collisional cascades in debris disks. With my numerical
code, I am able to follow the evolution of the distribution of particles in a collisionally evolving system. The collisional equation that I introduce in my thesis,
unlike previous ones, solves the full scattering equation. The model includes both
erosive and catastrophic collisions, but currently does not evolve the dynamical
state of the system. In Chapter 4, I compare the code to the previously published
similar numerical models, showing general agreement. My model does show
faster decays than previously published ones (Thébault et al., 2003; Löhne et al.,
2008; Wyatt et al., 2011) and also yields slightly higher dust-to-planetesimal mass
ratios. I attribute these characteristics to be a result of the accurate treatment of
collisional cascades. I place special emphasis on improving the fidelity of the code
by using solvers that achieve high numerical accuracy. The numerical methods I
use are extensively verified and tested in Appendix D.
In Chapter 5, I analyze the evolution of my numerical code and its dependence
on the model variables. I vary all twelve collisional, all six system and all three
numerical variables of the model and examine the effects of these variations on
the evolution of the particle mass distribution. The collisional system is surprisingly extremely robust against variations in its variables, with the strongest effects occurring from changes to the tensile strength curve (Holsapple et al., 2002;
Benz & Asphaug, 1999). Even significant variations seem to have only mild effects on the slope of the particle mass distribution. My reference model’s dust
mass distribution slope of 1.883 (3.65 in size space) is a good average representation of the expected dust distribution slope. I also derive a simple formula,
that gives the slope of the measured flux density in the Rayleigh-Jeans part of the
174
SEDs as
Fν ∝ λ1−ηa ,
where Fν is the flux density λ is the wavelength, and ηa is the size distribution
slope. This relationship shows that the flux density is only a function of the dust
distribution slope. This in turn means that the distribution slope itself, in principle, can be measured from long wavelength observations. I assemble a list of nine
debris disks that have been measured at the far-IR, submillimeter and millimeter
wavelengths and examine the Rayleigh-Jeans slope of their emissions. My predictions of a R-J slope of l = 2.65 ± 0.05 agrees well with the observations, which
have a global slope fit of l = 2.605 ± 0.0645.
6.2 Future Work
Debris Disk research has been ever expanding since their first discovery almost
thirty years ago. We have been able to resolve debris disk in optical scattered
and at thermal infrared and submillimeter wavelengths. Numerical models have
attempted to understand the physical phenomena that keeps these systems alive
at such old stellar ages and detailed analysis of their material composition have
been tested by spectroscopic observations. With the latest launches of mid/far
infrared satellites our sensitivity level to their excesses have significantly lowered
and the prospect of future launches of satellites as WFIRST and the JWST give
hope for new and exciting results to come.
Interpreting the high quality data requires evermore sophisticated numerical
models and theoretical predictions. We still do not know how to interpret certain
observational details, such as the extended halo around certain debris disks (like
Vega), the necessary large-mass cutoff in particle size in warm debris disk spectral
models, the variation in the predicted particle size distribution for various debris
175
disks, the timescale of stochastic events in debris disk evolution and the period
of the re-birth of debris disk excesses. I plan on addressing these problems with
my numerical code and elaborate on them below.
6.2.1 Evolution of the particle size distribution slope
In Chapter 4, I presented my numerical model of collisional cascades in Debris
Disks and in Chapter 5, I made predictions for the equilibrium size distribution
as a function of model parameters. I also showed that the Rayleigh-Jean tail of the
spectral energy distribution emitted by debris disks is in direct correlation with
the particle size distribution slope. A near-future plan I intend to execute expands
on this idea. In a recently submitted Herschel Space Observatory OT2 proposal I aim
at studying a number of young debris disks with the SPIRE detector at 250, 350
and 500 µm, in the hopes of detecting variations in the dust distribution when
compared to older, more tranquil systems. In my proposed sample, I include
systems with ages between 1 and 100 Myr, with more sources near the lower
limit. With already available submillimeter and millimeter data on a number
of older sources and more to appear in the Herschel data archive, I will be able
to study the evolution of the particle size distribution directly via observations.
This I will be able to compare with my model predictions.
6.2.2 Warm debris disk models
Our solar system has two remaining planetesimal belts, the asteroid and Kuiper
belt. When searching for solar system like extrasolar planetary systems, we hope
to find systems with similar components. The majority of the discovered debris
disks have been of Kuiper belt like cold debris disks, however lately there have
been successful observations of warmer components. These warm debris disks
have been found around many different type and age stars. Well known debris
176
disks stars, like Vega (Su et al., 2005; Peterson et al., 2006), β Pic (Boccaletti et al.,
2009), Fomalhaut (Absil et al., 2009), and η Tel (Smith & Wyatt, 2010) all show
signs of a warm component to their infrared excess, and even the 10 Gyr old τ
Ceti seems to have an inner, warm disk (di Folco et al., 2007). Observational data
of these warm debris disks yield spectral energy distributions that can be modeled only with a narrow range of particle sizes, which poses a problem. A general
property of collisional cascade numerical models is the formation of waves when
collisions are energetic (Campo Bagatin et al., 1994; Thébault & Augereau, 2007;
Löhne et al., 2008). This is a feature that my models show, which I present in
Chapter 5. A possible resolution to the problem of limited particle size ranges
seen in warm debris disks is the occurrence of waves in the particle size distribution in the higher orbital velocity and collisional system that forms in the close
orbits. I plan on assembling a uniform catalogue of warm debris disks, in collaboration with Kate Su and George Rieke, and give predictions to the particle size
distribution and observable SEDs of the systems. Comparison to actual observations will yield an opportunity to verify and fine tune my numerical model.
6.2.3 Stochastic events and debris disk evolution
All current observations and models agree that the timescale of terrestrial planet
formation is between 10 - 50 Myr, with the excess dust and gas removed from the
primordial circumstellar disk under the same timescale. It is within this timescale
that the runaway growth first turns into a slower, oligarchic accretion process and
then into a late-stage growth phase. Finally, planetesimal systems tranquilize.
According to all dating methods, the planet formation process in the solar system
ended completely after 200 Myr. However, as observational evidence shows, an
older system, even a Gyr old system, is able to produce a significant debris disk
signature, whilst the typical removal timescale for dust is on the order of a few
177
million years. We know that even our solar system underwent periods of intense
bombardment, the latest being the Late Heavy Bombardment (LHB) 3.8 Gyr ago
(Tera et al., 1974). All evidence shows that the ”re-birth” of circumstellar dust
disks is a common phenomenon, and more likely to occur at the earlier stages of
stellar evolution, but certainly after 10 - 20 Myr (Kenyon & Bromley, 2004). The
late episode of bombardment in our solar system, however, does not seem to be
a common phenomena (Gáspár et al., 2009).
Continuing my research, I will expand my theoretical framework to include
collisional dynamics, as it currently only solves the collisional Boltzmann equation for a system of interacting particles. This way, I will be able to follow the
spatial and time evolution of debris disks. This is necessary for us to understand
how a giant planet or planetary system initiates/effects the evolution of a collisional system and to understand if the debris disk systems we are detecting are
actually terrestrial planet bearing or not.
I am not the first to propose a dynamical simulation of debris disks, and many
interesting results have been published in the field. The collisional model ACE
(Krivov et al., 2005; Löhne et al., 2008) solves the collisional Boltzmann equation
like my current numerical model (Gáspár et al., 2011), but in a three dimensional
system, with the orbital elements of the dust particles evolved, thus correctly
accounting for the motion of β meteorites. A similar model was presented by
Thébault & Augereau (2007), who divide the debris disks into rings. However,
non of these models include individual large planets, just particle distribution
functions, meaning that they are unable to follow or predict the actual dynamical
formation and evolution of debris systems.
An opposite modeling philosophy was presented by Raymond et al. (2011),
who follow the dynamical evolution of a system of massive planets and a distri-
178
bution of planetesimals, assuming a variety of initial system configurations. Assuming the traditional (and now outdated) Dohnanyi (1969) particle size distribution slope, they predict the evolution of the mid-infrared excess in their model
systems. As intriguing as their models are, they employ a number of simplifications that question the validity of their results. They simplify their collisions as
being only mergers, while debris disks are the result of collisional cascades. Their
final spectral energy distributions are calculated by assuming that the planetesimal distribution particles are aggregates of a distribution of particles, that emit
as black bodies with distributions that can be described by traditional models.
A synthesis of these models will be able to provide an adequate tool to analyze the formation and evolution of debris disks and to finally understand the
connection between them and planetary systems. The model I am proposing
will be a hybrid Monte Carlo/Boltzmann solver, with the dynamical evolution
of the planets, and larger planetesimals followed by an orbital integrator, while
the distribution of the dust produced via collisions would be estimated from my
already existing numerical code. Since all the mass in collisions is in the largest
fragments, only the few largest fragments from larger collisions need to be followed, while the dust distributions can be scaled, extrapolated and accumulated
at certain orbital distances and also removed by their respective removal methods. This numerical code would properly fragment or merge planetesimals when
they collide, depending on their collisional energy, follow the orbital evolution
and location of the fragments, and predict the expected infrared signature of the
debris disks, assuming realistic particle size distributions and emission efficiencies.
179
6.2.4 Debris disk haloes
The very first debris disk discovered (Aumann et al., 1984), the one around Vega,
still presents unresolved problems for debris disk evolution models, as it harbors an extended halo. Similar, although not as impressive, halos have been
found around other early spectral-type, debris disk host stars, like Fomalhaut
(Espinoza, in prep.). The extended halo around Vega has initially been explained
as a stream of micron sized particles being ejected from the system via radiation forces (Su et al., 2005). This has been questioned by later numerical models
(Müller et al., 2010) that propose the halo to be a result of particles with sizes
larger than the blow-out limit on highly eccentric orbits.
Although my numerical model currently does not model or evolve the dynamical state of the collisional systems, it does keep track of the particles on eccentric orbits and is able to predict their number densities on their respective
orbits, meaning it is able to give a rough estimate for the surface brightness of the
extended halo of the disk. I plan on investigating this matter in more detail as
well.
180
A PPENDIX A
A GE
AND DISTANCE ESTIMATE OF
P RAESEPE
The precise value of the cluster age is important in constraining the debris disk
fraction as a function of stellar age. The age and distance of Praesepe have been
a matter of debate, especially since it is an important step in the galactic distance
ladder. The estimated ages spread from log t = 8.6 all the way to log t = 9.15 (400
Myr – 1.42 Gyr)1 . Most papers list it as a coeval cluster with the Hyades because
of their similar metallicities and spatial motions (see, e.g., Barrado y Navascués
et al., 1998). The Hyades on the other hand has a better defined age of log t ≈
8.8 (625±50 Myr)(Perryman et al., 1998; Lebreton et al., 2001). If the clusters are
coeval, their ages should agree within close limits.
Aside from using pulsating variables (Tsvetkov, 1993) or stellar rotation (Pace
& Pasquini, 2004) to estimate the age of the cluster, the only method is to fit theoretical stellar evolution turnoff points on the observed CMD. This procedure
involves a precise simultaneous fitting of the cluster distance, reddening, metallicity and age.
The metallicity of Praesepe has been revisited many times. The value of Boesgaard & Budge (1988) of [Fe/H] = 0.13 ± 0.07 is usually accepted. An et al. (2007),
with new spectroscopic measurements, obtained a value of [Fe/H] = 0.11 ± 0.03,
also showing that the cluster is slightly metal rich. This fact has been overlooked
in some studies that have used solar values for metallicity, and which therefore
underestimate the cluster distance and overestimate its age.
The distance to Praesepe has been determined with many methods, yielding
slight differences among the measured values. Gatewood & de Jonge (1994) used
1
Cox (2000); Vandenberg & Bridges (1984); Tsvetkov (1993); González-Garcı́a et al. (2006)
181
Table A.1. The distance-modulus of Praesepe in the literature.
Reference
Method used
m−M
[mag]
Nissen (1988)
Mermilliod et al. (1990)
Hauck (1981)
Vandenberg & Bridges (1984)
An et al. (2007)
Gatewood & de Jonge (1994)
Loktin (2000)
This paper
Photometric
Photometric
Photometrica
Photometric
Photometricb
Parallax
Geometric
Photometric
6.05
6.2
6.26±0.23
5.85
6.33 ± 0.04
6.42 ± 0.33
6.16 ± 0.19
6.267 ± 0.024
a
Using Lutz-Kelker corrections (Lutz & Kelker, 1973).
b
Using empirically corrected isochrones.
the Multichannel Astrometric Photometer (MAP) of the Thaw Refractor of the
University of Pittsburgh to determine a weighted mean parallax of π = 5.21 mas
for five cluster member stars. The geometric method used by Loktin (2000) determines the apparent variation of the angular diameter of the cluster as it moves
along the line of sight and estimates the distance to the cluster from it. The basic
idea of this method is very similar to that of the convergent point method. The
photometric distances (main-sequence fitting) seem to show a large scatter. We
summarize the previous distance measurements and the methods used to obtain
the values in Table A.1.
We determined the distance and age of the cluster by simultaneously fitting
the distance modulus and the age with isochrones. The photometry values we
used were our best SDSS g and r band data, with the corrections explained in §??.
We did not include reddening in our color values, because it can be neglected
182
towards Praesepe (E(B-V)=0.027 ± 0.004 mag; Taylor (2006)). Since the plotted
CMD of Praesepe clearly showed a vertical trend at the later spectral type stars
at g − r ≈ 1.2, we only fitted cluster member points with g − r < 1.2 (fitting the
distance modulus to a vertical trend is impossible and only adds errors to the
fit). The isochrones for the fit were obtained from the Padova group website2 ,
where isochrones of any age and metallicity can be generated for a large number
of photometric systems, such as the SDSS system (Girardi et al., 2004). These
isochrones are similar to the empirical isochrones produced by An et al. (2007).
Since the metallicity of the cluster is still debated, we fitted isochrone sets for
all metallicities in the literature. Assuming that metallicities are solar scaled, we
set [Fe/H]=[M/H]. We fitted the following: [Fe/H]=0.13 (Z=0.025; Boesgaard &
Budge, 1988), [Fe/H]=0.11 (Z=0.024; An et al., 2007) and [M/H]=0.2 (Z=0.03; An
et al., 2007). The two values from the An et al. (2007) paper are from [Fe/H],
determined from spectroscopy, and an [M/H] value from isochrone fitting. We
also fitted solar metallicity isochrones to show the errors they give in the age and
distance determinations.
We calculated the best fit via Monte Carlo (i.e. bootstrap) method. We generated 10,000 new samples with the same number of sources as in the original
cluster member list. As with the bootstrap method, the members in the new samples were randomly picked from the original, resulting in multiple picks of a few
sources and null of others. The best fitting isochrones (as a function of age and
distance) to these mock samples were found by χ2 minimization. We computed
χ2 from each fit as
χ2 =
X
(∆r 2 + ∆(g − r)2 ),
(A.1)
where ∆r 2 is the r magnitude difference while ∆(g − r)2 is the color difference
2
http://pleiadi.pd.astro.it/
183
from the closest point of the isochrone model. By finding the closest point of the
isochrones we not only fit the luminosity difference, but an actual distance from
the isochrone, thus allowing points to be horizontally offset. We did not weight
our fit by photometric errors, because the brightest members (that are most crucial in the age determination) did not have quoted errors, while the errors of the
SDSS data cannot be trusted brighter than 14th magnitude. The means and errors in age and distance modulus of the best fitting isochrone (for all metalicities)
were calculated from the distribution of solutions given by the bootstrap method.
Following an initial fit, we removed stars that were further than 3σm−M magnitude from the best fitting isochrones and reran the Monte Carlo code. The best fit
value is given as the arithmetic mean and its error as its standard deviation.
The 2D errors for the fits are shown in Figure A.1, both for the full and for
the clipped samples. The histograms of the distance modulus and age fits are
shown in Figure A.2, both for the full and for the clipped samples also. The
results of the fitting for the 3σm−M clipped sample are summarized in Table A.2
quoting the 1σ errors. These errors are purely from the fitting procedure, and
do not include possible systematic errors such as those from isochrone models,
reddening, extinction and photometry.
The best fitting isochrones for the four metallicities are shown in Figure A.3.
All isochrones seem to deviate from the observed trend at g − r > 1.2 magnitude.
This is either due to errors in the calculated isochrones or to the membership criteria of Kraus & Hillenbrand (2007), who used estimated Teff and luminosity values
from photometry fitted SEDs and theoretical Hertzsprung-Russell diagrams.
We adopted the metallicity of Z=0.03 (An et al., 2007) to give a final estimate of
the cluster’s age and distance. We chose this metallicity to ensure comparability,
since An et al. (2007) deduced it from isochrone fitting also. The distance modulus
184
Table A.2. The solutions for the fitting of isochrones via two parameters for
solar and the metallicities found in the literature given with 1σ errors. These
values are the ones determined after the 3σ clipping iteration.
Metallicity
m−M
[mag]
Age
[log t]
[Fe/H] = 0.00
[Fe/H] = 0.13a
[Fe/H] = 0.11b
[M/H] = 0.20b
6.012 ± 0.020
6.153 ± 0.022
6.179 ± 0.022
6.267 ± 0.024
8.952 ± 0.011
8.918 ± 0.018
8.908 ± 0.019
8.879 ± 0.020
a
Boesgaard & Budge (1988)
b
An et al. (2007)
of our best fit for this metallicity is m−M = 6.267 ±0.024 at 1σ confidence, within
errorbars of the value of An et al. (2007) (m − M = 6.33 ± 0.04 mag). The errorbars
on distance are small at 3σ and comparable to the diameter of the cluster’s central
region (∼ 6 pc). The age of the cluster is determined to be log t = 8.879±0.020 (757
± 36 Myr) at 1σ confidence. The errorbars on cluster age are significantly smaller
than in papers before and help to pin down the decay trend at ages between 0.5
and 1 Gyr. The bootstrap Monte Carlo isochrone fitting method we introduce
here turned out to be a very effective and successful way to determine cluster
distance and age, and to estimate the errors of these parameters.
185
Number of simulations producing a solution pair
6.5
0
50
100
150
200
250
Number of simulations producing a solution pair
6.5
0
Best fitting distance modulus [mag]
Z=0.019
40
60
80
100 120 140 160
Number of simulations producing a solution pair
6.5
0
Z=0.024
20 40 60 80 100 120 140 160 180 200
Number of simulations producing a solution pair
6.5
6.4
6.4
6.4
6.3
6.3
6.3
6.2
6.2
6.2
6.2
6.1
6.1
6.1
6.1
6
6
6
6
5.9
5.9
5.9
5.9
9
0
750
250
500
9.05 9.1
1000
8.7 8.75 8.8 8.85 8.9 8.95
6.5
0
100
200
300
400
9
500
9.05 9.1
600
700
8.7 8.75 8.8 8.85 8.9 8.95
6.5
Z=0.024
3σ clip
Z=0.019
3σ clip
0
100
200
300
400
9
500
9.05 9.1
600
700
6.5
Z=0.025
3σ clip
6.4
6.4
6.3
6.3
6.3
6.3
6.2
6.2
6.2
6.2
6.1
6.1
6.1
6.1
6
6
6
6
5.9
5.9
5.9
5.9
9
9.05 9.1
8.7 8.75 8.8 8.85 8.9 8.95
9
Best fitting log(t) [Myr]
9.05 9.1
60
80 100 120 140 160 180
0
9
9.05 9.1
100 200 300 400 500 600 700 800 900
Z=0.03
3σ clip
6.4
Best fitting log(t) [Myr]
40
8.7 8.75 8.8 8.85 8.9 8.95
6.4
8.7 8.75 8.8 8.85 8.9 8.95
20
Z=0.03
6.3
8.7 8.75 8.8 8.85 8.9 8.95
0
Z=0.025
6.4
6.5
Best fitting distance modulus [mag]
20
8.7 8.75 8.8 8.85 8.9 8.95
9
Best fitting log(t) [Myr]
9.05 9.1
8.7 8.75 8.8 8.85 8.9 8.95
9
9.05 9.1
Best fitting log(t) [Myr]
Figure A.1 The 2D probability maps show the number of solutions that were
given for certain solution pairs by the Monte Carlo isochrone fitting algorithm.
The top row shows the fitting for the full sample, while the bottom row gives the
solutions after a 3σ clipping iterational step. The fitted metallicities are Z = 0.019,
0.024, 0.025 and 0.03.
186
Z=0.019
full sample
Z=0.019
3σ clip
4000
Z=0.019
full sample
Z=0.019
3σ clip
Z=0.024
full sample
Z=0.024
3σ clip
Z=0.025
full sample
Z=0.025
3σ clip
Z=0.03
full sample
Z=0.03
3σ clip
1500
N
3000
1000
2000
500
1000
0
0
Z=0.024
full sample
Z=0.024
3σ clip
4000
1500
N
3000
1000
2000
500
1000
0
0
Z=0.025
full sample
Z=0.025
3σ clip
4000
1500
N
3000
1000
2000
500
1000
0
0
Z=0.03
full sample
Z=0.03
3σ clip
4000
1500
N
3000
1000
2000
500
1000
0
5.9 6.0 6.1 6.2 6.3 6.4 6.5
m-M [mag]
5.9 6.0 6.1 6.2 6.3 6.4 6.5 6.6
m-M [mag]
0
8.7
8.8
8.9
9.0
9.1
log(t) [Myr]
8.7
8.8
8.9
9.0
9.1
9.2
log(t) [Myr]
Figure A.2 These plots show the 1D representation of Figure A.1, separately for
m − M and log(t). The distributions get much narrower after the 3σ clipping
iterations. These plots clearly show that as you go to more metal rich isochrones,
the best fitting isochrones will be younger and more distant.
187
4
6
8
r [mag]
10
12
14
16
18
20
Z=0.019
Z=0.024
Z=0.025
Z=0.03
-0.5
0
0.5
1
g-r [mag]
1.5
Figure A.3 The best fitting isochrones for all metallicities in the literature plus
solar. The dotted line isochrone that deviates from the rest at high luminosities is
the solar (Z=0.019) isochrone.
188
A PPENDIX B
S TRENGTH
CURVES
The redistribution outcome of collisions depends almost solely on the energy of
the impact and the colliding masses. In experiments it is common to specify the
ratio of the kinetic energy of the projectile to the mass of the target. This ratio
is known as the specific energy Qimp of the impact. Gault & Wedekind (1969) already noticed that the fragment distribution of particles depends on Qimp (which
they called “rupture energy”) when firing aluminum projectiles into glass targets. Their experiments showed that the fragments will have a power-law distribution, with the largest fragment being a function of the specific energy of the
impact. This relationship was first given in equation format in Fujiwara et al.
(1977) for basalt targets. They note an offset from the Gault & Wedekind (1969)
results, likely due to material strength differences.
Two specific values of Qimp are used: Q∗S (the shattering specific energy) and
a somewhat larger Q∗D (the dispersion specific energy). The value of Q∗S gives the
energy required to shatter the target so that the mass of the largest fragment is no
more than half of the original target mass. However, if the target is large enough,
then self gravity pulls the fragments back together, leaving a remnant larger than
half of the original. The larger Q∗D gives the value of Qimp needed to disperse the
fragments, so that the largest remaining piece is half of the original target mass.
At lower target masses, where self-gravity can be neglected, Q∗D ≈ Q∗S . We use
Q∗D in our code and refer to it as Q∗ .
Determining the value of Q∗ is difficult, especially for such a large range of
particle sizes, from µm to km. The values for smaller bodies on the order of a few
kilograms are mostly determined from laboratory experiments, while the values
189
for larger bodies are determined from under-surface explosions, observations of
large asteroids and with experiments done under very high pressure (Holsapple
et al., 2002). However, material strength varies greatly as a function of material
type, object size, surface type and the number of shattering events an object has
gone through over its lifetime. An object that has gone through many collisions in
its lifetime, but still remains in one piece (descriptively called a “rubble pile”) can
endure harder collisions, which can actually be absorbed and help to compact the
object, rather than dispersing it into smaller particles. This may seem like an important parameter only for larger objects; however, the evolution of larger objects
significantly influences the evolution of smaller particles, and thus is important
in our study. We also lack experiments done with targets and impactors cooled
down to space temperatures of 100-150 K, where one would assume that objects
get more brittle and easier to shatter.
Experiments clearly show that Q∗ is a function of the target mass M, meaning that different mass targets will get shattered (with a 0.5M largest fragment)
by different specific energies. Holsapple et al. (2002) reviews experimental and
theoretical results on collisions and strength curves. A common result for all
of them is a minimum in the strength curve for bodies around 0.3 km in radius, where planetesimals are easiest to disperse (the number of cavities and
cracks weakening the bodies increases, while self-gravitation is not dominant
yet). As a result, there is a bump in the size distribution of minor planets in
the solar system around this size. Smooth particle hydrodynamic (SPH) models
give the Q∗ strength curve for larger bodies, while experiments help to anchor
the curve down to smaller rocks on the scale of a few cm in radius. It is still
not clear whether the power-law shape of the curve can be extrapolated down to
micron size particles, where experiments cannot be carried out. To study the col-
190
lisional evolution of the smallest particles, the exact value of the strength curve
must be known. In the absence of any models/experiments currently at those
sizes, the best that can be done is a simple extrapolation of the strength curve to
those regimes. Stewart & Leinhardt (2009) introduce a velocity-dependent tensile
strength curve, that is defined by variables such that it removes ambiguities over
material density and projectile-to-target mass ratio. Their tensile strength curve is
ideal for low-velocity (1-300 m s−1 ) collisions, such as those found during planet
formation or at large radii debris disks. However, their universal relationship
does not hold for conditions that depart from the catastrophic disruption regime.
In our models we use the Benz & Asphaug (1999) dispersion strength curve.
It is derived from SPH models, represents a reasonable average of all previous
strength curves, and is impact angle averaged. This curve can be written as (all
units are in SI)
a g i
h a s
J gerg−1 kg−1 ,
+ Gρ
Q∗ (a) = 10−4 Qsc S
1 cm
1 cm
where the fiducial values in the equation are given in Table 5.1.
(B.1)
191
A PPENDIX C
M ASS
CONSERVATION OF THE COLLISIONAL MODEL
A crucial test of any collisional code is for it to conserve the initial total mass of
the system. Since particles are removed at the low mass end, this behavior can
be complicated to verify. However, a system can only maintain its total mass
numerically, if its collisional equations are mass conserving analytically. Here,
we prove that our collisional equation is mass conserving.
The collisional equation can be written as
dn(m)
= −
dt
∞
Z
dm′ n(m)n(m′ )σ(m, m′ )
Z0 ∞
Z ∞
+
dµ
dMn(µ)n(M)σ(µ, M)δ [X(µ, M) − m]
0
µ
Z ∞
Z ∞
+
dµ
dMn(µ)n(M)σ(µ, M)R(m; µ, M) ,
0
(C.1)
µ
where R(m; µ, M) is the redistribution function to mass m from µ + M collisions,
such that
Z
∞
dmR(m; µ, M)m = µ + M − X(µ, M) ,
(C.2)
0
and δ is the Kronecker function. Multiplying Equation (C.1) by m and integrating
over dm gives
dM
dt
Z
∞
Z
∞
dm
dm′ n(m)n(m′ )σ(m, m′ )m
Z0 ∞
Z 0∞
Z ∞
+
dµ
dMn(µ)n(M)σ(µ, M)
dmδ [X(µ, M) − m] m
0
µ
0
Z ∞
Z ∞
Z ∞
+
dµ
dMn(µ)n(M)σ(µ, M)
dmR(m; µ, M)m ,
(C.3)
= −
0
µ
0
where
Z
0
∞
dmδ [X(µ, M) − m] m = X(µ, M) ,
(C.4)
192
resulting in
dM
dt
Z
∞
Z
∞
dm
dm′ n(m)n(m′ )σ(m, m′ )m
Z0 ∞
Z 0∞
+
dµ
dMn(µ)n(M)σ(µ, M)(µ + M)
= −
0
(C.5)
µ
The first integral can be separated into two sections as
Z
∞
dm
0
Z
∞
′
′
′
dm n(m)n(m )σ(m, m )m =
0
Z
∞
Z
m
dm′ n(m)n(m′ ) ×
0
0
Z ∞
Z ∞
′
σ(m, m )m +
dm
dm′ ×
dm
0
′
m
′
n(m)n(m )σ(m, m )m .
(C.6)
Since σ(µ, M) is a symmetric function, we can swap the limits of integration for
m and m′ in the second integral of Equation (C.6) and, after making a change of
variables of m = µ and m′ = M in the first and m′ = µ and m = M in the second
integral, the full equation becomes
dM
=
dt
Z
0
∞
dµ
Z
∞
dMn(µ)n(M) [σ(µ, M)µ + σ(M, µ)M − σ(µ, M)(µ + M)] .
µ
(C.7)
Since the collisional cross section is completely symmetric, the integral itself becomes zero, thus proving that our equation is mass conserving.
193
A PPENDIX D
N UMERICAL
EVALUATION OF THE COLLISIONAL MODEL AND VERIFICATION
TESTS
The integro-differential equation presented in §4.3 must be integrated over 40 orders of magnitude in mass space, contains a double integral whose errors can
easily increase if not evaluated carefully, and is bundled in a differential equation
that evolves the number densities of dust grains and boulders within the same
step. These characteristics demand attention in its numerical evaluation. In the
following subsections we explain the numerical methods used to evaluate each
integral and the ordinary differential equation (ODE). We also present verification and convergence tests for our code, which explain why such precisions are
really necessary.
D.1 Taylor series expansion of TI
First, we expand equation (4.13) to use a Taylor series when m′ ≪ m and m′ <
µX (m). For this, we rewrite M in terms of m and m′ as
M = m + m′ G(m, m′ ) ,
(D.1)
where m′ G(m, m′ ) equals the cratered mass, m is the largest X(M, m′ ) particle created and G(m, m′ ) can be found by root finding algorithms. As written, G(m, m′ )
can be related to the Γ parameter used by Dohnanyi (1969), for which he used a
constant value of 130 for 5 km s−1 collisions. We plot the value of G(m, m′ ) as a
function of µ and M in Figure D.1, with the thicker solid line giving the contour
of G(m, m′ ) = 100. This contour lies at sizes reasonable for experiments in laboratory conditions, which is why Dohnanyi (1969) used a value close to it. The
194
a(µ)
1 µm 1 mm 1 m
a(µ)
1 µm 1 mm 1 m
1 km
a(µ)
1 µm 1 mm 1 m
1 km
1 km
4
1m
2
0
1 mm
-10
-1
-1
(V = 0.5 km s )
-20
1 µm
-1
(V = 1.0 km s )
a(M)
log10(M)(kg)
1 km
10
0
log10[G(µm,M)]
6
20
-2
(V = 1.5 km s )
-4
-20
-10
0
10
log10(µ)(kg)
20
-20
-10
0
10
log10(µ)(kg)
20
-20
-10
0
10
log10(µ)(kg)
20
Figure D.1 The values of G(m, m′ ) as a function of the colliding masses. The
thick contour is for G(m, m′ ) = 100, which is roughly equal to the Γ value used in
Dohnanyi (1969). The panels give the contours as a function of collision velocities.
The collisional velocities of 0.5, 1.0, and 1.5 km s−1 correspond to debris ring radii
of 100, 25, and 10 AU around an A spectral type star, respectively. The G(m, m′ )
parameter is strongly dependent on the collisional velocity.
positions of the contours are a strong function of the interaction velocities. The
m′ < µX (m) integrand can be written as
2
I(m, m ) = f (m )w(m )σw (m )m a(m) f (m)w(m)σw (m) 1 + Z − f (M)×
n
1 o2
′
3 −η
′
3 3
w(M)σw (M) 1 + G(m, m )Z
Z + 1 + G(m, m )Z
(D.2)
′
′
′
′
′ −η
2
where Z = a(m′ )/a(m) and f (m) and f (m′ ) are dimensionless number densities
that can be expressed as
f (m) =
n(m)
.
Cm−η
(D.3)
195
We rewrite this integrand as
f (M)
×
f (m)
!
o2
n
1
w(M)σw (M) −η
(D.4)
.
1 + G(m, m′ )Z 3
Z + 1 + G(m, m′ )Z 3 3
w(m)σw (m)
I(m, m′ ) = f (m′ )w(m′ )σw (m′ )m′
−η
a(m)2 f (m)w(m)σw (m) (1 + Z)2 −
The Taylor series for the components are
(1 + Z)2 = 1 + 2Z + Z 2
≡ T1
(D.5)
and
′
1 + G(m, m )Z
3
h
i2
p
3
′
3
− η Z + 1 + G(m, m )Z
= 1 + 2Z + Z 2
2G(m, m′ )
′
− ηG(m, m ) Z 3
+
3
2G(m, m′ )
′
− 2ηG(m, m ) Z 4
+
3
− ηG(m, m′ )Z 5
≡ T1 + T2
(D.6)
Both f (M)/f (m) and w(M)/w(m) are close to 1, while σw (M)/σw (m) deviates
from 1 as m approaches mmax . In those cases, the ratio can be expressed as
−m
Exp − mmax
σw (M)
∂σw (m) Θ
m′ G(m, m′ ) ,
= 1+
(M − m) = 1 − P −m
σw (m)
∂M M =m
Θ 1 − Exp − mmax
Θ
(D.7)
since we know that M − m = m′ G(m, m′ ). We write this ratio as
σw (M)
= 1−F .
σw (m)
(D.8)
The integrand then takes the form
−η
I(m, m′ ) = f (m′ )w(m′ )σw (m′ )m′ a(m)2 f (m)w(m)σw (m) ×
w(M)f (M)
T1 − (T1 + T2 )
(1 − F ) .
w(m)f (m)
(D.9)
196
Rearranging it gives us
−η
I(m, m′ ) = f (m′ )w(m′ )σw (m′ )m′ a(m)2 f (m)w(m)σw (m) ×
( )
w(M)f (M)
w(M)f (M)
T1 1 − (1 − F )
− T2
(1 − F ) (D.10)
w(m)f (m)
w(m)f (m)
When
1−
w(M)f (M)
< 10−9 ,
w(m)f (m)
(D.11)
we use the approximate formula
′
′
′
′
′ −η
I(m, m ) = f (m )w(m )σw (m )m
a(m) f (M)w(M)σw (m) × T1 F − T2 (1 − F ) .
2
(D.12)
We use the Taylor series of the components to write the integrand below the limit
of Z < 10−3 (i.e., m′ /m < 10−9 ). This means that our full integral for the first term
(TI ) takes the final form


X (m)

 µZ
dfI (m, t)
dm′ I
= −V πC

dt

 mmin
+
m
Zmax
µX (m)
dm′ f (m′ , t)(m′ )−η f (m, t) (a(m) + a(m′ ))
2







(D.13)
197
where



f (m′ )w(m′ )σw (m′ )m′ −η a(m)2 f (m)w(m)σw (m)×



)
(


i
h


)f (M )
)f (M )

− T2 w(M
(1 − F )
T1 1 − (1 − F ) w(M


w(m)f (m)
w(m)f (m)






)f (M )


if m′ < m × 10−9 & 1 − w(M
≥ 10−9

w(m)f (m)


I = f (m′ )w(m′ )σ (m′ )m′ −η a(m)2 f (M)w(M)σ (m) × T F − T (1 − F )
w
w
1
2







)f (M )

< 10−9
if m′ < m × 10−9 & 1 − w(M

w(m)f (m)



h
i



M −η
′
′ −η
′ 2
′ 2

f
(m
,
t)(m
)
f
(m,
t)
(a(m)
+
a(m
))
−
f
(M,
t)
(a(M)
+
a(m
))

m






if m × 10−9 ≤ m′ < µX (m)
In Figure D.2, we show the mass of the X fragments created when particles of
mass µ and M collide. The m = X(µ, M) regions are well defined in our single
collisional velocity case. When using collisional velocity that depends on particle
size, more than one µX (m) boundary may exist.
D.2 Verification of the numerical precision of TI
To verify the precision of our integrator we set up an equation that is similar to
TI in behavior and that has an analytic solution and compare values given by our
code to it. The integral we evaluate both analytically and numerically with our
code is
T VI (m) =
Z
µX(m)
m + m′ Γ
m
dm′ (m′ )−η
mmin
−η )
(
+
Z
1
1
m 3 + m′ 3
mmax
µX(m)
2
i
h
1 2
1
− (m + m′ Γ) 3 + m′ 3 ×
1
1 2
dm′ (m′ )−η m 3 + m′ 3
,
(D.14)
where we have removed all the constants and the dimensionless number densities. We have also replaced G(m′ , m) with a constant Γ (which can be related to
the Γ parameter used by Dohnanyi 1969) to enable an analytic solution. To verify
198
a(µ)
1 µm 1 mm 1 m
a(µ)
1 µm 1 mm 1 m
1 km
a(µ)
1 µm 1 mm 1 m
1 km
1 km
20
20
1m
0
0
1 mm
-10
log10(X) (kg)
10
a(M)
log10(M) (kg)
1 km
10
-10
-1
-1
(V = 0.5 km s )
-20
-20
-10
0
10
log10(µ) (kg)
(V = 1.0 km s )
20
-20
-10
0
10
log10(µ) (kg)
1 µm
-1
(V = 1.5 km s )
20
-20
-10
0
10
log10(µ) (kg)
-20
20
Figure D.2 The largest X fragment produced by collisions between particles µ
and M as a function of collision velocities. The collisional velocities of 0.5, 1.0,
and 1.5 km s−1 correspond to debris ring radii of 100, 25, and 10 AU around an A
spectral type star, respectively.
our algorithm for the evaluation of this term we set the initial particle distribution
to a power-law (η=11/6). The integration boundary can be evaluated as



 m
if Γ < 1
1−Γ
µX (m) =


 mmaxΓ −m if Γ ≥ 1
(D.15)
′
)
The first integral is (setting Z ≡ m
m
(
1
2
1
2 − 2 ′ − 61 F = − Z 3m
15Z 3 + 10Z 3 + 3 − (1 + ΓZ) 6 ×
5
"
#)
2
1
10Z 3 (1 + 2ZΓ) 3 (1 + 6ZΓ) 3Z 3 (5 + 6ZΓ)
+
+
.
2
1
1 + ZΓ
(1 + ZΓ) 3
(1 + ZΓ) 3
When
m′
m
(D.16)
= Z < 10−9 , the equation does not describe the analytic result correctly
as the first three components completely cancel the last three components numerically; however, analytically the result is not zero and this non-zero component
gets multiplied by a large number. This is the same catastrophic cancellation that
199
affects the numerical evaluation of TI (m). To overcome this and correctly represent the analytic result of the integral in such cases, we rewrote this to a Taylor
series as well. The first three components cancel, and we are left with
"
2 −2/3 ′ −1/6 35
11
65
F = + Z
m
ΓZ + 15ΓZ 4/3 + ΓZ 5/3 − Γ2 Z 2
5
2
2
24
#
85
665 3 3
25
ΓZ .
− Γ2 Z 7/3 − Γ2 Z 8/3 +
4
24
432
(D.17)
The second integrand has a much simpler anti-derivative
F=−
6 m2/3
m1/3
6
−
4
− 1/6
5/6
1/2
5 m′
m′
m′
(D.18)
In Figure D.3, we show the computational error as a function of mass m, the
Γ constant and the number of grid points used (neighboring grid point mass ratio). In the actual model Γ is not a constant, but equal to the variable G(m, m′ )
which, as shown in Figure D.1, varies from 10−4 to 106 . Figure D.3 shows that
the errors do not improve much past N=1000 (δ = 1.1) and that, in general, errors are smaller for large values of Γ. This shows that errors originating from TI
are most likely to affect the smallest particles in the system. We expect the errors
to actually be completely symmetric, with the highest masses showing the same
quantitative errors as the lowest masses near the boundary. Offsets are due to
the fact that our analytic model includes targets larger than mmax . The maximum
error of 10−4 is not ideal, but acceptable. When running our code, we use N=1000
grid points, which corresponds to a δ = 1.1.
D.3 Numerical evaluation of TII
As a double integral, the second term, TII , poses a larger challenge to achieve
acceptable precision. A collision between masses µ and M will be able to produce
a mass m in the redistribution power-law, if m < Y (µ, M). In Figure D.4 we
200
a(m)
1m
Relative Error
1 mm
a(m)
1m
1 mm
1 km
Γ=0.001
Γ=0.01
Γ=0.1
Γ=1.0
Γ=10.0
Γ=100.0
Γ=1000.0
10-2
10-4
10-6
10-8
Relative Error
1 km
N=201 (δ=1.64)
N=601 (δ=1.18)
N=1001 (δ=1.10)
N=2001 (δ=1.05)
10-2
10-4
10-6
10-8
-10
-5
0
5
10
log10(m) (kg)
15
20
-10
-5
0
5
10
log10(m) (kg)
15
20
Figure D.3 The error in the integration of TI as a function of the mass m and the
value of the Γ constant for neighboring mass grid ratios of δ =1.64, 1.18, 1.10 and
1.05.
plot the iso-Y contours in the µ vs. M phase space, which shows that integrating
between exact boundaries is difficult for TII , especially if the collisional velocity
is not a constant but a function of the particle mass.
As a first step, we determine which m masses can be produced by the grid
points (µ, M) and their neighbors. For a grid point to be able to produce a particle
of mass m, Y (µ, M) has to be larger than m. We determine the limiting mass
that is produced by each grid point and all of their neighbors as well. Up to
that min(µ, M) value, all m masses are produced with the full weight of the grid
point. Between min(µ, M) and max(µ, M) - which is the largest m still produced
by the (µ, M) grid point itself - we analyze the areas divided into quadrants, and
assign integration weights appropriately. A simple plot is shown in Figure D.5 to
explain the weights given to each grid point.
201
a(µ)
1 µm 1 mm 1 m
a(µ)
1 µm 1 mm 1 m
1 km
a(µ)
1 µm 1 mm 1 m
1 km
1 km
20
20
1m
0
0
1 mm
-10
log10(Y) (kg)
10
a(M)
log10(M) (kg)
1 km
10
-10
-1
-1
(V = 0.5 km s )
-20
-20
-10
0
10
log10(µ) (kg)
(V = 1.0 km s )
20
-20
-10
0
10
log10(µ) (kg)
1 µm
-1
(V = 1.5 km s )
20
-20
-10
0
10
log10(µ) (kg)
-20
20
Figure D.4 Iso-size contours for the produced Y fragments as a function of the
colliding body sizes and interaction velocities. Fragments of size a(m) will be
produced within regions where a(m) < a(Y ). The largest fragments produced
are not heavily dependent on the interaction velocities. The collisional velocities
of 0.5, 1.0 and 1.5 km s−1 correspond to debris ring radii of 100, 25 and 10 AU
around an A spectral type star, respectively.
These minimum and maximum masses are usually at most 2-3 grid points
apart. This means that on average 2-3 numbers have to be stored for all (µ; M)
grid points, as below min(µ, M) all m masses have the same weight. The final
integration speed can be increased by factors of 5 as the integration loops can be
run in non-redundant ways.
D.4 Convergence tests
We run convergence tests on our code for both terms to find the least number
of grid points we can use and still keep an acceptable numerical accuracy. We
calculate convergence to the value given using 4000 grid points. Our convergence
plot for TI in Figure D.6 shows that, for such a large dynamic range in masses, one
202
Figure D.5 Description of the integration method used for TII . The blue line represents the boundary, within which collisions are able to produce a certain mass
m in the redistribution power-law. Resolution elements capable of producing m
on their full area are colored red. Boundary resolution elements (i.e., µ ≡ M or
M ≡ mmax ) will be able to produce m on a “half” area, colored blue. The tip of the
distribution (green) will be able to produce on an eighth of its full area. Partial
quarter contributions are given by the yellow areas. Increasing the number of
grid points used obviously increases not just the precision but the area used for
the integration as well.
needs a neighboring grid mass ratio of at most δ = 1.1 to reach a relative error of
10−5 .
Our convergence test for TII does not reach the same level of accuracy, as at
δ ≈ 1.1 we reach a relative error of 10−2 only. However, this error is driven by
the resolution dependent integration limits and not the method itself. As such
203
3.0
δ - Neighboring grid point mass ratio
2.0
1.5
1.2
1.1
1.05
3.0
1.05
-1
10
TII convergence
TI convergence
-1
δ - Neighboring grid point mass ratio
2.0
1.5
1.2
1.1
-2
10
-3
10
-4
10
-5
10
-6
10
50
100
500
Number of points
1000
10
-2
10
-3
10
-4
10
-5
10
-6
10
50
100
500
Number of points
1000
Figure D.6 Convergence test results of our code. The left panel gives the results
for TI , while the right panel for TII . We also plot a N −2 curve with a dashed line,
which is the effective accuracy of the trapezoid integration method. The accuracy
of the first term follows this trend, however, that of the second term is shallower,
due to the resolution dependent integration limits.
the number of particles added by TII will always be underestimated by a small
amount.
D.5 The ODE solver
Previous work (i.e., Thébault et al., 2003; Thébault & Augereau, 2007; Krivov
et al., 2000; Löhne et al., 2008) used only a first order Eulerian algorithm to solve
the differential equation. We are using a 4th order Runge-Kutta algorithm (RK4).
To verify the ODE solver we simply evolve the Poynting-Robertson drag term,
whose analytic solution is
n(m, t) = n(m, t = 0) exp −
t
τPRD
.
(D.19)
In the following, we verified the accuracy of the ODE integrator by setting β ≡
0.100 for the particles and using the solar system timescale of 400 years.
Using the results from the code, we define the ratio of the particle density at
some time t to the particle density at time zero, i.e., Rcode ≡ f (m, t)/f (m, t = 0),
204
and compare it to the analytic result, Rexp = exp(−t/τPRD ). We then compute the
fractional error between the numerical and the analytic solution. The left panel of
Figure D.7 shows the fractional error as a function of the time step ∆t, evaluated
at a time roughly t ≈ τP RD (4000 years) for both the RK4 and Euler method.
10-2
RK4
Euler
t-1
CPU time for a RK4 ODE step [s]
100
10-4
ε
10-6
t
-4
10-8
-10
10
10-12
10-14
Machine Precision
-16
10
100
10
1
0.1
0.01
0.001
0.01
0.1
1
10
Step size [yr]
100
1000
50
100
500
1000
Number of bins
5000
Figure D.7 Left Panel: The fractional difference between the numerical and analytical results at a time roughly equal to 4000 yr as a function of the time step used.
We show errors for both an Euler ODE solver and for our RK4 algorithm. We also
plot a t−1 and a t−4 curve to guide the eye. Right Panel: The amount of processor
time needed by our code to complete an RK4 step as a function of the number of
mass grid points used.
For this particular set-up, the optimal time step is ∼ 4 yr for the RK4. However, the optimal time step depends on the time t at which the fractional difference is evaluated because of the accumulation of round-off errors. Finally, the
right panel of Figure D.7 shows the amount of CPU time taken to calculate an
RK4 step as a function of the number of mass grid points used on a Mac Pro4
with 2 2.26 GHz Quad Core Intel Xeon processors.
205
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