WEATHER ON OTHER WORLDS IV: IN-DEPTH STUDY OF

WEATHER ON OTHER WORLDS IV: IN-DEPTH STUDY OF
WEATHER ON OTHER WORLDS IV: IN-DEPTH STUDY OF
PHOTOMETRIC VARIABILITY AND RADIATIVE
TIMESCALES FOR ATMOSPHERIC EVOLUTION IN FOUR L
DWARFS
by
Davin C. Flateau
A Thesis Submitted to the Faculty of the
DEPARTMENT OF PLANETARY SCIENCES
In Partial Fulfillment of the Requirements
For the Degree of
MASTER OF SCIENCE
In the Graduate College
THE UNIVERSITY OF ARIZONA
2015
2
STATEMENT BY AUTHOR
This thesis 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 thesis are allowable without special permission, provided
that accurate acknowledgment of the 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.
SIGNED:
Davin C. Flateau
Davin C. Flateau
APPROVAL BY THESIS DIRECTOR
This thesis has been approved on the date shown below:
Dániel Apai
Assistant Professor, Department of Planetary Sciences
and Department of Astronomy
Date
3
ACKNOWLEDGEMENTS
This work is based in part on observations made with the Spitzer Space Telescope,
obtained from the NASA/ IPAC Infrared Science Archive, both of which are operated by the Jet Propulsion Laboratory, California Institute of Technology under a
contract with the National Aeronautics and Space Administration. This research
has benefitted from the SpeX Prism Spectral Libraries, maintained by Adam Burgasser at http://pono.ucsd.edu/ adam/browndwarfs/spexprism. The author wishes
to acknowledge the faculty and staff of the University of Arizona Department of
Planetary Sciences/Lunar and Planetary Laboratory for their guidance and assistance.
4
DEDICATION
For Joori and Kyu
5
TABLE OF CONTENTS
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
CHAPTER 1 INTRODUCTION . . . . . . .
1.1 Brown Dwarf Overview . . . . . . . . .
1.2 Brown Dwarf Evolution . . . . . . . .
1.3 Brown Dwarf Spectra . . . . . . . . . .
1.4 Condensate Clouds in Brown Dwarfs .
1.5 Periodic Photometric Variability due to
1.6 Dynamic and Thermal Modeling . . . .
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Heterogenous
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Clouds .
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CHAPTER 2 STATEMENT OF WORK . . . . . . . . . . . . . . . . . . . . 22
CHAPTER 3 OVERVIEW . . . . . . . . . . . . . .
3.1 Target Selection . . . . . . . . . . . . . . . .
3.2 The Young, Low-Gravity L3 Dwarf 2M2208
3.3 The L5 Dwarf SDSS0107 . . . . . . . . . . .
3.4 The Young, Low-Gravity L6 Dwarf 2M0103
3.5 The L3 Radio Emitter 2M0036 . . . . . . .
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CHAPTER 4 OBSERVATIONS AND DATA REDUCTION . . . . . . . . . 31
4.1 Observations and Data Reduction . . . . . . . . . . . . . . . . . . . 31
4.2 Corrections for Intrapixel Sensitivity Variations . . . . . . . . . . . . 31
CHAPTER 5 SPITZER PHOTOMETRY RESULTS
5.1 Period Finding and Amplitudes . . . . . . . .
5.2 Searching for Correlations and Phase Shifts . .
5.3 Absolute Photometry . . . . . . . . . . . . . .
5.4 Discussion . . . . . . . . . . . . . . . . . . . .
5.4.1 Light Curve Evolution . . . . . . . . .
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TABLE OF CONTENTS – Continued
CHAPTER 6 MODELING AND SPECTRAL FITTING . . . . . . . . . . .
6.1 Description of the Models . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Model Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Heterogenous Clouds as a Cause of Photometric Periodic Variability .
6.3.1 Description of the Models . . . . . . . . . . . . . . . . . . . .
6.3.2 Comparing Model Flux Differences with Observed Amplitudes
6.3.3 Calculating Model Amplitudes . . . . . . . . . . . . . . . . . .
6.3.4 Discussion of Two-Component Cloud Models . . . . . . . . . .
6.4 Timescales of Light Curve Evolution (LCE) . . . . . . . . . . . . . .
6.4.1 Radiative Flux Variations and Time Scales . . . . . . . . . . .
6.4.2 Description of the Models . . . . . . . . . . . . . . . . . . . .
6.4.3 Thermal Perturbation Procedure and Results . . . . . . . . .
6.4.4 Radiative Flux Variations and Timescales Discussion . . . . .
6.4.5 Dynamical Processes and Timescales . . . . . . . . . . . . . .
46
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CHAPTER 7 CONCLUSIONS AND SUMMARY . . . . . . . . . . . . . . . 73
7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
7
LIST OF FIGURES
1.1
1.2
Representative spectra spanning the optical to infrared (0.65–14.5
µm) for the spectral sequence of brown dwarfs and Jupiter. The
spectra are normalized to unity at 1.3 µm and an offset applied for
clarity. Significant molecular absorption bands that characterize the
spectra are labeled, including CO, TiO, FeH, CH4 , H2 (dashed lines
represent a broad, smooth absorption over these regions), NH3 and
H2 O. Dwarf spectral data are from Cushing et al. (2006); Jupiter data
are from Rayner et al. (2009) and Kunde et al. (2004). Figure from
Marley and Leggett (2009). . . . . . . . . . . . . . . . . . . . . . . . 16
A schematic of condensate clouds as calculated for progressive pressure and temperature layers spanning from T dwarfs to the warmest
(spectral type M) brown dwarfs. Clouds of various condensate species
form at specific levels of pressures and temperatures, falling lower into
the atmosphere as the dwarf cools. Figure courtesy of Dániel Apai. . 17
4.1
Representative Ch1 frames for our four L dwarfs, overlaid with the
photometric flux apertures (red) and background annuli (yellow). All
non-target sources and bad array elements were masked out of background and flux calculations. Bad array elements near the edges of
the Ch1 array for 2M2208 were also masked out. . . . . . . . . . . . 32
5.1
Ch1 and Ch2 light curves for all four targets, each plotted with a vertical offset for the purpose of the figure. The grey horizontal lines represent the relative mean flux level for each dwarf. Note that 2M0036
is plotted on a smaller scale for clarity. . . . . . . . . . . . . . . . . . 36
The light curves for three of our targets in Ch1 are in good agreement
with the period found by epoch-folding (2M0036 does not fold to a
particular period in Ch1). Phase-folded Ch1 light curves for the three
targets for which a period could be derived from epoch folding are
plotted above. The folded phases are plotted twice for clarity, and
the respective periods are indicated for each light curve. . . . . . . . 37
5.2
8
LIST OF FIGURES – Continued
5.3
5.4
5.5
6.1
A possible correlation between Ch1 and Ch2 light curves in SDSS0107
is plotted, with shifted Ch2 data (red) plotted against Ch1 data
(blue). The shift in time for Ch2 was calculated by lag values corresponding to a maximum ZDCF (correlation) value. Red points represent Ch2 data with an applied shift, plotted both with one value
of shift and values of period+shift to overlap with Ch1 data. The
continuous curve is a 5-term Fourier fit to Ch1 data showing how the
shifted features lines up when two phases of Ch1 are plotted. Light
red points are original Ch2 data without the shift in time. . . . . . . 40
Left: Color magnitude diagram in Spitzer Ch1 and Ch2 using timemedianed photometry from this work for the four target dwarfs, with
a selection of M, L and T field dwarfs shown with parallaxes from
Dupuy and Liu (2012). The low-gravity L3γ dwarf 2M2208 has an
M[4.5] brightness similar to the L5 dwarf SDSS0107, but a brightness
even lower than the low-gravity L6 dwarf 2M0103. Right: Colormagnitude diagram in J and H bands, from compiled literature magnitudes for our targets, HR8799 planets, and 2M1207b. 2M0103 and
SDSS0107 are extremely red L dwarfs with colors comparable to the
HR8799 planets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Light curve evolution of the three photometrically periodic targets in
Ch1 that epoch-folded to a period show changes from the first phase
between 0–2%, with a general brightening trend in Ch1 of 2M0103 of
∼0.5%. For each target, the phase-folded light curve was subtracted
from the first phase after each subsequent phase was interpolated to
the same time-grid in phase space as the first. A medianed noise level
for the subtracted phases is shown in red for each dwarf. . . . . . . . 44
SpeX and [3.6] and [4.5] fluxes match well with spectral and flux
models for all four dwarfs. Each dwarf’s SpeX spectra and observed
Spitzer fluxes (black) are plotted with its best-matching mean spectral and flux models (red), with corresponding G values noted. A
median noise value for each SpeX observation is plotted in an inset;
the error bars in the observed Spitzer fluxes are within the plotted
symbols. The Tef f , fsed and log g of the best-fit model for each dwarf
are noted; all dwarfs matched models with a fsed =1. . . . . . . . . . 51
9
LIST OF FIGURES – Continued
6.2
6.3
6.4
6.5
A representative plot for one dwarf of the model amplitudes calculated from our model grid using Equations 6.4 and 6.5. A line is
interpolated from each model amplitude to the origin, representing
values of C2 from 0.5 to 0, respectively. In this model amplitude
space, the observed amplitude of the dwarf lies near the origin. . .
Observed Ch1 and Ch2 color changes for all four L dwarfs as described
as model Spitzer amplitudes calculated from cloud model pairs parameterized by a secondary surface covering fraction C2 . Secondary
models are noted by Tef f ,fsed , log Kzz parameters. Amplitude values
have been scaled in units of the uncertainty of the observed Ch1 and
Ch2 amplitudes for each dwarf, and the observed amplitudes from
the dwarf’s light curve (blue circle) are shown. Each model amplitude value is parameterized by a line of C2 values from C2 =0 (the
origin) to 0.5 (calculated model amplitude) as described in the text.
The best-matched secondary model to the observed amplitudes are
in bold. The 1σ uncertainty value is depicted as the dashed circle,
and we consider all models within this circle also likely matches. This
circle is outside the plot for 2M2208, 2M0036 and 2M0103. . . . .
Model pair combinations reproduce the observed Ch1–Ch2 color
changes for 2M2208 and SDSS0107. The parameter space is grouped
by fsed ={1,2,3,4,nc (no clouds)}, and may be best visualized as existing in 3D stacked on top of each other in this order. B is the base
model determined from spectra and flux matching, and green shows
the single best matching secondary model that reproduces the observed color change for each dwarf, with the corresponding secondary
surface area covering fraction C2 noted. Other matches within 1σ are
in blue, while secondary models we exclude (> 5σ) or that are likely
unphysical (see text) are in grey and brown, respectively. Models
within 1σ are constrained in Tef f to ±200K for SDSS0107 and +500K
−400K
for 2M2208. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Model pair combinations reproduce the observed Ch1–Ch2 color
changes for 2M0103 and 2M0036. Color key is identical to Figure 6.4.
The very small amplitudes of 2M0036 produce many likely matches
over a wide range of Tef f , and account for its extremely small value
of C2 . Similarly, the ACh1 / ACh2 ratio of 2M0103 (1.23) place the
observed amplitudes of that dwarf within a model amplitude space
where there are many likely model matches to observations. . . . .
. 57
. 58
. 59
. 60
10
LIST OF FIGURES – Continued
6.6
6.7
A linear combination of the best-fit SpeX-regime base and secondary
model with the best-fit C2 value are extremely close to the base model
for each dwarf. Very little observed change in Spex-regime spectra is
expected to be observed using the best-fit models that reproduce the
variability seen in Ch1 and Ch2. Combined models were calculated
according to Equation 6.3. The uncertainties from the observed SpeX
spectra at each wavelength are shown, and used in a reduced χ2
calculation between the combined and base models. . . . . . . . . . 63
Perturbed/unperturbed flux ratios (left column) and radiative
timescales ”t” (right column) for each thermal perturbation in purely
radiative cloudy models are shown. Hatched areas indicate the extent of pressure levels which produced peak flux ratios in Ch1 (blue)
and Ch2 (red) when perturbed. Radiative timescales for dissipating
perturbations initiated in the pressure level that produces peak perturbed/unperturbed flux ratios are ≤ 1.40 h for all dwarfs in both
channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
11
LIST OF TABLES
3.1
Target Properties and Observation Log . . . . . . . . . . . . . . . . . 27
5.1
Derived Properties for Targets from Photometry . . . . . . . . . . . . 38
6.1
6.2
SpeX Observing Log . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Best-Fit Dwarf Parameters from SpeX Spectra and Spitzer Flux
Matches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Best-Matched Model Pair Parameters for Ch1-Ch2 Color Changes . . 61
6.3
12
ABSTRACT
Rotational phase mapping of brown dwarfs allows exploration of different cloud and
photospheric properties within the same atmospheres, allowing a separation of these
parameters from global parameters, such as composition, surface gravity, and age.
This work presents an in-depth characterization of high SNR light curves from the
Spitzer Space Telescope with up to 13 hours of continuous monitoring of four dwarfs
spanning the L3 to L8 spectral type. An exhaustive exploration of currently available state-of-the-art models explains the observed color changes for two of these
dwarfs with a linear combination of two model cloudy surfaces differing in effective
temperature, cloud opacity and vertical mixing. Using state-of-the-art purely radiative convective atmospheric models, we calculate basic radiative timescales for
temperature perturbations in the atmosphere, and consider the effects of dynamics on these timescales. Along with dynamical atmospheric advection timescales,
we discuss the relationships between model timescales and the observed light curve
evolution.
13
CHAPTER 1
INTRODUCTION
Brown dwarfs are an excellent laboratory for studying cool, cloudy atmospheres that
overlap in physical and chemical properties with extrasolar planets. Brown dwarfs
are typically free-floating objects that are not irradiated by a host or companion
star, making them easier to study than both close-in and widely-separated planets.
The relatively recent observational discovery and characterization of thousands of
brown dwarfs throughout a range of temperatures has provided an opportunity to
study many different characteristics of cool, cloudy atmospheres in detail. With
recent advances in infrared detectors and telescopes, brown dwarfs present the opportunity to probe the composition of complex atmospheric cloud structures, as well
as characterizing rapid changes in cloud properties due to dynamical advection and
thermodynamic processes. The characterization of a wide variety of brown dwarf
atmospheres can provide important constraints to the physical and chemical models
of solar system and extrasolar planetary atmospheres.
1.1 Brown Dwarf Overview
The theoretical existence of free-floating substellar objects formed from the gravitational collapse of molecular gas with insufficient mass to sustain hydrogen fusion
was predicted as early as the 1960s (Hayashi and Nakano 1963; Kumar 1963). But
advances in optical and infrared instrumentation sensitivity would take another 30
years to lead to the announcement of the first generally accepted confirmed detection of a brown dwarf (Gliese 229 B, Nakajima et al. 1995). Since then, thousands of
brown dwarfs have been identified with the help of large surveys, including the Deep
Near-Infrared Southern Sky Survey (DENIS, Epchtein et al. 1999), the Sloan Digital
Sky Survey (SDSS, York et al. 2000) and the Two-Micron All-Sky Survey (2MASS,
14
Skrutskie et al. 2006). Observed spectral features in these objects throughout various
temperature regimes led to their subsequent classification into four spectral types,
late M (Tef f = 3500 − 2100 K, Kirkpatrick et al. 1991), L (Tef f = 2100 − 1300
K, Kirkpatrick et al. 1999), T (Tef f = 1300 − 500 K, Burgasser et al. 2006) and
“room temperature” (and below) Y dwarfs (< 500 K, Cushing et al. 2011). Despite
their historical elusiveness to detection, recent estimates of brown dwarf number
densities for the local stellar neighborhood (< 26 light years) indicate the existence
of perhaps one brown dwarf for every four stars (Andersen et al. 2008; Kirkpatrick
et al. 2011; Burningham et al. 2013).
1.2 Brown Dwarf Evolution
During initial contraction of the brown dwarf, gas density at the core is high enough
for free electrons to fill the lower Fermi energy states, creating pressure from the
partially degenerate electron gas that balances the gas’ gravitational force, keeping
the brown dwarf’s radius near that of Jupiter (7 × 107 m). Brown dwarfs will
typically fuse deuterium into lithium-3 for a few Myr before deuterium depletion
halts the reaction. More massive brown dwarfs (M > 65 MJup ) can additionally
fuse lithium (e.g. Dantona and Mazzitelli 1985; Stringfellow 1989). After nuclear
reactions have ceased, the brown dwarf continues to radiate away energy as it cools.
As the dwarf cools, its Tef f passes through the condensation points of various
chemical compounds. The spectral energy distribution (SED) of the dwarf is subsequently altered as gas-phase absorption features due to certain species (e.g. TiO,
VO) disappear as they condense into the cooler atmosphere, while the spectral features of condensates including silicates become visible. As the brown dwarf cools,
it evolves through the M-L-T-Y sequence. But since brown dwarfs can form with
various masses, thus starting off with different thermal energy from contraction,
they can begin their lives in different places along the M-L-T-Y sequence. This creates an ambiguity between the observed luminosity of the brown dwarf and its age,
mass and radius, which are calculated values. Additionally determining the object’s
15
surface gravity (g = GM/R2 , where G is the universal gravitational constant, M
the mass and R the radius of the dwarf, usually ranging from 10 − 3000m/s2 ) with
techniques such as spectral model fitting, can be an important characteristic used
to resolve this ambiguity in evolutionary models (Saumon and Marley, 2008).
1.3 Brown Dwarf Spectra
Figure 1.1 shows representative 0.65–14.5 µm spectra for brown dwarf spectral types
in comparison to Jupiter. With decreasing Tef f , the general blackbody-like shape of
the late M-dwarf’s SED is altered by increasingly prominent molecular absorption
bands. The warmest late-M type dwarfs are characterized by bands from metalhydrides and metal-oxides such as TiO and FeH, which are incorporated into grains.
In L dwarfs, lower temperatures allow iron and silicate grains to produce optically
thick clouds, which can obscure the molecular absorption bands from gases. These
condensate clouds also significantly redden the near-infrared JHK colors of these
dwarfs with decreasing temperature through later L types. Near the L and T spectral
type boundary (∼ 1, 400 K), a color shift toward the blue occurs over a narrow
temperature interval (Dahn et al. 2002; Tinney et al. 2003; Vrba et al. 2004), and
has been explained by the onset of patchiness in the global cloud structure (e.g.,
Ackerman and Marley 2001; Burgasser et al. 2002; Saumon and Marley 2008; Marley
et al. 2010; Radigan et al. 2012; Apai et al. 2013). In T dwarfs, the CH4 bands
become more prominent from their first appearance in mid L dwarfs (Noll et al.,
2000), and together with the sunken condensate clouds and collision-induced H2
opacity between 2.0–2.4 µm (for >T5), continue the blueward shift of NIR colors
for cooler dwarfs, even as the gradual cooling of the dwarf would naturally progress
the object towards redder NIR colors. Jupiter’s spectra < 4 µm are dominated by
scattered sunlight altered by NH3 and CH4 absorption features.
16
4
The Future of Ultracool Dwarf Science with JWST
103
Fig. 4.1 The most prominent signatures of the ultra cool dwarf spectral sequence are seen in these
Figure 1.1 Representative spectra spanning the optical to infrared (0.65–14.5 µm)
0.65 to 14.5
!m spectra of a mid-M, L, and T dwarfs as well as Jupiter (adapted from Cushing
for the spectral sequence of brown dwarfs and Jupiter. The spectra are normalized
et al. (2006)). The spectra have been normalized to unity at 1.3 !m and multiplied by constants.
to unity at 1.3 µm and an offset applied for clarity. Significant molecular absorption
Major absorption bands are marked. The collision-induced opacity of H2 is indicated as a dashed
bands that characterize the spectra are labeled, including CO, TiO, FeH, CH4 , H2
line because
it shows no distinct spectral features but rather a broad, smooth absorption. Jupiter’s
(dashed lines represent a broad, smooth absorption over these regions), NH3 and
flux shortward
of ∼ spectral
4 !m is predominantly
scatteredetsolar
light; thermal
at
H2 O. Dwarf
data are from Cushing
al. (2006);
Jupiteremission
data aredominates
from
longer wavelengths
(nearand
mid-infrared
Jovian
spectra
from
Rayner,
Cushing
&
Vacca
(in
Rayner et al. (2009) and Kunde et al. (2004). Figure from Marley and Leggett
preparation)
and Kunde et al. (2004), respectively)
(2009).
temperature as a function of spectral type from late M through late T. While the
general correlation of increasing spectral type with falling effective temperature
is unmistakable, a remarkably rapid set of spectral changes (as expressed in the
variation in spectral type) happens over a relatively small span of Teff near 1400 K.
As we will discuss, understanding this variation in expressed spectral signatures, the
‘L to T transition’, is a key subject of current brown dwarf research.
Most of the scientific inquiry into these ultracool dwarfs has focused on their
17
Figure 1.2 A schematic of condensate clouds as calculated for progressive pressure
and temperature layers spanning from T dwarfs to the warmest (spectral type M)
brown dwarfs. Clouds of various condensate species form at specific levels of pressures and temperatures, falling lower into the atmosphere as the dwarf cools. Figure
courtesy of Dániel Apai.
1.4 Condensate Clouds in Brown Dwarfs
Understanding the chemistry and structure of condensate clouds plays a critical role
in the L dwarfs’ observed SED and colors. Optically opaque cloud layers at altitude can absorb flux from deeper, warmer layers, and can be disrupted or altered
by processes including atmospheric dynamics, thermodynamic activity, and chemical processes. These processes may create a global patchwork of layered clouds
of varying properties including optical depth and grain sizes, that shape the observed SED from the dwarf (Marley et al., 2010). Figure 1.2 shows a schematic of
how the progression of pressure and temperature through a brown dwarf’s atmosphere allows the condensation of cloud decks of distinct chemical species. Higher
temperatures and pressures allow the formation of grains of refractory compounds
like metal-oxides and silicates, while lower pressures and temperatures at higher
altitudes allow volatile species to condense such as chlorides (Ackerman and Marley 2001; Lodders and Fegley 2006). For decreasing dwarf Tef f , these clouds decks
descend further into the atmosphere, following higher pressures and temperatures.
Modeling of clouds has been notoriously difficult, but there has been steady
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progress in modeling the emergent SED from brown dwarf condensate clouds towards
state-of-the-art models that have had success in reproducing the NIR colors and
spectra of L dwarfs (e.g. Lewis 1969; Fegley and Lodders 1994; Lunine et al. 1989;
Marley et al. 1999). In the models of Ackerman and Marley (2001), condensation
clouds are treated as longitudinally homogenous, whose vertical extent is balanced
between the downward transport of sediment and the upward transport of vertical
mixing of both condensate and vapor according to
− Kzz
∂qt
− fsed w∗ qc = 0
∂z
(1.1)
where Kzz is the vertical eddy diffusion coefficient, qc and qv are moles of condensate
and vapor (respectively) to moles of atmosphere, qt is total moles of condensate and
vapor (qt = qc + qv ), and w∗ is the convective velocity scale. The critical free
parameter in this model is the sedimentation efficiency factor fsed . Large values
of fsed correspond to rapid particle growth resulting in large mean particle sizes.
Condensates quickly settle in this case, leading to physically and optically thin
clouds. When fsed is small, particles grow more slowly, do not settle out, leading to
a larger atmospheric condensate load and thicker clouds. The cloud model is fully
coupled with the radiative transfer and self-consistent with the (P, T ) structure of
the model. These models have been used successfully in modeling the J − K color
shift of the L/T transition (Saumon and Marley, 2008), and in producing model
spectra that have matched well to the observed NIR spectra of L dwarfs (Cushing
et al., 2008).
1.5 Periodic Photometric Variability due to Heterogenous Clouds
The idea of disrupted clouds in an L/T dwarf led to the testable hypothesis that
a dwarf’s rotation could produce a modulated photometric variability as the heterogenous cloud structure rotates through the field of view. While hot and cold
spots in the atmosphere due to the coupling of a dwarfs’ magnetic field with the
atmosphere may be a possible mechanism for modulated photometric variability in
19
hotter, more magnetically-active late M or early L dwarfs, the neutral atmospheres
of L and T dwarfs are generally considered too electrically resistive for this kind of
coupling (e.g., Mohanty et al. 2002; Chabrier and Küker 2006), although there have
been some examples of observed flaring activity in radio bandpasses in objects as
cool as T dwarfs (e.g., Route and Wolszczan 2012).
Early searches for this variability, primarily with ground-based instruments in
optical red or NIR wavelengths, over timescales ranging from minutes to weeks,
led to largely ambiguous detections or detections that were not found on followup
observations (e.g. Bailer-Jones and Mundt 2001; Enoch et al. 2003; Koen 2005a,b;
Clarke et al. 2002a; Khandrika et al. 2013; Wilson et al. 2014, reevaluated in Radigan
2014). The first tentative detection of photometric periodic variability was the L2
dwarf Kelu-1 (Clarke et al., 2002b) with a 1.1% peak-to-peak amplitude at 860
nm, and a 1.8 h period. Although followup observations did not detect the same
variability in the I band, the Hα line intensity was observed to vary in intensity in the
same period (Clarke et al., 2003). The first unambiguous detection a variable brown
dwarf was the T2.5 dwarf SIMP J013656.5+093347, with a ∆J = 50 mmag and a
period of 2.4 h (Artigau et al., 2009). Subsequent notable detections include the
T1.5 dwarf 2MASS J21392676+0220226, with a maximum 26% variability in J and a
7.7 h period (Radigan et al., 2012), and the T0.5 dwarf SDSS J105213.51+442255.7,
with a 6% maximum variability and a 3 h period (Girardin et al., 2013).
In addition to ground-based studies, Spitzer Space Telescope IRAC and HST
WFC3 instruments (Buenzli et al. 2012; Heinze et al. 2013; Apai et al. 2013) observed
brown dwarf variations over 3.6 and 4.5 µm bands, and the 1.1–1.7 µm spectral range
(time-resolved spectroscopy), respectively. Apai et al. (2013) concluded that J and
H band color changes and spectral variations in two L/T transition dwarfs could be
explained by a heterogeneous mix of low-brightness, low-temperature thick clouds
and brighter, warmer, thin clouds, with the absence of deep cloud holes. Possible
mechanisms in that study for the change in cloud properties were suggested to be
large-scale vertical mixing and circulation. Subsequently, Buenzli et al. (2015a,b)
successfully described the mean spectrum and relative amplitudes of Luhman 16A
20
and B, two variable L/T transition dwarfs as observed with HST between 1.1–
1.66 and 0.8–1.15 µm (respectively), with sets of two-component cloud models that
linearly combined a warmer, thinner cloud with a cooler, denser cloudy model.
Yang et al. (2015) found that two L5 dwarfs with 1.1–1.7 µm spectral variability
as observed with HST were explained with models that utilized a spatially-variable
high-altitude haze layer that resided at very low pressures above the condensate
clouds.
The Apai et al. (2013) and Buenzli et al. (2012) works utilized their observations
at multiple wavelengths to find pressure-dependent phase shifts in the periodic photometric variability for a T6.5 dwarf, but no phase shifts for two T2 dwarfs. These
studies show how cloud structures can be explored vertically and longitudinally with
time-resolved observations. The Buenzli et al. (2014) and Metchev et al. (2015) HST
and Spitzer surveys show that most, if not all brown dwarfs have heterogeneous cloud
cover and show rotational modulations if observed with sufficient precision.
1.6 Dynamic and Thermal Modeling
Dynamic and thermal modeling have been important in understanding mechanisms
of cloud formation and stability as well as the role of possible temperature perturbations could play in understanding periodic photometric variability. Freytag et al.
(2010) used a 2-D radiative hydrodynamic model to show that gravity wave propagations in brown dwarf atmospheres are common above the radiative-convective
boundary, and should play an important role in cloud formation and evolution, at
least on small scales. Showman and Kaspi (2013) utilized 3-D global models to explore convection and global circulation in dwarfs. This work showed that stratified
turbulence and flows resulting from atmospheric waves could result in horizontal
temperature variations of up to ∼50K. This model also predicts large-scale organization of the atmospheric flow, and wind speeds of tens to hundreds of m/sec. Flow
organization will modulate the cloud structure, leading to large-scale patchiness in
the cloud structure as necessary to explain variability measurements. Robinson and
21
Marley (2014) investigated the role that the aforementioned atmospheric temperature variations could play as a possible source of observed NIR variations in brown
dwarfs. This work showed that in a purely radiative non-grey atmosphere, thermal
perturbations at pressures greater than 10 bar propagate to the upper atmosphere
by radiative heating through the windows in NIR water opacity. Such radiative
feedback mechanisms through various atmospheric depths are important for understanding other possible sources of brown dwarf variability. Zhang and Showman
(2014) performed one-layer shallow-water calculations of the atmospheric flow to
determine the extent to which small-scale turbulent forcing, associated with convective perturbations at the radiative-convective boundary, can lead to formation
of atmospheric vortices and zonal jets analogous to those occurring on Jupiter and
Saturn. Their models showed that, at long radiative time constants, zonal jets can
form, analogous to those on Jupiter. However, when the radiative time constant is
sufficiently short, the radiation damps the atmospheric turbulence before it can organize into jets, and in this case the circulation consists primarily of quasi-isotropic
turbulence without strong zonal jets.
In exploring the possible sources of periodic photometric variability, a variety
of complex interactions must be considered, from heterogeneous cloud properties
and surface coverage, to the dynamic and thermal properties of the atmosphere.
With high-precision uninterrupted NIR light curves spanning multiple wavelengths
over multiple dwarf rotation periods, we can assess the atmospheric origins of this
variability by constraining the cloud coverage, opacity and Tef f differences of condensate clouds in the dwarf. Such light curves can also reveal the timescales for
evolution of the periodic photometric variability. This allows us to discern possible
origins for this evolution by comparing these times to results from state-of-the-art
radiative and atmospheric dynamic models.
22
CHAPTER 2
STATEMENT OF WORK
In my research preceding this thesis, I contributed to five refereed publications in
The Astrophysical Journal closely aligned with the research summarized in this
thesis (listed below). Specifically, for Paper 1, I managed spacecraft observations,
created data reduction and characterization techniques, and conducted or assisted
with ground follow-up observations. In Paper 2, I reduced the Spitzer data and
characterized amplitudes and periods, and was involved in the target selection for
the two dwarfs that were the subject of the study. In Paper 3, I helped compare the
variable targets from HST with their variability from our Spitzer program. In Paper
4, I was involved in managing spacecraft observations as well as assisting in Spitzer
data reduction and overall Spitzer program execution, and in Paper 5, I reduced the
Spitzer data for the target object which had simultaneous HST observations.
In this thesis, I am presenting a work on four L dwarfs from the Weather on
Other Worlds Spitzer Exploration Program. The work described is being submitted
for publication in The Astrophysical Journal. This thesis is based on a work that
will have multiple authorship, with myself leading the investigation, and performing
spacecraft observation planning, data reduction and analysis. This work contains
contributions from Dániel Apai (University of Arizona, overall guidance and general contributions), Stanimir Metchev (Western University, Principal Investigator
for Weather on Other Worlds, general contributions), Mark Marley (NASA Ames
Research Center, spectral and flux models), Didier Saumon (Los Alamos National
Laboratory, spectral and flux models), Adam Showman (University of Arizona, atmospheric dynamics discussions), Aren Heinze (Institute for Astronomy, advice on
data analysis), Kelle Cruz (Hunter College, contributed spectrum), Nikole Lewis
(Massachusetts Institute of Technology, data reduction techniques), and Étienne
Artigau (Université de Montréal, general contributions).
23
Publications:
1. Metchev, S. A., A. Heinze, D. Apai, D. Flateau, J. Radigan, A. Burgasser,
M. S. Marley, É. Artigau, P. Plavchan, and B. Goldman (2015). Weather on
Other Worlds. II. Survey Results: Spots are Ubiquitous on L and T Dwarfs.
ApJ, 799, 154.
2. Yang, H., D. Apai, M. S. Marley, D. Saumon, C. V. Morley, E. Buenzli, É.
Artigau, J. Radigan, S. Metchev, A. J. Burgasser, S. Mohanty, P. J. Lowrance,
A. P. Showman, T. Karalidi, D. Flateau, and A. N. Heinze (2015). HST
Rotational Spectral Mapping of Two L-type Brown Dwarfs: Variability in
and out of Water Bands indicates High-altitude Haze Layers. ApJ, 798, L13.
3. Buenzli, E., D. Apai, J. Radigan, I. N. Reid, and D. Flateau (2014). Brown
Dwarf Photospheres are Patchy: A Hubble Space Telescope Near-infrared
Spectroscopic Survey Finds Frequent Low-level Variability. ApJ, 782, 77.
4. Heinze, A. N., S. Metchev, D. Apai, D. Flateau, R. Kurtev, M. Marley, J.
Radigan, A. J. Burgasser, É. Artigau, and P. Plavchan (2013). Weather on
Other Worlds. I. Detection of Periodic Variability in the L3 Dwarf DENIS- P
J1058.7-1548 with Precise Multi-wavelength Photometry. ApJ, 767, 173.
5. Buenzli, E., D. Apai, C. V. Morley, D. Flateau, A. P. Showman, A. Burrows, M. S. Marley, N. K. Lewis, and I. N. Reid (2012). Vertical Atmospheric
Structure in a Variable Brown Dwarf: Pressure-dependent Phase Shifts in
Simultaneous Hubble Space Telescope-Spitzer Light Curves. ApJ, 760, L31.
24
CHAPTER 3
OVERVIEW
In the following chapter, we will describe the observational background for 4 variable
L dwarfs from the Weather on Other Worlds program (hereafter WoW), and summarize the observational history and previous searches of variability in these targets.
In Chapter 4, we describe the data reduction methods for our warm Spitzer observations, including correcting for intrapixel sensitivities. In Chapter 5, we analyze
the reduced photometry for our targets, extracting relative amplitudes, rotation
periods, and examine light curve shapes. We also present a possible correlation
between the Ch1 and Ch2 light curves in SDSS0107, and present the results of absolute photometry of our dwarfs. In Section 5.4.1, we examine the light curves for
any observed evolution during the length of the observations, and in Chapter 6,
match model spectra with archived SpeX spectra and observed Spitzer fluxes for
our targets, determining atmospheric parameters that include the cloud opaqueness
of observed atmospheric layers and gravity. Also in Chapter 6, we investigate how
pairs of models of cloudy surfaces can be used in describing the colors changes of
our dwarfs due to variability. Finally, we explore the behavior of theoretical thermal
perturbations at atmospheric depths and atmospheric dynamics as a cause of our
variability and light curve evolution.
3.1 Target Selection
This work represents results from the Weather on Other Worlds program, a Spitzer
Space Telescope Cycle 8 Exploration Science program (GO 80179, Metchev et al.
2011, 2015), utilizing IRAC channel 1 ([3.6], Ch1) and channel 2 ([4.5], Ch2) during the spacecraft’s post-cryogenic period known as “warm Spitzer”. The WoW
campaign was an extensive campaign to characterize the variability in an unbiased
25
selection of 44 non-binary dwarfs ranging from L3 to T8, including both typical
and peculiar objects throughout this spectral type range. Most targets were continuously monitored for 7–14 hours to capture at least two typical brown dwarf
rotation periods (Metchev et al., 2015) in Ch1, and one additional period in a 6-7
hour observation in Ch2. Some targets in the 873 hour survey have simultaneous
or near-simultaneous ground-based observations in I or JHK bands. The survey results in Metchev et al. (2015) include that, after correcting for sensitivity, 80%+20%
−27%
of L dwarfs are photometrically variable with a relative amplitude ≥ 0.2%, and
36%+26%
−17% of T dwarfs vary by ≥ 0.4%. That work also found that a third of variable
L dwarfs have irregular light curves, which they attributed to multiple atmospheric
spots evolving over a single rotation. The work also revealed that a third of all of
the target dwarfs have rotation periods > 10 h, and presented a tentative association between low surface gravity and high-amplitude variability among L3–L5.5
dwarfs. Previous findings of this program were included in Heinze et al. (2013),
which described the L3 dwarf DENIS-P J1058.7-1548, which was found to have
a peak-to-trough variability amplitude of 0.388% ± 0.043 with P=4.25+0.26
−0.16 h, and
Heinze et al. (2015) which described the search for I–band variability of the Spitzer
program’s T-dwarfs, suggesting that high-amplitude photometric variability for T–
dwarfs is is more common in the optical than at longer wavelengths.
For this study, we chose to investigate a subsample of WoW targets spanning mid-L spectral types (L3–L8), whose atmospheres are characterized by persistent condensate clouds, as opposed to the patchy clouds expected in later L
and early T-type dwarfs. This provides a basic level of atmospheric homogeneity in our sample when creating modeling assumptions about recreating observed
variability. Within that sample, we included objects with a variety of previously
determined gravities. We also included one object that has been previously identified as an active radio emitter, which gives our sample an atmosphere that has
the possibility of magnetically-induced spots on the surface, allowing us to investigate whether cloud modeling results differ for that object. We also chose objects in this sample that have some of largest SNR for relative light curve am-
26
plitudes in both channels, enabling us to robustly determine periods and amplitudes for our objects. Our radio-emitting dwarf has the added advantage in this
sample of having a very small amplitude in both Spitzer channels, enabling us
to investigate the effects of very low amplitudes on our modeling approach. Our
targets are the L5.5 dwarf SDSSpJ010752.33+004156.1 (hereafter SDSS0107), the
known radio-emitting L3.5 dwarf 2MASSWJ0036159+182110 (2M0036), the lowgravity L6 dwarf 2MASSIJ0103320+193536 (2M0103) and the low gravity L3 dwarf
2MASSWJ2208136+292121 (2M2208).
Compiled literature data on our four targets can be found in Table 3.1. A brief
summary of each’s object’s key properties and results from previous searches for
variability follows.
3.2 The Young, Low-Gravity L3 Dwarf 2M2208
2M2208 was discovered by Kirkpatrick et al. (2000), and was identified as one of
only two spectroscopically peculiar L dwarfs at the time. Subsequent 0.6 − 1µm
Subaru FOCAS spectra identified 2M2208 as a low gravity L-dwarf due to weaker
K I doublets and hydride bands of CaH, CrH, and FeH when compared to other L
dwarfs of its type (Kirkpatrick et al., 2008), and identified it as spectral type L2.
That work also found strong spectral similarity to the L-dwarf G 196-3B, which has
an age estimate based on that object’s M2.5 dwarf primary companion between 20
and 300 Myr (Kirkpatrick et al., 2001), thereby estimating 2M2208’s age as ∼ 100
Myr. Cruz et al. (2009) reclassified 2M2208 as an L3γ dwarf, (γ designating low
gravity) and concluded that based on the behavior of low-gravity features of the
spectra of late M-dwarfs, 2M2208 had an age close to ∼10 Myr. Allers and Liu
(2013) confirm the low gravity of this object in their study of ultracool field dwarfs
using gravity-sensitive indices based on FeH, VO, K I, Na I, and H –band continuum shapes. Gagné et al. (2014) concluded that 2M2208 has a modest probability
(10.1%) of being a 9 − 11MJup planemo member of the β Pictoris moving group.
Enoch et al. (2003) searched for variability in Ks band in 2M2208 during a 29 day
27
Table 3.1. Target Properties and Observation Log
Quantity
SDSS0107
2M0036
2M0103
2M2208
Source
α (J2000)
01h 07m 52s .42
00h 36m 16s .17
01h 03m 32s .03
22h 08m 13s .63
3,3,3,3
δ (J2000)
+00d
+18d
+19d
+29d
3,3,3,3
41m
56s .3
21m
10s .4
35m
36s .1
21m
21s .5
J
15.824 ±0.058
12.466 ±0.027
16.288 ±0.080
15.797 ±0.085
3,3,3,3
H
14.512 ±0.039
11.588 ±0.029
14.897 ±0.056
14.793 ±0.071
3,3,3,3
K
13.709 ±0.044
11.058 ±0.021
14.149 ±0.059
14.148 ±0.073
3,3,3,3
J −H
1.31 ±0.070
0.878±0.040
1.391±0.098
1.004±0.111
3,3,3,3
H −K
0.803 ±0.059
0.530 ±0.036
0.748 ±0.081
0.748 ±0.102
3,3,3,3
SpT
L5.5 (NIR)
L3.5 (Opt)
L6 (Opt)
L3γ (Opt )
4,7,6,8
d (pc)
15.59 ±1.10
8.76 ±0.06
21.90 ±3.55
47.22 ±1.56
7,9,6,5
Ch1 Start (UT)
10 Oct 2012 15h06m
09 Oct 2012 16h14m
15 Oct 2012 06h06m
16 Sep 2012 10h31m
Ch1 Length (h)
13.83
7.88
13.83
13.83
12
12
12
12
3801
2166
3801
3801
Ch2 Start (UT)
11 Oct 2012 05h09m
10 Oct 2012 00h11m
15 Oct 2012 20h09m
09 Sep 2012 00h33m
Ch2 Length (h)
6.89
5.89
6.89
6.89
Ch2 texp (s)
12s
12s
12s
12s
1893
1619
1893
1893
Observing Log
Ch1 texp (s)
Ch1 Nexp
Ch2 Nexp
References. — 1. Geballe et al. (2002), 2. Skrutskie et al. (2006), 3. 2MASS Point Source Catalog, 4. Knapp et al. (2004),
4. Chiu et al. (2006), 5. Zapatero Osorio et al. (2014), 6. Faherty et al. (2012), 7. Reid et al. (2008), 8. McLean et al. (2003),
9. Dahn et al. (2002)
28
period with the Palomar 60-inch found no variability within their detection limits
of 0.09 mag. Subsequent Ks band monitoring found no periodic variability over a
2.7 year photometric sampling at their detection limit of 0.04 mag (López Martı́
and Zapatero Osorio, 2014). Companion searches in wide-fields (2-3100 , separations
of 101–1,656 AU, brightness limits of J ∼ 20.5 & K ∼ 18.5, (Allen et al., 2007) and
narrow-fields (HST, ≥ 0.00 06, projected separation (3AU), at ∆mF 814W =1, Bouy
et al. 2003) was also negative. Zapatero Osorio et al. (2014) measured its parallax
as 21.2 ± 0.7 mas, resulting in a distance of 47.22 ± 1.56 pc.
3.3 The L5 Dwarf SDSS0107
SDSS0107 was discovered by Geballe et al. (2002) who initially classified it as an
L5 based on its optical spectrum. It has been noted as a peculiarly red L dwarf (J –
K =1.31), and subsequently categorized with an optical spectral type of L8 (Hawley
et al., 2002) and as an L5.5 in the near-infrared by Knapp (2004). An HST/NICMOS
search for companions as close as 0.100 was negative at limits of m110 = 21.9 mag
and m170 = 20.0 mag (Reid et al., 2006). Its trigonometric parallax was measured
by Vrba et al. (2004), resulting in a distance of 15.59 ± 1.10 pc. SDSS0107 has been
identified as a member of the Hyades moving group by Jameson et al. (2008), which
has an age estimate of 625 ± 50 Myr (Perryman et al., 1998).
3.4 The Young, Low-Gravity L6 Dwarf 2M0103
2M0103 was discovered by Kirkpatrick et al. (2000) and classified with an optical
spectral type of L6. Faherty et al. (2012) measured the parallax of 2M0103 at
46.9 ± 7.6 mas, resulting in a distance of 21.90 ± 3.55 pc. Faherty also designated
this dwarf to be a low-gravity object, and underluminous in the near-infrared bands
compared to the average of its spectral subtype. Gagné et al. (2014) found that
2M0103 was a strong candidate (76.0% probability) to be a 10–11 MJup planemo
member of the Argus moving group. A search for planetary-mass companions using
HST/NICMOS high-resolution spectral differential imaging was negative (Stumpf
29
et al., 2010), as was a wide-field search for companions (2–3100 , separations of 57–877
AU, brightness limits of J∼20.5 & K∼18.5, Allen et al. 2007). Enoch et al. (2003)
reported a possible detection of variability in Palomar 60-inch Ks-band observations
with a peak-to-peak amplitude of 0.1 ± 0.02 mag over a 29-day period.
3.5 The L3 Radio Emitter 2M0036
2M0036 was discovered by Reid et al. (2000) and identified as an L3.5 by Kirkpatrick
et al. (2000) with optical spectra. Knapp et al. (2004) assigned a near-infrared
spectral type as L4 ± 1. Dahn et al. (2002) measured its parallax as 114 ± 0.8 mas,
corresponding to a distance of 8.80 ± 0.62 pc. A wide field companion search (2–3100 ,
separations of 18–273 AU, brightness limits of J ∼20.5 & K ∼18.5, Allen et al. 2007),
as well as a narrow-field search for companions (HST, ≥ 000 .06, projected separation
(0.5 AU), at ∆mF 814W =1, Bouy et al. 2003) was negative.
2M0036 is a known broad-band radio emitter (Berger 2002), with coherent
and observed unpolarized radio emissions at 4.88 GHz, and an identified period
of 3.08 ± 0.05 h (Hallinan et al., 2008). Hallinan concluded that the radio emission is due to cyclotron maser instability, and that the dwarf’s axis of rotation is
nearly perpendicular to the observed line-of-sight based on previous measurements
of vsin i by Zapatero Osorio et al. (2006) of 36 km s−1 . Previously, Lane et al.
(2007) reported I-band variability with a period of 3 h, with irregular amplitude.
Noting that the radio variability implied a strong magnetic field for 2M0036, Lane
postulated that the variability could be due to magnetically induced spots possibly
coupled with time-varying features such as dust clouds. Most recently, Blake et al.
(2010) measured this dwarf’s vsin i as 35.12 ± 0.57 km s−1 and its radial velocity as
32.84 ± 0.17 km s−1 .
With its previously known radio variability, 2M0036 was an added WoW target
that was not part of the survey’s unbiased target selection. Metchev et al. (2015)
selected this target for observation as a control for recognizing potential activityinduced photometric effects. In this work, we will investigate its Ch1 and Ch2
30
variability as the consequence of clouds and thermal perturbations in the same way
as our other three dwarfs.
31
CHAPTER 4
OBSERVATIONS AND DATA REDUCTION
4.1 Observations and Data Reduction
Spitzer observations for our four targets were conducted in 2012; an observing log
can be found in Table 3.1. The typical WoW observation lengths for each channel
(∼14 h for Ch1, ∼7 h for Ch2) were shortened for 2M0036, since its photometric
period in other bands had been previously found (Metchev et al., 2015).
All data reduction used the full-frame (256 × 256 pixels) post-calibration CBCD
data products released by the Spitzer Science Center, in addition to each frame’s
pixel uncertainty values. CBCD flux densities were converted into electrons for
photometry and error calculation using the EXPTIME, FLUXCONV and GAIN
keywords in the CBCD headers. 2M2208 was placed in the “sweet spot” near the
center of the well-characterized sub-array near pixel position [23,231]. 2M0036,
SDSS0107, and 2M0103 were placed near [126,130] for Ch1, and [127,128] for Ch2,
for purposes of assisting follow-up ground observations. We found no difference in
sensitivity, noise, or the ability to correct the pixel phase effect (described below)
between the two array positions for a variety of WoW targets that were imaged in
both locations. Representative Ch1 frames overlaid with the photometric apertures
used are shown in Figure 4.1.
4.2 Corrections for Intrapixel Sensitivity Variations
Spitzer IRAC Ch1 and Ch2 arrays in the warm mission provide flux measurements
that are correlated with the array position of the target. The correlated flux measurements are due to intra-pixel sensitivity across each array (e.g., Reach et al. 2005;
Charbonneau et al. 2005; Knutson et al. 2008). For time-series photometry, this has
32
SDSS0107
2M2208
2M0036
2M0103
Figure 4.1 Representative Ch1 frames for our four L dwarfs, overlaid with the photometric flux apertures (red) and background annuli (yellow). All non-target sources
and bad array elements were masked out of background and flux calculations. Bad
array elements near the edges of the Ch1 array for 2M2208 were also masked out.
the effect of a general increase in measured flux as the target centroid moves near
the pixel’s area of highest sensitivity during the observation, usually near the center
of the pixel, and a measured decreased flux as the target moves toward the edges of
the pixel. This effect can also be dependent on the intensity of the flux on the pixel.
In addition to the target’s x and y position on the pixel, another useful metric
that can be used to remove position-correlated changes in the data is the “noise
pixel” (NP) (Mighell 2005; Knutson et al. 2012; Lewis et al. 2013), a measure of the
width of the stellar point spread function, defined by:
P
2
i Ii )
P
β̃ =
2
i Ii
(
(4.1)
where Ii is the measured intensity in the ith pixel. We used two separate pixel
phase correction methods for each observation, and compared the results against
each other. The first method is the result of fitting a quadratic formula fit to the
centroid’s x and y position within the pixel (e.g., Reach et al. 2005; Knutson et al.
2008) with noise pixel values added in a linear combination to positional information.
33
Unlike the Spitzer exoplanet transit observations of the latter work, we do not have
a priori astrophysical reference signals to introduce into the reduction method, as
cloud activity can change or mask the brown dwarf’s rotational periodic variability
on short and long timescales (Artigau et al. 2009; Metchev et al. 2015).
The linear combination of quadratic terms was fit to each observation using a
least-squares method, after finding the target’s centroid position using IRAC’s IDL1
procedure box centroider.pro 2 using a box width of 5 pixels, and calculating the
noise pixel values with a circular photometry radius of 4 pixels. The second pixel
phase correction method was using “pixel maps” that use low-pass Gaussian spacial
filters (Knutson et al. 2012; Lewis et al. 2013), summing the gaussian contributions
from the nearest nearest neighbors (NN) in x, y, and noise pixel space. With this
method, we performed corrections for a small NN (50) and large NN (400), and had
very similar results for all four targets. We chose the 400 NN correction to use for
the corrected flux in this work for all of our targets due to their relative brightness
enabling reliable flux and position contributions from other neighbors in x,y, and
NP space.
Aperture photometry was performed on all eight time series using the IDL procedure aper.pro3 routine to return sky-subtracted flux values for each integration,
with sky calculations performed within an annulus centered on each star. The centroid of the target was calculated for each frame using box centroider.pro. This
position was then used to center the photometry aperture and sky annulus. A range
of fixed aperture radii from 2.0 to 3.0 pixels. as well as apertures that varied with
noise pixel values (with additive constants ranging from 0-0.5), along with sky annuli ranging [3-15,5-28] pixels were evaluated for scatter in the resulting light curve
when corrected for intrapixel phase sensitivity. For both Ch1 and Ch2 observations,
a fixed aperture radius of 2.0 pixels and a sky annulus set to an inner and outer
radius of [12,20] respectively was chosen to produce low scatter in the resulting time
1
the acronym IDL represents Interactive Data Language
provided on the Spitzer home page: http://irsa.ipac.caltech.edu
3
provided by the IDL Astronomy User’s Library at: http://idlastro.gsfc.nasa.gov
2
34
series photometry. Unrelated stars in the background annulus areas were masked
from flux and background level calculations, as was the amplifier glow and bad array
elements near the edges of the IRAC arrays. Photometry points with centroid positions outside of 5σ in x or y from a 25-point smoothed median value were rejected,
as were sky-subtracted photometry values outside of 5σ. Rejected flux points due
to clipping resulted in the rejection of < 3% of total flux points for each channel
run. Each time series was normalized to its own mean value, and binned in 5 minute
intervals to reduce noise. Spacecraft pointing stability during all observations was
assessed as excellent, with the position drift in x and y limited to < 0.2 pixels in
each direction, which resulted in relatively minimal pixel phase correction required.
35
CHAPTER 5
SPITZER PHOTOMETRY RESULTS
The final light curves for all eight time-series are presented in Figure 5.1, assembled on a timescale relative to the first Ch1 observation of each target. Peak-totrough variability amplitudes for the four dwarfs in Ch1 range from 0.44–1.80% and
from 0.33–1.51% in Ch2. Derived periods range from 2.56 h (2M0036) to 10.16
h (SDSS0107). Uncertainties for the 5 min binned data points range from 0.05%
(2M0036, Ch1) to 0.26% (2M0103, Ch2).
Derived periods, peak-to-trough amplitudes and the correlation search values
can be found in Table 5.1. In the following subsection, a brief explanation is given
for how these values were determined.
5.1 Period Finding and Amplitudes
Period searching was performed in both channels for all four targets using epoch
folding of the light curves (e.g., Davies 1990, Leahy et al. 1983, SchwarzenbergCzerny 1989). Each light curve was folded with a set of trial periods ranging from
1 h to the length of each channel’s observation, and a pulse profile was generated.
We calculated χ2 statistics for all trial periods to check for constancy, and the
period with the maximum χ2 (maximum deviation) was taken as the best possible
period. We determined uncertainties in the period with 105 Monte Carlo simulations,
with epoch folding the light curves of each trial with simulated Poisson errors and
subsequently computing the standard deviation of the results. Phase-folded light
curves to these periods are found in Figure 5.2.
Data from 2M0036 did not phase fold to a particular period in Ch1, and its
period in that channel was determined from evaluating a maximum power peak in
a Lomb-Scargle periodogram (Scargle 1982), with its uncertainty calculated by the
36
L−Dwarf Light Curves
SDSS0107, L5.5
1.02
Corrected Normalized Flux
3.6µm
4.5µm
2M0103, L6
1.00
0.98
2M2208, L3γ
0.96
1.004
1.000
2M0036, L3.5
0.996
0
5
10
15
Time from beginning of observation (hr)
20
Figure 5.1 Ch1 and Ch2 light curves for all four targets, each plotted with a vertical
offset for the purpose of the figure. The grey horizontal lines represent the relative
mean flux level for each dwarf. Note that 2M0036 is plotted on a smaller scale for
clarity.
37
Phase−Folded Ch1 Light Curves
Corrected Normalized Flux
1.02
SDSS0107 Ch1, P=10.16h
1.00
0.98
2M0103 Ch1, P=5.37h
0.96
2M2208 Ch1, P=3.58h
First Phase
Second Phase
Third Phase
Fourth Phase
0.94
0.0
0.5
1.0
Phase
1.5
2.0
Figure 5.2 The light curves for three of our targets in Ch1 are in good agreement
with the period found by epoch-folding (2M0036 does not fold to a particular period
in Ch1). Phase-folded Ch1 light curves for the three targets for which a period could
be derived from epoch folding are plotted above. The folded phases are plotted twice
for clarity, and the respective periods are indicated for each light curve.
38
Table 5.1. Derived Properties for Targets from Photometry
Quantity
SDSS0107
2M0103
2M2208
2M0036
Relative amplitudes, Ch1 (%)
1.58 ± 0.05
1.78 ± 0.09
1.80 ± 0.09
0.44 ± 0.02
Relative amplitudes, Ch2 (%)
1.51 ± 0.07
1.44 ± 0.09
1.02 ± 0.09
0.33 ± 0.02
Relative amplitude ratios, Ch1/Ch2
1.05 ± 0.06
1.23 ± 0.10
1.77 ± 0.18
1.35 ± 0.10
Periods
Period, Ch1 (h)
10.16 ± 0.26
5.37 ± 0.56
3.58 ± 0.30
2.63 ± 0.81
Period, Ch2 (h)
a
5.78 ± 0.80
3.44 ± 0.42
2.56 ± 0.21
90.50
0
0
0
Ch1 (mag)
12.34 ± 0.03
12.87 ± 0.03
13.07 ± 0.03
10.26 ± 0.03
Ch2 (mag)
12.17 ± 0.03
12.73 ± 0.03
12.87 ± 0.03
10.25 ± 0.03
Correlation Lag (deg)b
Absolute Photometry
Note. — a. No reliable period found in data b. Ch 2 relative to Ch 1
same method as above. 2M0036 Ch2 did phase-fold to a period, which corresponded
to the Ch1 period within the uncertainties of both channels.
We calculated maximum relative peak-to-trough amplitudes for all time series
from a 60-point (∼ 13.4 minutes) boxcar-smoothed version of the unbinned relative
light curve. We calculated the uncertainties in the relative amplitudes by taking the
standard deviation of the noise in the smoothed light curve.
5.2 Searching for Correlations and Phase Shifts
Phase shifts in the regular periodic signals between the Ch1 and Ch2 light curves
for each target were investigated using a Z-transformed Discrete Correlation Function algorithm (ZDCF, Alexander 1997) to calculate the cross correlation coefficient
between the Ch1 and Ch2 binned data for lags (Ch2–Ch1 times) ranging from 1
to 14 hours in 4 minute intervals. We choose this method over standard cross correlation methods to avoid altering the light curves by interpolating them in time;
the light curves have significant features that occur on relatively short time scales.
39
The ZDCF also removes known biases of the standard Discrete Correlation Function
(Alexander 1997), and has a method of calculating the uncertainties in any found
lag. The lag value with the maximum ZDCF value was taken as the most likely
match of the Ch2 data to the Ch1 data. Uncertainties on the best lag as well as
the ZDCFs for each lag were calculated by calculating the ZDCFs for 103 Monte
Carlo simulations. The most significant lags for 2M0103 and 2M2208 corresponded
to the length of their respective periods (5.37 and 3.58 h, with ZDCF=0.6 and 0.7,
respectively) within the period uncertainties. We therefore conclude that there is
no measurable phase shift between Ch1 and Ch2 for these two dwarfs.
SDSS0107 showed a large ZDCF value (0.7) at a lag of −12.96 h between Ch1
and Ch2, with Ch2 being ahead of Ch1 by 0.25 of the derived Ch1 period, or 90.50◦
if describing this as a phase shift. Upon inspecting the overlap of the Ch1 and Ch2
data when shifted with this lag, it appears that the shape of the Ch2 light curve
resembles the shape of the middle peak of the Ch1 data, but with some differences.
The dip in the Ch2 plateau at this feature is more pronounced than in Ch1, and
the Ch2 feature preceding the maximum appears to be damped when compared to
the aligned feature in Ch1. Figure 5.3 plots both channels of data with and without
the found lag to illustrate the similarity of the two light curves. While there are
some differences, the basic structure between the Ch1 light curve and the shifted
Ch2 light curve are similar.
However, SDSS0107 Ch1 observations last slightly longer than the calculated
single period, while Ch2 is less than one period in this dataset. Because of this and
the differences that do exist between the Ch1 and shifted Ch2 light curves, we do not
have enough information to conclusively describe these signals as a periodic signal
with a phase shift in Ch2. We conclude that these two correlated features may have
some relationship that resembles a phase shift between two periodic signals in our
data. Section 6.4.5 describes some of the atmospheric dynamics involved in brown
dwarf atmospheres that have been used to explain phase shifts found between other
NIR wavelengths in a brown dwarf.
Other possibilities that explain the SDSS0107 light curves include that at the
40
Corrected Normalized Flux
SDSS0107 Shifted Light Curves, Ch1, Ch2
Ch1, unshifted
Ch2, unshifted
Ch2, shifted
1.01
1.00
0.99
SDSS0107 Ch2 shift: 2.55 hr, 0.25 of period, 90.50 degrees
0
5
10
Elapsed Time (hr)
15
Figure 5.3 A possible correlation between Ch1 and Ch2 light curves in SDSS0107
is plotted, with shifted Ch2 data (red) plotted against Ch1 data (blue). The shift
in time for Ch2 was calculated by lag values corresponding to a maximum ZDCF
(correlation) value. Red points represent Ch2 data with an applied shift, plotted
both with one value of shift and values of period+shift to overlap with Ch1 data.
The continuous curve is a 5-term Fourier fit to Ch1 data showing how the shifted
features lines up when two phases of Ch1 are plotted. Light red points are original
Ch2 data without the shift in time.
20
41
moment when the Ch1 observation was ending and the Ch2 observation began, the
dwarf began undergoing activity that masked its regular periodicity, or that the
light curve of SDSSS0107 is quite complex, with the period of SDSS0107 actually
much longer than 10.2 h (the folding we find being coincidental), and that we have
not observed its true period after 13 h of continuous observations. We note that
a full period measurement in Ch2 would provide invaluable evidence for any phase
shift between these two channels.
5.3 Absolute Photometry
Absolute photometry was performed on all eight time series using the procedures
of Reach et al. (2005). For each integrated CBCD frame, we used the centroid
positions and noise pixel values found with the procedures described in the previous
section, with a fixed photometric aperture with a radius of 2 pixels and a background
annulus of [12,20]. We also applied array location-based and aperture corrections
from Reach et al. (2005), and pixel phase correction was performed as above. The
resulting photometry was then medianed over the length of the time series, and
converted to magnitudes using the zero-flux points of 280.9 and 179.7 Jy for Ch1
and Ch2 respectively. The results from the photometry can be found in Table 5.1.
Color-magnitude diagrams were generated for the four L dwarfs using the derived
absolute photometry and distance moduli from the parallaxes in Table 3.1. Figure
5.4 shows the Spitzer color-magnitude diagram for these objects plotted with L and
T field dwarfs that have both measured Spitzer fluxes and parallaxes compiled by
Dupuy and Liu (2012). The low-gravity L3 dwarf 2M2208 has an M[4.5] brightness
similar to the L5 dwarf SDSS0107, but a brightness even lower than the low-gravity
L6 dwarf 2M0103. Also plotted is a color-magnitude diagram of previously measured
JHK magnitudes compiled in Table 3.1, plotted similarly with field dwarfs with
known parallaxes from Dupuy and Liu.
42
2M0036
2M0036
2M0103
SDSS0107
2M2208
2M2208
2M0103
SDSS0107
Figure 5.4 Left: Color magnitude diagram in Spitzer Ch1 and Ch2 using timemedianed photometry from this work for the four target dwarfs, with a selection of
M, L and T field dwarfs shown with parallaxes from Dupuy and Liu (2012). The lowgravity L3γ dwarf 2M2208 has an M[4.5] brightness similar to the L5 dwarf SDSS0107,
but a brightness even lower than the low-gravity L6 dwarf 2M0103. Right: Colormagnitude diagram in J and H bands, from compiled literature magnitudes for our
targets, HR8799 planets, and 2M1207b. 2M0103 and SDSS0107 are extremely red
L dwarfs with colors comparable to the HR8799 planets.
43
5.4 Discussion
All four L dwarfs are clearly variable in both Spitzer channels, with distinct light
curve shapes. The light curve of 2M0036 has an irregular shape in Ch1, but a
more regular pattern in Ch2. The relative brightness of 2M0036 allows the robust
detection of its low amplitude of 0.33% in Ch1, representing the lowest amplitude
of our targets, with the highest amplitude being detected in the low-gravity L3γ
dwarf 2M2208 at 1.80%, also in Ch1. All targets have similar amplitudes in both
channels, but with marked light curve differences in each case. The amplitudes in
both channels for each object are similar, with a maximum Ch1/Ch2 amplitude
ratio of 1.77 (2M2208). The periods of these L dwarfs range from 2.56 ± 0.21 h for
2M0036, to 10.16 ± 0.26 h for SDSS0107. The light curves of SDSS0107 and 2M0103
show distinctive double-peaks.
5.4.1 Light Curve Evolution
Light curve evolution (LCE) is the change in light curves shape and characteristics
over time. One way to characterize LCE is to subtract the light curve of the first cycle
(the “reference” cycle) from subsequent cycles. We did this for each object by taking
the phase-folded light curves of binned data from Figure 5.2, and interpolating each
subsequent phase to the phase values of the reference cycle data. We used quadratic
interpolation for this purpose, although linear and spline interpolations gave similar
results. The residuals obtained after subtracting the reference cycle light curve are
shown in Figure 5.5. The long period of SDSS0107 (∼10.2 h) makes it difficult to
characterize any evolution in the incomplete second cycle of the light curve, but a
discussion on the evolution of 2M2208 and 2M0103 follows.
2M0103
Figure 5.5 shows changes in the light curve from the first period to be as high as
∼2%. A major evolution feature that can be seen in this LCE plot is the general
brightening of the source after the first period. We investigated if this brightening
44
Residual Normalized Flux from First Phase + Offset
Light Curve Evolution
SDSS0107
0.04
0.02
2M0103
0.00
−0.02
2M2208
−0.04
2.0
2.5
3.0
3.5
Phase
4.0
4.5
Figure 5.5 Light curve evolution of the three photometrically periodic targets in
Ch1 that epoch-folded to a period show changes from the first phase between 0–2%,
with a general brightening trend in Ch1 of 2M0103 of ∼0.5%. For each target, the
phase-folded light curve was subtracted from the first phase after each subsequent
phase was interpolated to the same time-grid in phase space as the first. A medianed
noise level for the subtracted phases is shown in red for each dwarf.
5.0
45
was due to uncorrected pixel phase effect, as the source is moving toward the center
of the IRAC pixel it was placed on during this observation. Such a movement could
result in increased flux values over time as the sensitivity of the pixel is generally
higher in the center of each pixel. Photometry for this object was repeated at a series
of shorter time intervals for this Ch1 time series, correcting pixel phase effects with
both noise-pixel and quadratic function-fitting to centroid positions. Evaluating
shorter intervals should allow for the more efficient correction of any brightness
ramp due to pixel phase effect, i.e. a better fitting to noise-pixel maps and x-y
space for any correlation of position and flux when the ramp is small. However,
the brightening feature remains in all smaller time-interval photometry reduction
and pixel phase effect correction, including those intervals for which there is no net
movement of the pixel over the array. We suggest that this brightening is physical,
and not due to intrapixel movement of the source on the array.
2M2208
This dwarf shows more modest light curve evolution nearer the noise level of the
light curves, although we can see instances of evolution from the reference period of
∼ 1% that are sustained for less than half a period.
46
CHAPTER 6
MODELING AND SPECTRAL FITTING
In this chapter, we describe our efforts to use one-dimensional spectral models to
match a mean model spectrum to each dwarf, determining best-matched Tef f , log g,
and cloud parameters used in further investigations in this work. We will first
describe the details of the models, followed by a description of the method used
in determining the best-matched mean model spectra to observed archived NIR
spectra. Finally, we will present the results of these matches for our four dwarfs.
6.1 Description of the Models
To evaluate possible physical mechanisms of periodic photometric variability as well
as any light curve evolution, we will consider the complex structure of clouds at
different pressures and temperatures in a given brown dwarf atmosphere. Using
principal component analysis, Apai et al. (2013) showed that for two L/T transition
dwarfs, over 99% of the observed periodic photometric variability can be reproduced
with the linear combination of just two different spectra, concluding that all of the
features in the visible photosphere share the same spectra. In that study, variations
in cloud thickness were found to account for the dwarfs’ periodic photometric variability. Buenzli et al. (2013, 2015a,b) performed a detailed fit to the spectra of L/T
variables and found that the varying NIR spectra of those dwarfs are well fit by the
linear combination of two models that differ in Tef f and cloud opacity parameters.
In this perspective, we will first find global atmospheric parameters for each of
our four dwarfs by fitting NIR spectra and the median observed Spitzer fluxes from
this work with one-dimensional cloudy atmosphere models. The fitted models then
serve as reference for the subsequent analysis of the Spitzer light curves.
The atmosphere models used here are described in Saumon and Marley (2008).
47
These brown dwarf atmospheric models give temperature-pressure structures in radiative/convective equilibrium and the corresponding synthetic spectra. The thermal radiative transfer is modeled with the source function method of Toon et al.
(1989) and allows the inclusion of arbitrary Mie scattering particles in the opacity
of each layer. All models used here use the solar elemental abundances of Lodders (2003) and the opacity database is described in Freedman et al. (2014). Our
baseline cloud model (Ackerman and Marley, 2001) parametrizes the efficiency of
sedimentation of cloud particles through an efficiency factor, fsed . Large values of
fsed correspond to rapid particle growth and large mean particle sizes. In this case,
condensates quickly settle, leading to physically and optically thin clouds. When fsed
is small, particles grow more slowly and the atmospheric condensate load is larger
and clouds thicker. The cloud model is fully coupled with the radiative transfer and
self-consistent with the (P, T ) structure of the model. Finally, the models consider
non-equilibrium chemistry driven by rapid vertical mixing, a process that has been
found to be prevalent in brown dwarf atmospheres (Griffith and Yelle, 1999; Saumon
et al., 2006; Stephens et al., 2009). Its main effect in late L dwarfs is to reduce the
atmospheric abundance of CH4 in favor of an increased fraction of CO. The mixing
time scale in the convective region is obtained from the mixing length theory and
it is parametrized by the coefficient of eddy diffusion Kzz in the overlying radiative
zone. We chose a value of log Kzz (cm2 s−1 ) = 4, which is representative of the values
found in previous studies. The Saumon and Marley atmosphere models have been
very successful when applied to brown dwarfs and Jupiter (Ackerman and Marley
2001; Knapp et al. 2004; Golimowski et al. 2004; Saumon et al. 2006; Leggett et al.
2007; Cushing et al. 2008; Stephens et al. 2009).
6.2 Model Fitting
We utilized spectra obtained with SpeX spectrograph (Rayner et al., 2003) at the
NASA InfraRed Telescope Facility (IRTF), with a spectral range of 0.6–2.6 µm,
initially downloaded from the SpeX Prism Spectral Library for SDSS0107, 2M0103,
48
and 2M0036 (Burgasser et al. 2010; Cruz et al. 2004; Burgasser et al. 2008). Unnormalized versions of these spectra were also obtained (Burgasser, A., private correspondence), and were used throughout this work. A SpeX spectrum for 2M2208
(Cruz et al., in prep) is presented for the first time in this work. The observing log
for these spectra can be found in Table 6.1.
We noted that the JHK flux values for the spectra for 2M0036, 2M0103 and
SDSS0107 did not match literature 2MASS JHK flux values calculated from the
values found in Table 3.1; 2M0036 in particular deviated from these values by more
than a factor of 8. Such proper calibration is necessary to properly fit relative
values between the SpeX spectra, spectral models and the model and observed
Spitzer fluxes. We adjusted each of these spectra by multiplying data from the
spectra by a factor that was calculated from Fλ,2M ASS /Fλ , where Fλ,2M ASS is the
flux from literature 2MASS magnitude values at the respective effective wavelength,
and Fλ is the flux from the SpeX spectra at the effective wavelength of the 2MASS
fluxes. These factors were calculated for the 2MASS H and K bands, and a weighted
average of the two factors was taken using the SpeX and 2MASS uncertainty values
at both respective effective wavelengths. This weighted average was the factor used
to multiply the flux values in each spectrum.
Each corrected, reduced spectrum was normalized to a flux value calculated from
integrating the flux density of each spectrum from 1.50–1.80 µm; the same value
was also used to normalize corresponding observed [3.6] and [4.5] flux values. Model
spectra and model Ch1 and Ch2 flux values were normalized together in the same
manner.
Each spectra and Spitzer flux pair was matched to a grid of model spectra. The grid of model spectra ranged from Tef f =1,200–2,400 K in 100 K steps,
log g(cm s−2 )=4–5.5 in steps of 0.5. Cloudy models with fsed ={1,2,3,4} as well as
models with no clouds (nc) were considered. The parameter space covered in this
grid of models was not complete, especially for models in the low gravity (log g=4.0)
regime. The models that were available are noted in the final results of this section
in Figures 6.4 and 6.5.
49
Table 6.1. SpeX Observing Log
Date
R
Reference
SDSS0107
2M0103
2M2208
2M0036
12 Oct 2007
19 Sep 2003
14 Dec 2007
7 Sep 2004
120
120
120
120
Burgasser et al. (2010)
Cruz et al. (2004)
Cruz et al. (in prep.)
Burgasser et al. (2008)
For each model spectra and Spitzer flux pair set, a goodness of fit statistic
was calculated between each model spectra, k, and the SpeX spectra based on the
procedure of Cushing et al. (2008),
n
1X
Gk =
wi
n i=1
fi − Dk Fk,i
σi
2
,
(6.1)
where n is the number of data points; wi is the weight of ith wavelength and is
given by the length of the wavelength bin wi = ∆λi , and whose total is normalized
to unity; f and Fk are the flux densities of the SpeX and model spectra respectively;
σi is the noise in the observed flux densities, and Dk is an unknown multiplicative
constant for each model that is equivalent to (R/d)2 , where R is the radius of the
dwarf and d is its distance. Dk is calculated by minimizing Gk with respect to Ck
and is calculated by
P
wi fi Fk,i /σi2
P
Dk =
.
2
wi Fk,i
/σi2
(6.2)
G values can be large due to small uncertainties (bright targets like 2M0036),
and will also tend to be dominated by the fit of the Spitzer fluxes over the fit of
the SpeX spectra, due to the relatively wide spectral windows in the [3.6] and [4.5]
integrations. The challenges of calculating a goodness-of-fit statistic that does not
follow these extremes when fitting spectra and photometry has been the subject of
much discussion and ongoing experimentation (e.g., Cushing et al. 2008).
For SDSS0107 and 2M2208, the lowest G value was taken as the best-matched
model; when visually inspecting the matches resulting from the lowest five G values,
50
Table 6.2. Best-Fit Dwarf Parameters from SpeX Spectra and Spitzer Flux
Matches
Tef f (K)
SDSS0107
2M0103
2M2208
2M0036
1,300
1,400
1,400
1,700
log g
4.0
4.0
4.0
4.5
fsed
1
1
1
1
these spectral matches produced good fits to the 1.3, 1.7, and 2.2 µm features in
the SpeX spectrum as well as the two Spitzer photometric points. Both of these
dwarfs matched models well with similar temperatures, and the same fsed =1, and
low gravity (log g=4.0), although such low gravity for 2M0103 is lower than its
previous designation as a moderately low-gravity object (Faherty et al., 2012). This
best-fit low gravity model is consistent with the previous “γ” classification of 2M2208
(Cruz et al., 2009). According to the evolutionary models of Saumon and Marley
(2008), this best-fit model also supports the seemingly low best-fit parameter of
Tef f = 1, 400 K for this L3γ dwarf. The models calculate an age of ∼10 Myr,
while also indicating a mass of ∼8 MJup , which are values generally consistent with
findings from Gagné et al. (2014) and Cruz et al. (2009).
6.3 Heterogenous Clouds as a Cause of Photometric Periodic Variability
In this section, we detail our efforts to evaluate heterogenous clouds as a source
of the observed periodic photometric variability in our four L dwarfs. We will first
describe the 1-D spectral models we will use to compute Spitzer Ch1 and Ch2 fluxes,
and the effects of different model parameters on those fluxes. We then introduce a
linear parameterization between two models of cloudy surfaces. Using the best-fit
mean spectrum found for each dwarf in the previous section, we investigate what
second cloud model has to be fit to best describe the observed Ch1 and Ch2 periodic
photometric amplitudes for those dwarfs.
51
7
6
5
4
3
2
1
5
4
2M0036
Teff= 1700 K
log g= 4.5
fsed= 1
log Kzz= 4
G =52.5
2M2208
Teff= 1400 K
log g= 4.0
fsed= 1
log Kzz= 4
G =0.018
2M0103
Teff= 1400 K
log g= 4.0
fsed= 1
log Kzz= 4
G =0.062
SDSS0107
Teff= 1300 K
log g= 4.0
fsed= 1
log Kzz= 4
G =0.782
Normalized Fλ
3
2
1
5
4
3
2
1
5
4
3
2
Observations
Models
1
1
2
3
4
5
λ (µm)
Figure 6.1 SpeX and [3.6] and [4.5] fluxes match well with spectral and flux models
for all four dwarfs. Each dwarf’s SpeX spectra and observed Spitzer fluxes (black)
are plotted with its best-matching mean spectral and flux models (red), with corresponding G values noted. A median noise value for each SpeX observation is plotted
in an inset; the error bars in the observed Spitzer fluxes are within the plotted symbols. The Tef f , fsed and log g of the best-fit model for each dwarf are noted; all
dwarfs matched models with a fsed =1.
52
6.3.1 Description of the Models
The Ackerman and Marley (2001) cloud model successfully reproduces the observed
NIR color of L dwarfs. The dramatic shift in the NIR colors through the L/T
transition is widely understood as due to a reduction of the cloud opacity as Tef f
decreases. At the cool end of the transition, the NIR colors of mid-T dwarfs are well
explained by cloudless models. The transition can be reproduced in the context of
the Ackerman & Marley cloud model with an increase in the fsed parameter (i.e.
the cloud particle size, Saumon and Marley, 2008), or alternatively by a decrease in
fractional cloud cover (Marley et al., 2010). These results suggest that horizontal
variations in cloud properties, modeled by variations in fsed across the surface of
the brown dwarf, can produce model flux differences that are realistic enough to be
compared to the observed photometric variability amplitudes. For a given Tef f and
log g, increasing fsed decreases the cloud opacity, resulting in larger Ch1 and Ch2
fluxes, with the Ch1−Ch2 color becoming bluer, and vice-versa for increases in fsed .
In addition to fsed , we also consider variations in the coefficient of eddy diffusion
Kzz as it affects the chemistry of carbon in L dwarfs and the abundances of two
important absorbers: CO and CH4 . Vertical mixing in L dwarf atmospheres tends
to increase the mole fraction of CO and the expense of that of CH4 as these two
molecules contain nearly all the carbon available in the atmosphere. The very strong
3.3 µm band of CH4 falls within the IRAC Ch1 bandpass and the 4.5 µm band of CO
is centered on the Ch2 bandpass. The deviation of the CO/CH4 ratio from chemical
equilibrium caused by vertical transport dramatically increases the Ch1 flux for Tef f
less than approximately 1,500 K, but the models predict a minimal decrease in the
Ch2 flux. Thus, increasing Kzz makes the Ch1−Ch2 color bluer.
The results of Apai et al. (2013) suggest that our Spitzer light curves can be modeled with a rotating brown dwarf that has longitudinal variations in cloud structure
described by areas corresponding to two different values of fsed . We express the two
extremes of the cloud surfaces as
53
Fcombined = C1 F1 + C2 F2
(6.3)
where F1 represents a rotational-phase dependent model flux from the best-matched
spectrum which we will consider the primary atmospheric component, and F2 is the
flux from a secondary atmosphere component, which is a model with the same log g
but different cloud parameters, and can be thought of as a perturbation of F1 . The
contribution from each surface to the combined flux, Fcombined , is parameterized by
a Ci , which is a surface area covering fraction, or the surface area covered by a photosphere with a given spectrum divided by the complete surface area of the object.
C2 is the surface area covering fraction of the secondary atmospheric component
(e.g., Marley et al. 2010; Apai et al. 2013), while C1 = 1 − C2 is the surface area
covering fraction of the primary atmospheric component.
Our most important assumption in this model is that C2 C1 . This sets the
expectation that the overall spectrum of the object is very similar to the spectrum
from the F1 surface, thereby allowing us to treat F2 as a perturbation. We expect
C2 to be less than approximately 0.1, if we expect this model to match the relatively
small variability observed in our dwarfs in Spitzer Ch1 and Ch2. In addition, we
assume that we are not observing the dwarf pole-on, as such a viewing geometry
would not result in rotationally-modulated periodic photometric variability with
persistent atmospheric features in the atmosphere. We additionally assume that the
distribution of the surface associated with F2 is not axisymmetric over the surface
of the dwarf, which could also lead to no rotationally-induced variability as no net
flux differences would be observed as the dwarf rotates these features into and out
of view.
6.3.2 Comparing Model Flux Differences with Observed Amplitudes
We attempted to reproduce the Ch1 and Ch2 amplitudes of the photometric periodic
variations (Ch1–Ch2 color changes) in each of our four targets by describing them
as a linear combination of fluxes from two model atmospheres. The first model in
54
this pair (henceforth “base model”) is the best-fit to the SpeX NIR spectrum and
Spitzer fluxes as described above, while the second model is a model that results
in amplitudes from flux differences with the first model that reproduces the closest
values to the observed Ch1 and Ch2 amplitudes.
We utilized the model Ch1 and Ch2 fluxes from a similar model grid as the
previous fit to spectra and photometry. The best-fit values of Ch1 and Ch2 fluxes
found in the previous section were used, and designated as the base fluxes. To search
for a best secondary model, a grid of model flux values was utilized that spanned the
parameter space of Tef f =900–2,400 K in 100 K steps, fsed ={1,2,3,4,nc (no clouds)},
but we also now consider a range of Kzz values that include log Kzz (cm2 s−1 )={2,4,6}
as well as equilibrium models with Kzz =0. The log g values for this model grid were
the kept the same as found from the base model fitting. The parameter space
covered in this grid of models was not complete, especially for models in the low
gravity (log g=4.0) regime.
Eliminating Unphysical Secondary Model Possibilities and Limitations of
Models
When considering the best secondary model that would reproduce the observed
Spitzer amplitudes in our light curves, we must also consider the physical likelihood of a particular secondary model’s atmospheric parameters residing in an atmospheric column adjacent to a surface with the parameters with the base model.
1D Temperature-pressure (T-P) profiles are not expected to be extremely different for two different cloud types in these adjacent columns. Marley et al. (2010)
modeled two adjacent atmospheric columns of different clouds parameterized with
a cloudy contribution fraction into a single consistent T-P profile. However, such
an approach is more appropriate for partly-cloudy dwarfs, not for a homogeneous
cloud cover with few patches of different cloud parameters, such is our approach
here. A more realistic comprehensive model for assuring the physicality of adjacent
cloud models would take into account the global circulation of the atmosphere and
its dynamics. This intrinsically 3D approach is far more complicated than our 1D
55
static approach in this work, and while there is progress toward realistic circulation
models of brown dwarfs (Showman and Kaspi 2013; Zhang and Showman 2014), future work remains on unifying dynamical and cloud models with respect to observed
photometric variability.
Comparing T-P profiles of our models for determining possible physical compatibility is beyond scope of this work; we do not have a meaningful metric for comparing
the true likelihood of their physical compatibility in adjacent atmospheric columns.
But we will make one basic assumptions about the likelihood of our model pairs residing near each other in an atmosphere for a given set of model parameters. When
cloud opacity decreases from our base model to an adjacent secondary model, we
would generally expect the Tef f of the secondary model to increase, as the reduced
opacity allows radiation from deeper levels in the atmosphere where temperatures
are warmer. Such a process is seen on Jupiter and its “hot spots” as observed at 5
µm, which have among the lowest cloud opacity anywhere on Jupiter, but also the
highest flux in the 5 µm spectral window of anywhere on the planet (e.g., Westphal
1969; Bjoraker et al. 1986; Carlson et al. 1994; Atreya et al. 1999; Showman and
Dowling 2000). Therefore, we exclude any secondary models that have an equal or
lower Tef f and a higher fsed than the base model of each dwarf.
6.3.3 Calculating Model Amplitudes
With the exception of the models we deem unphysical, this model grid was then
evaluated for each change in Tef f , fsed and Kzz whose associated fluxes, when differenced from the base fluxes, created Ch1 and Ch2 model amplitudes according to
the equations
ACh1,model = 100 × (F 2Ch1 − F 1Ch1 )/F 1Ch1
(6.4)
ACh2,model = 100 × (F 2Ch2 − F 1Ch2 )/F 1Ch2
(6.5)
where F 1 and F 2 are the channel specific fluxes from Equation 6.3, representing
the primary and secondary model fluxes (respectively) that are being evaluated,
and ACh1,model and ACh2,model are the resulting Ch1 and Ch2 model amplitudes,
56
respectively, expressed as a relative percentage that can be compared to the observed
amplitude values listed in Table 5.1.
With model amplitudes calculated, we investigated which secondary model, when
combined with the base model, is the most likely to reproduce the observed amplitudes in both channels. A key factor in evaluating how close the calculated model
amplitudes are to our observed amplitudes is determining C2 , the secondary surface
area covering fraction that determines the contribution fractions between the base
and secondary model fluxes.
For each model amplitude, we interpolated a line in ACh1 and ACh2 space that
extends from each calculated amplitude to the origin. A representative example
is found in Figure 6.2. This line parameterizes the calculated amplitudes with
C2 , the secondary surface area covering fraction, where the slope of the line is
ACh2,model /ACh1,model . The amplitude values (black diamonds) at the top of each
line represent C2 =0.5, the amplitude that results from an equal contribution from
both model fluxes F 1 and F 2. Model amplitudes at the origin are zero, representing
C2 =0 (model amplitudes here are calculated only from 100% of the F 1 (base model)
fluxes, 0% of the F 2 (secondary model) fluxes, and thus no model amplitude results).
Points on each line represent values of C2 , and have their own value in ACh1 and ACh2
space. Our dwarfs’ observed amplitudes are ≤ 1.8%, and in this model amplitude
space, reside very near the origin of each dwarf’s Figure 6.2 equivalent.
Figure 6.3 shows a detailed region of the respective Figure 6.2 equivalent for
each dwarf, but scaled near the observed amplitudes values for each object. In this
region, the observed amplitude values are far out of the plot’s boundary, with only
the C2 line associated with each amplitude visible. The secondary model parameters
associated with each line are labeled. In Figure 6.3, all amplitude values are scaled in
units of the uncertainties of the observed Spitzer light curve amplitudes to evaluate
the significance of any matches to observations, and the 1σ boundary is plotted as
a dashed circle. For clarity, the plots for 2M2208, 2M0036 and 2M0103 are scaled
well inside the 1σ circle to show only the closest secondary model matches.
Each interpolated C2 line was evaluated for its closest approach to the observed
57
S0107 Model Color Changes
50
ACh2 (%)
40
30
20
10
0
0
10
20
30
ACh1 (%)
40
50
Figure 6.2 A representative plot for one dwarf of the model amplitudes calculated
from our model grid using Equations 6.4 and 6.5. A line is interpolated from each
model amplitude to the origin, representing values of C2 from 0.5 to 0, respectively.
In this model amplitude space, the observed amplitude of the dwarf lies near the
origin.
58
Model Color Changes
2M2208
SDSS0107
23.0
1700K,nc,0
11.6
1000K,1,4
22.5
1900K,nc,0
+
ACh2 (σA, Ch2)
11.5
ACh2 (σA, Ch2)
900K,1,6
1700K,4,0
f
1100K,1,0
C2 = 1.2%
11.4
-
1100K,1,6
22.0
+
21.5
11.3
21.0
11.2
1500K,1,0
1700K,3,2
1900K,nc,2
11.1
-
1500K,1,0
C2 = 1.8%
1500K,2,0
f
20.5
1800K,4,2
1100K,1,4
1500K,1,2
11.0
19.8
20.0
20.2
20.4
ACh1 (σA, Ch1)
20.0
30.0
20.6
30.5
31.0
31.5
32.0
ACh1 (σA, Ch1)
32.5
33.0
2M0103
2M0036
16.60
1000K,1,0
16.1
1300K,1,0
2100K,2,2
2300K,4,4
C2 = 0.21%
2300K,4,6
ACh2 (σA, Ch2)
ACh2 (σA, Ch2)
16.55
16.50
2000K,2,0
2000K,2,4
-
1700K,2,0
f
2000K,2,4
+
16.2
2000K,2,0
2300K,4,2
-
f
16.40
21.8
21.9
2000K,2,2
2000K,2,4
2200K,3,0
2000K,2,6
22.0
22.1
ACh1 (σA, Ch1)
1200K,1,2
1900K,2,0
C2 = 0.87%
2000K,2,2
2000K,2,6
+
16.45
16.0
22.2
15.9
1900K,2,4
1900K,2,0
1700K,2,6
1700K,2,4
1700K,2,2
19.6
19.7
19.8
ACh1 (σA, Ch1)
19.9
Figure 6.3 Observed Ch1 and Ch2 color changes for all four L dwarfs as described
as model Spitzer amplitudes calculated from cloud model pairs parameterized by a
secondary surface covering fraction C2 . Secondary models are noted by Tef f ,fsed , log
Kzz parameters. Amplitude values have been scaled in units of the uncertainty of
the observed Ch1 and Ch2 amplitudes for each dwarf, and the observed amplitudes
from the dwarf’s light curve (blue circle) are shown. Each model amplitude value
is parameterized by a line of C2 values from C2 =0 (the origin) to 0.5 (calculated
model amplitude) as described in the text. The best-matched secondary model to
the observed amplitudes are in bold. The 1σ uncertainty value is depicted as the
dashed circle, and we consider all models within this circle also likely matches. This
circle is outside the plot for 2M2208, 2M0036 and 2M0103.
20.0
59
2M2208
Secondary Model Matches to Observed Color Changes
Base Spectral Model Fit:
fsed, log Kzz
Model Teff (K) 1,eq 1,2 1,4 1,6
900
1000
1100
1200
1300
1400
B
1500
1600
1700
1800
1900
2000
2100
2200
2300
2400
Teff=1400K, log g=4.0, fsed=1, log Kzz=4
best C2=1.2%
2,eq 2,2 2,4 2,6 3,eq 3,2 3,4 3,6
4,eq 4,2 4,4 4,6
nc,eq nc,2 nc,4 nc,6
SDSS0107
Base Spectral Model Fit:
Teff=1300K, log g=4.0, fsed=1, log Kzz=4
fsed, log Kzz
best C2=1.8%
Model Teff (K) 1,eq 1,2 1,4 1,6 2,eq 2,2 2,4 2,6 3,eq 3,2 3,4 3,6 4,eq 4,2 4,4 4,6
900
1000
1100
1200
1300
B
1400
1500
1600
1700
1800
1900
2000
2100
2200
2300
2400
B Base Model
eq Kzz=0
nc Cloudless
Closest Match
<1
1< <3
3< <5
Excluded (>5 )
nc,eq nc,2 nc,4 nc,6
Excluded (unphysical)
No Model
Figure 6.4 Model pair combinations reproduce the observed Ch1–Ch2 color changes
for 2M2208 and SDSS0107. The parameter space is grouped by fsed ={1,2,3,4,nc (no
clouds)}, and may be best visualized as existing in 3D stacked on top of each other
in this order. B is the base model determined from spectra and flux matching, and
green shows the single best matching secondary model that reproduces the observed
color change for each dwarf, with the corresponding secondary surface area covering
fraction C2 noted. Other matches within 1σ are in blue, while secondary models
we exclude (> 5σ) or that are likely unphysical (see text) are in grey and brown,
respectively. Models within 1σ are constrained in Tef f to ±200K for SDSS0107 and
+500K
−400K for 2M2208.
60
Secondary Model Matches to Observed Color Changes
2M0103
Base Spectral Model Fit:
Teff=1400K, log g=4.0, fsed=1, log Kzz=4
best C2=0.87%
fsed, log Kzz
Model Teff (K) 1,eq 1,2 1,4 1,6 2,eq 2,2 2,4 2,6 3,eq 3,2 3,4 3,6 4,eq 4,2 4,4 4,6
900
1000
1100
1200
1300
1400
B
1500
1600
1700
1800
1900
2000
2100
2200
2300
2400
2M0036
Base Spectral Model Fit:
Teff=1700K, log g=4.5, fsed=1, log Kzz=4
fsed, log Kzz
best C2=0.21%
Model Teff (K) 1,eq 1,2 1,4 1,6
900
1000
1100
1200
1300
1400
1500
1600
1700
B
1800
1900
2000
2100
2200
2300
2400
B Base Model
eq Kzz=0
nc Cloudless
2,eq 2,2 2,4 2,6
3,eq 3,2 3,4 3,6
Closest Match
<1
1< <3
4,eq 4,2 4,4 4,6
3< <5
Excluded (>5 )
nc,eq nc,2 nc,4 nc,6
nc,eq nc,2 nc,4 nc,6
Excluded (unphysical)
No Model
Figure 6.5 Model pair combinations reproduce the observed Ch1–Ch2 color changes
for 2M0103 and 2M0036. Color key is identical to Figure 6.4. The very small
amplitudes of 2M0036 produce many likely matches over a wide range of Tef f ,
and account for its extremely small value of C2 . Similarly, the ACh1 / ACh2 ratio of
2M0103 (1.23) place the observed amplitudes of that dwarf within a model amplitude
space where there are many likely model matches to observations.
61
Table 6.3. Best-Matched Model Pair Parameters for Ch1-Ch2 Color Changes
Tef f (K)
2M2208
SDSS0107
2M0036
2M0103
1, 400 + 1, 900
1,400 + 1,100
1, 500 + 1, 100
1, 700 + 2, 300
fsed
1+1
1+1
1+4
1+2
Kzz
104 + 0
104 + 0
104 + 104
104 + 0
C2
1.2%
1.8%
0.21%
0.87%
Closest distance (σ)
0.12
0.23
0.06
0.08
Ch1 and Ch2 amplitude point in ACh1 and ACh2 space. The model with the closest
approach to the observed amplitudes point was chosen as the best match, but other
model pairs that produce amplitudes within 1σ were also noted as being likely
matches. Lines that had their closest approach to the observed amplitude point
greater than 5σ were designated model pairs that could not produce the observed
amplitudes. The results for all models, including the parameter spaces that were
not covered by our model grid, are summarized in Figures 6.4 and 6.5, and the single
best-matching secondary models for each dwarf are listed in Table 6.3.
In addition to considering how well the model amplitudes created from each
individual secondary model in our grid compares to the observed periodic variability
amplitudes of our dwarfs, we must consider if any models with parameters between
our parameter grid spacing could also match the observed Spitzer amplitudes. For
an example of this, consider the upper left panel of Figure 6.3. For 2M2208, a
line above the blue dot (observed color changes) represents a cloud-free model of
Tef f =1,900 K with Kzz = 0 is plotted. Below the blue dot is a line representing the
cloud-free model of Tef f =1,900 K and log Kzz = 2. It’s possible that there exists
a cloudless model with Tef f =1,900 K, with the same gravity, but a Kzz between 0
and log Kzz =2 that would produce fluxes that would intersect the blue dot and be
best-matched for our observations. We will consider any models in our parameter
space that have the same Tef f and fsed that straddle the observed amplitudes of the
dwarf (blue dot) in Kzz in this way as likely matches.
62
With the best-fit secondary model parameters and C2 for each dwarf determined,
we can calculate how the combination of these two spectra would effect observed near
infrared spectra such as ones obtained with SpeX. We do not expect observed SpeX
spectra (which represents our base model) to exhibit large changes over the course
of a dwarf’s rotation with the assumptions we’ve made with our linear combination
model. Figure 6.6 shows that to be the case for each of our dwarfs when combining
the two best-matched model Spex-regime spectra using Equation 6.3.
6.3.4 Discussion of Two-Component Cloud Models
We present several results from our Spitzer color change model investigations by
first summarizing the results for each dwarf
2M2208
The secondary models that matched within 1σ of the observed amplitudes are constrained in Tef f to
+500K
−400K
from the base model. The single best-matched secondary
model has its closest approach to the dwarf’s observed values at 0.12σ, with the
same fsed = 1 as the base model, a Tef f =1,110K (–300K from the base model),
and a very small C2 value of 1.2%. All likely models have C2 values similar to the
best-matched models. Excluded models include Tef f > 1600K for fsed = 2 and the
higher temperature regime for all cloudless models.
SDSS0107
SDSS0107 has only five models that are within 1σ of the observed amplitudes,
which are constrained in Tef f to ±200 K. Four out of five of these models are of
the same fsed =1 as the base model, even when additionally examining the models
we designated as unphysical. The single best-matched model has a Tef f =1,500K,
fsed =1, Kzz =0, and a small C2 value of 1.8%. All best-matched secondary models
had similarly small values of C2 . One reason for the fewer likely secondary model
matches is that SDSS0107’s Ch1/Ch2 amplitudes (1.05) place the dwarf’s amplitude
63
Spectral Combinations
SDSS0107
2M2208
5
C2=1.2%, log g=4.0
C2=1.8%, log g=4.0
4
3
Normalized Fλ
Normalized Fλ
4
2
Combined Model
SpeX Noise
Base Model
1
3
2
Combined Model
SpeX Noise
Base Model
1
(Teff=1300K,fsed=1,log Kzz=4)
(Teff=1400K,fsed=1,log Kzz=4)
Secondary Model
Secondary Model
(Teff=1500K,fsed=1,Kzz=0)
(Teff=1100K,fsed=1,Kzz=0)
0
1.0
1.5
2.0
λ (µm)
0
2.5
1.0
2M0036
2.5
2M0103
7
7
6
Combined Model
SpeX Noise
Base Model
6
Secondary Model
5
Combined Model
SpeX Noise
Base Model
(Teff=1400K,fsed=1,log Kzz=4)
(Teff=2300K,fsed=4, log Kzz=4)
4
3
Normalized Fλ
(Teff=1700K,fsed=1,log Kzz=4)
5
Normalized Fλ
1.5
2.0
λ (µm)
Secondary Model
(Teff=1900K,fsed=2,Kzz=0)
4
3
2
2
1
1
C2=0.2%, log g=4.5
C2=0.9%, log g=4.0
0
0
1.0
1.5
2.0
λ (µm)
2.5
1.0
1.5
2.0
λ (µm)
2.5
Figure 6.6 A linear combination of the best-fit SpeX-regime base and secondary
model with the best-fit C2 value are extremely close to the base model for each
dwarf. Very little observed change in Spex-regime spectra is expected to be observed using the best-fit models that reproduce the variability seen in Ch1 and Ch2.
Combined models were calculated according to Equation 6.3. The uncertainties from
the observed SpeX spectra at each wavelength are shown, and used in a reduced χ2
calculation between the combined and base models.
64
values in our model amplitude space near the edge of most of the models (as seen
qualitatively in Figure 6.2). Other dwarfs have observed amplitude ratios higher
than this, which put them in the range of more secondary models. If the dwarf’s
amplitudes were lower, it would also put more models in range of this dwarf.
2M0103
2M0103 has a range of best-matching secondary models that span range of Tef f that
encompass much of the Tef f range of our model grid (+900K
−400K from the base model).
The single best matching model was a model with Tef f =1900K, fsed =2, Kzz =0 and
a C2 =0.87%, which was a similar C2 to all likely secondary model matches. All but
the highest Tef f values for Kzz =0 models can be excluded as being likely secondary
model matches.
2M0036
Like 2M0103, 2M0036 has a wide range of best-matching < 1σ secondary models
that span the Tef f range of our model grid (+700K
−800K from the base model). The very
small observed amplitudes (0.33 and 0.44%) for this dwarf place it even closer to the
origin in ACh1 and ACh2 space, enabling many more models to fall within the < 1σ
range. However, with a more complete model grid in the log g=4.5 model space for
this dwarf, we can see that secondary model matches generally become increasingly
constrained to higher temperatures as fsed increases.
In addition to the dwarf-specific results above, we also present the following
general results:
1. When considering possible models that are between our Kzz model grid spacing that could reproduce our observed color changes, for SDSS0107, we find
that 17 out of 18 of these possible models fall within the same ∆Tef f as our
discretely computed models, with the outlier model at Tef f = 900K, fsed = 1.
For 2M2208, all of these possible model matches fall within the temperature
range of our matches from the computed model grid, with ∆T =+500K
−400K from
65
the base model. We do not have these possibly interpolated models to confirm
their fluxes, so we will include them in the best-fit range of < 1σ, but will still
designate our best single matched secondary model from our discretely computed grid. For 2M0036 and 2M0103, we find that these possible interpolated
models also span the ranges of Tef f as described above.
2. For SDSS0107 and 2M2208, the Tef f differences for our best-matched model
pairs, as well as the secondary models that produce color changes within 1σ
of the observations are generally consistent with the model pair temperature
differences for cooler, periodic photometrically variable L/T dwarfs found in
Apai et al. (2013, ∆Tef f =300 K) and Buenzli et al. (2015a,b, ∆Tef f =100–300
K).
3. Two key factors for a dwarf having more constraints on possible best-matching
secondary models are the observed variability amplitudes and the Ch1/Ch2
amplitude ratio. Amplitude ratios that put the dwarf on the edges or in a
less populated area of a model amplitude grid such as Figure 6.2 will result in
many models being excluded from being best-matches, and higher amplitudes
will also distance the dwarf’s value from the origin in model amplitude space
where all C2 lines converge.
4. Values of C2 that represent the closest secondary model matches to the observed color changes for all dwarfs occur at very small values, ranging from
0.21% for 2M0036 to 1.8% for SDSS0107. This indicates that a very small
fraction of the secondary spectra is contributing to the observed Spitzer variations for each dwarf, and that the observed photometric periodic variability of
these two L dwarfs, if due to two different cloudy surfaces with differing Tef f ,
cloud opacities and vertical mixing, is dominated by a single surface. The
domination of one model spectra over the contributions of a secondary model
was also found in model fitting variable L/T transition dwarfs (Buenzli et al.,
2015a,b). From Figure 6.6 and the resulting low χ2 statistic between the base
and the best-fit secondary models in combination with the noise from SpeX
66
observations, we can see that the assumptions from our linear combination of
two atmosphere models from Equation 6.3 hold; we would expect to see very
little change in the NIR spectra for our dwarfs when combining the primary
and secondary models that reproduce the amplitudes as seen in Spitzer, at
these very low values of C2 .
5. No universal differences in Tef f , fsed , or Kzz parameters in model pairs can
reproduce the observed color changes across both of our L dwarfs. Flux differences arise from changes in many combination of these parameters between
the base models and possible secondary models.
6.4 Timescales of Light Curve Evolution (LCE)
In this section, we will discuss aspects of possible physical explanations for the
observed timescales of light curve evolution. First we, will explore a basic model
involving purely radiative dissipation in a dynamics-free atmosphere that can provide insights into future realistic brown dwarf atmosphere models. We will then
separately investigate general properties of atmospheric dynamics in brown dwarfs
that could result in the observed changes of our light curves.
6.4.1 Radiative Flux Variations and Time Scales
Any physically realistic model of cloudy brown dwarf atmospheres must consider the
complex interaction of dynamics with the radiative properties throughout the dwarf.
One source of atmospheric disturbances are changes in heating rates at various
depths, which can be caused by any number of processes that include the interaction
of dynamics and radiation, such as wave breaking and cloud effects. These thermal
perturbations can propagate radiatively and dynamically through the atmosphere,
including being altered by dynamic processes that created the perturbation, or ones
that directly result from it, such as Rossby waves propagating upward through the
atmospheric column. Such perturbations can lead to observed periodic photometric variability due to brightness temperature differences and cloud changes, while
67
also resulting in the alteration of any existing variability. The timescales that the
resulting temperature differences remain in the atmosphere in these models can be
compared to those in the observed periodic photometric variability and variability
evolution to begin to inform us as to the responsible atmospheric processes.
Changes to dynamical wave fluxes or other details of the dynamics could easily
lead to variations in the thermal perturbations and the length they can be maintained from just considering purely radiative properties of the atmosphere. (Showman and Kaspi, 2013). However, we can inform future complex models of brown
dwarf atmospheres and investigations into periodic variability by exploring the relatively narrow subject of radiative time scales in a purely radiative cloudy atmosphere
model.
6.4.2 Description of the Models
Robinson and Marley (2014) used a 1-D time-stepping model of the thermal structure of a dwarf’s atmosphere to calculate how a theoretical change in heating rates at
various atmospheric pressures would result in integrated flux changes at the surface
at various bandpasses. This model was free of any dynamics or clouds. Temperature
variation rates from 10 to 500 h, temperatures from 8 to 20 K that corresponded
to pressure levels from 100 to 1 bar. The timescales of temperature propagation
from depth ranged from 10 to 100 h, and it was noted that deep thermal perturbations lead to brightness variations at nearly all NIR wavelengths. Corresponding
timescales to fully radiate away the perturbation ranged in the hundreds of hours.
In this work, we will utilize a similar, but even simpler model to investigate the
radiative timescales due to purely radiative processes; our 1D model only investigates
radiative dissipation vertically outward from lower to higher atmospheric depths.
6.4.3 Thermal Perturbation Procedure and Results
We selected thermal perturbation models that matched parameters of the best-fit
Tef f , log g, fsed , and Kzz parameters from the previous part of this work. We then
68
introduced a temperature perturbation into the atmospheric pressures ranging from
10−4 to ∼15 bar, which was divided into quarter scale height segments. For each
trial for each dwarf, these models begin with the initial perturbed layer, calculating
the perturbed temperature profile one quarter scale height at a time, moving to
lower atmospheric pressures (outward) until the initial temperature perturbation has
completely radiated from all layers. For each trial, a perturbed mean spectra at the
top of the atmosphere was generated. Artificial photometry was then performed on
the perturbed spectra for Ch1 and Ch2 using IRAC filter profiles (Hora et al., 2008),
which was ratioed with flux values calculated from the unperturbed mean spectra.
The amount of time needed for each temperature perturbation trial to dissipate
completely from all layers was calculated. Subsequent trials for each dwarf had the
perturbation beginning in a different atmospheric pressure, with new perturbation
levels changing in quarter scale height increments.
The plot of these flux ratios throughout each dwarf’s atmosphere is found in
Figure 6.7. The peak perturbed/unperturbed flux ratio for each Spitzer channel is
marked in the plots by hatched areas representing the corresponding pressure levels.
These pressure levels represent the atmospheric level at which peak Spitzer Ch1 and
Ch2 perturbed/unperturbed flux ratios would be generated from if a temperature
perturbation was introduced there (flux from other non-peak pressure levels would
also contribute if perturbations existed there, as seen in the plot). The radiative
timescales for those peak perturbed/unperturbed flux ratios are noted on the plot.
6.4.4 Radiative Flux Variations and Timescales Discussion
In these radiative models, the timescales for thermal perturbations to completely
dissipate from pressure levels that produce peak perturbed/unperturbed flux ratios
Spitzer Ch1 and Ch2 (∼0.1 bar) are ≤ 1.40 h for both Ch1 and Ch2 in all of our
dwarfs. For all dwarfs, the radiative timescales for each perturbed level are similar
for a regime of lower pressures, before entering a distinct pressure regime where the
timescales begin to increase with increasing pressure (Figure 6.7, right column). For
each of our dwarfs, the pressure level that produces the peak perturbed/unperturbed
Peak Perturbation
Flux Ratios
Perturbation
Radiative Time Scales
SDSS0107,Teff=1300 K,log g=4.0,fsed=1
S0107,Teff=1300 K,log g=4.0,fsed=1
t=1.40 h
1.04
10
t=1.37 h
10
[3.6]
[4.5]
1.03
1.02
10
1.01
10 0
1.00
10 1
10
10
[3.6]
[4.5]
10
P (bar)
Pert./Unpert. Flux Ratio
69
10
10
10 0
10 2
10
10 1
P (bar)
10 0 10 3 10 5 10 8 10 11 10 14
t (h)
2M0103 & 2M2208,Teff=1400 K,log g=4.0,fsed=1 0103 & 2208,Teff=1400 K,log g=4.0,fsed=1
10
1.03
t=1.36 h
t=1.23 h
10
[3.6]
[4.5]
1.02
10
1.01
10 0
1.00
10 1
10
10
[3.6]
[4.5]
10
P (bar)
Pert./Unpert. Flux Ratio
1.04
10
10
10
0
10
10 2
10
1
10 0
10 5 10 10 10 15 10 20
P (bar)
t (h)
2M0036,Teff=1700 K,log g=4.5,fsed=1
2M0036,Teff=1700 K,log g=4.5,fsed=1
10
[3.6]
[4.5]
1.02
[3.6]
[4.5]
10
P (bar)
Pert./Unpert. Flux Ratio
10
t=0.31 h
1.01
10
10 0
1.00
10 1
10
10
10
10
P (bar)
10 0
10 1
10 2
10
10 0 10 2 10 4 10 6 10 8 10 10
t (h)
Figure 6.7 Perturbed/unperturbed flux ratios (left column) and radiative timescales
”t” (right column) for each thermal perturbation in purely radiative cloudy models
are shown. Hatched areas indicate the extent of pressure levels which produced
peak flux ratios in Ch1 (blue) and Ch2 (red) when perturbed. Radiative timescales
for dissipating perturbations initiated in the pressure level that produces peak perturbed/unperturbed flux ratios are ≤ 1.40 h for all dwarfs in both channels.
70
flux ratio as observed with Spitzer Ch1 and Ch2 is at the boundary of these two
pressure regimes. Although these radiative timescales in these purely radiative models are short (≤ 1 rotation period for our dwarfs), and too short to be the cause
of sustained rotationally-induced variability, it’s very likely that various dynamic
processes will lengthen these timescales (Showman and Kaspi, 2013). The extent to
which these perturbation timescales are lengthened, and the resulting atmospheric
levels these perturbations are sustained in is the subject of the intersection of future
dynamic and radiative modeling. If the timescales for these perturbations are not
lengthened by dynamic and other processes more than an order of magnitude, such
perturbations at these depth could still be responsible for the light curve evolution
as seen in our Spitzer light curves.
6.4.5 Dynamical Processes and Timescales
Our light curves exhibit variations of ∼0.3–2% relative amplitudes with structure
that is sufficiently periodic to suggest rotational modulations on timescales of ∼3–
10 hours, and yet the light curve structure nevertheless evolves from one period
to the next. To provide context for understanding this variability, it is interesting
to consider the types of dynamical behavior expected on brown dwarfs. Showman
and Kaspi (2013) presented an analytic theory for the dynamics of the stratified
atmosphere of brown dwarfs which allowed predictions for the wind speeds and
temperature perturbations associated with the circulation. In their theory, interior
convection triggers waves and small-scale turbulence in the atmosphere, which interact with the mean flow to generate large-scale atmospheric structures such as
vortices and/or zonal jet streams. Depending on the assumed efficiency of the wave
driving, their theory predicts wind speeds of tens to hundreds of m/sec and horizontal temperature perturbations (on isobars) of several to ∼ 50 K. A large-scale
overturning (e.g., meridional) circulation will accompany this stratified turbulence,
with the regions of ascent and descent organized at large scale. Because regions of
ascent would lead to cloudiness, whereas regions of descent would tend to be less
cloudy, their theory implies the presence of cloud patchiness. Since radiation escapes
71
to space from significantly greater pressure in the less-cloudy regions, there can be
large differences in brightness temperature between cloudy and cloud-free regions,
allowing the existence of significant rotational modulation in light curves.
Dynamical studies to date provide guidance on the key length scales of the
atmospheric circulation. The Rossby deformation radius, LD , represents a natural
scale for the interaction of buoyancy and rotation, and as a result, vortices and
turbulent eddies often have a characteristic size similar to the deformation radius.
On brown dwarfs, LD is typically a few thousand km (Zhang and Showman, 2014).
Another key flow length scale is the Rhines scale, which is a typical size scale at
which zonal banding becomes prominent (for reviews, see Vasavada and Showman
2005, Showman et al. 2010). The Rhines scale is π(U/β)1/2 , where β = 2Ω sin φ/a
is the gradient of Coriolis parameter with northward distance, Ω is rotation rate, φ
is latitude, and a is the brown dwarf radius. Given a rotation period of 5 hours,
a Jupiter radius, and wind speeds ranging from 30 to 300 m/sec, one obtains a
Rhines scale of ∼5,000 to 20,000 km. This is the likely scale of any zonal jets that
exist. Nevertheless, Zhang and Showman (2014) showed that, for sufficiently short
radiative timescales, it is possible for the radiation to damp turbulent structures
before they have time to reorganize into zonal jets. Thus, the question of whether
zonal banding exists on our L dwarfs remains open.
To provide guidance in understanding the evolution of our light curves, estimating relevant dynamical timescales is useful. The advection timescale for air to advect
over a brown dwarf radius, a/U , where a is the radius and U is the wind speed, is
∼2 × 105 –2 × 106 sec for wind speeds of 30–300 m/sec. In contrast, the horizontal
advection time across a deformation radius is significantly shorter — 104 –105 sec for
wind speeds of 30–300 m/sec. This is the timescale over which individual vortices
could change significantly in shape or structure. Moreover, mass continuity then
suggests that the vertical advection time over a distance comparable to the vertical
scale of the circulation (likely to be of order a scale height) is also ∼104 –105 sec.
Thus, significant changes to the cloud structure of individual eddies or vortices could
occur over these timescales, causing gradual changes to observed IR light curves on
72
these timescales. Given a rotation period of 5 h, this implies gradual changes to
the light curve shape on timescales of 0.5 to 5 rotation periods. Complete “loss of
memory” in the light curve (i.e. complete scrambling of the phase and shape of the
light curve from some earlier state) is likely to take place on timescales comparable
to advection over a brown-dwarf radius, which is 10–100 brown dwarf rotation periods (Showman and Kaspi, 2013; Zhang and Showman, 2014). These estimates are
consistent with, and help to explain, the changes in shape of our light curves and
other dwarf light curves (e.g. Artigau et al. 2009) on timescales of several rotation
periods. Further detailed modeling of the connection between dynamical structure
and light curve evolution would be extremely beneficial.
Another issue concerns the differing behavior in Ch1 and Ch2 light curves for
SDSS0107 and the seeming lagged correlation between the two curves. This behavior implies that the spatial pattern of patchiness at the time of our observations
must differ significantly between the two channels. Buenzli et al. (2012) considered
a number of scenarios by which this could come about, including the possibility
of stacked circulation cells and multiple cloud decks that exhibit differing spatial
patterns of patchiness, with differing wavebands (e.g., Ch1 and Ch2) sampling different vertical regions within this complex structure. Nevertheless, these possibilities
remain qualitative and further work on the question is clearly warranted.
73
CHAPTER 7
CONCLUSIONS AND SUMMARY
All four of the L dwarfs in this study are clearly photometrically variable in the
Spitzer [3.6] and [4.5] bandpasses, with relative amplitudes ranging from 0.3–1.8%,
with measured rotation periods from 2.6–10.2 h. The light curves in the two channels
for a given object are similar, but with shape and amplitude differences. Two of the
four objects have double-peaked light curves. We also observe light curve evolution
for two of our L dwarfs, with timescales ranging from as short as a few hours, to the
observed longer term brightening trend of 2M0103 in Ch1 of ∼ 0.5%. Similarities in
the Ch1 and Ch2 light curves for SDSS0107 with the strongest correlation occurring
at a 90.5 degrees difference of the Ch1 period is observed, indicating a possible
correlation of the same global cloud structures through the different pressure-layers
probed by the Spitzer bandpasses.
Observed Ch1–Ch2 color changes are well explained for SDSS0107 and 2M2208 as
a combination of two model cloudy surfaces (for SDSS0107, 21/22 possible matches
∆Tef f = ±200K from the base model and for 2M2208, ∆Tef f =+500K
−400K from the base
model). Best-matched model pairs with various differences in Tef f , fsed and Kzz are
able to reproduce the observed periodic photometric amplitudes. The large number
of possible secondary model matches over a wide range of Tef f for 2M0036 and
2M0103 can be attributed to relatively higher observed [3.6]/[4.5] amplitude ratios
and in the case of 2M0036, very low observed amplitude values. Small secondary
model contribution fractions between all base and secondary models indicate that
the atmospheres are dominated by one surface, with only very small contributions
from a secondary spectral model needed to produce the observed amplitudes in both
Spitzer channels.
Fit to SpeX spectra and Spitzer fluxes provide further evidence of the low-gravity
nature of the L3γ dwarf 2M2208, with evolutionary models indicating that 2M2208
74
is a young ∼10 Myr planetary mass object of ∼8 MJup .
Exploration of purely radiative cloudy models can inform us about the very
basic radiative timescales that are associated with thermal perturbations within
these model atmospheres. Thermal perturbations using these models indicate that
these basic radiative timescales are short (≤ 1.40 h) for thermal perturbations at
the atmospheric levels that produce peak perturbed/unperturbed flux ratios in both
Spitzer channels. Dynamic and radiative processes are tightly coupled in real brown
dwarf atmospheres, and those processes are likely to alter these timescales. Future
models that incorporate dynamics and radiative properties of cloudy atmospheres
can ultimately investigate the final form of these perturbations as they propagate
through a more realistic model atmosphere. However, these radiative timescales can
be lengthened by an order of magnitude and still be a candidate as a source for some
of our short-term light curve evolution.
Recent dynamical studies in brown dwarf atmospheres provide insight into the
atmospheric sources of variability due to cloud patchiness and the evolution of light
curves over time. Large-scale overturning circulations and stratified turbulence can
lead to patchy clouds and differences in brightness temperatures between observed
surfaces. Light curve evolution can be the result of horizontal and vertical advection,
which have similar timescales of ∼3–28 h, which overlaps the rotational periods
of all of our periodic L–dwarfs and the observed short-term light curve evolution.
Qualitative possibilities have been previously put forward for previous NIR phase
shifts in brown dwarfs that may help to explain the light curve correlations found in
SDSS0107, including stacked circulation cells and multiple cloud decks that exhibit
differing spatial patterns of patchiness.
We note that the mechanisms of any long-term changes in periodic photometric
variability and overall luminosity changes of ultracool dwarfs over a period of years
can be explored, now that there exists a 10+ year photometric history of some
variable dwarfs. One of the most robust investigations into the evolution of periodic
photometric variability that can address these issues is the Spitzer Extrasolar Storms
campaign (GO:90063, PI: Apai), which, when completed, will have monitored six
75
known variable brown dwarfs spanning a wide range of spectral types and rotation
periods with high precision Ch1 and Ch2 photometry for up to a year.
7.1 Summary
We summarize our key results as follows:
1. Photometric Ch1–Ch2 color changes are well described for two of our dwarfs
as a combination of two model cloudy surfaces, with possible secondary model
matches with ∆Tef f = ±200 K and +500K
−400K . Model pairs that differ in Tef f , fsed
and Kzz are able to reproduce the observed amplitudes. Dwarfs with higher
amplitudes and lower [3.6]/[4.5] amplitude ratios are more easily described
by a small number of secondary models. Small secondary model contribution
fractions between base and secondary models indicate that the atmosphere is
dominated by one cloud type with only small contributions from a secondary
spectral model needed to produce the observed Spitzer color changes.
2. Through spectral and flux fitting and with evolutionary models, we provide
additional evidence of the low gravity nature of L3γ dwarf 2M2208, and note
it as a young ∼10 Myr planetary mass object of ∼8 MJup
3. Purely radiative model temperature perturbations at atmospheric depths that
produce peak perturbed/unperturbed flux ratios in [3.6] and [4.5] channels
have short radiative timescales (≤ 1.40 h) for our L dwarfs. Dynamic and radiative processes are likely to lengthen these dissipation timescales from these
simple models, but they can be lengthened by an order of magnitude and still
be in the range of some of the observed light curve evolution of our dwarfs, indicating that thermal perturbations that arise from a variety of sources could
result in short-term light curve evolution in brown dwarfs. Atmospheric dynamic mechanisms that can effect cloud composition, temperature and distribution can also be likely sources of this light curve evolution.
76
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