Dissertation Eva Meyer 2010

Dissertation Eva Meyer 2010
Dissertation
submitted to the
Combined Faculties of the Natural Sciences and Mathematics
of the Ruperto-Carola-University of Heidelberg, Germany
for the degree of
Doctor of Natural Science
Put forward by
Diplom-Physikerin Eva Meyer
born in: Essen, Germany
Oral Examination: 21st July 2010
High Precision Astrometry with
Adaptive Optics aided Imaging
Referees:
Prof. Dr. Hans-Walter Rix
Prof. Dr. Joachim Wambsganß
Abstract
Currently more than 450 exoplanets are known and this number increases nearly every
day. Only a few constraints on their orbital parameters and physical characteristics can
be determined, as most exoplanets are detected indirectly and one should therefore refer
to them as exoplanet candidates. Measuring the astrometric signal of a planet or low
mass companion by means of measuring the wobble of the host star yields the full set
of orbital parameters. With this information the true masses of the planet candidates
can be determined, making it possible to establish the candidates as real exoplanets,
brown dwarfs or low mass stars. In the context of this thesis, an M-dwarf star with
a brown dwarf candidate companion, discovered by radial velocity measurements, was
observed within an astrometric monitoring program to detect the astrometric signal.
Ground based adaptive optics aided imaging with the ESO/NACO instrument was
used with the aim to establish its true nature (brown dwarf vs. star) and to investigate
the prospects of this technique for exoplanet detection. The astrometric corrections
necessary to perform high precision astrometry are described and their contribution to
the overall precision is investigated. Due to large uncertainties in the pixel-scale and
the orientation of the detector, no detection of the astrometric orbit signal was possible.
The image quality of ground-based telescopes is limited by the turbulence in Earth’s atmosphere. The induced distortions of the light can be measured and corrected with the
adaptive optics technique and nearly diffraction limited performance can be achieved.
However, the correction is only useful within a small angle around the guide star in
single guide star measurements. The novel correction technique of multi conjugated
adaptive optics uses several guide stars to correct a larger field of view. The VLT/MAD
instrument was built to demonstrate this technique. Observations with MAD are analyzed in terms of astrometric precision in this work. Two sets of data are compared,
which were obtained in different correction modes: pure ground layer correction and
full multi conjugated correction.
i
Zusammenfassung
Mehr als 450 extrasolare Planeten sind zurzeit bekannt und diese Zahl wird fast täglich
grösser. Da die meisten Exoplaneten indirekt entdeckt werden, können nur wenige Einschränkungen bezüglich ihrer Bahnparameter und physikalischen Eigenschaften gemacht
werden und sie sollten daher vorläufig als Exoplanet-Kandidaten bezeichnet werden.
Misst man das astrometrische Signal eines planetaren oder massearmen Begleiters,
indem man die Reflexbewegung des Hauptsterns vermisst, so erhält man den vollen
Satz an orbitalen Parametern. Mit dieser Information kann die genaue Masse der
Kandidaten bestimmt werden und es ist somit möglich, die Planetenkandidaten als
wahre Exoplaneten, Braune Zwerge oder massearme Sterne einzustufen. Im Rahmen
der vorliegenden Doktorarbeit wurde ein Zwergstern der Spektralklasse M, der einen
mittels Radialgeschwindigkeitsmessungen entdeckten wahrscheinlichen Braunen Zwerg
als Begleiter hat, innerhalb eines fortlaufenden Beobachtungsprogramms zur Detektion
des astrometrischen Signals beobachtet. Bodengebundene Beobachtungen mit dem
Adaptiven Optik (AO) Instrument ESO/NACO wurden durchgeführt, um die wahre
Natur des Begleiters zu bestimmen (Brauner Zwerg oder massearmer Stern) und die
Aussichten dieser Technik im Bereich der Planetenendeckung zu untersuchen. Die
astrometrischen Korrekturen, notwendig um hochpräzise Astrometrie zu betreiben,
werden in diesem Zusammenhang beschrieben und ihr Beitrag zur Gesamtmessgenauigkeit untersucht. Die großen Unsicherheiten in der Messgenauigkeit der Änderung
der Pixel-Skala und der Ausrichtung des Detektors verhinderten jedoch, das Signal des
astrometrischen Orbits zu messen.
Die Abbildungsqualität eines bodengebundenen Teleskopes ist begrenzt durch die Turbulenz in der Atmosphäre der Erde. Die dadurch hervorgerufenenen Verformungen
der Lichtwellen können mit Hilfe der Technik der Adaptiven Optik vermessen und korrigiert werden und somit beinahe beugungsbegrenzte Abbildungen erzeugt werden. Im
Fall der klassischen AO mit nur einem Referenzstern ist die Korrektur jedoch nur in
einem engen Bereich um den Referenzstern möglich. Multikonjugierte Adaptive Optik verwendet mehrere Referenzsterne, um ein grösseres Gesichtfeld zu korrigieren. Das
MAD Instrument wurde gebaut und am Very Large Telescope installiert, um diese neue
Technik zu demonstrieren. Beobachtungen mit MAD wurden im Rahmen dieser Arbeit
auf ihre astrometrische Genauigkeit hin ausgewertet. Dabei wurden zwei Datensätze
verglichen, die in unterschiedlichen Korrektur-Modi aufgenommen wurden: zum einen
wurde nur die Turbulenzschicht nahe am Boden korrigiert, zum anderen die volle multikonjugierte Konfiguration des Instrumentes genutzt.
ii
for my father
in loving memory
iii
iv
Contents
1 Introduction
1
1.1
Orbital Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2
Detection Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.2.1
Pulsar Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.2.2
Radial Velocity Measurements . . . . . . . . . . . . . . . . . . .
4
1.2.3
Transits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.2.4
Gravitational Microlensing . . . . . . . . . . . . . . . . . . . . .
8
1.2.5
Direct Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.2.6
Astrometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
Brown Dwarfs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
1.3.1
Brown Dwarf Formation Processes . . . . . . . . . . . . . . . . .
17
1.3.2
The Brown Dwarf Desert . . . . . . . . . . . . . . . . . . . . . .
17
Goal of this Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
1.3
1.4
2 Introduction to Adaptive Optics
2.1
2.2
2.3
21
Atmospheric Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2.1.1
Fried-Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.1.2
Time Dependent Effects . . . . . . . . . . . . . . . . . . . . . . .
25
Principles of Adaptive Optics . . . . . . . . . . . . . . . . . . . . . . . .
25
2.2.1
General Setup of an AO System . . . . . . . . . . . . . . . . . .
26
2.2.2
Strehl Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
2.2.3
Anisoplanatism . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
NACO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.3.1
30
Our Observation Configuration . . . . . . . . . . . . . . . . . . .
v
3 Observations and Data Reduction
31
3.1
The Target Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
3.2
The Reference Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
3.3
Adaptive Optics Observations of GJ 1046 . . . . . . . . . . . . . . . . .
35
3.4
Data Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
3.4.1
Sky Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
3.4.2
50 Hz Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
3.4.3
Shift and Add
40
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Analysis and Astrometric Corrections
4.1
4.2
4.3
4.4
41
Position Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
4.1.1
Positional Error Estimate - Bootstrapping . . . . . . . . . . . . .
43
Astrometry with FITS-Header Keywords . . . . . . . . . . . . . . . . . .
43
4.2.1
World Coordinates in FITS . . . . . . . . . . . . . . . . . . . . .
43
4.2.2
Celestial Coordinates in FITS . . . . . . . . . . . . . . . . . . . .
46
4.2.3
Transformation from xy-Coordinates into RA/DEC . . . . . . .
46
Astrometric Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
4.3.1
Theory of Atmospheric Refraction . . . . . . . . . . . . . . . . .
47
4.3.2
Differential Atmospheric Refraction . . . . . . . . . . . . . . . .
50
4.3.3
Correction for Differential Refraction . . . . . . . . . . . . . . . .
51
4.3.4
Errors from Differential Refraction Correction . . . . . . . . . . .
55
4.3.5
Theory of Aberration . . . . . . . . . . . . . . . . . . . . . . . .
56
4.3.6
Correction for Differential Aberration . . . . . . . . . . . . . . .
59
4.3.7
Errors from Differential Aberration Correction . . . . . . . . . .
60
4.3.8
Light Time Delay . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
4.3.9
Differential Tilt Jitter . . . . . . . . . . . . . . . . . . . . . . . .
61
4.3.10 Parallax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
4.3.11 Proper Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
Plate-scale and Detector Rotation Stability . . . . . . . . . . . . . . . .
66
4.4.1
69
Plate-scale Correction . . . . . . . . . . . . . . . . . . . . . . . .
5 The Orbit Fit
73
vi
5.1
Preparing the Coordinates for the Orbital Fit . . . . . . . . . . . . . . .
73
5.2
Theory of Deriving the Orbital Elements . . . . . . . . . . . . . . . . . .
76
5.2.1
The Thiele-Innes Constants . . . . . . . . . . . . . . . . . . . . .
77
The Astrometric Orbit Fit . . . . . . . . . . . . . . . . . . . . . . . . . .
79
5.3
6 Results
83
6.1
The Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
6.2
Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . .
86
7 Introduction to MCAO and MAD
7.1
91
MCAO - The Next Generation of Adaptive Optics . . . . . . . . . . . .
91
7.1.1
Star Oriented Approach . . . . . . . . . . . . . . . . . . . . . . .
92
7.1.2
Layer Oriented Approach . . . . . . . . . . . . . . . . . . . . . .
92
7.1.3
Ground Layer Adaptive Optics . . . . . . . . . . . . . . . . . . .
93
7.1.4
Current and Future MCAO Systems . . . . . . . . . . . . . . . .
93
7.2
MAD - Multi conjugated Adaptive optics Demonstrator . . . . . . . . .
94
7.3
Goal of this Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
8 Astrometry with MAD
8.1
97
Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
8.1.1
GLAO - 47 Tuc . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
8.1.2
MCAO - NGC 6388 . . . . . . . . . . . . . . . . . . . . . . . . .
99
8.2
Data Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
8.3
Strehl Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
8.4
PSF Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
8.5
Position Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
8.6
Ensquared Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
8.7
Distortion Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
9 Results
117
9.1
Separation Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 118
9.2
Residual Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
9.3
Mean Positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
vii
9.4
Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 126
A Strehl plots
129
B Acronyms
137
Bibliography
148
viii
List of Figures
1.1
Definition of the orbital elements. . . . . . . . . . . . . . . . . . . . . . .
3
1.2
Radial velocity curve of the system HD 209458 . . . . . . . . . . . . . .
6
1.3
Transit light curves for the planetary system HD 209458 . . . . . . . . .
7
1.4
Light curve of a gravitational lensing event of a star with a planet . . .
9
1.5
Image of the three planetary companions to the star HR 8799 . . . . . .
10
1.6
Examples of astrometric motions of stars due to planets . . . . . . . . .
13
1.7
Brown dwarf desert in mass and period . . . . . . . . . . . . . . . . . .
19
2.1
The Point-Spread-Function (PSF) of a point source . . . . . . . . . . . .
22
2.2
Effect of atmospheric turbulence on the image of a star . . . . . . . . .
23
2.3
Structure of the atmosphere with typical turbulence profile. (Hardy, 1998) 24
2.4
Principle setup of an AO system. . . . . . . . . . . . . . . . . . . . . . .
26
2.5
Angular Anisoplanatism . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
3.1
RV time series of GJ 1046 . . . . . . . . . . . . . . . . . . . . . . . . . .
33
3.2
Time series of the movement of GJ 1046 on the sky . . . . . . . . . . . .
34
3.3
NACO image of GJ 1046 and the reference field in the globular cluster
47 Tucanae from July 2008. . . . . . . . . . . . . . . . . . . . . . . . . .
35
3.4
GJ 1046 observations overplotted over a simulated orbit . . . . . . . . .
37
3.5
Jitter pattern for the target and reference field . . . . . . . . . . . . . .
39
4.1
Stepwise description of the corrections applied to the measures pixel
coordinates of the stars. . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
Conversion of pixel coordinates to world coordinates (after Calabretta
and Greisen (2002)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
4.3
Refraction in the atmosphere. For more details see text. . . . . . . . . .
48
4.4
Differential refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
4.2
ix
4.5
Sign of the differential refraction correction during different times of
observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
4.6
Stellar aberration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
4.7
Parallax movement and correction of a star’s position . . . . . . . . . .
62
4.8
Stars in the reference field in 47 Tuc used to calculate the change in
pixel-scale and rotation between the different epochs. . . . . . . . . . . .
67
Calculated distortion parameters of all epochs . . . . . . . . . . . . . . .
68
4.10 Relative change in positions due to the plate-scale correction . . . . . .
70
4.9
5.1
Effect of separation between the target and reference star on the precision
obtainable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
Principle elements of an ellipse with the definition of the eccentric anomaly
E and the true anomaly ν. . . . . . . . . . . . . . . . . . . . . . . . . .
76
Simulated change in declination versus right ascension of the orbit of
GJ 1046 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
Measured separation between GJ 1046 and the reference star vs. time
for the 10 observed epochs . . . . . . . . . . . . . . . . . . . . . . . . . .
84
χ2 contour map for fitting the orbital motion to the separation and
position angle measurements . . . . . . . . . . . . . . . . . . . . . . . .
85
6.3
χ2 of the astrometric fit as a function of only the inclination . . . . . . .
86
7.1
Principle of Multi Conjugated Adaptive Optics correction . . . . . . . .
92
7.2
Examples for jitter offsets with MAD . . . . . . . . . . . . . . . . . . . .
94
7.3
3-dimensional view of the MAD bench . . . . . . . . . . . . . . . . . . .
95
8.1
MAD image of the core of the globular cluster 47 Tuc . . . . . . . . . .
98
8.2
MAD image of the globular cluster NGC 6388 . . . . . . . . . . . . . . . 100
8.3
Jitter pattern of the observations of NGC 6388 . . . . . . . . . . . . . . 102
8.4
Arc in the MAD frames produced by unshielded and unfiltered reflected
light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
8.5
Strehl-maps for some of the mad frames of the cluster NGC 6388 . . . . 105
8.6
Strehl-maps for some of the mad frames of the cluster 47 Tuc . . . . . . 106
8.7
Distribution of the orientation and eccentricity of the PSF in one of the
MCAO frame of the NGC 6388 data . . . . . . . . . . . . . . . . . . . . 108
8.8
Correlation of the eccentricity and orientation of the PSF in one of the
MCAO frames of the NGC 6388 data . . . . . . . . . . . . . . . . . . . . 109
5.2
5.3
6.1
6.2
x
8.9
Fitted Moffat functions, displayed as ellipses (enlarged) at the positions
of the stars, showing the orientation and shape of the PSFs . . . . . . . 110
8.10 Distribution of the orientation and eccentricity of the PSF in one frame
of the 47 Tuc data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
8.11 Stars used for PSF extraction in the fields 47 Tuc and NGC 6388 . . . . 113
8.12 Example of a with StarFinder extracted PSF and a saturated PSF . . . 114
8.13 Example of the development of the ensquared energy with box size and
relation to the FWHM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
9.1
FWHM over seeing for the 47 Tuc and NGC 6388 data . . . . . . . . . . 118
9.2
Separation between pairs of stars over frame number . . . . . . . . . . . 119
9.3
Example contour and arrow plots of the residuals in the positions of the
stars after the distortion correction . . . . . . . . . . . . . . . . . . . . . 121
9.4
Applied distortion parameters over frame number . . . . . . . . . . . . . 122
9.5
Mean absolute positional residuals over the radius of 50% ensquared energy123
9.6
MAD positional RMS over Magnitude . . . . . . . . . . . . . . . . . . . 124
xi
List of Tables
1.1
Examples for astrometric signals . . . . . . . . . . . . . . . . . . . . . .
12
2.1
List of available cameras for CONICA with plate scales, field of view and
spectral range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
3.1
Stellar and orbital parameters of GJ 1046 . . . . . . . . . . . . . . .
32
3.2
Overview over the observations, with exposure time, number of jitter
positions and quality indicator of the obtained data. . . . . . . . . . . .
36
3.3
Performance estimates for the middle epoch observation . . . . . . . . .
38
3.4
Positions of the bright target star, the ghost it is producing closest to
the reference star and the distance of the ghost to the reference star. . .
38
4.1
Positional error from the PSF fit calculated with bootstrap re-sampling
44
4.2
Differential refraction corrections . . . . . . . . . . . . . . . . . . . . . .
55
4.3
Uncertainties in the pixel-scale . . . . . . . . . . . . . . . . . . . . . . .
71
5.1
Summary of the measured separation and position angle together with
their uncertainties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
6.1
Parameters derived from astrometry . . . . . . . . . . . . . . . . . . .
87
8.1
Summary of the observations of the cluster 47 Tuc . . . . . . . . . . . .
99
8.2
Summary of the observations of the cluster NGC 6388 . . . . . . . . . . 101
9.1
Summary of the expected and achieved astrometric precisions . . . . . . 125
xii
Chapter 1
Introduction
With a few 100 billion stars in our own Milky Way Galaxy and just as many galaxies in
the universe, it would be very ignorant to believe mankind is alone in this tremendous
and beautiful universe. Even though detecting other life-forms is still far far away,
detecting planets orbiting other stars is already reality. More than 450 so-called extrasolar planets around more than 380 different stars have already been discovered, with
more and more to come every day.
Ever since mankind can remember, the heaven - sprinkled with stars, galaxies and
planets - has fascinated humanity and led to the desire to explore, understand and
explain what can be seen in the endless space. Since the first use of a telescope for
astronomical observations by Galileo Galilei over 400 years age, ever better telescopes
and instruments were, are and will be developed. Information achieved of an object on
the sky is brought to the observer on Earth via light coming from this object. After
travelling through space for thousands and millions of years, this information is altered
on the last milli-seconds when passing through Earth’s atmosphere and part of the
information is lost. One of the most sophisticated methods for telescopes on Earth to
retrieve this lost information and to 'turn off' the twinkling of the stars is a method
called adaptive optics. Images of celestial objects, blurred out by the atmosphere to the
resolution of a backyard telescope, are sharpend to unveil the tiny but great mysteries
of the universe.
The main goal of this work aims at investigating a technique to detect those planets
and low mass companions orbiting other stars than our Sun. High precision astrometry,
supported by adaptive optics, is used to go for the detection of the tiny motion of a
star due to its unseen companion. Astrometry alone has not been successful to find
new worlds yet, and only few known exoplanets could be further characterized by
astrometric measurements, but the future is promising with new space missions and
instruments to come.
Furthermore the technique of adaptive optics is investigated for future astrometric high
precision measurements. Data obtained with an instrument based on a novel concept
of adaptive optics instrumentation is analyzed and the astrometric precision achievable
is determined.
1
2
CHAPTER 1. INTRODUCTION
Thesis Outline
This thesis consists of two main parts. In this chapter, an introduction to the orbital
elements of the motion of a planet/companion around a star (or vice versa) and the
various planet detection methods is given. Brown dwarfs and the brown dwarf desert
are presented at the end of this chapter. In chapter 3 the adaptive optics technique
and the NACO instrument, which is used in this work, are introduced. Chapters 4
and 5 describe the observed target star and its known companion together with the
observing strategy, as well as the necessary astrometric corrections applied to the data
to obtain high precision astrometry. In chapters 6 and 7 the final orbit fit, the resulting
companion mass and its significance are summarized and discussed.
The second part of the thesis deals with a new instrument and observing technique
in the adaptive optics field with respect to astrometric precision. In chapter 8 multi
conjugated adaptive optics and the MAD instrument, which is used in this context,
are introduced, followed by the analysis of the stability and precision of astrometric
measurements with MAD in Chapter 9. The results and a discussion are given in
chapter 10.
1.1 ORBITAL ELEMENTS
3
Periastron P
Orbital plane
n
w
i
W
Line of nodes
n
a
ae
Tangential plane
Observer
Figure 1.1: Definition of the orbital elements.
1.1
Orbital Elements
The orbit of a planet or every other companion round a star is defined by six orbital elements (a, e, P, ω, Ω, i). Fig. 1.1 shows the definition of the orbital elements:
The dynamical elements characterizing the size and shape of the orbital ellipse are the
semi-major axis a, the eccentricity e and the period P . Often the time of periastron
Tp is also used to specify the timing of the orbit.
The position of the orbit in space with respect to the local coordinate system is characterized by three solid angles. The intersection of the orbital plane with the plane
perpendicular to our line of sight is called the line of nodes. It connects the two points
where the orbit intersects the tangential plane. These points are called ascending node
and descending node, depending on whether the companion passes the tangential plane
from South to North or North to South, respectively. The angle from the coordinate
zero point of the reference plane, the projection of the North celestial pole, to the ascending node is the longitude of the ascending node Ω. It is measured Eastward.
The angle between the reference plane and the orbital plane is called the inclination i.
If the orbit of the companion is direct, i.e. the position angle increases with time, then
i < 90◦ , in the case of an retrograde orbit 90◦ < i < 180◦ . For an inclination of 0◦ or
180◦ the orbit is seen face-on, for i = 90◦ the orbit is seen edge-on.
The third angle is the longitude of periastron ω. It specifies the orientation of the orbit
in the orbital plane and defines the angle of the direction to the periastron Π from the
line of nodes.
4
1.2
1.2.1
CHAPTER 1. INTRODUCTION
Detection Methods
Pulsar Timing
By surprise the first planetary-mass objects detected were orbiting a pulsar and had
masses close to the terrestrial mass. Two planetary objects with 2.8 M ⊕ and 3.4 M ⊕
with 98.88 days and 66.54 days period, respectively, were found to orbit the millisecond
pulsar PSR B1257 + 12 (Wolszczan, 1994; Wolszczan and Frail, 1992). Later also a
third component was found in this system. Pulsars are extremely rapidly rotating
neutron stars which emit mostly radio emission in a very narrow light-cone. If the
alignment with the observer is favorable, a pulse effect can be observed similar to a
lighthouse. These pulses are very precise and stable in time, which makes it possible to
detect small variations in the periodicity. Such a variation can occur when a companion
is orbiting the pulsar, causing a positional shift of the pulsar around the barycenter of
the pulsar-companion system. The motion of the pulsar around the barycenter leads
to a change in light travel time of the incoming pulses, which becomes manifest in a
delay or early arrival of the pulse signals τ .
τ = sin i
³a ´ µ m ¶
p
c
p
M∗
(1.1)
Here ap is the semi-major axis of the planet’s orbit, i the inclination of the orbit, Mp
the planetary and M∗ the pulsar mass and c stands for the speed of light. With this
method one can measure the period of the planet, its eccentricity and the projected
planet to star mass ratio. Because of the projection of the true motion of the pulsar
onto the radial direction between the observer and the pulsar, one only measures a
minimum mass for the companion which is still dependent on the inclination of the
orbital plane, assuming a known mass for the pulsar.
1.2.2
Radial Velocity Measurements
The most successful detection method so far has been the radial velocity (RV) method.
The first extrasolar planet around a solar type star was discovered this way by Michel
Mayor and Didier Queloz (Mayor and Queloz, 1995). The radial velocity method measures, as the pulsar timing method, the movement of the star due to an unseen planet in
the direction of the line of sight. In this process the Doppler-shift of the spectral lines is
measured. High precision spectral line measurements can be performed by comparing
the stellar spectrum with a set of reference lines. This reference lines are superimposed
on the stellar spectrum and can be produced for example by an iodine cell in the light
path of the spectrograph. If the target star has a planet, it will exhibit a Doppler
shift ∆λ/λ = v/c, with the same period as the planetary orbit. The spectral lines will
move redward when the star is moving away from the observer and bluewards when it
is approaching. These variations only measure the component of the motion projected
onto the line of sight of the observer and hence only a minimum mass, mp sin i of the
planet orbiting the star can be measured. The semi-amplitude of the radial velocity
1.2 DETECTION METHODS
5
variation is given by:
µ
K=
2πG
P
¶1/3
mp sin i
1
√
(M∗ + mp )2/3 1 − e2
(1.2)
where P is the planetary orbital period, e the eccentricity and G the gravitational
constant. K, P and e can be derived from the shape of the Doppler curve. Also the
argument of periastron, ω, and the time of periastron, Tp , can be derived from the RV
curve. Estimating the stellar mass M∗ from stellar models and assuming mp ¿ M∗ one
can determine mp sin i. With Kepler’s third law P [yr] = (ap [AU])3/2 (M∗ /M¯ ), where
M ¯ is one solar mass, one can also derive the semi-major axis ap of the planet.
If the mass of the companion cannot be neglected, one cannot derive mp sin i but has
to use the mass function for the star-planet system instead:
f (m) =
(mp sin i)3
P K 3 (1 − e2 )3/2
=
2
(M∗ + mp )
2πG
(1.3)
and a minimum semi-major axis of the stellar wobble:
√
1 − e2
(1.4)
a∗ sin i = K · P
2π
The fact that the RV measurements only yield the component of the orbital motion in
the direction of the observer’s line of sight, can lead to the case that a low-mass star
orbiting another star with a small inclination of the orbit is interpreted as the signal
produced by a planetary companion orbiting at a high inclination. However, this is very
unlikely if one assumes random orientation of the orbits. The most likely observable
inclination would be close to edge-on, with a median inclination of 60◦ (Kürster et al.,
1999).
The RV technique has the advantage of being mostly independent of the distance. The
only distance related limitation is that the more distant the stars are the fainter they
are, leading to a lower signal to noise ratio in the spectra. The precision possible for
the RV detection method is about 1 m/s, limited by intrinsic stellar turbulence and
activity in even the most stable stars. Because of this, spectral types of F, G, and
K are preferred for this technique, as later type stars are often too faint for adequate
signal to noise and early type stars have much less spectral lines to measure and are
limited in the line positioning accuracy due to the spectral line broadening. The RV
measurements are strongly biased towards close-in orbits and high masses, as the RV
semi-amplitude K is higher for shorter periods P , i.e. smaller separations from the host
star and higher masses of the companion, explaining the high number of Hot-Jupiter
detections.
For this reason most of the large radial velocity surveys target non-active main sequence
stars (e.g. Tinney et al., 2001; Queloz et al., 2000), but also M-stars (e.g. Kürster et al.,
2006; Zechmeister et al., 2009; Bonfils et al., 2004) and young stars (Setiawan et al.,
2008) are being monitored. Low mass planets are thought to be found more easily
around M-stars, as their stellar mass is smaller and the effect of perturbations of smaller
planets is easier to detect. But on the other hand M-stars are fainter and therefore the
precision obtained in the RV measurements is not as high as for solar-type stars.
6
CHAPTER 1. INTRODUCTION
Figure 1.2: Radial velocity curve of the system HD 209458 (see Mazeh et al., 2000). The
planetary companion has a mass of 0.685 MJ and orbits its parent star in 3.525
days. The different symbols show the data taken with the different instruments
and the solid line is the best fit Keplerian orbit.
1.2.3
Transits
If a planet passes between its host star’s disk and the observer, the observed flux
drops by a small amount. The amount of the dimming depends on the relative sizes
of the planet and star and its maximum depth is given by (Rp /R∗ )2 , where Rp is the
radius of the planet and R∗ that of the star. So, if one can estimate R∗ , one has
a direct measure for the radius of the planet, something one can only measure with
this method. From the periodicity of the transit event one gets the orbital period P
and if one can estimate the stellar mass M∗ one can derive the semi major axis of
the planetary orbit from Kepler’s third law. The shape of the dip in the light curve
depends on the inclination of the system, which has to be close to 90◦ to observe a
transit. Due to simple geometric reasons, this is the case only for a small minority of
planets. Additionally the probability of a transit is proportional to the ratio of the
diameter of the star and the diameter of the orbit. The longer the orbital period, the
smaller is the chance of a proper alignment. Also the chance of seeing the transit by
measuring at the right time is decreasing with longer orbital periods. Nevertheless
around 70 planets have already been detected using this method with likely more to
come from the ongoing surveys of the KEPLER (e.g. Basri et al., 2005; Borucki et al.,
2010) and CoRoT (e.g. Deleuil et al., 2010; Bordé et al., 2003) missions, which observe
large areas on the sky with thousands of stars.
The transit method for detecting exoplanets is also biased to close-in orbits, as is the
radial velocity method. If one can combine the two methods and solve the degeneracy
1.2 DETECTION METHODS
7
Figure 1.3: Transit light curves for the planetary system HD 209458. Left the first measurements from ground (Charbonneau et al., 2000) with the STARE Project Schmidt
camera and right a light curve obtained with the STIS instrument aboard the
Hubble Space Telescope (Brown et al., 2001). The solid lines show the transit
shape for the best fit model and the dashed lines in the left panel show additionally the transit curves for a planet with a 10% larger and smaller radius than
the one from the best fit, respectively.
of the orbital inclination, one can compute the true mass of the planet, and, together
with the radius determined from the transit, the density of the planet.
Comparing observations of spectra of the star during transit and outside of transit can
yield spectral features of the transmission spectrum of the planetary atmosphere if the
signal to noise ratio of the spectra is high enough. Such observations were conducted
for the first transiting planet detected, HD 209458 (Charbonneau et al., 2000, 2002).
Likewise one can use this so-called secondary eclipse, when the planet is behind the
star, to measure the thermal flux emitted from the planet. Hot Jupiters typically have
a thermal flux which is ’only’ about 10−4 times smaller than that of the star, which
makes it possible to indirectly measure it. When the planet is behind the star, one has
the unique opportunity to measure the true brightness of the star. Subtracting this
from the combined planet + star brightness one can derive the thermal flux, Fp of the
planet assuming a known distance of the system. Under the assumption of blackbody
radiation Fp , is given by:
Fp = 4πRp2 σTp4
(1.5)
where σ is the Stefan-Boltzmann constant and Tp the effective temperature of the
planet. Deducing Rp from the depth of the transit light curve, one can infer the
effective temperature of the planet.
Comparing the spectrum of the star + planet with the one of the star observed during
secondary eclipse, provides the opportunity to carry out infrared spectroscopy of the
planet. The space telescope Spitzer has been used mainly for this purpose and the
upcoming James Webb Space Telescope (JWST) will provide even more progress in
this field. But also from the ground first approaches have started to examine the
secondary eclipse and its measurands (Swain et al., 2010).
8
1.2.4
CHAPTER 1. INTRODUCTION
Gravitational Microlensing
Due to general relativistic effects a light path is bent in the presence of a gravitational
field. In principle any massive object can act as a lens, bending the light of a background
object and causing a temporary magnification of the brightness of the background
object. Such lensing can be observed on a galactic scale, where for example a massive
cluster in the foreground is acting as a lens for distant galaxies. In the case of a perfect
alignment the lens would cause the background object to appear as a ring, the so-called
Einstein-Ring, with an angular radius of:
s
4GM∗ (DS − DL )
θE =
(1.6)
c2 DL DS
where DL is the distance to the lens, DS the distance to the background source and
M∗ the mass of the lens. In the case of an imperfect alignment several single images of
the background object are imaged around the lens.
In the case relevant for planetary detection with the gravitational lensing method, a star
with a planetary companion acts as the lens and the background object is a distant
star. The probability of an alignment among two stars is very small, but increases
towards the galactic center. But even there it is only about one in 106 . Contrary to the
lensing on a galactic scale it is not possible with current instrumentation to resolve the
Einstein ring on the stellar scale. Instead one measures the total magnification which
depends on the angular separation between the lens and the background object u and
its change with time. The magnification factor Q of the event is given by:
Q(t) =
u2 (t) + 2
p
u(t) u2 (t) + 4
(1.7)
As the lens passes the background star, Q changes with time, and measuring the light
curve in a close enough time sample during the event yields information about the lensing star. If a planet is in orbit around the lensing star and the already magnified image
of the background star comes close to this planet, then the planet’s own gravitational
potential, distorting the star’s potential, becomes also a visible effect. An additional
brightening will occur on top of the brightening due to the lensing star, causing a sharp
peak in the light curve, see Fig. 1.4. This detection method is sensitive down to very
low-mass planets and also to planets orbiting very distant stars.
As these events do not repeat and two stars need to be aligned, this approach is challenging. The current approach is to monitor a large number of planets and alert other
collaborating observatories and institutes as soon as a lensing event is detected, which
then also observe the event if possible. This provides a good time sampling of the light
curve. A very successful survey is the OGLE survey (Optical Gravitational Lensing
Experiment), which has detected several planets to date (see e.g. Udalski et al., 1993).
1.2.5
Direct Imaging
Direct imaging of an exoplanet yields a wide range of information about the planet.
One can characterize it spectroscopically, providing information about the atmosphere
1.2 DETECTION METHODS
9
Figure 1.4: Light curve of a gravitational lensing event of a star with a planet. On top the
brightening due to the lensing of the star and later the planet is shown. Image
taken from the Microlensing Planet Search Project homepage1 .
and measure its astrometric motion to derive information about the orbit. But to
directly image a planet next to a bright star is a challenging task, given the brightness
contrast and the small angular separation typical for exoplanetary systems. Jupiter
for example would only have a 4 arcsecond separation from the sun when viewed from
Alpha Centauri and typical angular distances of known exoplanets are much smaller.
Exoplanets are cool objects with temperatures in the range of a few 100 K, which makes
their thermal brightness in the visual negligible. The light observable from the planet
in this wavelength range is reflected light from the primary star. For an Earth-like
planet the flux ratio between planet and star is ∼ 10−10 . This is an almost impossible high contrast for today’s instruments, given the very small separation between the
star and the planet of 0.1 − 100 for nearby stars, and additional techniques have to be
used. One possibility to nevertheless image the planet is coronography, where most
of the light from the star is blocked, so the planetary signal becomes visible. Other
methods include spectral differential imaging, the system is imaged simultaneously in
two different filters and the two images are subtracted afterwards, angular differential
imaging, the system is imaged with two different position angles and the two images
are subtracted afterwards, and nulling interferometry, see below.
In the infrared wavelength regime, the thermal emission of the planets is higher and
peaks in the mid-infrared. The younger and hotter a planet is, the higher is its IR flux.
For an Earth-like planet the star-planet flux ratio goes down to about 10−7 . But at
the same time, the spatial resolution is getting worse with longer wavelengths. Using
interferometry is one solution, as it is easier at longer wavelengths and improves the
spatial resolution in addition to the lower flux contrast. Supplementary nulling inter1
http://bustard.phys.nd.edu/MPS/
10
CHAPTER 1. INTRODUCTION
Figure 1.5: Image of the three planetary companions to the star HR 8799 produced by
combining J-, H-, and Ks-band images obtained at the Keck telescope in July
(H) and September (J and Ks) 2008 (Marois et al., 2008). The three planets b,
c, d have masses around 7, 10 and 10 MJ respectively, inferred from photometry
and fitting evolutionary tracks.
ferometry is possible. The light from two telescopes is brought together with a shift of
λ/2 in one light path, so the two beams interfere destructively on-axis where the star
is centered. In the ideal case, this cancels out all light at zero phase, but keeping the
flux at other phases and hence is working like a coronograph.
Most of the extrasolar planets detected with direct imaging so far have bigger separations from their host stars than the ones detected with radial velocity or transits. But
with the already installed and near future instruments for direct imaging with adaptive
optics correction and coronography and/or interferometry more and more planets will
be detected closer to their stars.
One of the first extrasolar planetary systems whose orbital motions were confirmed via
direct imaging is the system HR 8799. HR 8799 is a young (∼ 60 million year old) main
sequence star located 39 parsecs away from the Earth in the constellation of Pegasus.
The system contains three detected massive planets and also a debris disk. The planets were detected by Christian Marois with the Keck and Gemini telescopes on Mauna
Kea, Hawaii (Marois et al., 2008). Just recently the first spectrum of an exoplanet was
obtained from the middle one of the three planets orbiting HR 8799 with the NACO
instrument at the VLT (Janson et al., 2010). This is the first step into a new and
amazing area of extrasolar planet characterization. In Fig. 1.5 the direct image of the
three exoplanets is shown.
1.2 DETECTION METHODS
1.2.6
11
Astrometry
Astrometry is the oldest measurement technique in astronomy. The method consists
of precise measurements of the position of a star on the tangent plane on the sky
relative to a reference frame and has been used for centuries to measure proper motions,
parallaxes and astrometric orbits of visual binaries. The gravitational influence of an
orbiting planet causes the star to move around their common center of mass in a small,
down-scaled orbital movement. Assuming M∗ À mp a combination of the law of the
lever for the two-body problem with Kepler’s 3rd law gives the semi-amplitude a∗ of
the stellar wobble due to the companion in radians:
µ
¶
µ
¶
mp
G 1/3
P 2/3
a∗ =
(1.8)
r
4π 2
M∗
Here r is the distance of the system, M∗ is the stellar mass , mp the companion’s mass,
P the orbital period and G the gravitational constant. The astrometric signal becomes
stronger the more massive the companion, and/or the less massive the primary and the
bigger the separation between the two components. This makes this detection method
complementary to RV and transit detection, which are most sensitive to close-in planets.
As one can see from Equ. 1.8, the astrometric detection of a planetary companion is
very sensitive to the distance of the system, which limits this technique to applications
to nearby stars. The astrometric signals for some of our own solar systems planets seen
from 10 pc distance and examples for a hot Jupiter and an Earth-like planet orbiting
a solar-type star, as well as several other examples are listed in Tab. 1.1. The values
are calculated for the case the full major axis a∗ is measured and are given as the
peak-to-peak astrometric signal, α = 2 ∗ a∗ . If the orientation of the system is such,
that not the full major axis is measurable, the astrometric signal is even smaller. No
matter how the system is oriented, one can always measure at least the signal of the
minor axis.
If one has obtained astrometric data points which cover a sufficient part of the orbit,
all orbital parameters can be determined. Especially with a known distance and stellar
mass, the true mass of the companion can be calculated. Astrometric planet detection
gives therefore more information about the detected companion than RV does. But
to derive the orbital motion of the star due to the companion, one has to disentangle
this motion from the proper motion of the star and the parallactic movement of the
Earth-bound observer. Astrometric position determination always needs a reference
system to which the position of the target star is referenced. Preferable would be a
fixed system, but this is rather difficult to set up and sometimes not possible. To a
much higher precision, positions can be determined relative to another system. For
planet detection this can be a star asterism in the same field of view (FoV) as the
targeted star. Since the stars used to set up the reference frame have their own proper
motion and parallactic movement, the proper motion and parallax of the target star
can only be derived relative to this reference frame and do not need to be the same as
the absolute ones. If one is only interested in the orbital movement due to a companion,
one does not need to know the absolute proper motion and parallax, but of course, it
is always beneficial to know the absolute parameters, e.g. to calculate the distance of
the target.
12
CHAPTER 1. INTRODUCTION
Planet
ap [AU]
α
Primary mass
Jupiter X
5
0.96
mas
1 M ¯ (sun-like star)
Jupiter X
1
0.19
mas
1 M¯
hot Jupiter
0.05
9.5
µas
1 M¯
Neptune [
1
1.3
µas
1 M¯
Neptune [
30
0.31
mas
1 M¯
Earth ♁
1
0.60
µas
1 M¯
Brown dwarf (30 MJ )
0.5
2.9
mas
1 M¯
Brown dwarf (30 MJ )
30
171.9
mas
1 M¯
Brown dwarf (30 MJ )
1
11.5
mas
0.5 M ¯ (M Dwarf)
Brown dwarf (30 MJ )
15
171.9
mas
0.5 M ¯
Jupiter X
1
0.38
mas
0.5 M ¯
Jupiter X
15
5.7
mas
0.5 M ¯
Earth ♁
1
1.2
µas
0.5 M ¯
Earth ♁
0.1
0.12
µas
0.5 M ¯
Table 1.1: Examples for astrometric signals for planets with different masses and semi major
axes, orbiting a sun-like star or a dwarf star with 0.5 M ¯ . All values are calculated
for a distance of 10 pc and the case that the full major axis is measured.
Astrometric measurements have been used to determine astrometric binary systems
for quite a long time. One of the first comments about the detection of an unseen
companion was made by William Herschel in the late 18th century, when he claimed an
unseen companion being responsible for the position variations of the star 70 Ophiuchi.
Other systems were announced in the coming two centuries, but all of them were later
vitiated or are still under discussion.
The only measurements of an astrometric signal due to an unseen companion were
obtained with the Hubble Space Telescope (HST). In 2002 Benedict et al. succeeded in
detecting the astrometric motion of the previously with RV discovered planet around
the star Gliese 876, and in 2006 the signal of the planet orbiting ² Eridani (Benedict
et al., 2006) (see also Bean et al., 2007; Martioli et al., 2010; Benedict et al., 2010, for
more examples).
In 2009 a planet orbiting the ultracool dwarf VB 10 (= GJ 752B), spectral class M8 V,
was discovered from ground with astrometry using the wide-field seeing limited imager
at the Palomar 200-inch telescope within the Stellar Planet Survey program (STEPS)
(Pravdo and Shaklan, 2009). The reflex motion of VB 10 around the system barycenter
compared to a grid of reference stars in the same FoV was monitored over 9 years. The
best fit Keplerian orbit yields a 6.4+2.5
−3.1 MJ planet in an 0.74 year almost edge-on orbit
(i = 96.9◦ ). Unfortunately, lately obtained RV observations with the high precision
1.2 DETECTION METHODS
13
Figure 1.6: Left: Astrometric motion of the Sun’s center around the barycenter of the solar
system due to all planets over 50 years, viewed face-on from a distance of 30
light years (Jones, 2008). The dashed circular line shows the motion of the Sun
due to only Jupiter. The size of the disk of the sun is shown in the upper right
corner for comparison. Right: Astrometric orbit of HD 33636, a G0 V star at
28.7 pc distance (Bean et al., 2007). Open circles are the HST position for each
epoch connected with a line to the positions calculated by the fit model. The
open square shows the predicted position of periastron passage.
spectrograph HARPS at the ESO 3.6 m telescope in La Silla, Chile, were not able to
confirm this planet, but instead ruled out the astrometric orbit solution (Bean et al.,
2010).
Measuring an astrometric signal from ground is very difficult, as the changes of the
stellar position are very small and the atmospheric and systematic distortions, such
as plate-scale variations between different observations, may be larger. The usage of
a correction of the atmospheric distortions with adaptive optics and determining the
plate-solution with great care, as done in the context of this work, can reduce these
difficulties and enable one to go down to the desired ∼ 1 mas precision needed to detect
and characterize companions in wider orbits. However, to detect Earth-like planets orbiting solar-like stars, one needs a higher precision in the astrometric measurements.
Interferometry makes it possible to achieve precisions about a few microarcseconds.
The already installed and soon available instrument PRIMA at the VLT will boost
the number of planets detected with astrometry. The spacebound astrometric mission
GAIA, planned launch in August 2011, will also make a huge contribution in stellar
position measurements and with this yield a huge number of newly detected Jupitersize planets (Sozzetti et al., 2001). Special designed missions, such as SIM PlanetQuest
(e.g. Catanzarite et al., 2006; Unwin et al., 2008), which are specialized for astrometric
measurements both for parallax movements and exoplanet detections around stars in
the solar neighborhood, will reach accuracies about one microarcsecond and will be
sensitive down to Earth-like masses. But also the astrometry obtained from the pho-
14
CHAPTER 1. INTRODUCTION
tometric transit mission KEPLER may be used for astrometric planet detection, given
its stable and precise pointing.
Combination of Radial Velocity Data and Astrometric Data
Planet detections with radial velocities only yield a minimum mass for the companion
as one cannot determine the inclination of the orbital plane with respect to the observer.
The measured semi-amplitude of the velocity change is the projection of the true motion
onto the line of sight from the observer to the star. Companions detected with radial
velocities should therefore be seen as exoplanet candidates for the time being.
Astrometry yields the whole parameter set necessary to describe the full orbital motion
in space, but typically a large number of high precision measurements need to be
obtained. The complete astrometric fit for the motion of a star with a companion in
space includes:
α0 , δ0
two coordinate zero points
µα , µδ
proper motion in right ascension and declination
π
parallax
a∗
semi-major axis of the orbit
e
eccentricity
P
orbital period
ω
longitude of periastron
Ω
longitude of the ascending node
i
inclination
To solve for these 11 parameters one has to obtain at least 11 epochs of measurement,
preferentially more for reasons of robustness of the fit and taken the present noise in
the data into account.
Combining radial velocity and astrometric measurements yields the whole three dimensional orbit as one combines measurements which are sensitive to the movement in the
direction of the line of sight and in the plane perpendicular to it. Taking the orbital
parameters inferred from radial velocities, P, e, ω, a, one is left with only seven parameters to solve for with the astrometric fit. If one additionally can infer the proper motion
and parallax independently, only the inclination and longitude of the ascending node,
as well as the coordinate zero points, need to be determined. However, most of the time
this is not possible, as either the precision of these parameters is not sufficient, or one
has to determine the proper motion and parallax relative to the local reference frame.
Combination of the two measurement techniques has advantages and disadvantages.
The advantage of this approach is that normally the radial velocity measurements are
obtained with much higher precision than what is possible for the astrometric measurements. One can therefore hold the parameters from the RV fit fixed. A disadvantage or
1.3 BROWN DWARFS
15
rather a constraint is the circumstance that RV detections are most sensitive to close-in
companions and astrometry to companions in wider orbits. Therefore the astrometric
signals for most of the planets detected by radial velocities are very small and difficult
to detect. But with the upcoming specialized astrometric instruments, these will come
more and more into range.
The astrometric measurements of radial velocity exoplanet candidates conducted with
the HST Fine Guidance Sensor, pushed three out of five candidates from the planetary
regime into the brown dwarf or low mass star regime:
Planet cand.
M sin i[MJ ]
M [MJ ]
inclination
Reference
HD 33636 b
9.3
142 ± 11
4.1◦ ± 0.1◦
Bean et al., 2007
M dwarf star
HD 136118 b
12
42+11
−18
163.1◦ ± 3.0◦
Martioli et al., 2010
48.3◦ ± 0.4◦
Benedict et al., 2010
Brown Dwarf
HD 38529 c
13.1
17.6+1.5
−1.2
Brown Dwarf
This shows how important it is to obtain complementary measurements to solve for the
whole 3D orbit and calculate the true mass of the exoplanet candidates.
Combination of radial velocity data with astrometry provided by the HIPPARCOS
satellite has also been done (Perryman et al., 1996; Mazeh et al., 1999; Zucker and
Mazeh, 2000). Han et al. (2001) used the HIPPARCOS intermediate astrometric data
to fit the astrometric signal of stars known to have a possible planetary companion
found by radial velocity. Most of the expected astrometric perturbations were close
to or lower than the precision obtained from the HIPPARCOS data. The conclusion
of this statistical study was that a significant fraction of the exoplanetary systems
are seen edge-on, which would push their masses to also significantly higher masses.
Pourbaix (2001) later showed that the high inclinations found by Han et al. are artifacts
of their adopted fitting procedure. As the reason for that he named the size of the
orbit with respect to the precision of the astrometric measurements, meaning, the
’measured’ astrometric signal was not large enough compared the astrometric precision.
Future astrometric missions, such as GAIA will lead to much higher precisions in the
astrometric measurements, thus leading to better constraints on the orbital inclination.
1.3
Brown Dwarfs
In the same year, at the same conference, the first widely accepted brown dwarf (BD),
Gliese 229B (Nakajima et al., 1995), and the first extrasolar gas giant planet, 51 Peg b
(Mayor and Queloz, 1995), were announced to the astronomical community. Now there
16
CHAPTER 1. INTRODUCTION
are around 720 BDs known to exist as companions to nearby stars, in young clusters
and most frequently as faint isolated systems within a few hundred parsecs in the solar neighborhood. Brown dwarfs are star-like objects with a maximum mass between
0.07 and 0.08 M¯ , depending on their metallicity. This mass limit for BDs is defined
by the disability to sustain stable hydrogen fusion reactions in their cores and sets a
division between stars and brown dwarfs. But BDs are massive enough to be able to
burn deuterium in their cores at the beginning of their evolution, followed by a steady
decline in their luminosity and effective temperature with time, once their supply of
deuterium is exhausted.
The division between brown dwarfs and giant planets is yet not clear and still under
debate. Two possible ways of defining brown dwarfs and giant planets are under discussion. One widely used definition is based on the mass limit to burn deuterium, which
would define an object with less than 13 MJup as a planet (Saumon et al., 1996; Chabrier
et al., 2000). This is also the IAU2 definition for brown dwarfs, which considers objects
above the deuterium burning mass limit as brown dwarfs. The definition can be applied
to both companions and isolated objects and is the reason why very-low mass objects
in clusters are sometimes called free-floating planets. A drawback of the definition of
the border between BDs and giant planets over the deuterium burning limit is that
unlike the hydrogen burning limit, the ability to fuse deuterium is insignificant for the
physical properties of BDs and therefore describes no meaningful boundary for the evolution of low mass objects (Chabrier et al., 2007). In fact there are more differences
in stellar structure and evolution between high and low mass stars than for low mass
brown dwarfs and giant planets. Also the mass determination with evolutionary models based on the luminosity of the objects is often uncertain, so a definitive conclusion
whether an object is above or below the deuterium burning mass limit is difficult. For
example the best mass estimate for the object GQ Lup b is 10 − 40 MJup (McElwain
et al., 2007; Seifahrt et al., 2007), hence both definitions, planet and brown dwarf, are
possible. Determining the mass dynamically and therefore independent of the models
puts more constraints on the evolutionary theories for these low mass objects and helps
to better understand their mass distribution and formation processes, but this is only
measurable for brown dwarfs which are bound in a system.
The other definition to distinguish brown dwarfs and giant planets is based on their
formation processes. Here a planet is a substellar object formed in a circumstellar disk
and a brown dwarf formed through cloud fragmentation like a star. This definition also
has its drawbacks, as it is obviously difficult to tell the formation process of a given substellar object. It is for example most likely for wide massive companions to be formed
by cloud fragmentation rather than in a stellar disk, but for lower mass companions
in intermediate orbits (8 MJup , @ 20 AU) it will be more difficult to determine the
formation mechanism (Luhman, 2008). So neither of the definitions provides a clear
distinction between brown dwarfs and giant planets. The most commonly accepted
one at the moment is the one of the IAU, that distinguishes substellar objects by their
mass. Independent of their formation process, objects below 13 MJup which orbit a star
or stellar remnant are planets and objects with masses above this deuterium burning
mass limit and below the hydrogen burning limit (13 MJup − ∼ 80 MJup ) are defined as
brown dwarfs.
2
www.dtm.ciw.edu/boss/definition.html
1.3 BROWN DWARFS
1.3.1
17
Brown Dwarf Formation Processes
Similar to the difficulty to define a distinctive boundary between brown dwarfs and
giant planets, it is challenging to put constraints on the formation process of brown
dwarfs. Several scenarios are under discussion, but none of them can explain all of the
available observations. A star-like formation by direct collapse and fragmentation of a
molecular cloud, a scaled version of the Jeans model, is one possibility for BD formation
(Bate and Bonnell, 2005). Another possibility is the ejection of protostellar embryos
that form in a cluster environment, but are ejected due to dynamical interactions before they can accrete enough material to become a star (Reipurth and Clarke, 2001;
Umbreit et al., 2005). This scenario has difficulties to explain young wide BD binaries,
because these should be ripped off during the ejection, likewise large disks should not
survive an ejection, but are nevertheless observed (e.g. Scholz et al., 2006). Photoevaporation of their accretion envelope by the radiation of a nearby massive and hot
star (Whitworth and Zinnecker, 2004), as well as instabilities in massive disks, followed
by a pulling off of the still substellar companions through dynamical encounters with
other stars can produce isolated free-floating brown dwarfs (Goodwin and Whitworth,
2007; Stamatellos and Whitworth, 2009). Turbulence in a molecular cloud can produce local over-densities that can collapse and form a compact low mass object. This
turbulent fragmentation could explain the formed brown dwarfs as the low-mass tail
of the regular star formation process. Also the continuity in disk fraction, the ratio of
stars or BDs without a disk to those having a disk, from the stellar to the BD regime
(Caballero et al., 2007) favors this formation scenario. None of these scenarios can
be seen as the dominant contributor to the brown dwarf population, neither one can
explain all observations. Most likely the formation processes of brown dwarfs depend
on the stellar environment and its varying initial conditions from case to case, leading
to a changing relevance of the individual scenarios.
1.3.2
The Brown Dwarf Desert
Radial velocity surveys to find substellar companions orbiting their host stars have led
to the detection of a variety of exoplanets and brown dwarfs. They also led to the
definition of the “brown dwarf desert”, the absence of BD companions relative to giant
planets and stellar companions to low-mass stars at separations less than 3 AU (Marcy
and Butler, 2000). The BD companion frequency at separations > 1000 AU is at least
10 times higher than that of separations of only a few AU, which is ≈ 0.5 % (Gizis
et al., 2001; Neuhäuser and Guenther, 2004). The frequency of stellar companions at
small separations is 13 ± 3% (Duquennoy and Mayor, 1991; Mazeh et al., 1992), at
least a factor 30 larger than the BD frequency, whereas at larger separations the ratio of
frequencies of stellar and substellar companions is between ∼ 3 and 10 (McCarthy and
Zuckerman, 2004). At wider separations to solar-type stars it appears therefore that
the brown dwarf desert is no longer present (Luhman et al., 2007). Fig. 1.7 depicts the
brown dwarf desert in mass and period. The plot is taken from Grether and Lineweaver
(2006) and shows the estimated companion masses versus the period of the companions.
The lack of brown dwarf companions in short-period orbits can clearly be seen. For
more details about the sample and the companion mass estimates see Grether and
18
CHAPTER 1. INTRODUCTION
Lineweaver (2006).
The rareness of close-in BD companions is highly significant since the commonly employed RV searches for sub-stellar companions to stars are very sensitive to such objects
as the RV semi-amplitude K is higher for shorter periods P, e.g. smaller separations
from the host star (see also Equ. 1.2). Because the RV measurements only result in a
minimum companion mass and the HIPPARCOS astrometric precision is mostly not
sufficient to distinguish BDs from stellar companions, the masses of the few known
close-in BD candidates are often uncertain (Pourbaix, 2001; Pourbaix and Arenou,
2001). RV studies of M dwarfs indicate that the BD desert continues also into the early
M dwarf population3 .
1.4
Goal of this Work
Most planets have been discovered by radial velocity measurements. But RV measurements only yield a minimum mass for the companions and are most sensitive to planets
in close-in orbits. Transit measurements can yield the true mass of a companion when
combined with RV measurements. But they are also most sensitive to close-in planets,
as the probability of an occurring transit is higher for those companions.
However, astrometry yields the full set of orbital parameters and therefore the true
mass of a companion. It also opens a new parameter space of planets with a longer
orbital period.
The goal of this work is to measure the astrometric signal of the wobble of a star due
to its unseen companion. Using adaptive optics aided imaging to detect such a signal
I want to show the feasibility of this approach for planet and low mass companion
detection and characterization. Seeing limited astrometry has mostly a larger field of
view than an adaptive optics imager, but the achieved accuracy is most times not high
enough to measure the tiny signal of a planet.
To test whether the astrometric signal is measurable, we chose an object known to
harbor a candidate brown dwarf companion found by RV measurements (Kürster et al.,
2008). When astrometry can be combined with precision RV measurements the number
of orbital parameters to be derived from the astrometric data can be strongly reduced
(see Sect. 1.2.6). Also the astrometric signal is higher for such an object than for an
exoplanet. Additionally, the companion is a very interesting object as it is a brown
dwarf desert candidate. Deriving its true mass, confirming it as a BD or pushing it
into the low mass star regime, would be a very interesting result.
3
M. Kürster, private communication
1.4 GOAL OF THIS WORK
19
Figure 1.7: Brown dwarf desert in mass and period. Shown are the companion masses M2
versus the period of the companions to sun-like stars. The solid rectangle defines
companions with periods smaller than 5 years and masses between 1 MJup and
∼ 1 M¯ . The stellar (open circles), brown dwarfs (blue circles) and planetary
(filled circles) companions are separated by the dashed lines at the hydrogen and
deuterium burning limit. The brown dwarf desert for companions with orbital
periods P < 5 years can clearly be seen. The detected, being detected and not
detected regions of the mass-period space mark the limit to which high-precision
radial velocity measurements were able to detect companions at the time of the
analysis leading to this plot (Lineweaver and Grether, 2003). For more details
about the plot and the sample see Grether and Lineweaver (2006).
20
CHAPTER 1. INTRODUCTION
Chapter 2
Introduction to Adaptive Optics
Stellar wavefronts are assumed to be spherical waves and are treated as plane waves
when they reach Earth because of the immense distance of the stars. But turbulences
in our atmosphere lead to random deformations of the incoming wave. Telescope optics
collect the light and transform it into an image. In the ideal case, without any distortions, the angular resolution of this image is determined by the bending of the light,
this is called the diffraction limit case. For circular apertures the angular diameter of
the image of a point source is given by:
θ = 1.22
λ
D
(2.1)
where λ is the observed wavelength and D is the telescope diameter. Such an image
of a point source is called Point-Spread-Function (PSF). The intensity distribution of
a PSF has the form as shown in Fig. 2.1. The inner part of the distribution is called
λ
Airy-disc and it has the radius 1, 22 D
. Two close point sources can only be imaged
separately if the maximum of the one PSF falls onto the first minimum of the other PSF.
Due to the atmosphere, part of the intensity contained in the PSF is moved from the
maximum to the outer parts of the light distribution. Details are smeared out and
cannot be resolved anymore. In Fig. 2.2 the effects of the atmosphere on an image of
a close binary star are demonstrated. In the ideal case without any distortions, one
obtains a perfect image for each star with a diameter of 2θ = 2, 44λ/D (Fig. 2.2a). If
the exposure time is very short, less than 1/50 second, one derives an image consisting
of many speckles, which all have a diameter of the diffraction-limited PSF (Fig. 2.2c).
This speckle pattern is changing randomly in subsequent images and can be seen as
a sketch of the momentary atmospheric distortions. If the exposure time is increased
to a multiple of the timescale of the turbulence change in the atmosphere, the speckle
pattern is smeared out and one obtains an image of the point source, the so-called
seeing disk, with a diameter which is no longer determined by the ratio λ/D but by
λ/r0 (Fig. 2.2b). The factor r0 , called-Fried-Parameter, is an important parameter
to describe the atmospheric turbulence. It will be explained in more detail in Chapter 2.1.1. The consequence of the above circumstance is, that even for the bigger but
uncompensated telescopes the resolution limit is between 0.5-1 arcseconds or more.
Also the sensitivity of a telescope does not increase in the same way as in the diffraction
21
22
CHAPTER 2. INTRODUCTION TO ADAPTIVE OPTICS
Imax
2.24 l/d
1.22 l/d
3.24 l/d
x
airy-disk
Figure 2.1: The Point-Spread-Function (PSF) of a point source. The diameter of the main
maximum is called Airy-disk and has the radius θ = 1.22λ/D (from Stumpf
(2004)).
limit case, which is again due to the atmosphere. Generally, the bigger an aperture, the
more light it can collect and the fainter objects it can detect. In the ideal, diffraction
limited case the capability of a telescope to detect a point source is ∝ D4 . But this
capability is degraded to ∝ D2 in the seeing limited case, because the image size is no
longer determined by the telescope diameter.
There are two possibilities to overcome the image degradations caused by the atmosphere:
• One can go to space and leave the atmosphere behind. Building space based
telescopes has the advantage that one can also observe in the wavelength range
shorter than 0.3 µm (UV) where Earth’s atmosphere is opaque, and in the visual
and infrared between 0.5 and 2.5 µm where emission lines from OH-molecules
compromise observations. Disadvantages of satellites and space telescopes are
their immense costs and technical as well as logistic requirements. The transport
into orbit limits their size and weight. They need to be shock-resistent and maintenance or upgrades with new instruments are, if performed at all, expensive and
dangerous. But there are certain wavelength ranges which can only be observed
from space.
• The other possibility to obtain diffraction-limited images is the usage of an adaptive optics (AO) system. This system consists of two main components, a wavefront sensor, which measures the aberrated wavefront, and a deformable mirror
which corrects the wavefronts. Such a system is cheaper to build and to maintain
than a space telescope and can be upgraded or modified more easily. In the wavelength range observable from the ground, the ground based telescopes equipped
with an AO system are comparable or superior to space based ones, because their
primary mirrors can be made considerably larger.
A more detailed description of an AO system is given in Chap. 2.2.
2.1 ATMOSPHERIC TURBULENCE
23
Source
Turbulence
recieved
wavefront
Telescope
Image plane
(b) long exposure
Intensity
profile
(c) short exposure
(a) perfect image
Figure 2.2: Effect of atmospheric turbulence on the image of a close double star (right). Perfect image with no atmosphere (a). Short exposures produce a speckle pattern,
which shows a snapshot of the momentary distortions (c). In longer exposures
the pattern is smeared out and results in the seeing disk (b). (from Hardy (1998))
2.1
Atmospheric Turbulence
The atmosphere is neither static nor homogeneous. Temperature, density and humidity
change continuously on different scales of time and space. With it comes a permanent
change of the refractive index n of the atmosphere (Clifford, 1978):
µ
¶
1 P
n(λ, P, T ) = 1 + 7, 76 · 10−5 1 + 7, 52 · 10−3 2
(2.2)
λ
T
where λ is the wavelength of the light in µm, P the pressure of the air in mbar and
T the temperature of the air in Kelvin. The dependency on the wavelength is small
for a broad range of λ and the fluctuations of the pressure balance with the speed of
sound. But the fluctuations of the temperature are more inertial, and thus dominate
the changes of the refractive index. For two parallel light beams which pass through
the atmosphere, a difference of the refractive index leads to an adjournment of their
24
CHAPTER 2. INTRODUCTION TO ADAPTIVE OPTICS
Stratosphere
Altitude
in km
Wind
Shear
Tropopause
Troposphere
Wind
Shear
Surface Layer
Log of turbulence strength Cn2
Figure 2.3: Structure of the atmosphere with typical turbulence profile. (Hardy, 1998)
relative phases and therefore to a distortion of the previously plane wavefront.
The air masses in different layers of the atmosphere move because of different reasons.
In Fig. 2.3 the structure of the atmosphere and a typical turbulence profile is shown.
The vertical distribution of the turbulence varies strongly with height (right panel). Cn2
is a measure for the vertical turbulence strength. On average it is strongest near the
ground, the so-called ground layer (GL), because of thermal heating of the ground from
sunshine. In the change-over layer between the Troposphere and the Tropopause, at an
altitude of around 10 km, the turbulence is dominated by strong winds. A measurement
to describe the movement of the air is the Reynolds-number Re. It describes whether
a flow is laminar or turbulent. Re is given by:
Re =
L0 υ0
η
(2.3)
Here L0 is the characteristic size of a turbulence cell, υ0 its characteristic velocity and η
the kinematic viscosity of the medium, here the air. If the Reynolds-number is smaller
than the critical value 103 , the flow is laminar, otherwise it is turbulent. In air the
viscosity is η = 15 · 10−6 m2 s−1 and a typical turbulence cell has a size of L0 = 15m and
a velocity of υ0 = 1ms−1 . This yields an average Reynolds-number for the atmosphere
of Re = 106 , which is clearly above the critical value. So the air flow in the atmosphere
is mostly turbulent.
To describe and analyze a system as complex as the atmosphere, elaborate models
are needed. The most commonly used one in astronomy is the Kolmogorov model
(Kolmogorov, 1961). This model describes the turbulence as originating in energy
input in large air structures, eddies, with a typical size, the so-called outer scale. This
large structures transport the energy loss-free to smaller and smaller structures, the
inner scale, till the Reynolds-number is getting smaller than the critical value. The air
flow is then laminar and the energy dissipates into thermal heating. Typical sizes of
the outer scale are between a few meters at ground level and up to 100 m in the free
atmosphere. The inner scale lies between a few millimeters at ground and up to a few
2.2 PRINCIPLES OF ADAPTIVE OPTICS
25
centimeters close to the Tropopause.
2.1.1
Fried-Parameter
The Fried-Parameter, also called correlation-length, defines the length over which the
mean divergency of the phase-difference to a plane wavefront does not exceed the standard deviation of one radiant (1 rad). It was first introduced and calculated by David
L. Fried (Fried, 1965):

r0 = 0.423
µ
2π
λ
¶2
Z∞
−3/5
Cn2 (h)dh
(cos γ)−1
(2.4)
0
Cn2 (h) is a measure for the vertical strength of the turbulence profile and depends on the
height h. γ is the angle between the line of sight and the zenith. But one can also interpret r0 as the size of an aperture that has the same resolution as a diffraction-limited
aperture without any turbulence. That means that the resolution of a telescope with
a diameter D larger than r0 , the Full Width at Half Maximum (FWHM), is limited to
F W HM ∝ λ/r0 and not anymore ∝ λ/D as is the case for D < r0 . The VLT (Very
Large Telescope) has a diffraction-limited resolution of λ/D = 0.05700 at λ = 2.2µm.
But the resolution is lowered by the atmosphere to λ/r0 ≈ 0.700 . One can also see from
the equation, that r0 ∝ λ6/5 , which means the area over which the wavefront error is
negligible is growing with wavelength. An r0 of typically 10 cm at 0.5 µm in the visual
corresponds to an r0 of 360 cm at 10 µm in the infrared and at 2.2µm r0 is typically
60 cm.
2.1.2
Time Dependent Effects
In the same way as for the spatial case, a time can be defined over which the variance
of the wavefront changes account to 1 rad. This time scale is called coherence time τ0 .
The relation to r0 is given over the wind speed v:
τ0 =
r0
|v|
(2.5)
With an average r0 of 60 cm and v = 10m/s this yields τ0 ≈ 60 ms in the infrared at
2.2µm . To derive a diffraction-limited image, one needs to correct the wavefront errors
in this frequency range.
Both τ0 and r0 are critical parameters and the larger they are, the more stable the
atmosphere is.
2.2
Principles of Adaptive Optics
A powerful technique in overcoming the degrading effects of the atmospheric turbulence
is real-time compensation of the deformation of the wavefront by adaptive optics.
26
CHAPTER 2. INTRODUCTION TO ADAPTIVE OPTICS
Science
object
Telescope
tip/tilt - mirror
Reference
Star
Beam splitter
Turbulent
layers
Deformable
mirror
Wavefront
sensor
Wavefront
reconstruction
Control loop
Compensated
image
Figure 2.4: Principle setup of an AO system.
2.2.1
General Setup of an AO System
The task of an adaptive optics system is the measurement of the wavefront distortions
caused by the atmosphere, the calculation of a correction and its execution. In Fig. 2.4
the principle setup of an AO system is depicted. The light coming from a star is a
plane wavefront (WF) until it is distorted by the Earth’s atmosphere. It passes trough
the telescope optics and then reaches the AO system, where it first passes through
the correction unit, so that after the first correction only the differential wavefront error between the single measurement cycles has to be corrected. In the ideal case, the
wavefront is plane again after the correction. The correction unit often consists of two
elements, a tip-tilt mirror that is tiltable in two directions and acts for image stabilization, which is the largest disturbance generated by the turbulence, and a deformable
mirror (DM) which is deformed inversely to the incoming WF and compensates for the
higher order aberrations.
After the correction unit the light is divided by a beam splitter into two parts. One
part, mostly the infrared part of the light, is directly led to the science camera. The
other part, mostly the visual light, is led to the wavefront sensor (WFS), which measures the distortions of the WF. If it is necessary to observe and sense in the infrared,
for very red objects for example, a intensity-filter is used instead of a dichroic for the
beam splitting. With the WF reconstruction unit, wavefront errors are translated into
control signals for steering the deformable mirror and the tip/tilt mirror. With sending
these steering signals the loop is closed.
Typical frequencies for one cycle of the loop are 1-2 kHz. The frequency depends on
the wavelength λ in which the observation is conducted, because the time dependent
change of the atmospheric disturbance, τ0 , is depending on r0 and therefore on λ, see
Sec. 2.1. The longer the wavelength, the easier it is to correct the wavefront distortions.
Therefore most current AO systems correct for instruments which work in the infrared
or even longer wavelengths.
2.2 PRINCIPLES OF ADAPTIVE OPTICS
2.2.2
27
Strehl Ratio
A good way to describe the quality of a partially corrected PSF is the Strehl ratio,
which basically corresponds to the amount of light contained in the diffraction-limited
core relative to the total flux, (Strehl, 1902). Due to the wavefront distortions, part of
the intensity of a PSF (Fig. 2.1) is moved from the peak to the halo. Therefore, the
maximum intensity of the PSF is decreased. The Strehl ratio is defined as the ratio
between the actual maximum intensity of a point source, Ip , and the intensity which
would be reached by a perfect, diffraction-limited telescope of the same aperture, I ∗ .
The Strehl ratio can be calculated analytically, if the aberrated wavefront Φ(ρ, θ) is
known, following (Hardy, 1998):
¯ 1 2π
¯2
¯Z Z
¯
¯
Ip
1 ¯¯
S= ∗ = 2¯
exp(ikΦ(ρ, θ)) ρdρ dθ¯¯
(2.6)
I
π ¯
¯
0
0
where, k is the wave number and ρ, θ are polar coordinates. But if the mean quadratic
error of the wavefront only amounts up to 2 rad, the Maréchal-Approximation (Maréchal,
1947) is used commonly in astronomy:
S ≈ exp−σ
2
(2.7)
Here σ stands for the mean-square wavefront error. Typically the Strehl-ratio is less
than one percent for a uncorrected system. In case of good conditions and a bright,
nearby reference source, the correction is good and the resulting PSF is close to the
diffraction limit. A good correction in the K-Band typically corresponds to a Strehl
ratio larger than 30%.
2.2.3
Anisoplanatism
All adaptive optics systems need a point source to measure the wavefront distortions
caused by the atmosphere. Generally this is a suitable reference star, whose wavefront
’records’ on its way through the atmosphere an image of the actual wavefront distortions. Suitable means that the source needs to have a certain magnitude to serve as
a guide star. This magnitude is dependent on the AO system and the wavelength in
which one observes. Additionally the reference source should be close to the target,
so the wavefronts of both objects pass through the same turbulences. If possible the
distance of the target and the reference star should not exceed the isoplanatic angle
θ0 , which is defined as the angle over which the mean quadratic wavefront error σ 2
amounts 1 rad2 (Fried, 1982). The resulting error varies as a function of the angle
α between two light rays (Fig. 2.5). Assuming a single turbulence layer at height h,
θ0 = 0.31r0 /h, with r0 being the correlation-length. The isoplanatic angle is very small,
on average ∼ 200 in the visual and up to 2000 in the near infrared.
The correction of the target degrades with the distance to the guide star and the PSF
of the target star is elongated in the direction to the guide star.
In some cases the observed target can be used as a guide star itself. In astronomy
though, most of the scientific interesting objects are faint or extended objects, like protoplanetary disks, star clusters or galaxies. In the case of protoplanetary disks as well
28
CHAPTER 2. INTRODUCTION TO ADAPTIVE OPTICS
Turbulent
layer
Isoplanatic angle
for a single
tubulent layer
Isoplanatic
error
Figure 2.5: Angular anisoplanatism (Hardy, 1998).
as with observing exoplanets, the host star can mostly be used as guide star. Also in
clusters one normally finds a suitable bright star. In the cases where no suitable guide
star is close enough, artificial laser guide stars can possibly be used if implemented.
This artificial sources are replacing the natural guide star as reference objects for the
AO. So the sky-coverage can be enhanced by a huge factor. But laser guide stars are
also introducing new problems, for example the cone-effect, which arises because of the
finite distance of the laser focus point, which is produced in a height of 90 - 100 km
by resonant fluorescence of a layer which is enriched with natural sodium atoms, and
the therefore only cone-like sampling of the atmosphere (Bouchez et al., 2004). This
effect is even more prominent in the case of the so-called Rayleigh laser beacons. These
are produced in a height of 5 - 10 km by Rayleigh scattering of the laser radiation on
atmospheric molecules (Fugate et al., 1991).
2.3
NACO
The instrument NACO is the adaptive optics system at the VLT and it is mounted at
the UT4, Yepun, at the Nasmyth B focus (Lenzen et al., 2003; Rousset et al., 2003).
It consists of two parts, the Nasmyth Adaptive Optics System NAOS and the HighResolution Near IR Camera CONICA.
The overall tip and tilt of the wavefront is compensated by the tip-tilt plane mirror of
NAOS. The higher order aberrations, including static aberrations of NAOS and CONICA, are compensated by the 185 actuators deformable mirror. A dichroic acts as the
beam splitter to separate the light between CONICA and the WFS. There are several
different dichroics, depending on whether the WF correction is done in the visible or
the IR, how bright the guide star is and so on.
NAOS has two WF sensors, one operating in the visible and one in the near-IR. This
2.3 NACO
29
Camera
px-scale
FoV
Spectral
[mas/px]
[arcsec]
range
S13
13.27
14 x 14
1.0 − 2.5 µm
S27
27.15
28 x 28
1.0 − 2.5 µm
S54
54.6
56 x 56
1.0 − 2.5 µm
SDI
17.32
5x5
1.6 µm
L27
27.19
28 x 28
2.5 − 5.0 µm
L54
54.6
56 x 56
2.5 − 5.0 µm
Table 2.1: List of available cameras for CONICA with plate scales, field of view and spectral
range
kind of sensor consists of a lenslet array that samples the incoming WF. Each lens
forms an image of the object and the displacement of this image from a reference position gives an estimate of the slope of the local wavefront at that lenslet. This WFS
works with white light and also with extended sources and faint stars, although with
a lower performance. For the visible sensor, two Shack-Hartmann sensors, one with a
14x14 lenslet array, with 144 valid sub-apertures and one with a 7x7 lenslet array and
36 valid sub-apertures are available and for the infrared sensor three Shack-Hartman
sensors are available.
Anywhere within a 11000 diameter field of view an off-axis natural guide star can be
selected, leading to a maximum distance between guide star and target of 5500 . But
due to the anisoplanatism the Strehl ratio is going down to ∼ 9 % for a V = 10 mag
target with a separation to the guide star of 3000 , observed in λ = 2.2 µm and a seeing
of 0.800 , compared to a Strehl ratio of ∼ 47 % for an on-axis target observed in the same
wavelength and under the same seeing conditions.
If no suitable guide star is close enough to the target, a laser guide star (LGS) can be
used. The PARSEC instrument is based on a 4W sodium laser guide star. The laser
which is tuned on the D2 -line (λ = 589 nm) of sodium, is focused at 90 km altitude
where a thin, natural layer of atomic sodium exists. The backscattered light produces
a V ≈ 11 mag artificial guide star which is used for the AO correction.
The CONICA instrument is an IR imager and spectrograph in the wavelength range
1 − 5 µm and it is fed by NAOS. It is capable of imaging, long slit spectroscopy and
coronographic and polarimetric observations. Several different plate scales are available, too.
For imaging, a variety of filters and pixel scales are offered. In principle three different
plate scales and field of views (FoV) are available, plus one small FoV for spectral
differential imaging (SDI). Table 2.3 lists the available cameras with their FoV, pixel
scale and spectral range. Five broad band filters (J, H, Ks , L0 , M 0 ) and 29 narrow and
intermediate band filters are available, plus two neutral density filters which can be
30
CHAPTER 2. INTRODUCTION TO ADAPTIVE OPTICS
combined with the other filters to reduce the flux of very bright sources.
The detector of CONICA is a Santa Barbara Research Center InSb Aladdin 3 array,
with a net 1024 x 1024 pixels and a 27 µm pixel size. The wavelength range is from
0.8 − 5.5 µm and it has a quantum efficiency of 0.8 - 0.9. For bright objects, a number
of ghosts become apparent (see Chap. 3.3).
A single integration with the detector corresponds to the Detector Integration Time
(DIT) and the pre-processor averages N of these exposures, NDIT, before the result is
transferred to the disk. The number of counts in the final images always corresponds
to DIT and not to the total integration time NDIT x DIT.
Three readout modes are offered. In case of the Uncorr mode the detector is reset and
then read once. This mode is used when the background is high. The Double RdRstRd
mode is used when the background is intermediate between high and low and the detector array is read, reset and read again. If the background is low, the FowlerNsamp
mode is used. Here the array is reset, read four times at the beginning of the integration
ramp and four times at the end of the integration ramp.
A detailed description of the above mentioned characteristics of NACO and more information can be found in the NACO User’s Manual:
http://www.eso.org/sci/facilities/paranal/instruments/naco/doc/
2.3.1
Our Observation Configuration
In our observations, I used the visible dichroic with the 7x7 optical wavefront sensor
and near-IR imaging with the narrow-band filter N B 2.12 and the S27 camera. Due to
the special arrangement of the stars in our FoV I calculated an own jitter pattern. For
readout I chose the Double RdRstRd mode with an integration time of 0.9 seconds for
the target field and 0.4 seconds for the reference field. For more details see Chap. 3.3.
Chapter 3
Observations and Data Reduction
3.1
The Target Field
The target star I observed is an M2.5 V dwarf star in the solar neighborhood. GJ 1046
has an apparent magnitude of V = 11.61 mag and K = 7.03 mag and a stellar mass
of 0.398 ± 0.007 M¯ . It is a high proper motion star with µα = 1394.10 mas and
µδ = 550.05 mas and a parallax of 71.56 mas (i.e. a distance of 13.97 pc) (Perryman
et al., 1997).
The companion orbiting GJ 1046 was found by radial velocity measurements with the
UVES/VLT spectrograph within a search for planets around M dwarfs (Kürster et al.,
2008, 2003; Zechmeister et al., 2009). In Figure 3.1 (from Kürster et al., 2008) the RV
time series and best-fit Keplerian orbit of GJ 1046 is plotted. Assuming a stellar mass,
from K-band mass-luminosity relationship (Delfosse et al., 2000), of M = 0.398 M¯ ,
a minimum companion mass mmin = 26.9 MJup can be calculated for an inclination
i = 90◦ , corresponding to an edge-on view of the orbit and therefore measuring the
maximum RV amplitude, from the mass function f (m) = (m sin i)3 /(M + m)2 =
9.5 · 10−5 M ¯ . Together with Equ. 1.2 for the RV semi-amplitude K an minimum
orbital semi-major axis a = 0.42 AU was inferred. Its mass and distance to the host star
makes this companion a promising candidate for the brown dwarf desert. In Tab. 3.1
the stellar parameters and orbital characteristics inferred from the RV measurements
are listed. These parameters are kept fixed later in the astrometric orbit fit.
To exceed the upper brown dwarf mass limit of 0.08 M¯ the orbital inclination would
have to be smaller than 20.4.◦ (or larger than 159.6◦ ). But the probability that the
inclination angle i is by chance smaller than this value, p(90◦ ≥ i ≥ θ) = cos θ is
only 6.3%, assuming random orientation of the orbit in space. Combining the RV data
with the HIPPARCOS astrometry of GJ 1046 (Kürster et al., 2008); (see also (Reffert
and Quirrenbach, 2006)) a 3 σ upper limit to the companion mass of 112 MJup was
determined. The probability for the companion to exceed the star/BD mass threshold
(0.08 M ¯ ) is just 2.9 %. These two constraints make it very unlikely that the companion
is stellar, but it rather is a true brown dwarf desert companion.
The expected minimum astrometric signal (see Equ. 1.8) due to the companion is
3.7 mas peak-to-peak, which corresponds to 0.136 pixel on the NACO S27 detector.
This value is calculated using the HIPPARCOS parallax of 71.56 mas and holds for
31
32
CHAPTER 3. OBSERVATIONS AND DATA REDUCTION
Table 3.1: Stellar and orbital parameters of GJ 1046
RV -derived parameters
value
uncertainty
units
RV semi-amplitude K
1830.7
±2.2
ms−1
Period P
168.848
±0.030
days
Eccentricity e
0.2792
±0.0015
Longitude of periastron ω
92.70
±0.50
degree
3225.78
±0.32
BJD-2 450 000
9.504
±0.024
10−5 M ¯
Stellar Mass M
0.398
±0.007
M¯
Minimum companion mass mmin
26.85
±0.30
MJup
Min. semi-major axis of companion orbit a
0.421
±0.010
AU
Critical inclinationa icrit
20.4
Probability for i < icrit
6.3%
Time of periastron Tp
Mass function f(m)
Inferred parameters
a
degree
for m = 0.08 M ¯
the case that the orientation of the system is such that one only sees and measures the
minor axis, 2 ∗ b1 of the orbit. But the true effect is possibly much larger. For an object
at the brown dwarf/star border the full minor axis would extend 11.5 mas or 0.42 px
and the full major axis 12.1 mas, as the orbit is not very eccentric. At the HIPPARCOS
derived upper limit for the mass of 112 MJup , it would be 2 ∗ b1 = 15.4 mas or 0.57
px. This is one of the rare cases where a spectroscopic star-substellar compaion system
comes into reach for astrometric observations.
For astrometric measurements a reference star, preferably close to the observed star is
needed. By chance GJ 1046 is located at ∼ 3000 separation from a suitable reference
star, see Fig. 3.3 and 3.2. This reference star, 2MASS 02190953 -3646596, has V = 14.33
and K = 13.52 (colors taken from the 2MASS/SIMBAD3 4 catalogues) which makes it
from its color V-K = 0.81 an F2 star with an effective temperature of ∼ 6750 Kelvin
(Tokunaga, 2000). Assuming the star to be a main sequence star, one can infer an
absolute magnitude in the visual of MV = 3.7 mag from theoretical ¡isochrones
(Marigo
¢
et al., 2008) and use the distance modulus m − M = −5 + 5 log r[pc] to estimate
a distance for the reference star of ∼ 1337 pc. At this distance the star would have
a parallax movement of only ∼ 0.75 mas. Also interstellar reddening occurs at such
distances, which makes the color of the star redder, so in reality it is even bluer and
therefore further away. So I do not expect a strong influence on the relative parallax
3
4
2MASS: Skrutskie et al. (2006)
SIMBAD: http://simbad.u-strasbg.fr/simbad/
3.2 THE REFERENCE FIELD
33
Figure 3.1: RV time series of GJ 1046 from Kürster et al. (2008). The solid line corresponds
to the Keplerian solution with a period of 169 d. Note that the average measurement error is 3.63 ms−1 , much smaller than the plot symbols. Additional
RV measurements with ESO/FEROS from the last years2 fit very well with this
orbit, showing that the signal is indeeed due to a companion orbiting the star.
between the two stars compared with the parallax of GJ 1046 alone in my measurements
within the aimed precision. Also the proper motion of the reference star is not expected
to be very high and I therefore use the HIPPARCOS values for the proper motion and
parallax of GJ 1046 as a good first estimate for the results in the fit later.
3.2
The Reference Field
Adaptive optics corrections during an observation are not constant. It is a dynamical
process, whose performance depends on the atmospheric conditions during the observation and changes of these conditions. Also the telescope focus may have changed
between two observing epochs, inducing a slightly changed platescale. I rotated the
FoV with the derotator, to fit the star asterism onto the detector, but this rotation
only has a finite accuracy. To check and calibrate for such effects, I observed a reference field in the rim of the globular cluster 47 Tucanae. This field contains more stars
than our target field and is observed very close in time to the target field. It is used to
measure the change in platescale between the different epochs. I do not need to know
the absolute platescale, but its change between the single epochs must be determined
to attain subpixel accuracy. Also the accuracy of the rotation of the detector was monitored with the reference field, to adopt a reasonable error for the rotation angle to our
data in the target field. To correct for the uncertainty in the rotation is not possible,
34
CHAPTER 3. OBSERVATIONS AND DATA REDUCTION
Figure 3.2: Time series of the movement of GJ 1046 on the sky over nearly 30 years. Due to
its high proper motion it passes the star which now is used as the reference star
in this work. The first, third and fourth image is taken from the SuperCosmos
Sky Survey (SSS), the second one from the ESO archive and the last image is
one of my own observations with NACO.
because the de-rotator is turned back to the zero position when moving the telescope
to the reference field and a fixation of the rotated instrument to the angle of the target
field was not possible in service mode, either.
I chose the reference field in the old globular cluster 47 Tucanae, because of its large
distance of 4.0 ± 0.35 kpc (McLaughlin et al., 2006) and accordingly with it the small
intrinsic movement of the single stars in the field. The velocity dispersion of the inner
parts of the cluster is 0.609 mas in the plane of the sky and the dispersion in the outer
parts being slightly smaller (McLaughlin et al., 2006). The reference field contains
three bright stars and several fainter ones suitable to check the image scale and the
field rotation, see Fig. 3.3, right.
As far as possible I checked the cluster membership of the stars in the field. But for
some, especially the faint ones, it was impossible as no 2MASS magnitudes exist, so that
I could not confirm their membership via the color-magnitude diagram. McLaughlin
et al. observed 47 Tuc with the Hubble Space Telescope (HST) and calculated proper
motions and stellar dynamics for the stars in the core of the cluster (McLaughlin et al.,
2006). Unfortunately my field lies just outside their radius around the core, where they
obtained their high precision measurements. However, they also observed the outer
part of the cluster where our field lies and did not mark any of the stars in their tables,
meaning they have not measured an uncommon high velocity compared to the mean
cluster motion for these stars. Also, comparing the positions of the single stars in the
different epochs in our NACO images did not show any unusual or large motion of one
of the stars in one direction. I therefore assume all the stars in the observed reference
field to be cluster members with common proper motion.
The asterism in the reference field was chosen to be similar to the configuration of
the AO and reference star in our target field plus additional stars for computing the
necessary field distortions between the single epochs. For that I had to rotate the FoV
by 42◦ anticlockwise. Altogether I used 11 stars for the final fit of x and y-shift, -scale
and rotation between the epochs.
3.3 ADAPTIVE OPTICS OBSERVATIONS OF GJ 1046
35
Figure 3.3: NACO image of GJ 1046 (left) and the reference field in the globular cluster
47 Tucanae (right) from July 2008. GJ 1046 is the bright star in the upper
left corner, the reference star is located in the lower right corner. Also visible
in the frames are ghosts produced by the bright star (elongated features, see
Chap. 3.3).
3.3
Adaptive Optics Observations of GJ 1046
The adaptive optics observations of GJ 1046 analyzed here were carried out with the
adaptive optics instrument NAos COnica (NACO) at the Very Large Telescope (VLT)
of the European Southern Observatory (ESO) with the UT4 Yepun telescope. All observations were executed in Service Mode within a monitoring program from July 2008
till October 2009 to derive the true mass of the companion via astrometric measurements. Altogether I obtained 10 epochs, where the first nine are in roughly three week
intervals between July and December 2008 and the last epoch was obtained end of
September 2009, see table 3.2 for an overview of the observations. The last row is an
indicator for the quality of the data: A = good, B = mostly within specifications, C =
outside specifications; with the observing specifications of: seeing better than 0.800 and
airmass lower than 1.6. In the target field, GJ 1046 itself was used as the AO guide star,
in the reference field the brightest star in the upper left corner was used for wavefront
sensing. The target star is not visible from the Paranal Observatory between February
and June, or only at very high airmass. But as the orbital period of the companion is
169 days, I covered a full orbit between July and December 2008 (epochs 1-9). This
was very important to better distinguish the parallactic motion from the half year orbital motion. It was planed to obtain two more epochs in the period between July and
September 2009 to better constrain the orbit, but I only got the last epoch in October
2009, which was still important to extend the time baseline for constraining the relative
proper motion between GJ 1046 and the reference star. In Fig. 3.4 the observations
are shown distributed over one year in the left panel and over one orbital period in
the right panel. The plotted ellipses are simulations of the motion of GJ 1046 with
36
CHAPTER 3. OBSERVATIONS AND DATA REDUCTION
Epoch
Date
Target Field
Ref. Field
# jitter
# images per
DIT
NDIT
DIT
NDIT
position
position
Quality
1
03/07/08
0.9
110
0.4
10
5
5
A
2
02/08/08
0.9
110
0.4
10
5
5
B
3
22/08/08
0.9
110
0.4
10
5
5
C
4
24/08/08
0.9
110
0.4
10
5
5
A
5
27/09/08
0.9
110
0.4
10
5
5
B
6
30/10/08
0.9
122
0.4
10
5
5
A
7
18/11/08
0.9
122
0.4
10
5
5
B
8
07/12/08
0.9
122
0.4
10
5
5
B
9
27/12/08
0.9
122
0.4
10
5
5
B
10
30/09/09
0.9
120
0.4
15
5
5
B
Table 3.2: Overview over the observations, with exposure time, number of jitter positions
and quality indicator of the obtained data.
parallactic motion, but without proper motion: in the left panel for inclination i = 30◦
and ascending node Ω = 150◦ (dashed line), i = 45◦ , Ω = 60◦ (solid line) and pure
parallax movement without any orbital motion (dotted line). The red squares represent
the times of observations overplotted over the simulated motion with i = 45◦ , Ω = 60◦ ,
with the number denoting the corresponding epoch. This plot shows how important it
is to have a proper sampling of observations to distinguish the orbital motion from the
parallax movement. The right panel shows the times of observations overplotted over a
simulated orbit with i = 45◦ , Ω = 60◦ and proper motion and parallax subtracted. As
one can see, the full orbit is coverd by the observations. The open blue square marks
the periastron passage, T0 = 54745.41.
The DIT values were chosen, such that the peak counts for the bright star in each field
are ≤ 80% of the detector linearity limit of 110, 000 e− for a small seeing of 0.400 . For the
star in the field with the highest separation from the AO guide star I estimated a signalto-noise (S/N) value integrated over the stellar disk and a formal astrometric precision
using photon statistics (σ/(S/N )) for a seeing of 0.800 . All the performance estimates
were made with the NAOS preparation software and the NACO ETC (Exposure Time
Calculator). In Tab. 3.3 the estimated performance for an observation in the middle of
the epochs is listed.
The observations were made with a narrow-band filter centered on 2.12 µm and with
a FWHM of 0.022 µm to minimize the effects of differential atmospheric dispersion.
The reference field in 47 Tuc was observed each epoch immediately before the science
field. I observed the reference field before the target field and not after it, for reasons
of visibility of the two fields. 47 Tuc culminates roughly half an hour before GJ 1046 at
3.3 ADAPTIVE OPTICS OBSERVATIONS OF GJ 1046
37
Figure 3.4: GJ 1046 observations distributed over one year in the left panel and over one
orbital period in the right panel. In the left panel, the ellipses are simulations
with $ = 71.56 and i = 30◦ , Ω = 150◦ (dashed line), i = 45◦ , Ω = 60◦ (solid
line) and pure parallax movement without any orbital motion (dotted line),
proper motion has been subtracted. The right panel shows a simulation of an
orbit with i = 45◦ , Ω = 60◦ and parallax and proper motion subtracted. The red
squares represent the times of observations overplotted on the simulated orbits,
with the number denoting the corresponding epoch. The open blue square marks
the periastron passage, T0 = 54745.41.
the location of the Paranal Observatory, hence the conditions were best for both fields
when observed in this order.
As one can see in Fig. 3.3 left, the target star GJ 1046 and the reference star just fit
in the FoV of the S27 camera of the NACO instrument. GJ 1046 lies in the upper left
corner of the detector and the astrometric reference star in the lower right corner when
the detector is rotated by 6◦ anticlockwise. I chose this camera because of its, for the
purpose of this work, suitable pixel-scale of 27 mas/px. I aimed for a precision of 0.5
mas and with the presumption of an accuracy of 1/50 pixel this camera was the one
best suited of the three offered ones, as the FoV of the S13 camera is too small.
To avoid a badpixel coincident at the pixel positions of the stars and to better calculate the sky-background, a jitter pattern is necessary for observations in the infrared.
Preferable is a pattern with an offset of several arcseconds in different directions, which
is difficult with this arrangement of the stars on the detector.
Another problem occurred, because of the brightness of the target star. Bright stars
are known to produce a number of electronic and optical ghost features on the NACO
detector, depending on their position on the detector. When the position of a bright star
is (xs, ys), the electronic ghosts5 appear approximately at the positions (1024 − xs, ys),
(xs, 1024−ys) and (1024−xs, 1024−ys). The last one of these would appear very close
to our faint reference star and could therefore influence the positional measurement.
An optical ghost which looks like a set of concentric rings may also appear, but I did
not see that in the data.
Special care has been taken to calculate a jitter pattern which made sure both stars are
always within the FoV and at least 5 sigma afar from the detector edges or any ghost.
Because the whole configuration plus the additional jitter pattern brings the two stars
5
http://www.eso.org/sci/facilities/paranal/instruments/naco/doc/
38
CHAPTER 3. OBSERVATIONS AND DATA REDUCTION
Strehl[%]a
Field
0.4
00
00
FWHM[mas]a
00
00
Encircled energy[%]a
0.6
00
00
0.8
0.4
0.6
0.8
0.4
00
0.6
00
00
S/N
0.8
@ 0.8
∆
00
[mas]
GJ1046
53.1
47.8
41.1
72
72
73
59.4
54.6
48.6
-
-
ref
39.7
21.3
7.6
76
87
132
48.6
31.7
16.0
142
0.39
47 Tuc
54.4
49.9
44.4
71
72
73
60.6
56.5
51.7
-
-
field
45.5
30.1
14.6
74
79
94
53.3
39.6
23.7
351
0.11
a
@ indicated seeing
Table 3.3: Performance estimates for the middle epoch observation. Calculated for the AO
star and the star furthest away from it. The last column lists the astrometric
precision ∆ of the fainter star which is 30.1400 separated from GJ 1046 in the
target field and 21.2800 separated from the AO star in the reference field 47 Tuc.
All the performance estimates were made with the NAOS preparation software
and the NACO ETC.
close to the edges of the detector, I had to make sure that the reference star was always
located on the very same pixel at the beginning of the observations, so the calculated
jitter pattern assures a successful observation. In Table 3.4 the position of GJ 1046,
the reference star and the closest ghost are listed together with the final jitter pattern,
which is shown in Fig. 3.5. I had to differ the last jitter point in the observations of the
reference field due to the different positions of the stars and the ghosts of the bright
star in this field.
3.4
Data Reduction
I did not use the NACO-pipeline products for these astrometric measurements. Instead
I reduced the data on my own. The data reduction is the same for the target and the
reference field. So I only describe the principle chain here and name differences directly
when they occur.
rel. jitter offset
00
x[ ]
00
y[ ]
GJ 1046
Ghosts
Ref. star
Dist. to
Dist. to
x [px]
y [px]
x [px]
y [px]
x [px]
y [px]
ref star [px]
ref star [σ]
0
0
156
854
868
170
917
65
115.8
9.3
1
0.5
193
873
831
151
954
83
140.1
11.2
0
2
193
946
831
78
954
157
145.6
11.7
-1
0.5
156
965
868
59
917
175
125.5
10.0
0
-1.5
119
946
905
78
880
157
82.4
6.6
Table 3.4: Positions of the bright target star, the ghost it is producing closest to the reference
star and the distance of the ghost to the reference star.
3.4 DATA REDUCTION
39
47 Tuc
GJ 1046
1”
4
4
5
3
3
5
2
1
2
1
Figure 3.5: Jitter pattern for the target and reference field.
First I created a badpixel-mask using the badpixel routine included in the eclipse data
reduction package6 . This routine uses the flatfield images to create the badpixel mask.
If possible and available I used the badpixel masks created by ESO from sky-flats taken
in the same night and in the same filter as my observations. But this was only the case
for two observations. In the other cases the dome-flats were used. The mask was then
combined with the hotpixel mask, created by ESO during the pipeline data reduction.
The so derived strange-pixel mask is then used to correct the flaged pixels with the
median of their 8 nearest neighbors in the science frames as well as the dark current
frames.
Then the dark frames were multiplied by the gain factor of the S27 camera detector,
which is 11 e− /ADU. Masterdarks were created by median combining darks with the
same exposure times as the target and reference frames respectively.
Also the dome-flats are multiplied by the gain. The masterdark for the target field and
the reference field are then subtracted for the corresponding data reduction. Flats with
lamps on and flats with lamps off were median combined, respectively, and the off-flat
was subtracted from the on-flat. Finally the obtained flat field image is normalized by
dividing it through its mean value.
Now I have everything to reduce the science data following the standard technique.
The single frames are multiplied by the gain, masterdark subtracted and divided by
the normalized flat field. Finally the frames are corrected for the strange-pixels.
3.4.1
Sky Subtraction
I investigated the subtraction of the sky-background in different ways. As I do not have
additional sky images I created a sky frame from median combining all science frames
in the first step. But this left me with some ’holes’ close to the stars after subtracting
the sky frame from the science frames. This is due to the small jitter pattern I used
6
http://www.eso.org/sci/data-processing/software/eclipse/
40
CHAPTER 3. OBSERVATIONS AND DATA REDUCTION
during the observations and the fact that I only have 5 different jitter positions. A
bigger and therefore better pattern was not possible, see Chap. 3.3. Light from the
wings of the stars is not averaged out and produces somewhat higher values in the sky
frame, which then appear as holes in the sky subtracted science frames.
The second step was to remove the influence of the stars by using κ − σ clipping. If the
mean of a single science frame is significantly higher than the median value of the sky
created before in step one, pixels in the science frame with values higher or lower than
a certain range around the mean, the stars, are substituted by the mean of the pixels
left after the κ − σ clipping. These frames, cleaned for stars, are then used to calculate
the sky by median combination of them.
I tested this procedure for different ranges around the mean value, which define the
pixels which are substituted. The holes indeed were less pronounced, but they also
were more irregular, which introduced a strange pattern into the wings of the PSF of
the bright star. This degrades the possibility to measure accurately the position of the
stars and I decided not to create the sky this way.
Another test was made by creating the sky only by median combining an inner part of
the science frames, where there are no stars. But this did not lead to a huge improvement in the sky subtraction either.
I decided finally to not subtract the sky background at all, as it is not very high and
does not show any slope over the frames. It can be therefore assumed as a local constant
background which should not disturb the astrometric measurements.
3.4.2
50 Hz Noise
The reduced frames showed a strong noise pattern along the rows. This phenomenon
is known as 50 Hz noise and was found to be caused by the fans in the front end
electronics of the Infrared Array Control Electronics (IRACE). One can decrease the
effect by subtracting the median of each row from the very same row. I created a frame
with the same size as the science frames in which the values in the rows have the median
value of the corresponding science row. I created the median value not over the whole
row, but detector quadrant wide. With this I took care of the fact, that the NACO
detector of the S27 camera is read out quadrant wise.
3.4.3
Shift and Add
Now the frames are fully reduced. The frames obtained on the same jitter position are
added. This is possible because the telescope was not moved between the single frames
of one jitter position. I am left with five frames per epoch and field. These frames are
added by simple shift and add using the jitter routine (Devillard, 1999). So I have one
frame per target and reference field for every epoch. I also tested shifting and adding
all single frames, but this gave no better result than adding the five already stacked
frames from the different jitter positions.
Chapter 4
Analysis and Astrometric
Corrections
Several corrections have to be applied to the measured positions of the stars before
fitting the astrometric orbit. These include converting pixel coordinates to celestial
coordinates, differential refraction, differential aberration and change in plate-scale and
detector rotation. Logically, these corrections are executed in the inverse order as they
appear. The detector distortions have to be corrected as first step in principle, as it is
the last effect which results in a displacement in position of the stars on the detector.
But to avoid cross-talk between this effect and displacements from aberration and
refraction, I corrected for the differential refraction and aberration before. First the
differential refraction has to be corrected as it deflects the light rays after the aberration
already occurred, then the aberration is corrected. In an iterative way the correction
of the detector distortion measured as third step, is applied to the originally measured
detector positions of the stars before correction for refraction and aberration. After
that, the correction for differential refraction and differential aberration is performed
again. Fig. 4 shows the stepwise and iterative corrections which are described in the
following sections in detail.
4.1
Position Measurements
The positions of the stars were measured by fitting a Moffat-function to the individual
stars using the non-linear least square fitting package MPFIT2DPEAK, written in
the IDL language and provided by Craig Markwardt (Markwardt, 2009). The Moffat
function is a modified Lorentzian with a variable power law index β (Moffat, 1969). It
better represents the form of a PSF corrected by AO than a simple Gaussian-function,
but the Gaussian-function is contained in the Moffat-function as a limiting case with
β → ∞:
#−β
"µ
¶
µ
¶
y − y0 2
x − x0 2
+
+1
(4.1)
f (x, y) = c + I0
ρx
ρy
c is a constant, x0 , y0 are the center of the function and ρx , ρy define the FWHM of
the PSF: F W HMx = ρx (2 β1 − 1)1/2 . The Moffat-function has two advantages over the
41
42
CHAPTER 4. ANALYSIS AND ASTROMETRIC CORRECTIONS
Position
measurement
Gnomonic
transformation
Differential refraction
correction
Differential aberration
correction
First
iteration
Platescale change
determination
Second
iteration
Orbit fit
Figure 4.1: Stepwise description of the corrections applied to the measures pixel coordinates
of the stars.
Gaussian. It is numerically well behaved in the treatment of narrow PSFs, because of
the use of polynomials instead of exponential expressions. But it also allows the wings
of the PSF to be fitted. This makes it a better fitting function for AO corrected PSFs.
For the PSF-fit, the stars in the stacked frames were marked and a small box of 15×15 px
(407×407 mas) was cut around them. After finding the px with the maximum, a bigger
box with 31×31 px (842×842 mas) was cut around the star. Then the Moffat was fitted
to the PSF. I also tried to just fit the Center of Weight (CoW), but the fitted CoW of
the distribution is dependent on the box size, whereas the fitted Moffat peak was very
stable for all kinds of box-sizes. I therefore decided to fit a rotated Moffat-function to
determine the peak position of the PSFs.
A phenomenon in adaptive optics images which occurs for sources which are not the
guide star, is known as radial breathing. The PSFs of the objects are elongated in the
direction to the guide star, but not in the direction perpendicular to it. The effect gets
stronger the farther away a source is from the guide star, so the shapes of the PSFs get
more and more elliptical. This is important to know for photometry, but should not
influence the center of the distribution.
The fit routine also outputs values for the Full Width at Half Maximum (FWHM).
I could have used this for an estimate of the positional error, if I assumed photon
statistics. But to also cover the error, which may arise from the fit itself, I adopted
another method for the positional uncertainty.
4.2 ASTROMETRY WITH FITS-HEADER KEYWORDS
4.1.1
43
Positional Error Estimate - Bootstrapping
To estimate the positional error resulting from the fit, I re-sampled the intensity distribution in the cut boxes, PSF plus background, 100 times with the bootstrap method
and fitted each single realization again with a Moffat-function. The bootstrap method
is a general technique for estimating for example standard errors for estimators and
was first introduced by Bradley Efron (Efron, 1979). I numbered all photons in each
PSF distribution and generated the same amount of random numbers. The random
numbers defined which of the photons was picked and put at the same position in the
re-sampled PSF. But after that, the photon is 'put back ', so it could be picked up again.
To do so, one needs to know the total number of photons, Nph , in the box. To get that
number, one has to take into account the gain-factor of the detector, (11 e− /ADU for
NACO), and the fact that the images are averaged twice. First during the acquisition,
when N frames with the exposure time DIT are averaged, and then when the frames
are stacked to obtain the final frame (see Chaps. 2.3 and 3.4.3). Because of the massive
rising computational amount when multiplying the averaged image by NDIT, the number of photons is a lot larger, and because I did not want to underestimate the error,
I decided to stay conservative and worked on the averaged image without multiplying
by NDIT. Only the gain and the average factor from adding the single frames is taken
into account. In this way I generated the re-sampled PSFs by executing the pick and
put-back procedure as many times as there were photons in the original distribution.
This was done 100 times for each star in the target field as well as in the reference field.
The standard deviations in x and y of the 100 fits from the mean position are used as
an estimate for the errors.
To check whether the obtained fitting errors are reasonable, I also fitted a rotated Gaussian distribution to the same PSFs in the boxes and calculated the positional error if
we only assume photon statistics, dx = √σx . The errors are of the same order as
Nph
the ones from the bootstrap re-sampling, but due to the fact that a Gaussian does not
represent the form of the AO corrected PSFs very well and that I wanted to include
the error contribution of the fit, I used the errors from the bootstrap re-sampling as
the positional uncertainties. In Table 4.1 the errors from the bootstrapping and the
Gaussian approximation are listed.
Additionally to this error one has to take into account the errors originating from the
differential refraction, differential aberration and the plate-scale and rotational errors.
These contributions to the error budget are described in the following sections.
4.2
4.2.1
Astrometry with FITS-Header Keywords
World Coordinates in FITS
World coordinates are coordinates that serve to locate a measurement, as for example frequency, wavelength or longitude and latitude, in a multidimensional parameter
space. The representation of world coordinates in the Flexible Image Transport System (FITS), which is used by all observatories since the General Assembly of the IAU
(resolution R11), was first introduced by Wells et al. (1981). This initial description
44
CHAPTER 4. ANALYSIS AND ASTROMETRIC CORRECTIONS
Bootstrap error
AO star
GJ 1046
ref star
AO star
47 Tuc
ref star
Gauss error
x
y
x
y
[px]
2.62 · 10−4
2.48 · 10−4
2.15 · 10−4
2.02 · 10−4
[mas]
7.11 · 10−3
6.73 · 10−3
5.84 · 10−3
5.48 · 10−3
[px]
0.70 · 10−2
0.73 · 10−2
1.64 · 10−2
1.61 · 10−2
[mas]
0.19
0.70
0.45
0.44
[px]
5.48 · 10−4
5.96 · 10−4
3.70 · 10−4
3.47 · 10−4
[mas]
0.01
0.02
0.01
9.42 · 10−3
[px]
1.63 · 10−2
1.60 · 10−2
1.31 · 10−2
1.14 · 10−2
[mas]
0.44
0.43
0.36
0.31
Table 4.1: Positional error from the PSF fit calculated with bootstrap re-sampling and
simple photon statistics. In the 47 Tuc reference field, the star furthest away in
the opposite corner of the detector is used as the reference star (ref star).
was very simple and Greisen and Calabretta (2002) later described it in more detail
with more possible extensions. Keywords in the FITS header are used to describe the
parameters necessary to convert the x and y coordinates from the detector into the
world coordinates. Each axis of the image has a certain coordinate type and a reference point, for which the coordinate value, the pixel coordinate and an increment are
given. The basic keywords are:
CRVALn
coordinate value at reference point
CRPIXn
pixel coordinate at reference point
CDELTn
coordinate increment at reference point
CTYPEn
axis type
To convert pixel coordinates to world coordinates a multi-step process is needed, whose
principle steps are shown in Fig. 4.2. The first step is a linear transformation via matrix
multiplication from the pixel coordinates pj to intermediate pixel coordinates qi :
qi =
N
X
mij (pj − rj )
(4.2)
j=1
rj are the pixel coordinates of the reference point, given by the CRPIXj elements. In
the following notation the index j refers to the pixel axis and i to the world coordinate
axis. The mij matrix is a non-singular square matrix with the dimension of N ×
N . N is given by the keyword value of NAXIS, which gives the dimension of the
data array, but not necessarily that of the world coordinates. If the dimensions of
the World Coordinate System (WCS) are different, the keyword WCSAXES is used,
which then gives the maximum value of the index of any WCS keyword. The resulting
4.2 ASTROMETRY WITH FITS-HEADER KEYWORDS
Pixel
coordinates
45
pj
linear transformation
matrix rotates, skews,
and (optionally) scales
{
CRPIX j
PCi_j
or
CDi_j
Intermediate pixel
coordinates
rescale to physical
units
mij
qi
{
CDELTi
Intermediate world
coordinates
coordinate computation
per agreement
rj
si
xi
{
CTYPEi
CRVALi
PVi_m
World
coordinates
Figure 4.2: Conversion of pixel coordinates to world coordinates (after Calabretta and
Greisen (2002)).
intermediate pixel coordinates qi are offsets from the reference point along axes that
are coincident with those of the intermediate world coordinates xi . The transformation
to the corresponding xi is then a simple scaling:
xi = si qi
(4.3)
There are two formalisms that describe the transformation matrix in FITS. One is the
PCi j, where the matrix elements of mij are given by the PCi j header keywords and
si by CDELTi. If PCi j and CDELTi are not given in the FITS header, the second
formalism is used. This one combines Eqs. 4.2 and 4.3 to :
N
X
xi =
(si mij )(pj − rj )
(4.4)
j=1
with the keywords CDi j = si mij . This formalism omits the second step on the transformations.
The last step depicted in Fig. 4.2 from the intermediate world coordinates to the world
coordinates applies a non-linear transformation, depending on the world coordinate
system one is aiming at.
46
4.2.2
CHAPTER 4. ANALYSIS AND ASTROMETRIC CORRECTIONS
Celestial Coordinates in FITS
For the transformation to celestial coordinates, the intermediate world coordinates are
interpreted as Cartesian coordinates in the plane of the projection. The final step is
divided into two steps which are a spherical projection defined in terms of a convenient
coordinate system, the native spherical coordinates, and a spherical rotation of these
coordinates to finally obtain the required celestial coordinates (Calabretta and Greisen,
2002). Depending on the kind of projection, the equations for the transformation to
the native longitude and latitude (φ, θ) are specified in the keyword CTYPEi. There is
a wide variety of projecting a sphere onto a plane and vice versa. The one used in data
set of this work is the so-called gnomonic projection. It is described in more detail in
Sec. 4.2.3. The values of the keywords are:
CTYPE1
= ’RA - - - TAN’
CTYPE2
= ’DEC - - TAN’
The last three characters define the projection type and the leftmost four characters
are used to identify the celestial coordinate system. In this case a gnomonic projection
(4.2.3) and equatorial coordinates.
For the last step, the spherical rotation from native coordinates to celestial coordinates,
one needs three Euler angles which specify the rotation. The celestial coordinates for
the reference point, (α0 , δ0 ), specified in the header via the CRVALi keyword, are associated with a native coordinate pair (φ0 , θ0 ), which are defined explicitly for each
projection. The difference of these native coordinates for different projections is due
to the fact that the projections diverge at different points. The Mercator projection
diverges for example at the native pole, while the gnomonic one diverges at the equator. Therefore they cannot have the reference point at these points, because that would
mean infinitive values for CRVALi. The projection equations are then constructed in
a way so that (φ0 , θ0 ) transform to the reference point (x, y).
There are several other FITS keywords which define the necessary astrometry:
RADECSYS
defines the reference frame, e. g. FK5
EQUINOX
defines the coordinates epoch, e. g. 2000.0
MJD-OBS
defines the Modified Julian Date of the observation start
The combination of CTYPEi, RADECSYS and EQUINOX define the coordinate system of the CRVALi and of the celestial coordinates resulting from the transformations.
4.2.3
Transformation from xy-Coordinates into RA/DEC
To convert the pixel values measured for the peak positions of the stars in the frames to
celestial coordinates, I used the astrometry in the FITS header. The projection given
by the keyword CTYPEi is a special form of a zenithal projection, the gnomonic projection. The name deduces from the Greek word Gnomon (γν óµoν), which stands for
a shadow-stick used as an astronomical instrument. The shadow cast by the tip of the
stick was already used to measure the time with a sundial in antiquity. The gnomonic
4.3 ASTROMETRIC CORRECTIONS
47
projection is the oldest map projection, developed by Thales in the 6th century BC.
The gnomonic projection displays all great circles on a sphere as straight lines. The
surface of the sphere is projected from its center, hence perpendicular to the surface,
onto a tangential plane and the least distortions occur at the tangent point. In Astronomy, the Earth’s radius is small compared to the distance to the stars and can be
neglected, so the observatory can be seen as being in the center of the projection. This
directly implies the negligence of the diurnal parallax (see Chap. 4.3.10).
A problem occurred after stacking the images with the jitter routine, as described in
Chap. 3.4.3. The image obtained is slightly larger in x- and y-direction than the single
images before. The new size of the image is updated in the FITS header by the jitter
routine, but not the resulting coordinate shift of the reference point. Additionally, no
information is given on how much the zero-point of the array is shifted. To compute
the new pixel coordinates of the reference point in the stacked image, I had to execute
some more steps.
As the star I used as a guide star for the AO system is quite bright and also very well
corrected in the single frames I could measure its pixel coordinates very accurately. I
measured its position in the first stacked image of the first jitter position in the same
way as described in Chap. 4.1. Then I calculated the distance to the given coordinates
for the reference point. After measuring all positions in the final image, I recalculated
the pixel coordinates of the reference point by using the beforehand calculated distance
to the AO guide star and updated the CRPIXi values in the FITS header.
The transformation of the coordinates was then performed with the IDL routine xyad.pro
from the IDL astrolib1 which uses the formalism described in detail in paper II of the
series of papers describing coordinates in the FITS formalism (Calabretta and Greisen,
2002).
4.3
Astrometric Corrections
The exact position where an object appears on the sky does not only depend on the
coordinates of the observed object, but also on various effects which are connected to
the relative velocity of the observer, i.e. the aberration, and the atmosphere of the
Earth, i.e. the atmospheric refraction.
4.3.1
Theory of Atmospheric Refraction
The atmospheric refraction decreases the true zenith distance of an object. Light,
passing a surface that separates two layers with different refractive indices n and n + dn
is refracted in a way described by Snell’s Law. Let η be the angle of incident in a
medium with refractive index n. The angle of refraction in a medium with refractive
index n + dn is then η + dη following
(n + dn) sin (η + dη) = n sin η
1
http://idlastro.gsfc.nasa.gov/
(4.5)
48
CHAPTER 4. ANALYSIS AND ASTROMETRIC CORRECTIONS
n -dn
n
zenith, Z
n0
O
L
z
z0
dr
r0
r
f
E
X
df
h
Y
M
Y
h + dh
Figure 4.3: Refraction in the atmosphere. For more details see text.
The total bending of the light can be expressed as
R = zt − z0
(4.6)
where R is the total refraction angle experienced by the light ray, zt the true zenith
distance in degrees and z0 the apparent observed zenith distance. In the infrared
wavelength regime, the refraction can reach tens of arcseconds, where the effect is
larger in the J-Band than in the K-Band.
In theory one needs to know n at all points of the light-path through the atmosphere
to determine R. In practice this is normally not possible and one has to represent the
atmosphere by a model which leaves R only as a function of atmospheric conditions.
Using a spherical model for the atmosphere means assuming iso-refractive index spheres
around the center of the Earth.
In figure 4.3, r0 is the geocentric distance of the observer O from the center of the Earth
E, n0 the refractive index at ground level and z0 the apparent distance of an celestial
object to the local zenith Z.
If one considers a small layer with width dr and the refractive index n, then the range
X Y is the linear path of the light ray coming from the observed object through this
layer. At the point Y one can apply Snell’s law to the incoming light ray which comes
from a medium with the refractive index n − dn. η + dη is then the incident angle in
the point Y and Ψ the refractive angle. This leads to:
n sin Ψ = (n − dn) sin (η + dη)
(4.7)
Naming the angle between the two zenith directions of the observer and the one in
point X , φ, the increment of this angle towards the direction M, dφ and the radius
vector from E to X , r, one can write:
r sin η = (r + dr) sin Ψ
(4.8)
4.3 ASTROMETRIC CORRECTIONS
49
This equation comes from the sine relation in the triangle EX Y.
If one now multiplies respectively the right and left sides of 4.7 and 4.8 one gets
nr sin η = (r + dr)(n − dn) sin (η + dη)
(4.9)
Again with Snell’s law this proves nr sin η to be invariant. Transformed to the observer’s
position and together with Equ. 4.7 one can write the important relation:
nr sin Ψ = n0 r0 sin z0
(4.10)
If one looks at the triangle LEX one can see that z = φ + Ψ and from the triangle
X MY one gets tan Ψ = rdφ
dr . Combining these two equations one gets
tan Ψ
dr
r
Now one first differentiates 4.10, which yields
dz = dΨ +
nr cos ΨdΨ + r sin Ψdn + n sin Φdr = 0
(4.11)
(4.12)
or written in a different way
¶
µ
dr
tan Ψ = −r sin Ψdn
nr cos Ψ dΨ +
r
(4.13)
Together with Equ. 4.11 one has:
dn
tan Ψ
(4.14)
n
Now one can look at Equ. 4.10 in a different way and write it, with nr cos Ψ =
q
r2 n2 − r02 n20 sin2 z0 , as:
dz = −
r0 n0 sin z0
tan Ψ = q
r2 n2 − r02 n20 sin2 z0
(4.15)
This one substitutes into the previously obtained Equ. 4.14 and finally derives after
integration over dz
Z n0
dn
R = r0 n0 sin z0
(4.16)
2
2
2
n(r n − r0 n20 sin2 z0 )1/2
1
This general refraction formula requires a detailed knowledge of the refractive index
n and therefore for the whole atmosphere itself, as n depends on the wavelength λ of
the incoming light and the temperature T , pressure P and water pressure Pw of the
atmosphere. This formula is an exact description of the refractive index of the air
and can be calculated numerically if the variation of the refractive index with height,
r = r(n), is postulated. However, a simplification is possible when only modest zenith
distances are allowed. One can then expand Equ. 4.16 to (Green, 1985; Gubler and
Tytler, 1998):
R = A tan z0 + B tan3 z0
(4.17)
A and B are constants and depend on the wavelength λ, T , P and the relative humidity
H 2 . This expansion makes the calculation of the differential refraction a lot easier, see
next section.
2
H = Pw /Psat with Psat the saturation water vapor at temperature T
50
4.3.2
CHAPTER 4. ANALYSIS AND ASTROMETRIC CORRECTIONS
Differential Atmospheric Refraction
The differential atmospheric refraction between two stars, ∆R, is due to two main
effects: The different color of the stars (meaning different temperature of the stars
and therefore different effective wavelength in the used filter) and the different zenith
distance of the two stars. The color dependent effect is smaller (around 20 times) than
the effect due to the differing zenith distances for moderate zenith distance differences
of ∼ 1500 and temperature differences of ∼ 3000 Kelvin in the K, K’ and H band
(Gubler and Tytler, 1998). Assuming two stars with refraction R1 and R2 respectively,
∆R = R1 − R2 , and an apparent zenith distance difference of ∆z0 = z1 − z2 , one can
write the refraction experienced by star 2 expressed by z1 and ∆z0 and (4.17) as:
R2 (z2 ) = R2 (z1 − ∆z0 )
= A2 tan (z1 − ∆z0 ) + B2 tan3 (z1 − ∆z0 )
(4.18)
Expanding this equation around z1 to second order in ∆z0 then yields
∆R = ∆Rλ + ∆Rz
(4.19)
where ∆Rλ denotes the color dependent effect and ∆Rz the differential zenith distance
dependent effect. The two components of ∆R have the form:
∆Rλ = (A1 − A2 ) tan z1 + (B1 − B2 ) tan3 z1
¡
∆Rz = (1 + tan2 z1 ) (A2 + 3B2 tan2 z1 )∆z0
£
¤
¢
− A2 tan z1 + 3B2 (tan z1 + 2 tan3 z1 ) ∆z02
(4.20)
(4.21)
Usually the constants A and B are determined empirically. Gubler and Tytler did this
by first changing the integration variable in Equ. 4.16 from n to z which yields
Z z0
rdn/dr
R=−
dz
(4.22)
n + rdn/dr
0
Full integration of this integral for two different zenith distances yields the refraction
indices corresponding to these two values of z. A curve fit to Equ. 4.22 then gives the
two constants A and B. I am using characteristic but fixed values for the atmospheric
parameters hereafter, so the constants A and B are only dependent on the wavelength:
T0 = 278 K
P0 = 800 mbar
H0 = 10 %
One can then derive an expression for A and B in seconds of arc with only a wavelength
dependence. This was done by Gubler and Tytler (Gubler and Tytler, 1998) empirically
for the K-Band (centered on 2.2 µm):
0.26147
λ2
0.0002622385
B(λ) = −0.05083862 −
λ2
A(λ) = 45.95126 +
(4.23)
(4.24)
4.3 ASTROMETRIC CORRECTIONS
51
I used during the observations a narrow-band filter centered at 2.12µm. So we could
suppress and therefore neglect the color dependent part of the differential refraction,
∆Rλ , and only corrected for the effect due to the different zenith distances of the stars,
∆Rz . With λ = 2.12µm, the two constants then take the values A = 46.009400 and
B = −0.05089700 .
Looking at Equ. 4.21 one can see that even with the most extreme values during the
observations, ∆z0 = 17.200 and z1 = 48◦ 89, the quadratic term never exceeds ∼ 1.26 µas.
The following expression for ∆Rz is therefore a good approximation:
(1 + tan2 z1 )(A2 + 3B2 tan2 z1 )
∆z0
(4.25)
206265
The factor in the denominator comes from the fact that ∆z0 is now expressed in arcseconds instead of radians. Also, ∆Rz is now expressed in seconds of arc.
∆Rz =
As stated above, the values for A and B are calculated for fixed standard values of
atmospheric temperature, pressure and humidity, T0 , P0 , H0 . But in reality, the actual
circumstances during the observations are different. Depending on how much the actual
temperature, pressure and humidity vary from these standard values, the differential
refraction changes, too. Gubler and Tytler (1998) investigated this phenomenon and
gave values for the change in ∆R due to changes differences of temperature, pressure
and humidity form the standard values. The relation between the parameters is linear, the differential refraction goes as the inverse of the ground-level temperature and
changing the pressure by a given fraction changes ∆R by the same fraction. A change
in the humidity does not change the differential diffraction significantly. Changing the
humidity from 10% to 100% only leads to a change in the differential refraction of a
few micro-arcseconds.
While I took the change in temperature and pressure into account, I did not correct for
any change in the humidity, as it was close to the standard value of 10% anyway during
all observations. The correction factors were taken into account as follows. Taking
the values given by Gubler and Tytler (tables 3 and 4) I made a linear least-squares
approximation in one-dimension to the data, yielding values for the slope m and the
interception point with the y-axis b for the relation between temperature and diff.
refraction and pressure and diff. refraction, respectively:
∆RT = mT · ∆T + bT
(4.26)
∆RP = mP · ∆P + bP
(4.27)
Here ∆T is the difference to T0 in Kelvin, ∆P the difference to P0 in mbar and ∆R
is given in percent. The derived values for m and b are: mT = −0.363 ± 0.049,
bT = 0.086 ± 0.409, mP = 0.125 ± 0.018 and bP = 0.000 ± 0.380. After converting the
changes in ∆R from percentage to arcseconds, the final differential refraction is given
by:
∆R = ∆Rz + ∆RT + ∆RP
(4.28)
4.3.3
Correction for Differential Refraction
With the results from Equs. 4.25 and 4.28, one can start to correct the measured positions of the stars for differential refraction. As this is the last distortion before the light
52
CHAPTER 4. ANALYSIS AND ASTROMETRIC CORRECTIONS
a) DR = R 1 - R 2> 0
b) DR = R 1 - R 2< 0
Z
Z
z 0,2
z 0,2
R2
z t,2
R1
R1
z 0,1
z t,1
z 0,1
z t,1
O
R2
z t,2
Horizon
O
Figure 4.4: Differential refraction
arrives on the detector, it has to be corrected first and before the differential aberration.
The celestial coordinates given for the reference point in the FITS header are already
refraction and aberration corrected3 . So one only has to correct for the differential
part of the refraction between the reference point and the stars. This is done by first
undoing the aberration correction for the reference point (see Chapter 4.3.5), so that
one knows at which point the light rays really entered the atmosphere. With these ’new’
coordinates for the reference point, the celestial coordinates of the stars are calculated
in the way described in Chapter 4.2. Then the zenith distances z0 of the reference point
and the stars are given by:
cos z0,i = sin δi sin ϕ + cos δi cos ϕ cos HAi
(4.29)
Here ϕ denotes the local latitude of the observatory, which is for the VLT/UT4 ϕ =
−24.◦ 6270, δ the declination of the star and HA = LST − α the local hour angle of the
star. i stands for the different stars. As the zenith distance changes with time, HA
and z0 were calculated over the whole observing time of one epoch. This means for
every single frame of the field in that epoch, which leaves me with 25 (35 for epoch 1,
see Tab. 3.2) values for HA and z0 for each night. I therefore introduce another index,
j, which stands for the different frames at an individual epoch.
Now one can calculate ∆z0i,j = z0ref,j − z0i,j , and after that ∆Ri,j for each star, using
Equ. 4.28 and z1 = z0ref,j in Equ. 4.25. ∆T was in the range between 3.72 and 8.71 K,
and ∆P between -57.3 and -54.7 mbar for all epochs. I calculated ∆RT i,j and ∆RP i,j
for each frame in each epoch with the parameters mT , mP , bT and bP derived from the
linear fits. The sign of ∆z0 and therefore ∆R depends on the time the observation was
3
Communication with Claudio Mela via ESO USD help
4.3 ASTROMETRIC CORRECTIONS
53
conducted and therefore the orientation of the star asterism on the local sky relative
to the reference point. If the zenith distance z0ref of the reference point is bigger than
the zenith distance of any star z0i , then ∆z0i > 0 and also ∆Ri > 0 for this star.
If z0ref < z0i then ∆z0i , ∆Ri < 0. In Figure 4.4 a and b, the two cases are shown.
Depending on whether the observation was carried out before, after, close to or even
during the passing of the stars through the local meridian, the time-depending behavior
of ∆z0i and all the following parameters is different. In Fig. 4.5 one can see different
cases during different phases of the visibility of the stars. Plotted as the solid curve
is the difference in altitude of the two stars in the field: Alt(reference star) - Alt(GJ
1046). The dotted curve shows the visibility of the objects above the local horizon at
Paranal Observatory in mid-October. The right y-axis gives the altitude in degrees.
The vertical dashed line marks the time of the local Meridian Passage of GJ 1046. Due
to the slightly different coordinates of the two objects, they overtake each other during
the night, leading to a change in the sign of the differential refraction correction. The
reference point of the detector, to which the differential refraction is measured, lies
between the two stars. The small inlets show the configuration of the two stars at
different phases of their visibility: the left one, when the reference star is at a lower
altitude, the middle one when they are nearly at the same height above the horizon
and the right one when GJ 1046 is at a lower altitude. Also shown are the times of
observation in LST (in blue) for all individual epochs.
The true zenith distance of the stars is then:
zti,j = z0i,j + ∆Ri,j
(4.30)
Now one needs to recalculate α and δ of the stars. Here the fact that the refraction
changes the zenith distance, but not the azimuth of the stars helps. The azimuth AZ
of a given frame in a given observation for a star is:
cos AZ =
(sin δ − cos z0 sin ϕ)
sin z0 cos ϕ
(4.31)
Rearranging this equation and replacing z0 with zt lets one calculate the declination
corrected for differential refraction:
¢
¡
δcorri,j = sin−1 cos zti,j sin ϕ + sin zti,j cos ϕ cos AZi,j
(4.32)
To compute the right ascension one needs to recalculate the hour angle HA first, as
αi,j = LSTj − HAi,j :
µ
¶
−1 cos zti,j − sin δcorri,j sin ϕ
HAi,j = cos
(4.33)
cos δcorri,j cos ϕ
From this I calculated offsets from the positions in right ascension and declination (α, δ)
without the differential refraction correction:
∆αi,j = αi,j − αcorri,j
(4.34)
Again I had 25 (35) values of ∆α for each star. To get one value for correction I
calculated the mean change in position, ∆αi =
PN
j=1 ∆αi,j
,
N
N = 25 (35), same for
CHAPTER 4. ANALYSIS AND ASTROMETRIC CORRECTIONS
0.06
Time of local
Meridian passage
epoch 7
0.04
epoch 2
epoch 6
epoch 5
epoch 1
0.02
GJ 1046
Ref. star
80
epoch 9
epoch 8
60
epoch 4
epoch 10
0.00
epoch 3
40
-0.02
Altitude [deg]
Altitude ref. star - Altitude GJ 1046 [deg]
54
20
-0.04
-0.06
-5
0
2 h 19 min
5
0
10
LST [hour]
Figure 4.5: Sign of the differential refraction correction during different times of observations.
Plotted as the solid curve is the difference in altitude of the two stars in the field:
Alt(reference star) - Alt(GJ 1046). The dotted curve shows the visibility of the
objects above the local horizon at the Paranal Observatory in mid of October.
The right x-axis gives the altitude in degree. The vertical dashed line marks
the time of the local Meridian Passage of GJ 1046. The small inlets show the
orientation of the two stars at different phases of their visibility; the left one,
when the reference star is at a lower altitude, the middle one when they are
nearly at the same height above the horizon and the right on when GJ 1046 is
at a lower altitude. Also shown in blue are the times of observation in LST for
all epochs.
√
√
δ, and the corresponding error of the mean value, σαi / N , σδi / N , where σαi and
σδi are the standard deviations of the ∆αi,j and ∆δi,j . The coordinates corrected for
differential refraction are then derived by:
αcorri
= αi − ∆αi
(4.35)
δcorri
= δi − ∆δi
(4.36)
In Tab. 4.2 the corrections in right ascension and declination are listed for the two stars
in the target field for each epoch. Now one has the coordinates of the stars corrected
for differential refraction. The next step is then to correct for differential aberration.
4.3 ASTROMETRIC CORRECTIONS
GJ 1046
epoch
∆α [mas]
55
reference star
∆δ [mas]
∆α [mas]
∆δ [mas]
1
0.678 ± 0.141
−0.126 ± 0.023
−1.891 ± 0.149
0.510 ± 0.009
2
−0.006 ± 0.066
0.043 ± 0.037
−0.441 ± 0.010
0.195 ± 0.046
3
3.553 ± 0.099
−0.034 ± 0.019
−5.740 ± 0.149
0.057 ± 0.030
4
1.782 ± 0.061
−0.272 ± 0.004
−3.114 ± 0.092
0.478 ± 0.010
5
1.998 ± 0.059
−0.275 ± 0.006
−1.898 ± 0.087
0.531 ± 0.005
6
0.349 ± 0.068
−0.122 ± 0.025
−0.993 ± 0.103
0.393 ± 0.028
7
0.248 ± 0.141
2.123 ± 0.068
−0.358 ± 0.184
−2.732 ± 0.109
8
2.902 ± 0.021
1.186 ± 0.046
−4.251 ± 0.041
−1.733 ± 0.063
9
2.780 ± 0.039
1.540 ± 0.057
−3.960 ± 0.066
−2.186 ± 0.047
10
3.475 ± 0.101
−0.172 ± 0.017
−4.743 ± 0.132
0.237 ± 0.024
Table 4.2: Differential refraction corrections applied to the two stars in the target field.
Values are given for correction in right ascension and declination plus the error
arising from the change in zenith distance of the stars during the observation.
The total differential effect between the two stars is the difference between their
values.
4.3.4
Errors from Differential Refraction Correction
Correcting for differential refraction can introduce several error terms:
• The first one, which I already showed, is the error coming from neglecting the
quadratic term in Equ. 4.21. This error is even for the most extreme observation
configurations only ∼ 1.26 µas and well below the aimed precision.
• Another error source are the not perfectly known celestial coordinates of the stars
to calculate z0 . Although the given absolute coordinates may not have a precision
better than a few arcseconds, this is no problem as the error in the differential
refraction is only about ∼ 5 µas if the zenith distance z0 is only known to a precision of 0◦ 01 = 3600 (Gubler and Tytler, 1998). This is well within the pointing
accuracy of the VLT. Also, I am not aiming for absolute astrometry to obtain the
orbital motion due to the companion, but instead only need relative astrometry
to the reference star in the image.
• The biggest error term comes from the linear fit made to calculate the correction
factors due to different temperature and pressure during the observations. The
deviation of the data from a linear relation can be used to calculate errors for the
derived slope and intersection values, which then lead to errors of the correction
56
CHAPTER 4. ANALYSIS AND ASTROMETRIC CORRECTIONS
factor in ∆R of ∼ 0.5 − 1.0%. This translates into a small error of the calculated
∆αij and ∆δijq
. The errors for the coordinate offsets ∆αi and ∆δi are then
P
1
calculated by 1/ N
j=1 σi,j , where σi,j stands for the single errors of the ∆αij
and ∆δij . The errors arising from this fit are between 0.129 - 1.779 mas for ∆αi
and between 0.050 - 0.313 mas for ∆δi . The large upper limit results from one
epoch, where the temperature and pressure values were deviating a lot stronger
than at the other observing epochs. The errors of all other epochs are below
the milli-arcsecond range. Still the errors resulting from the correction due to
differing atmospheric conditions are pretty high and probably need a better model
to correct for if one wants to go down to small micro-arcsecond precision.
• Additional errors arise from the change of the zenith distance during the observation. The observations took around 45 min for the target field, which lead to
a change in zenith distance depending on the time of observation, before, close
to or after the crossing of the targets through the local meridian. The result is a
variation of z0 between 0.5◦ in the best case (epoch 7) and 16.1◦ in the worst case
(epoch 1), which translates into 0.35 − 2.29 mas peak-to-peak difference in the
correction ∆αij and 0.07 − 0.98 mas in ∆δij for the different epochs. The larger
the zenith distance the stronger the refraction, because of the higher airmass the
light has to travel through. The error from calculating the mean value for ∆αi
and ∆δi varies therefore between 21 − 184 µas and 4 − 109 µas for right ascension
and declination, respectively, see also Tab. 4.2.
This error and the error from the correction necessary due to the different temperature and pressure are taken into account by quadratic addition with the error
from the PSF-Fit.
4.3.5
Theory of Aberration
The observer’s velocity through space relative to an observed object is responsible for
a phenomenon called Bradley aberration or Stellar aberration. It was discovered by
James Bradley in 1727 (Bradley, 1727). He was trying to measure the stellar parallax
of g Draconis, in order to confirm the Copernican theory of the Solar System, as it
proves the motion of the Earth. Instead he measured an annual variation which was
not consistent with the expected parallax. The variation was strongest for stars in
the direction perpendicular to the orbital plane of the Earth. As γ Draconis passes
right through the zenith in Greenwich, where Bradley did his observations, it was by
chance a perfect target to detect stellar aberration. Bradley concluded rightly that the
displacement he saw was not due to changes in the Earth’s position, but rather due
to changes in Earth’s velocity. The aberration is caused by the relative velocity of the
source, e.g. star, and the observer and the finite speed of light during the light travel
time. The relative positions between the observer on Earth and the source change and
the light seems to be coming from a direction different from the direction from which it
was emitted. The movement of the observer can be divided into three different motions
leading to the following three effects:
• The diurnal aberration caused by the daily rotation of the Earth
4.3 ASTROMETRIC CORRECTIONS
57
• The annual aberration, due to the movement of the Earth around the barycenter
of the Solar System
• The secular aberration due to the motion of the Solar System barycenter in space
The secular aberration is a displacement due to the relative motion of the stars and
the Solar System barycenter and is equal to the proper motion of the stars multiplied
by the light time. Since this is rarely well known and the barycentric position of a star
is mostly of marginal interest, this effect is normally ignored.
The absolute effect of the annual aberration can be approximated by
Ayear ≈ kyear sin θ
(4.37)
Here k = V /c ≈ 30 km s−1 /300000 km s−1 ≈ 20.500 is the annual aberration constant
and θ is the angle between the velocity vector of the Earth and the direction of the
light coming from the star, see Fig. 4.6. The differential effect between two stars at a
separation of 3000 can then reach up to 3 mas over one year and needs to be corrected
for. The differential effect of the diurnal aberration is smaller, around 42 µas per day.
But to derive and correct the effect of the aberration precisely one needs to take the
relativistic addition of velocities into account, where the distinction between annual and
diurnal aberration is not possible. This is done by applying the Lorentz transformation
between two sets of coordinates, x, y, z, (r) and time t in a given reference frame S, and
x0 , y 0 , z 0 , (r0 ) and t0 in another frame S 0 which is moving with the constant velocity V
with respect to S. The generalized Lorentz transformation from the moving frame to
the stationary one can then be written as:
r = r0 + γVt0 + (γ − 1)
µ
¶
V · r0
0
t=γ t + 2
c
V(V · r0 )
V2
here γ = (1 − V 2 /c2 )−1/2 , is the Lorentz factor. A point r in the stationary
S has the velocity U = dr/dt. In the moving system its coordinates are r0 , t0
velocity would be U0 = dr0 /dt0 . Now one can differentiate Equ. 4.39:
µ
¶
dr0 dt0
dt0
dr0 dt0 V
dr
= 0
+γ V
+ (γ − 1) V · 0
dt
dt dt
dt
dt dt V 2
µ
µ
¶
¶
dr0
dt = γ dt0 + V · 0 dt0 /c2
dt
(4.38)
(4.39)
system
and its
(4.40)
(4.41)
And with the new notation
dt0
1
=
dt
γ(1 + V · U0 /c2 )
(4.42)
one can than write:
U=
U0 + γV + (γ − 1)(V · U0 )V/V 2
γ(1 + V · U0 /c2 )
(4.43)
This is the formula one needs to correct for aberration. Now one can apply this result
58
CHAPTER 4. ANALYSIS AND ASTROMETRIC CORRECTIONS
S
V
S
c
c
p
dq p
q
V
E
Figure 4.6: Stellar aberration
to the case of the moving Earth. The velocity of the Earth, E with respect to the
stationary stars is V. In Fig. 4.6, the unit vector p is in the direction to the geometric,
true position of an object S at time t in the stationary frame. The velocity vector of
the light coming from this object in the very same frame is then −cp. But the observer
on Earth will see the object at time t at the position S 0 whose direction is defined by
adding the velocities of the Earth and the light coming from S, cp0 = V + cp, with the
unit vector p0 . Summarizing the important notations
• velocity of light in the geometric direction: −cp
• velocity of light in the apparent direction: −cp0
• velocity of the observer on Earth: V
and applying them to the previously derived equation 4.43 one obtains:
cp =
cp0 − γV + (γ − 1)(V · cp0 )V/V 2
γ(1 − cp0 · V/c2 )
(4.44)
Dividing by c and expanding by γ/γ one gets:
p=
γ −1 p0 − V/c + (1 − γ −1 )(V · p0 )V/V 2
(1 − p0 · V/c)
(4.45)
Using the definition of γ = (1 − V 2 /c2 )−1/2 and writing
(1 − γ −1 )(1 + γ −1 ) = V2 /c2
(4.46)
one finally gets the vector in the geometric direction of the aberration corrected coordinates:
γ −1 p0 − V/c + (1 + γ −1 )(p0 · V/c)(V/c)
p=
(4.47)
(1 − p0 · V/c)
4.3 ASTROMETRIC CORRECTIONS
59
The inverse formula is:
p0 =
γ −1 p + V/c + (1 + γ −1 )(p · V/c)(V/c)
(1 + p · V/c)
(4.48)
In the following correction for aberration these last two equations are used.
4.3.6
Correction for Differential Aberration
Looking at the coordinates given for the reference point in the FITS-Header, the celestial coordinates given by the keyword CRVALi are already corrected for aberration
and refraction. So the only thing I have to take care of, is the correction for the differential effect between this reference point and the stars. The correction for differential
refraction was already done in the previous step, see Sec. 4.3.3. I did not correct for
the differential effect of the aberration directly, as this is not possible, but calculated
for each star the celestial coordinates without aberration correction and then corrected
for each star for total aberration. This directly corrects also for the differential effect.
This is done in two steps.
First I calculated the celestial coordinates of the reference point before they were corrected for aberration. So in practice, I undid the correction for aberration. This was
done the following way:
A set of three linearly independent unit-vectors, X, Y, Z was calculated from the celestial coordinates, α1 , δ1 of the reference point in the aberration corrected frame:
X = cos δ1 ∗ cos α1
Y = cos δ1 ∗ sin α1
(4.49)
Z = sin δ1
X, Y, Z describe the equatorial system, therefore X and Y lie in the equatorial plane and
X points to the vernal point with α = 0 hours and Y points to the direction of α = 6
hours. Z points to the north pole. Then the current velocity of the Earth is extracted
from the JPL DE405 ephemerides4 . This set of ephemerides is the most recent and
precise one. It includes nutations and librations and is tied to the International Celestial
Reference Frame (ICRS) through VLBI observations of the Magellan spacecraft in orbit
around Venus. The origin of the coordinate system is the barycenter of the Solar
System.
To calculate the current velocity of the observer on Earth’s surface, one needs to take
into account the position of the observer on the Earth. The observer moves eastwards,
as the Earth rotates. For points on the celestial sphere which pass the local Meridian,
the right ascension is equal to the Local Siderial Time (LST). As the direction to the
East, the direction of the movement of the observer on Earth, is perpendicular to the
Meridian, a factor of six hours has to be added to the LST. The velocity in x and y
direction then amounts to:
Vx = Vx + Vr ∗ cos (LST + 6)
Vy = Vy + Vr ∗ sin (LST + 6)
4
http://ssd.jpl.nasa.gov/, http://cow.physics.wisc.edu/∼craigm/idl/ephem.html
60
CHAPTER 4. ANALYSIS AND ASTROMETRIC CORRECTIONS
where Vr denotes the rotational velocity of the Earth at the position of the observer.
The z direction is not influenced. Taking Equ. 4.48 and splitting p into its components
X, Y, Z, we can calculate the new vector p0 = (X 0 , Y 0 , Z 0 ). After dividing X 0 , Y 0 , Z 0
with the length of the calculated vector, | p0 |= X 0 ∗ X 0 + Y 0 ∗ Y 0 + Z 0 ∗ Z 0 , one has the
now unaberrated linearly independent unit vectors X 0 , Y 0 , Z 0 for the reference point.
The declination and right ascension can then simply be computed by:
δ0 = sin−1 Z 0
Y0
(4.50)
X0
Here the coordinates α0 , δ0 are the celestial coordinates of the reference point before
aberration correction.
I updated the CRVALi values in the FITS header with these coordinates and calculated
the celestial coordinates of the other stars in the field. These coordinates are then taken
to be corrected for differential refraction afterwards as described in Sec. 4.3.3.
Now one can proceed with the correction for differential aberration. Each star’s set of
celestial coordinates is transformed into their corresponding set of linearly independent
vectors X 0 , Y 0 , Z 0 . But this time Equ. 4.47 is applied to calculate the vector p =
(X, Y, Z) in the geometric direction of the velocity of light. These Cartesian coordinates
are finally transformed with Equ. 4.50 (with only X 0 , Y 0 , Z 0 exchanged by X, Y, Z) into
α, δ of celestial coordinates now corrected for differential refraction and differential
aberration.
α0 = tan−1
4.3.7
Errors from Differential Aberration Correction
To calculate the effect of aberration on the coordinates of an object and then correct
for it, one has to know the apparent position of the object. The pointing accuracy of
VLT/NACO is at least accurate to one second of arc. To estimate the error of the correction of the differential aberration effect I looked at the effect of the total aberration.
During our observations the total amount of aberration correction was dα ∼ 1000 in
right ascension and dδ ∼ 1200 in declination. I took these values to estimate the error
in the differential effect originating from the inaccuracy of knowing the exact apparent
celestial coordinates. I changed the reference coordinates from the reference point by
α0 ± dα and δ0 ± dδ and calculated the correction again. The absolute values of the
coordinates for the stars changed by roughly the amount I added and subtracted, as
expected. But the measured distance between the stars almost did not change. The
error in the differential aberration correction, introduced by an error in the absolute
coordinates is therefore very small. It is in the range of a few micro arcseconds in both
directions of right ascension and declination.
4.3.8
Light Time Delay
Light emitted at a time T1 takes longer or shorter to reach the observer than light
emitted at an earlier time T0 :
∆T = (D(T1 ) − D(T0 ))/c
(4.51)
4.3 ASTROMETRIC CORRECTIONS
61
Where D is the distance of the star and c the speed of light. At the time of the second
observation the star has moved a little bit closer or further away due to its radial
motion in space. Assuming two stars in a field with different radial velocities, then
their distances to us change by a different amount. The light reaching us at the same
time during the observation was emitted at two different times in the past and also this
time difference changes over time, as the two stars move with their own spatial velocity.
Together with the proper motion, this leads to a steady change in angular separation
observed on the sky. This light time delay typically is of the order of hours or days,
leading to a change in angular separation of about 10 − 100µas over the course of a few
years5 .
Let us assume a radial velocity difference for my target star and the reference star of
20 km/s and a maximum epoch difference of the observations of 1.5 yrs. That would
mean that the two stars are 9.5 · 108 km further apart (or closer together) in the radial
direction at the last observation compared with the first one. This then corresponds to
a light travel time of ∆T = 53 min. With a proper motion difference of ∼ 1500 mas/yr
one star travels within this 53 min only 150µas with respect to the other. So this is
the changing angular separation between the two stars over 1.5 yrs and it is way below
the aimed measurement precision, especially as the effect is even smaller between the
single epochs. I can therefore neglect this effect in my observations and data analysis.
4.3.9
Differential Tilt Jitter
As described previously (Sec. 2.2) a tip/tilt mirror is used in adaptive optics observations to compensate for the image motion of the guide star. The image of the guide
star is stabilized with high accuracy with respect to the imager. But as the guide star
is not necessarily the target star, or like in my case more than one star in the field is of
interest, an effect known as Differential Tilt Jitter (DTJ) comes into play. The difference of the tilt component of turbulence along any two lines of sight in the FoV causes
a correlated, stochastic change in their measured separation. Light from the target
and the reference star passes through different columns of atmospheric turbulence that
are sheared. Arising from this shearing effect, the decorrelation in the tilt component
of the wavefront phase aberration yields the DTJ. This leads to a fluctuation in the
relative displacement of the two objects, which is random, achromatic and isotropic
(Cameron et al., 2009). Turbulence at higher altitudes contributes most to the differential tilt jitter as the light from the two stars traverses more through common parts
of the atmosphere near the ground. The effect is bigger along the separation axis than
perpendicular to it. The longer the exposure time of the observation, the more the
effect of differential tilt jitter averages out. Cameron et al. showed that in their 1.4
seconds exposure already part of the effect averaged out, as the measured magnitude
of the tilt jitter was smaller than the expected one from theoretical models.
As I have exposure times of 99 sec for the target field and 4 sec for the reference field,
plus I add all single frames to one image before measuring the positions and distances
between the stars, I assume the effect to be averaged out in the images.
5
Numbers taken from the lecture Modern Astrometry: Methods and Applications given by S. Reffert
at the University of Heidelberg, Germany in 2008
62
CHAPTER 4. ANALYSIS AND ASTROMETRIC CORRECTIONS
Fixed reference frame
z
r
v
rg
rg
r
d
rE
a
rE
x
E
y
Figure 4.7: Parallax movement and correction of a star’s position
4.3.10
Parallax
The stars one observes are not at infinity and as a result of their finite distance, the
motion of the observer through space produces a displacement of the position of the
star with respect to the fixed reference system. This parallactic displacement is due to
several motion components of the observer:
• The motion of the geocenter around the barycenter of the Earth-Moon system
• The motion of the barycenter of the Earth-Moon system around the barycenter
of Solar system. This combination of motions results in a yearly periodic displacement called annual parallax and a small monthly component and is due to
the change in viewing angle from different positions on Earth’s orbit.
• The motion of the observer around the center of mass of the Earth, caused by
the daily rotation of the Earth. This motion is called the diurnal parallax.
The stellar or trigonometric parallax $ is known as the motion of a star with respect
to a fixed reference system, e.g. background stars, over one year.
$=
k
r
(4.52)
with k = π/648000 and r, the distance of the star, expressed in astronomical units
(AU) and $ the angle between the vectors ~rg and ~r in Fig. 4.7, left side. The denominator of k erases from the transformation of $ from radians into arcseconds. More
conveniently is the expression of r in parsec, 1 pc = 648000 AU/π = 206264.81AU. The
4.3 ASTROMETRIC CORRECTIONS
63
parallax can then be written as $ = 1 AU/r and one parsec corresponds to the distance under which an 1 AU displacement is seen as a 100 angle. The apparent geocentric
motion of the star is an ellipse with its semi-major axis equal to the parallax if in a
first approximation the Earth orbit is described as an ellipse with its semi-major equal
to one AU. As one can see, the parallax decreases with the distance of the source. The
approximation tan $ ≈ $ made here is good to about 2 · 10−17 radians ≈ 10−7 mas6 , so
in practice it can be used for all parallax applications.
The diurnal parallax effect is smaller than the annual one by a factor equal to the ratio
of Earth’s radius (RE = 6378km) and the astronomical unit (1 AU = 1.496 · 108 km).
For the parallax motion of my target star (71.56 mas) this would mean a correction of
3µas, which is well below the aimed precision in the measurements, so I can use the
geocentric coordinates for the parallax correction.
To correct for all motion effects at once,¡ one should use precise
Earth ephemerides, giv¢
ing the exact position of the Earth, ~r = xE (t), yE (t), zE (t) in the above defined system
of unit vectors (Equ. 4.49), at the time t of observation with respect to the barycenter
of the Solar system. The movement of the Earth should never be approximated as a
circle. The position of a star in barycentric, ~r, and geocentric, ~rg , coordinates is given
by:


α: right ascention
cos δ cos α
~r = r
cos δ sin α
δ: declination


sin δ
r: distance


cos δg cos αg
~rg = rg sin αg cos δg
g: geocentric


sin δg
From Fig. 4.7 one can see that
~r = ~rg + ~rE
(4.53)
With the approximation of r ≈ rg and r = 1/$ and plugging this into Equ. 4.53 one
obtains:
cos δ cos α = cos δg cos αg + xE $
(4.54a)
cos δ sin α = cos δg sin αg + yE $
(4.54b)
sin δ = sin δg + zE $
(4.54c)
Introducing the parallactic corrections, ∆α and ∆δ, which need to be added to the
geocentric coordinates to derive the barycentric coordinates (α, δ) we have:
6
α = αg + ∆α
⇔
∆α = pα $
(4.55a)
δ = δg + ∆δ
⇔
∆δ = pδ $
(4.55b)
Numbers taken from the lecture Modern Astrometry: Methods and Applications given by S. Reffert
at the University of Heidelberg, Germany in 2008
64
CHAPTER 4. ANALYSIS AND ASTROMETRIC CORRECTIONS
with pα , pδ called the parallax factors. These factors define the shape of the parallax
ellipse, depending on the position on sky and $ defines the size of this ellipse. To
derive them one has to do some transformations and small angle approximations to
Equations 4.54 which yield
∆α =
yE $ cos α − xE $ sin α
cos δg
(4.56)
∆δ = −(xE $ cos α + yE $ sin α) sin δ + zE $ cos δ
Comparison with Equations. 4.55a and 4.55b yield for the parallax factors the relation:
pα =
yE cos α − xE sin α
cos δg
pδ = −(xE cos α − yE sin α) sin δ + zE cos δ
(4.57a)
(4.57b)
The parallax factors change, depending on where on the Earth orbit the observation
was taken. One can see this in the dependency of pα and pδ on xE , yE , zE .
As one notices, to calculate the parallax factors one needs the coordinates one wants to
compute. So an iterative approach is necessary, but in practice the difference between
the geocentric and barycentric coordinates is mostly negligible for the computation of
the parallax factors and one iteration is sufficient.
If one only corrects for the parallax, the position (α, δ) of a star at epoch T is given by:
α = α0 + pα $
δ = δ0 + pδ $
(4.58)
where α0 , δ0 are the coordinates of the star at some reference epoch T0 .
As we will fit the parallax motion in our orbit fit we need the parallax factors, which
we can compute with our measured geocentric coordinates for each epoch. We then
have to fit only the size of the parallax, $. A detailed description of the orbit fit will
be given in Sec. 5.3.
4.3.11
Proper Motion
Each star has its own proper motion µ, leading to a displacement on the celestial sphere
when observed at an epoch T compared with an observation at epoch T0 . This displacement can be in any direction. A few stars have a high proper motion of the order
of 100 / per year or more, but mostly the annual movement is only a small fraction of
that. A star’s proper motion depends on its space motion relative to the center of the
celestial sphere, the barycenter, but also on its distance. So in general the further away
a star with a given space velocity, the smaller its proper motion. The velocity of a star,
resulting in its proper motion is assumed to be uniform. The path of the star is then
an arc of a great circle on the celestial sphere.
Ideally, proper motion is measured by comparing the position of a star in observations
many years apart. The longer the baseline, the more accurate the proper motion can
4.3 ASTROMETRIC CORRECTIONS
65
be derived. But the positions measured are only relative to the other stars observed
with the target star and one has to assume that their positions and proper motions
are known for each epoch. The best solution is to measure the positions relative to a
fixed reference frame. Most satisfying would be to use objects outside our Milky Way
as reference, such as other galaxies or quasars (QSO = Quasi-Stellar Object). QSO are
preferable as they are more point like and many of them are also radio sources, whose
position can be measured to a very high accuracy with radio interferometry. They
are galactic nuclei at enormous distances and are therefore unaffected by any galactic
motion.
Taking equatorial coordinates the proper motion in right ascension and declination is
the time derivative of the coordinates at epoch T0
µ ¶
dα
µα =
dt T =T0
µ ¶
dδ
µδ =
(4.59)
dt T =T0
µδ corresponds to a full (great circle) angle on the sky, but µα is reckoned on the
equator. Its actual component along the local small circle is given by µα cos δ. µα and
µδ are normally expressed in arcseconds per year, however if µα is given in seconds of
time per year, one has to multiply it by a factor of 15 to get arcseconds.
A star’s motion can be separated into a radial motion Vr (see Sec. 4.3.8) and a transverse
motion on a tangentqplane to the celestial sphere. The transverse motion equals the
proper motion µ = µ2α cos2 δ + µ2δ . With φ being the position angle of this motion
measured from North over East, the components of the proper motion in right ascension
and declination can be written as:
µα cos δ = µ sin φ
µδ = µ cos φ
(4.60)
One can now calculate a star’s position for any given epoch, if one only assumes proper
motion and parallactic motion. The position (α, δ) of a star at epoch T is then given
by:
α = α0 + (T − T0 )µ∗α + pα $
δ = δ0 + (T − T0 )µδ + pδ $
(4.61)
with µ∗α being short for µα cos δ.
Secular Acceleration
The secular acceleration is a variation in the proper motion and is a purely geometric
effect, due to an object which is approaching or receding. Writing the proper motion as
a function of the object’s space velocity V , its angle Θ with the direction barycenter-star
and its distance r
V sin Θ
µ=
(4.62)
r
66
CHAPTER 4. ANALYSIS AND ASTROMETRIC CORRECTIONS
and calculating its derivative with respect to time t, one gets:
dµ
V
dr V
dΘ
= − 2 sin Θ + cos Θ
dt
r
dt
r
dt
With
dΘ
dt
= −µ and V cos Θ = Vr =
dr
dt
(4.63)
and using Equ. 4.62 one can write
dµ
2µ
= − Vr
dt
r
Expressing µ in arcseconds per year , the distance $ =
the secular acceleration can be calculated by:
(4.64)
1
r
in arcseconds and Vr in km/s,
dµ
= −2.05 · 10−6 µ$Vr
dt
(4.65)
Using the values for proper motion (µ = 1.500 /year) and parallax ($ = 71.56 mas) from
the HIPPARCOS catalog and a radial velocity Vr = 63 km/s for our target star7 , the
secular acceleration amounts to −13.84·10−6 arcsec yr−1 . This would result in a change
in proper motion of 0.02 mas/yr over the 1.5 yrs time baseline of our observations, which
we will not be able to measure. We can therefore stay with the truncation of the Taylor
series of the star’s motion after the first term and assume the proper motion of our
target as constant in time. As the reference star is much further away and its proper
motion therefore is way smaller, the effect for it will be even smaller.
4.4
Plate-scale and Detector Rotation Stability
Doing astrometry with NACO, one has to take care of two effects. The global pixel- or
plate-scale can change and field distortions can be present, both effects can change with
time. To monitor and correct the possible change of pixel-scale and also the rotation
of the detector, I observed a reference field in the globular cluster 47 Tuc every time
right before the target field. To minimize these effects, I centered the stars on the same
pixel positions at the beginning of each observation in both, the target and reference
field, and executed the same jitter pattern each time. As long as the jitter pattern is
the same in each observation, I do not have to take care about the absolute distortions,
but have to monitor, if they change with time. Experience from other observers using
NACO, indicates that the distortion pattern is stable over time8 . If the distortions
are constant with time, they should be the same for each observation, and only the
change in pixel-scale has to be corrected. Unfortunately, it is not so easy to separate
the change in the globular pixel-scale and the field distortions, which create different
local pixel-scales. Also, the final image, on which I work, is the sum of the shifted and
added single images. The distortions, present in each single image, are smeared out in
the final image, making it very difficult to model them, as they are dependent on the
size and distribution of the jitter offsets. NACO is at the Nasmyth focus, but it does
not have an optical de-rotator. It rotates as a whole instrument, instead. Therefore
7
Mathias Zechmeister (private communication; based on measurements with the ESO FEROS spectrograph)
8
Andreas Seifahrt (private communication)
4.4 PLATE-SCALE AND DETECTOR ROTATION STABILITY
67
Figure 4.8: Stars in the reference field in 47 Tuc used to calculate the change in pixel-scale
and rotation between the different epochs.
variable flexures exist in this Nasmyth instrument and the distortions from the reference
field can probably not be used to correct the target field.
The only thing left is to model a global pixel-scale, assuming it only changes on a global
scale and has no direct cross-talk with the distortions.
To calculate the changes between the different epochs, I chose the same 11 stars for
each epoch in the observed reference field. In Fig. 4.8 the stars used for the distortion
fit are marked. As I do not need to measure an absolute plate-scale or rotation of
the detector, but only relative values, I chose epoch 9 as the reference epoch to which
all other epochs are mapped. This epoch was chosen, because it is the middle one in
time of all observations. I then mapped the positions of the stars in each epoch to the
positions measured in epoch 9 with a linear coordinate transformation, calculating a
shift in x and y, scale in x and y, and a rotation and skew. The skew is realized by
allowing a different rotation for the x and y axes. The parameters were calculated with
the geomap program, which is part of the IRAF reduction and data analysis software9 .
The calculated values for each epoch are shown in Fig. 4.9 The panels show from top
to bottom the calculated shift, scale and rotation for the x and y direction versus the
9
http://iraf.noao.edu/
68
CHAPTER 4. ANALYSIS AND ASTROMETRIC CORRECTIONS
x−shift relative to epoch 9
y−shift relative to epoch 9
y−shift [px]
x−shift [px]
0
−20
−40
−60
−80
600 700 800 900 1000 1100 1200
MJD − 54000
120
100
80
60
40
20
0
600 700 800 900 1000 1100 1200
MJD − 54000
y−scale changes to epoch 9
x−scale changes to epoch 9
1.002
y−scale
x−scale
1.002
mean = 0.9994
stddev = 0.0003
1.000
1.000
0.998
0.998
0.996
600 700 800 900 1000 1100 1200
MJD − 54000
0.996
600 700 800 900 1000 1100 1200
MJD − 54000
x−rotation changes to epoch 9
0.6
y−rotation changes to epoch 9
0.8
mean = 0.3001°
stddev = 0.2733°
0.6
y−rot [deg]
0.8
x−rot [deg]
mean = 1.0006
stddev = 0.0004
0.4
0.2
0.0
mean = 0.2887°
stddev = 0.2657°
0.4
0.2
0.0
−0.2
600 700 800 900 1000 1100 1200
MJD − 54000
−0.2
600 700 800 900 1000 1100 1200
MJD − 54000
Figure 4.9: Calculated distortion parameters of all epochs. The parameters for shift, scale
and rotation in x and y, respectively, are shown from top to bottom versus the
MJD of the observation. For the scale and rotation parameters the mean value
and the standard deviation are given, for an impression of the stability of these
parameters over time.
Modified Julian Date (MJD), for each epoch mapped to epoch 9. The much larger
values for the shift of the first epoch are due to a rotation of the frame about 90◦ ,
due to an error in the value of the applied rotation of the detector. I corrected the
coordinates of the stars to this rotation, but their positions on the frames nevertheless
differ by roughly +75 pixel on the x-axis and -100 pixel on the y-axis. But the shift
is not the important parameter, as it does not change the distance between the stars.
The more interesting values are the scale and rotation. Here the mean value and the
standard deviation are also given in the respective panels. The standard deviation
gives an impression of the stability of the parameters over time. It seems the larger
4.4 PLATE-SCALE AND DETECTOR ROTATION STABILITY
69
the separation in time is for the first nine epochs, the more different is the plate-scale,
but this would have to be explored in more detail with more data doing a distortion
analysis for NACO. The calculated rotation between the different epochs does not show
a trend with time. For the 47 Tuc observations, the detector was rotated by 42◦ anticlockwise. The scatter of the measured rotation between the epochs gives the accuracy
with which the applied rotation of the detector can be performed. The scatter is quite
large, 0.26 - 0.27 degree, but there is no large difference for the two axes, indicating
only a very small skew. Because the detector is again rotated between the observations
of the reference and the target field, and I cannot assume that the rotation error is the
same for both fields in the same night, I cannot correct for the rotation between the
epochs. As there are only two stars in the target field, an independent estimate of the
differential rotation between the epochs for this field is not possible. I added the scatter
of the rotation parameter over time, measured as a mean of the x- and y-rotation, in
the reference field as the uncertainty of the later calculated position angle between the
two stars in the target field.
4.4.1
Plate-scale Correction
The detector distortions are the last effect changing the true positions of the stars.
Therefore one should correct for them first. I assumed that the change in pixel-scale
at any epoch with respect to the one in epoch 9 is the same in the reference field and
the target field. To correct for the change in pixel-scale I multiplied the FITS header
keywords CDi j = si mij (Equ. 4.4) with the scale factors derived from the distortion
fit:
new CD11 = CD11 ∗ x-scale
(4.66)
new CD12 = CD12 ∗ x-scale
new CD21 = CD21 ∗ y-scale
new CD22 = CD22 ∗ y-scale
This was done for every epoch (for the first epoch the x- and y-scale are exchanged
due to the 90◦ rotation) and following this, the measured pixel coordinates of the
stars were transformed to celestial coordinates, as described in Chap. 4.2.3, using the
new plate-scale. After that, the corrections for differential refraction and aberration
were calculated and applied again (Chap. 4.3.3 and 4.3.6). That the celestial positions
changed indeed due to the plate-scale correction can be seen in Fig. 4.10 where the
change, after the transformation from x/y coordinates to celestial coordinates, in right
ascension and declination is plotted in milli-arcseconds for each epoch.
A scaling of the image itself before measuring the positions would involve interpolation,
which again can introduce errors. I therefore chose the applied way of correction,
because here I do not have to work on the images directly to change the pixel-scale.
I therefore corrected directly the coordinates of the stars, which is more precise. The
RMS given by the fit for the coordinate mapping takes all transformations into account.
I cannot use this value as an estimate for the plate-scale correction alone, as it also
takes for example the rotation into account. To estimate the precision of the platescale correction I took star pairs in the reference field in one epoch and calculated
70
CHAPTER 4. ANALYSIS AND ASTROMETRIC CORRECTIONS
30
∆α GJ 1046
∆δ
∆α Ref. star
∆δ
∆α, ∆δ [mas]
20
10
0
−10
−20
−30
0
2
4
6
Epoch
8
10
Figure 4.10: Relative change in positions due to the plate-scale correction. Shown are the
values for ∆ α = α − αcorr and ∆ δ = δ − δcorr for both stars in the target field.
The opposite directions of the changes for GJ 1046 and the reference star are
expected, due to the fact the the coordinates are calculated from the reference
point, which is almost in the middle of the detector. The large displacement
i n epoch 1 is due to the larger change in pixel-scale compared with the other
epochs. The visible trends correspond to ones already visible in Fig. 4.9.
their separation. Taking the separation of the same star pairs in epoch 9, the reference
epoch, as the separation with a plate-scale of the theoretical value of 27.15 mas/px, I
could calculate the pixel-scale of the other epochs by comparing the separations:
plate-scale i =
mas
separation epoch 9
∗ 27.15
separation epoch i
pixel
(4.67)
I did this for 100 random star pairs for each epoch to have good statistics. The standard
deviation of the distribution of pixel-scales calculated this way is then taken as the
uncertainty in the pixel-scale correction for the later separation measurement of the
two stars. In Tab. 4.3 the uncertainties in the pixel-scale are listed for each epoch.
These first results in this chapter show, that in multi-epoch astrometry the most challenging task is to measure and correct the pixel-scale very precisely to obtain milli- or
even micro-arcsec astrometric solutions.
4.4 PLATE-SCALE AND DETECTOR ROTATION STABILITY
Epoch
pixel-scale
pixel-scale
uncertainty [ mas
px ]
uncertainty [%]
1
0.046
0.171
2
0.035
0.127
3
0.027
0.100
4
0.018
0.066
5
0.016
0.058
6
0.045
0.164
7
0.014
0.051
8
0.032
0.117
9
0.000
0.000
10
0.029
0.105
71
Table 4.3: Uncertainties in the pixel-scale. Calculated by randomly taking 100 star pairs
in the reference field, calculating the changed pixel-scale relative to epoch 9, and
taking the standard deviation of the distribution of derived pixel-scales as uncertainty.
72
CHAPTER 4. ANALYSIS AND ASTROMETRIC CORRECTIONS
Chapter 5
The Orbit Fit
5.1
Preparing the Coordinates for the Orbital Fit
To fit an orbit to the positions of the two stars derived so far, one needs to apply
some last transformations and calculations. In a first approach I re-transformed the
celestial coordinates of the target star GJ 1046 and the reference star 2MASS 02190953
-3646596 to pixel coordinates with the IDL routine adxy.pro from the IDL astrolib1 .
This routine is the inverse of xyad.pro, which I used to transform pixel coordinates into
celestial coordinates in Chap. 4.2.3. I then finally corrected the coordinates for the
nominal rotation of 6◦ of the detector, relative to the North direction. The x and right
ascension axes, as well as the y and declination axes are now parallel. One can now
work with the separations of the two stars in x and y and their change with time.
Additionally, I calculated the separation and position angle of the two stars. Because of
the large separation of the stars, one cannot just calculate the separation in Cartesian
coordinates. One has to calculate the separation ρ of the two stars along a great circle
using the cosine formula (Green, 1985, p. 12):
ρ = cos−1 (sin δ1 sin δ2 + cos δ1 cos δ2 cos ∆α)
(5.1)
Where δ1 and δ2 are the declination of the two stars, star 1 = reference star and star 2
= GJ 1046, and ∆α is the difference in their right ascension.
The position angle Θ of GJ 1046 relative to the reference star, measured from North
through East from the Meridian containing the reference star is calculated using the
four-parts formula (Green, 1985, p. 12):
¶
µ
sin ∆α
−1
(5.2)
Θ = tan
cos δ1 tan δ2 − sin δ1 cos ∆α
The errors of the separation and position angle, resulting from the positional uncertainties of the stars, are between 0.22 - 1.45 mas for the separation and between 0.00045◦ 0.00252◦ for the position angle. However, a much larger uncertainty due to the pixelscale variability and the detector rotation uncertainty had to be added for each epoch.
The resulting uncertainties in the separation are between 15.30 mas and 50.90 mas and
1
http://idlastro.gsfc.nasa.gov/
73
74
CHAPTER 5. THE ORBIT FIT
0.26072◦ in the position angle with only very small differences for the single epochs.
A summary of the calculated separation and position angle values together with their
errors is given in Tab. 5.1 for each epoch. In column two the separation and position
angle are noted, column three lists the combined errors due to the position measurement and the correction for differential refraction and aberration. In column four the
error due to the pixel-scale uncertainty and the detector rotation are given and column
five finally lists the combined errors of all these effects.
This already shows, how crucial it is to measure the pixel-scale and the rotation of the
detector very accurately. The large separation of the target and reference star yields
this large separation and position angle uncertainty. In Fig. 5.1 the effect of the separation between the target and reference star on the obtainable precision is shown. The
left panel shows the error in the separation in arcseconds as a function of pixel-scale
uncertainty and separation. The lines represent the errors in the separation between
the stars for three different values of uncertainty (±0.05%, ±0.11%, ±0.17%) in the
pixel-scale. The red diamonds represent my measured uncertainties. The right panel
shows the error in the separation in right ascension (∆α) and declination (∆δ) due
to the uncertainty in the position angle. Shown is the case for a position angle of
42◦ (solid line) with an uncertainty of ±2.5◦ (dashed lines). This 10 times larger error
than measured was taken for reasons of better depiction of the effect. The separation
between the stars is indicated along the lines by marks at 1000 , 2000 and 3000 . Projecting
the error onto the directions of right ascension and declination leads to large errors in
these two values. In the small inlet the size of the uncertainty of the position angle,
Θ ± 0.25◦ (red in the big panel), along the direction perpendicular to the separation
(blue in the big panel) is shown in seconds of arc as function of the separation.
30
0.06
0.04
± 0.11 %
0.02
± 0.05 %
0.00
0
10
20
30
Separation [arcseconds]
40
25
∆δ [arcseconds]
± error [arcseconds]
± 0.17 %
20
± error ["]
0.15
measurements
0.10
0.05
0.00
0
∆Θ = ± 0.25°
10
20
30
40
Separation ["]
30"
15
20"
10
5
0
0
10"
10
20
∆α [arcseconds]
30
Figure 5.1: Effect of separation between the target and reference star on the precision
obtainable for the separation (left panel) and the position angle (right panel).
For more details and explanation see text.
29.7709
29.8758
29.9273
29.9330
30.0251
30.1134
30.1570
30.2174
30.2864
31.3265
1
2
3
4
5
6
7
8
9
10
38.62
38.53
38.47
38.54
38.38
38.78
38.62
38.57
38.54
37.96
Θ [◦ ]
1.45
0.40
0.36
0.33
0.47
0.40
0.28
1.06
0.54
0.23
∆ρ1 [mas]
∆ρ2 [mas]
50.91
37.94
29.93
19.76
17.41
49.39
15.38
35.35
0.00
32.89
∆Θ1 [◦ ]
4.4 · 10−4
9.9 · 10−4
19.0 · 10−4
5.0 · 10−4
7.5 · 10−4
8.6 · 10−4
6.1 · 10−4
6.7 · 10−4
7.1 · 10−4
25.2 · 10−4
0.26
0.26
0.26
0.26
0.26
0.26
0.26
0.26
0.26
0.26
∆Θ2 [◦ ]
pixel-scale, de-rotator
33.00
0.40
35.42
15.32
49.42
17.43
19.70
29.98
38.04
50.90
[mas]
2
i ∆ρi
qP
0.26
0.26
0.26
0.26
0.26
0.26
0.26
0.26
0.26
0.26
[◦ ]
2
i ∆Θi
qP
errors due to the PSF fit and the correction for differential refraction and aberration. In column four the errors due to the pixel-scale
uncertainty and the detector rotation are given and column five finally lists the combined errors of all these effects.
Table 5.1: Summary of the measured separation ρ and position angle Θ together with their uncertainties. The third column list the combined
ρ [00 ]
Epoch
Pos., Refrac., Aberr.
5.1 PREPARING THE COORDINATES FOR THE ORBITAL FIT
75
76
CHAPTER 5. THE ORBIT FIT
y
P’
P
b
r
n
E
C
F
H
P
x
a
ea
Figure 5.2: Principle elements of an ellipse with the definition of the eccentric anomaly E
and the true anomaly ν.
5.2
Theory of Deriving the Orbital Elements
The orbit of a planet around its host star, as well as the motion of the star due to a
companion, are ellipses with the common center of mass of the two bodies at one of
the focal points of both ellipses. To describe the orbital motion of the companion or
the host star, one needs to know where the body is at any time t. The orbital elements
are the same for the two bodies, except the size of the semi-major axis. To derive the
orbital elements one can regard the ellipse as a projection of a circle. In Fig. 5.2 the
principal components of such an ellipse are depicted. The denotation is as follows:
C
F
Π
a
b
e
ea
P0
P
r
E
ν
center of the ellipse
focal point of the ellipse
periastron
semi-major axis
semi-minor axis
eccentricity
distance from C to F
point on ellipse at time t
point on circle which is projected onto the ellipse
radius vector
eccentric anomaly
true anomaly
The eccentric anomaly E is defined as the angle between the semi-major axis a and
0
the direction from the center of the ellipse to the point P on the circle. It is zero at
the periastron Π and increases by 2π during one orbit. The true anomaly is the angle
between the distance F Π = a − ae = a(1 − e) and the distance F P = r, which is the
5.2 THEORY OF DERIVING THE ORBITAL ELEMENTS
77
radius vector at time t. Looking at Fig. 5.2, one can see that for a Cartesian coordinate
system with its origin at F , the x and y values of the point P in polar coordinates in
the plane of the orbit are given by:
X = r cos ν = a(cos E − e)
p
Y = r sin ν = a 1 − e2 sin E
(5.3)
and the distance of P from the focal point F is:
r = a(1 − e cos E) =
a(1 − e2 )
1 + e cos ν
Defining the mean anomaly M at time T by the mean motion n =
the period and Tp the time of periastron:
M = n(T − Tp ) = E − e sin E
(5.4)
2π
P ,
with P being
(5.5)
one can calculate the eccentric anomaly E at any time using an iterative approach. The
last expression M = E − e sin E is known as Kepler’s equation. Once E is determined
one can compute r and ν from Equ. 5.4 and from:
r
1+e
E
ν
tan
(5.6)
tan =
2
1−e
2
In the simple case of visual binaries, one centers the primary in the local coordinate
system to describe the orbital motion of the companion with respect to the primary
star, but the equations are also true for the motion of the primary around the secondary.
The coordinates of the companion on the tangential plane on the sky with respect to
the primary, expressed in seconds of arc, are then defined by the separation ρ and the
position angle θ, starting from North on eastward:
x = ρ sin θ
y = ρ cos θ
5.2.1
(5.7)
The Thiele-Innes Constants
Each observation of a visual binary yields a pair of coordinates at a given time T . These
coordinates are the separation of the two stars and the position angle. After a hopefully sufficiently long time interval of observations one has a series of values (ρ, θ, T ) or
equivalently (x, y, T ). These measured values give the apparent orbit in the x/y plane
on the sky.
The true orbit of the companion about the primary, which one wants to obtain, is an
ellipse with the primary situated at one focus F . Projecting this ellipse onto the plane
of the sky yields again an ellipse, the apparent orbit. This apparent ellipse is not a Keplerian ellipse anymore. The focus F of the true ellipse does not coincident with a focus
of the apparent ellipse, meaning the primary is not seen at a focus. The semi-major
axis of the apparent orbit does not correspond to the semi-major axis of the true orbit
78
CHAPTER 5. THE ORBIT FIT
either and therefore does not lie along the projection of the diameter that contains the
periastron Π. Only the projection of the center of the true orbit appears at the center
of the apparent orbit and the areas on the apparent ellipse are projections of the areas
of the true ellipse. Kepler’s 2nd law of proportionality of the areas swept by the radius
vector still holds.
With the orbital parameters defined in Chap. 1.1 one can transform the local apparent
coordinates (ρ sin θ, ρ cos θ) in the coordinate system OXY to the true coordinates in
the coordinate system Oxy. With the scale of the true orbit in arcseconds, a00 = a$
with the parallax $ and the semi-major axis a, one can write the true coordinates in
the plane of the orbit, also expressed in arcseconds, as
(
X = $r cos ν
B
Y = $r sin ν
To transform B into r = (ρ sin θ, ρ cos θ) one has to perform three transformations.
The first one is a rotation by −ω around the Z-axis which is perpendicular to the
X/Y -plane, the true orbital plane. After this, one has to account for the projection
angle i, the inclination, between the true orbital plane and the plane perpendicular to
the line of sight. The intermediate coordinates can be written as:
Xinter = X cos ω − Y sin ω
Yinter = cos i(X sin ω − Y cos ω)
(5.8)
One then has to rotate these coordinates by −Ω around the z-axis, which is in the
direction of the line of sight and perpendicular to the tangential plane. Finally a
permutation of the abscissa and ordinate needs to be performed to obtain the coordinate
system Oxy in the tangential plane
x = Xinter sin Ω + Yinter cos Ω
y = Xinter cos Ω + Yinter sin Ω
Combining this with Equ. 5.8 one finally can write the apparent coordinates expressed
by the true coordinates as:
x =X(cos ω sin Ω + sin ω cos Ω cos i)
+ Y (− sin ω sin Ω + cos ω cos Ω cos i)
y =X(cos ω cos Ω − sin ω sin Ω cos i)
+ Y (− sin ω cos Ω − cos ω sin Ω cos i)
With the Thiele-Innes constants defined as:
XT I1 = a(cos ω sin Ω + sin ω cos Ω cos i)
YT I1 = a(− sin ω sin Ω + cos ω cos Ω cos i)
XT I2 = a(cos ω cos Ω − sin ω sin Ω cos i)
YT I2 = a(− sin ω cos Ω − cos ω sin Ω cos i)
(5.9)
5.3 THE ASTROMETRIC ORBIT FIT
79
one can write in short form:
r
r
cos νXT I1 + sin νXT I2
a
a
r
r
y = cos νYT I1 + sin νYT I2
a
a
x=
5.3
(5.10)
The Astrometric Orbit Fit
In the context of this work, I do not measure the companion’s motion around the star
directly, but instead the star’s motion about the common center of mass of the system.
Therefore the orbital motion (xorbit , yorbit ) I see, is related to the star’s motion and
semi-major axis. xorbit and yorbit are the expressions derived in Equ. 5.10 with an
index orbit added to point out that this is the orbital motion of the star due to the
companion in the following equations.
In my case I already have some of the orbital elements from the radial velocity measurements. These are: the period P , the eccentricity e, the longitude of periastron ω
and the time of periastron Tp . The semi-major axis a is related to the inclination i over
Equ. 1.4, so the only two orbital parameters one has to solve for, after fixing the spectroscopic parameters, are the inclination of the orbit and the longitude of the ascending
node Ω. These two angles of the orbit of the star are the same for the companion. I
could fix the values of P, e, ω, and Tp to the values derived in the spectroscopic orbit
fit, because the accuracy from this fit is much higher than the accuracy with which
they could be derived in the astrometric fit.
One can write the position of the star on the sky depending on proper motion and
parallax of the center of mass and the reflex motion of the star at any time T as:
α = α0 + (T − T0 )µα + pα $ + xorbit
δ = δ0 + (T − T0 )µδ + pδ $ + yorbit
(5.11)
where α0 , δ0 are the position of the star at epoch T0 , µα and µδ are the proper motion
and pα , pδ are the parallax factors in right ascension and declination, respectively. As
I measure relative positions and not absolute ones, the positions α0 , δ0 are replaced by
the separation of the two stars in right ascension and declination ∆α0 , ∆δ0 and α, δ
then correspond to separations, too.
To solve this equation one also has to fit for the proper motion and parallax of the target
star relative to the reference star. As seen in the previous sections, the uncertainty in
the pixel-scale and the orientation of the detector lead to large errors in the final
separation and position angle, with the error in the separation being still a lot smaller
than that of the position angle. The orbital period of the companion of GJ 1046,
and therefore of GJ 1046 itself, is 169 days, almost half a year. Special care has to
be taken to disentangle the orbital motion from the parallax motion. For this, the
observations were timed in such a way that measurements over almost a full orbit were
obtained. In Fig. 5.3 an example of a simulated astrometric signal of GJ 1046 with
proper motion, parallax and orbital motion is shown as a change in right ascension and
80
CHAPTER 5. THE ORBIT FIT
Figure 5.3: Simulated change in declination versus right ascension of the orbit of GJ 1046.
The left panel shows a simulated astrometric signal for GJ 1046 (see text for
details). The other two panels show the simulated astrometric signal without
proper motion and with (middle) and without (right) parallax, respectively. Note
the different scales in the changes of right ascension and declination in the three
different cases.
declination over one year. The HIPPARCOS proper motion and parallax are taken
and i = 45◦ and Ω = 60◦ are assumed. As the proper motion of GJ 1046 is very high,
it totally dominates its motion and pulls the parallax + orbital ellipse apart into a
wave-like motion (left panel). Subtracting the proper motion in the simulation, one
can see the orbital motion due to the companion, but still dominated by the parallax
(middle panel). The curve is not a perfect ellipse as in the case of pure parallax motion
and is not closing after one full orbit of the Earth around the sun. Finally the pure
orbital motion without parallax and proper motion is displayed (right panel). As one
can see, it is very important to know/measure the parallax motion to distinguish the
orbital motion from it. Note the different scales in the changes of right ascension and
declination in the three different cases.
The fit for the orbital parameters was performed by Rainer Köhler from the Landessternwarte Heidelberg, who is experienced in deriving orbital solutions, using his
orbit fit program. A model is calculated based on Equations 5.11 which is then compared with the actual measurements by minimizing χ2 , using a Levenberg-Marquardt
algorithm (LMA) (Press et al., 1992).
A first fit was performed, accounting only for the measurements of the separations of
GJ 1046 from the reference star, because of the relatively smaller error of this parameter. No orbital motion was taken into account at this step, so Equations. 5.11 reduce
to:
α = ∆α0 + (T − T0 )µα + pα $
δ = ∆δ0 + (T − T0 )µδ + pδ $
(5.12)
We took the values for the proper motion and parallax from the HIPPARCOS catalog
for a first test, µα = 1394.10 mas/yr, µδ = 550.05 mas/yr, $ = 71.56 mas, so the only
free parameters are the separation zero-points ∆α0 and ∆δ0 . The calculated separations from the model parameters for α and δ are then compared with the measured
ones for each epoch.
5.3 THE ASTROMETRIC ORBIT FIT
81
The second approach was to include the orbital movement in the fit. This was done
by scanning the possible angles for the inclination i and the longitude of the ascending
node Ω in one degree steps and calculating a χ2 value for each (i, Ω) pair. Again, the
only free parameters in Equ. 5.11 for the LMA were ∆α0 and ∆δ0 , as i and Ω were
given at each step. For each (i, Ω) pair a model was calculated for the separation ρ and
this time also for the position angle θ. Inclination i and ascending node Ω are needed
to calculate the Thiele-Innes constants (Equ. 5.9) to derive the orbital influence on the
separation and position angle. After scanning i from 0-180◦ and Ω from 0-360◦ we have
a χ2 map for the full parameter space of these two angles. We again used the proper
motion and parallax from the HIPPARCOS catalog. We can now search the resulting
χ2 map for a minimum.
82
CHAPTER 5. THE ORBIT FIT
Chapter 6
Results
Due to the large uncertainties in the detector orientation and pixel-scale correction I
was not able to detect any astrometric motion of the star due to its brown dwarf companion. Only a formal value for the inclination of the system and thus for the mass
of the companion could be derived, but with low significance of the results, see Sect. 6.1.
Using the HIPPARCOS values for proper motion and parallax and not taking any
orbital motion into account for the model fit, results in a good agreement of the model
with the measured separation changes. In Fig. 6.1 the measured separation between
the stars is plotted versus time. Overplotted as the solid line is the model with the
motion calculated from the HIPPARCOS values for parallax and proper motion. The
very good agreement indicates that the assumption of a small or negligible parallax
and proper motion of the reference star was right and that we only see and detect the
motion of GJ 1046 itself. The χ2 value of this model fit is χ2 = 7.63, with 18 degrees
of freedom (DoF) the chance probability of this value is p(χ2 ) = 0.984. Therefore the
confidence for rejecting the model without orbital motion is only 1.6%. In principle
one should stop at this point and accept the model, which only takes proper motion
and parallax movement of the star into acoount, as precise enough to represent the
data. However, from the radial velocity measurements, I know about the presence of
a companion, hence also a model including orbital motion is calculated and fit to the
data.
6.1
The Orbit
Due to the large uncertainty in the alignment of the detector a transformation of the
separation onto the right ascension and declination axes results in large error bars in
these two directions. An attempt to fit a model based on Equ. 5.11 with the separation
at epoch zero ∆α0 , ∆δ0 , parallax $, proper motion µα and µδ , inclination i and
longitude of the ascending node Ω as free parameters led to resulting values of the
proper motion and parallax strongly diverging from the HIPPARCOS values. Even
though a small deviation is in principle possible, as only relative motions to the reference
star are measured, a deviation of up to 200 mas, as derived in the fit, is not possible.
83
84
CHAPTER 6. RESULTS
31.5
Separation [arcsec]
31.0
30.5
30.0
29.5
2008.5
2009.0
Time [years]
2009.5
2010.0
Figure 6.1: Measured separation between GJ 1046 and the reference star vs. time for the
10 observed epochs. The error bars represent the uncertainty in the separation
due to the pixel-scale uncertainty. Epoch 9 has only a very small error bar as
it is the reference epoch for the measurement of the pixel-scale changes (see
Sect. 4.4). Overplotted is the model for the separation changes calculated with
the HIPPARCOS values for proper motion and parallax (solid line). No orbital
motion is taken into account. The good agreement of the measurements with
the model shows that the reference star has indeed only a very small intrinsic
parallax and proper motion.
Additionally, the results from the fit without orbital motion using the HIPPARCOS
values showed a very good agreement with these values. Therefore, the proper motion
and parallax were fixed to the HIPPARCOS values in the following approach to fit the
orbital motion, as we are not able to improve these values.
In Fig. 6.2 the resulting χ2 contour map from the model fit with orbital motion to
the measured separation and position angle is depicted. The white contour represents
the 1σ (68.3%) and the black contours the 2σ (95.4%) and 3σ (99.7%) confidence
levels. The formally best fit is achieved with an inclination i = 145.0◦ (180◦ − i = 35◦ )
and a longitude of the ascending node Ω = 180◦ , leading to a formal mass of the
companion of mp = 48.5 MJup , calculated with the mass function (Equ. 1.3). This
would mean that the companion is indeed a brown dwarf, residing in the brown dwarf
desert. However, this result cannot be considered significant, as already the simplier
model without orbital motion could not be rejected with sufficient confidence. Likewise
6.1 THE ORBIT
180
85
χ2 Map
> 13.46
Inclination i [deg]
x
11.46 = 2σ
135
9.46
90
8.46 = 1σ
45
7.96
7.46 = min
0
90
180
270
ascending node Ω [deg]
360
Figure 6.2: χ2 contour map for fitting the orbital motion to the separation and position angle
measurements. The spectroscopic parameters were fixed, as well as the parallax
and proper motion (HIPPARCOS values). The inclination i and the ascending
node Ω were looped through, leaving the zero-point separation in declination
and right ascension as the only free fit parameters. The white contour represents
the 1σ (68.3%) confidence level and the black contours the 2σ (95.4%) and 3σ
(99.7%) confidence levels. The white cross marks the formally best fit solution
found at i = 145◦ and Ω = 180◦ .
the model including orbital motion cannot be rejected. The confidence for rejection
of this model is only 3.7% (DoF = 16, χ2 = 7.46), but setting better constraints on
the orbital parameters is not possible. This is also indicated by the large formal errors
derived for the inclination i and the ascending node Ω. Already within 1σ, inclinations
from 8.5◦ − 170.5◦ are possible. The 1σ, 2σ and 3σ levels span the entire parameter
space for the ascending node, which is therefore completely undetermined. In Fig. 6.3
the χ2 of the fit as a function of only the inclination is plotted together with the 1σ
and 3σ confidence levels. Only inclinations smaller than 3◦ and larger than 175◦ can
be excluded with a 3σ confidence. The formally best mass estimate is at the same
time the lower mass limit for the companion, as both the upper and lower 1σ limit
for the inclination yield higher masses. The 1σ upper limit for the companion mass is
236 MJup for an inclination of 8.5◦ . Calculating the 3σ upper limit of the companion
mass leads to the very unlikely case that the 'companion' is more massive than the
primary, which is already excluded by the upper limit of 112 MJup derived from the
86
CHAPTER 6. RESULTS
3σ
16
χ2
14
12
10
1σ
8
x
0
45
90
Inclination i [deg]
135
180
Figure 6.3: χ2 of the astrometric fit as a function of only the inclination together with the 1σ
and 3σ confidence levels, represented as the dashed and dash-dotted horizontal
lines, respectively. Ω is treated as uninteresting, therefore the confidence levels
correspond only to the single parameter inclination and are located at the levels
χ2 + 1 and χ2 + 9, respectively. The cross marks the formally best fit solution.
combination of the RV data with the HIPPARCOS astrometric data (Kürster et al.,
2008). All masses are calculated with the mass function (see Equ. 1.3 and Tab. 3.1).
In Tab. 6.1 the parameters derived from the astrometric orbit fit are summarized.
6.2
Discussion and Conclusion
I have observed an M Dwarf with a known Brown Dwarf desert candidate companion
with adaptive optics aided imaging to detect and measure the astrometric signal of the
star due to its unseen companion. The observations were conducted with the NACO
instrument at the VLT UT4 telescope. A star at 3000 separation was used as a reference
for the astrometric measurements. The expected astrometric peak-to-peak signal is
minimum 3.7 mas and up to 15.4 mas for an object at the upper mass limit derived
from HIPPARCOS astrometry (see Chap. 3).
To measure the relative positions of the target star and the reference star very precisely,
one has to correct for several effects which change the true positions of the stars. After
6.2 DISCUSSION AND CONCLUSION
87
Table 6.1: Parameters derived from astrometry
inclination i
145◦
formal optimum
180◦
formal optimum
48.52 MJup
formal optimum
longitude of the
ascending node Ω
companion mass mp
minimum i
3◦
3σ limit
maximum i
175◦
3σ limit
measuring the positions on the detector by fitting a Moffat function to the PSFs, I
calculated the error due to the fit with the bootstrap method. The position accuracy
is well below milli-arcsecond accuracy at this step.
To be very precise in the relative positions between the stars, I corrected for differential
refraction, which takes the different zenith distances of the two stars into account. As
I could not work on the single frames, because of the faintness of the reference star, I
calculated a mean correction for the positions in right ascension and declination. The
main error contributions in this case come from the long duration of the observations
and the correction for different temperature and pressure in the individual observing
epochs. Depending on when the observations were conducted, close to, before or after
the local Meridian passage, the zenith distance and therefore the differential refraction
changes faster or slower with time. I could not correct for the time dependent effect,
but rather applied the mean change in right ascension and declination as the correction
factor and the standard deviation of the measured values as the error. Additionally,
the error from the correction for the different temperature and pressure is taken into
account. The positional precision after this correction is still below the milli-arcsecond
range, except in one case, where the temperature and pressure during the observation
deviated strongest from the standard conditions.
The next correction I applied is the one for differential aberration. Due to the movement of the Earth through space the positions observed are different from the true ones.
Depending on the position of the star on the celestial sphere relative to the observer, a
correction for aberration has to be applied which is different for each star.
The by far biggest uncertainty in the relative separation of the two stars comes from
the change in the plate-scale between the different epochs and the uncertainty of this
change. Because of the large separation of the two stars an uncertainty of 0.05% adds
a large error to the overall error budget. Also the error in the applied detector rotation
amounts to large error bars of the position angle. The detector rotation is stable to
about 0.26◦ as measured in the reference field 47 Tuc. Again, the large separation of
the target and reference star leads to large errors.
The formally best fit with fixed spectroscopic parameters and including orbital motion
yields an inclination i = 145◦ and therefore a companion mass mp = 48.5 MJup . The
1σ upper limit for the mass is 236 MJup . Unfortunately the result has no significance,
as already the model without orbital motion could not be rejected. The 1σ limit almost
88
CHAPTER 6. RESULTS
spans the whole parameter space for the inclination, only angles smaller than 3◦ and
larger than 175◦ can be excluded with 3σ confidence.
The formal inclination value derived in this work is in rough agreement with the value
(i = 125.9◦ ) derived by the combination of the radial velocity data with the HIPPARCOS astrometric data (Kürster et al., 2008). Even though the HIPPARCOS astrometry
could also only yield a formal best fit value for the inclination and the ascending node
with low significance, they could set stronger constraints on the inclination. A 3σ upper limit for the companion mass of 112 MJup could be set with a lower limit for the
inclination of 15.6◦ and an upper limit of i = 161◦ .
I therefore could not detect the astrometric signal of the companion to GJ 1046, nor
could I set stronger constraints on its true mass.
Another point one has to keep in mind is the possible light contamination from the
companion. When determining the position of GJ 1046, I measured the photocenter
of the light distribution. If the companion itself has a certain brightness, the position
of the photocenter is shifted from the primary to a position between the two objects.
The direction of this shift changes with the orbital motion of the companion and can
scale down the observable orbital motion of the primary. The amount of flux the
companion contributs to the combined flux distribution depends on the mass and age
of the companion. As the energy distribution of very low mass stars and brown dwarfs
peaks in the near-IR and my observations are obtained in the K band, the contribution
of the companion will be more severe than for example in the V band, where the
spectroscopic measurement were conducted by Kürster et al. (2008). For a given age
one can calculate flux ratios as a function of the mass ratio of the two objects with
theoretical models (see e.g. Burrows et al., 2001, Fig. 1). But as the companion is
probably a brown dwarf which cools and dims with time, its age also plays an important
role in terms of its brightness. The age of GJ 1046 is likely >1 Gyr, but the brightness of
a 1 Gyr and a 5 Gyr old brown dwarf already differs notedly (see e.g. models by Baraffe
et al., 2003). This makes it so difficult to conclude masses of isolated brown dwarf from
photometry alone. As my measurement are not precise enough to detect any orbital
motion, I did not take the possible contamination by the companion into account in
my analysis. If, on the other hand, my measurements would have been precise enough
to detect the orbital motion, I would have needed to simulate the possible effect on the
measurements of the position of GJ 1046 and include the results either by correction
of the measured position or by adjusting the error budget.
Current AO imaging instruments have only small FoVs of a few tens of arcseconds. This
makes it difficult to find targets for astrometry which have several suitable reference
stars close by. The results from this work have shown, how important it is to have
more than one star in the same FoV which one can use as astrometric reference points.
Even a third star can already help to constrain the rotation of the detector in the very
same field and therefore offer the ability to correct for a differential rotation between
different observing epochs. Additional reference stars enhance the precision with which
the position of the target star is measured with respect to the other stars. The so-called
plate-solution can be derived in the target field itself, making it possible to correct for
differential distortions and plate-scale changes between the epochs.
Also a smaller separation between target and reference star is preferable. As shown, the
uncertainties in the final fitting parameters scale with the separation between target
6.2 DISCUSSION AND CONCLUSION
89
and reference source. Measurements of other groups have shown a similar precision
of the plate-scale and detector rotation. Neuhäuser et al. (2008) measured a pixelscale uncertainty of ±0.05 mas/px (= 0.38%) and a rotation stability of ±0.25◦ . The
calibration was made with a HIPPARCOS astrometric binary and the smaller camera
S13 of NACO, which has a FoV of 1400 × 1400 . Köhler (2008) used images obtained
in the Orion Trapezium cluster to monitor and calibrate the pixel-scale and detector
orientation. The results are a scatter of the pixel-scale smaller than 1% and a rotator
precision of a few tenths of a degree. Here the S13 camera was used, too. In the case
of the calibration with the binary, the change in pixel-scale is only measured in one
direction and the calibration measurements of both groups are obtained on a smaller
FoV than the measurements presented in this work. The scatter of the pixel-scale is
roughly the same in mas/px in all measurements, but due to the larger FoV and pixelscale of the S27 camera the relative scatter is smaller. This shows that a correction
for differential refraction and aberration before measuring the change in pixel-scale
enhances the precision. On the other hand, my measurements with different scales
fitted to the x- and y-axes show different values for the two directions (Fig. 4.9) and a
fit with only one scale yields bigger overall fitting errors. This indicates a differential
scale change for different directions in the field, hence the length of a given distance
on the detector depends on its location on the detector. The relatively large scatter
in the pixel-scale suggests that a global pixel-scale is probably not the correct ansatz
for obtaining high precision astrometry over larger separations, as there probably are
local pixel-scale changes due to distortions present in the frames. Knowledge of and
correction for these distortions, and correction of a left over change in pixel-scale will
lead to a higher precision in the separation.
At one epoch, without correction for a change in the pixel-scale to another epoch,
the positional precision achieved in this work is already promising. The change-over
to multi-epoch astrometry and the involved need for very well known pixel-scales and
distortions is the most challenging task. A very well characterized distortion analysis,
both spatial and temporal, would enhance the precision additionally.
A group analyzing the galactic center nicely shows the possible precision, but also the
limitation of astrometry with NACO (Trippe et al., 2008; Gillessen et al., 2009; Fritz
et al., 2010). They use the inner few arcseconds of the S13 camera for their analysis and
the S27 camera to set up a reference frame. As they have sufficient reference stars in
their FoV (∼ 100 − 200), they are able to calculate and perform a distortion correction,
already including a correction for the scale. With this method they are able to obtain
position uncertainties for the stars down to 0.6 mas for the S27 camera (Trippe et al.,
2008), for the S13 camera and the smaller FoV they get even smaller uncertainties. This
shows that with a good distortion correction between the frames a precision sufficient to
detect astrometric signals of large planetary companions is possible, but it also shows
that astrometry on large scales with NACO will probably not be able to detect or
characterize Earth-like planets.
90
CHAPTER 6. RESULTS
Chapter 7
Introduction to MCAO and
MAD
7.1
MCAO - The Next Generation of Adaptive Optics
As seen in the introduction to classical adaptive optics correction with one guide star
(Chap. 2.2), the FoV is limited by the effect of anisoplanatism, because only the integrated phase error over the column above the telescope in the direction to the guide
star is measured. Turbulence outside this column, e.g. in the direction of the target, if
it cannot be used as guide star, is not mapped and cannot be corrected. In the case of
a laser guide star as reference source this problem is even more severe, due to the low
focussing altitude and the resulting cone-effect (Chap. 2.2.3).
Multi Conjugated Adaptive Optics (MCAO) (Beckers, 1988; Ellerbroek et al., 1994) is
an approach to achieve diffraction limited image quality over bigger FoVs of up to 2
arcminutes and hence overcome anisoplanatism. Moderate Strehl-ratios, 10-25%, can
be achieved, but with a higher uniformity of the PSF shape over the FoV. This is desired
for resolving structures of extended sources, such as galaxies or cores of star clusters. In
MCAO the 3-dimensional structure of the turbulence is reconstructed by means of the
information coming from several guide stars, i.e. natural or laser guide stars. Instead
of correcting the turbulence integrated over the column above the telescope, which size
is defined by the isoplanatic patch, at once, turbulence from different layers is corrected
with several deformable mirrors conjugated to these layers. Typically two layers are
being corrected, the ground layer close to the telescope and a higher layer at around
8 - 10 km height. A full correction for the higher layer is only guaranteed if this layer
is fully covered by the footprints of the columns of the beam-paths from the reference
stars to the telescope. So the number of needed guide stars depends on the altitude
one wants to correct. Two different approaches exist to combine the signals from the
different reference stars, the Star Oriented (SO) approach and the Layer Oriented (LO)
approach.
91
92
CHAPTER 7. INTRODUCTION TO MCAO AND MAD
High-Layer
Ground-Layer
Telescope
GL - DM
HL - DM
WFC
WFC
HL - WFS
GL - WFS
Figure 7.1: Left: Principle of Multi Conjugated Adaptive Optics correction. Several guide
stars are used to probe the 3-dimensional structure of the atmosphere By illuminating the volume within the FoV and distinctive layers are corrected. A full
correction for the higher layer is only guaranteed if this layer is fully covered by
the footprints of the guide star metapupils. (Image taken from Kellner (2005)).
Right: Schematics of the layer oriented MCAO approach with two layers. WFS,
DM, and WFC are labels for the wavefront sensor, deformable mirror and wavefront computer for the high layer (HL) and ground layer (GL), respectively.
7.1.1
Star Oriented Approach
In the star oriented mode, each reference star is observed by one wavefront sensor and
typically one detector. The information from the different directions of the guide stars
is combined to generate 3D information of the atmosphere within the mapped FoV. In
this approach of turbulence tomography (Tallon and Foy, 1990) the influence of a single
layer can be computed and corrected with one deformable mirror conjugated to this
layer. The first verification of this approach was done in an open loop measurement at
the Telescopio Nazionale di Galileo (TNG) (Ragazzoni et al., 2000b).
7.1.2
Layer Oriented Approach
In the layer oriented approach (Ragazzoni et al., 2000a), each WFS and detector is
conjugated to one layer in the atmosphere instead to a single star. The light of several
guide stars is optically co-added to increase the SNR on the detector, such that also
fainter stars can be used as guide stars. This increases the sky coverage, the fraction
of regions on the sky that can offer a suitable asterism, substantially for this approach.
Also the number of needed wavefront sensors and detectors is reduced, reducing the
detector read-out-noise and the needed computing power compared to the SO approach.
Only as many detectors are needed as layers are being corrected and not as many guide
stars are used. Information from one WFS, and therefore one layer, can directly be fed
to the corresponding DM. This results in a combination of independent control loops
7.1 MCAO - THE NEXT GENERATION OF ADAPTIVE OPTICS
93
for the different layers and allows to adjust the integration time and bin size on the
detector independently to the characteristics of the conjugated layer.
Distortions introduced by turbulence in layers close to the layer for which correction
is attempted contribute stronger to the measurements than those from layers further
away. The further away a layer, the more its introduced aberrations are smoothed out
(Diolaiti et al., 2001).
A limitation of this approach can be the different brightnesses of the used guide stars.
This can lead to an overestimation of the turbulence from a certain direction, when the
light is co-added (Nicolle et al., 2004). An asterism of stars with similar brightness is
therefore preferable.
7.1.3
Ground Layer Adaptive Optics
As already pointed out in Chap. 2.1 most of the turbulence in the atmosphere is generated in the ground layer. Correcting only this layer, one can remove the major
contributor to the phase aberrations of the incoming wavefronts (Rigaut, 2002). Additionally, the correction is valid over a large FoV, as the light coming from different
directions passes through the same region of turbulence near the ground, because of its
small distance from the telescope pupil. In principle Ground Layer Adaptive Optics
(GLAO) corrections can be operated by any MCAO system, operating just one single
correction loop conjugated to the ground layer. In this context a Rayleigh Laser Guide
Star (LGS), which is focused to about 5 - 10 km altitude, can be used, as it automatically illuminates only the ground layer and is not usable for correction of higher layers
but still valid for ground layer corrections (Morris et al., 2004).
7.1.4
Current and Future MCAO Systems
Several MCAO instruments are planned and being built for different telescopes. The
first on-sky tested MCAO system is the MAD instrument at the ESO/VLT. It will be
described in detail in the next section.
At the Gemini South observatory on Cerro Pachon, Chile, a laser guide star assisted
MCAO System, GeMS, is being installed with first light expected soon. The system
will consist of 5 laser guide stars and additionally up to three natural guide stars. The
single 50 W laser is split into five beams which are launched from behind the secondary
mirror. The laser beacons will be located in the middle and the four corners of the
1.200 ×1.200 FoV. Three deformable mirrors are going to be used to correct for turbulence
in three layers: at ground level, 4.5 km and 9 km altitude. Shack-Hartmann wavefront
sensors will be used in this star oriented MCAO approach.
The Fizeau-Interferometer LINC-NIRVANA for the Large Binocular Telescope (LBT)
on Mt. Graham in Arizona, will be equipped with four layer oriented correction units,
two for each telescope, which will correct the ground layer and a high layer (e.g. Farinato
et al., 2008). In the so-called LINC mode the instrument will work with a classical single
guide star AO system. In the later implementation phase called NIRVANA, the MCAO
system will be able to use up to 12 NGS per side for the GLAO plus up to 8 NGS for
94
CHAPTER 7. INTRODUCTION TO MCAO AND MAD
the high layer adaptive optics correction (HLAO). The stars for GL correction can be
located anywhere in a ring with an inner diameter of 20 and an outer diameter of 60
around the central science field of 10.500 ×10.500 and the guide stars for the HLAO inside
a circle with a diameter of 2 arcminutes.
7.2
MAD - Multi conjugated Adaptive optics Demonstrator
The ESO Multi conjugated Adaptive optics Demonstrator (MAD) is a prototype MCAO
instrument, which was used to test different MCAO reconstruction techniques in the
laboratory and on sky (Hubin et al., 2002; Marchetti et al., 2003; Arcidiacono et al.,
2006). After extensive testing in the laboratory it was installed at the Nasmyth-focus
platform of the ESO VLT UT3 telescope Melipal in the beginning of 2007. Because
the instrument bench is fixed to the Nasmyth-platform, the pupil co-rotates with the
field and an optical de-rotator at the entrance of the adaptive optics system is needed,
as well as for the science camera.
MAD is designed to characterize the performance of both MCAO approaches, the star
oriented one and the layer oriented one. It is optimized for corrections in the Ks
band (2.2µm) over a circular 20 × 20 FoV using natural guide stars (NGS). For the
SO approach a multi Shack-Hartmann wavefront sensor, consisting of three movable
Shack-Hartmann WFS (SHS) to look at any star present in the FoV, is implemented.
SHS consist of a lenslet array that samples the incoming WF. Each lens forms an image
of the guide star and the displacement of this image from a reference position gives an
estimate of the slope of the local wavefront at that lenslet. The guide stars used for
sensing the distortions should not be fainter than mv = 14. The layer oriented approach
uses a multi Pyramid wavefront sensor (PWS) which is capable to sense up to 8 NGS
simultaneously. The PWS consists of a four-sided glass pyramid which tip is located
in the focal point of the telescope and it measures, similar to the SHS, the local slope
2
5
1
4
3
Figure 7.2: Examples for jitter offsets with MAD to scan the 20 × 20 FoV. Left: Mosaic of
5 pointings covering the full 20 × 20 FoV. Right: Jitter offsets for 5 pointings
around the center of the FoV.
7.2 MAD - MULTI CONJUGATED ADAPTIVE OPTICS DEMONSTRATOR
95
Figure 7.3: 3-dimensional view of the MAD bench with the major instrument components
marked. The light beam enters at the de-rotator, passing the two deformable
mirrors (DM) and is split by the dichroic into the IR part which is led to the
science camera and the visible part which is led to the wavefront sensor objective.
The layer oriented wavefront sensor is located above the star oriented multi
Shack-Hartmann (SH) sensor.
of the wavefront. A uniform distribution of these stars is preferable but they can be
everywhere in the 20 × 20 FoV. The two different WFSs cannot be used simultaneously,
a selector folds the light either to the SO-WFS or the LO-WFS.
Two deformable mirrors conjugated to the ground layer and a layer at 8.5 km altitude
correct for the turbulence induced phase errors. A dichroic splits the light into an IR
part (1.0 − 2.5 µm), led towards the science detector and a visual part (0.45 − 0.95 µm)
led towards the wavefront sensor path. The CAMCAO (CAmera for MCAO) IR camera, built by the Faculdade de Ciências da Universidade de Lisboa (FCUL), is the
science camera of MAD (Amorim et al., 2004). It consists of a 2k × 2k Hawaii2 IR
detector with a pixel scale of 0.028 arcsec/pixel and a FoV of 5700 × 5700 . The CAMCAO optics provide diffraction limited images down to the J (1.25 µm) band, and it
is equipped with standard IR band filters for J (1.25 µm), H (1.65 µm), Ks (2.2 µm),
Br-gamma (2.165 µm) and Br-gamma continuum. In Fig. 7.3 a detailed sketch of the
layout of the instrument is shown. Unlike other instruments, with MAD the camera
and not the telescope is moved to jitter within the field of view. Jitter movements are
small offsets from the central pointing to avoid bad pixel coincidences on always the
same position in the science field, to better estimate the sky background and/or to
scan a larger FoV. The detector can cover the full 20 × 20 FoV by moving into the four
adjacent 10 × 10 quadrants using linear x-y stages, see Fig. 7.2 for examples of jitter
96
CHAPTER 7. INTRODUCTION TO MCAO AND MAD
offsets for observations with MAD.
On sky testing of the star oriented mode was started in February/March 2007 with
the first closing of the MCAO loop on March 25th (Marchetti et al., 2007). This first
demonstration run consisted of 8.5 effective nights spread over 12 nights in total, with
another demonstration run following for the SO mode. During the third demonstration
run in September 2007 the LO mode was tested (Arcidiacono et al., 2008; Ragazzoni
et al., 2008; Falomo et al., 2009). After the very successful demonstrations of the
performance of MAD, three public science demonstration runs for the star oriented
mode followed in November 2007, January 2008 and August 2008, where high resolution
galactic as well as extragalactic science was performed (see e.g. Bouy et al., 2009; Wong
et al., 2009).
7.3
Goal of this Work
The reason for analyzing MAD data was the uniqueness of this very first MCAO data.
I want to analyze the stability of the multi conjugated adaptive optics correction regarding the potential achievable astrometric precision. MCAO observations offer the
advantage of a big field of view with a resolution close to the diffraction limit. Although
most of this advantage is seen in the enhancement in photometric studies, there are
also some very interesting applications in the field of astrometry. Cluster dynamics,
measuring velocity dispersions and common proper motions is just one of the many
possibilities. Another very interesting case is the possibility of detecting the astrometric signal of a planetary companion orbiting its star. Limitations of today’s AO
based searches are often the small FoV of classical AO imagers. Having an 10 × 10 FoV
with several reference stars will enable more precise astrometric measurements. But
before starting with such science, one should check and analyze the performance of
these systems, to see which effects this special kind of AO system has on astrometric
measurements. When the first observations with such a system, the MAD instrument,
were performed I had the chance to analyze some of these data. I decided to work with
the data obtained in the layer oriented mode, as this will be the approach the future
LBT instrument LINC NIRVANA will work with.
The goal of this work is to see how stable the AO performance of MAD is over time in
terms of astrometric stability and to measure the achievable positional precision in the
first MCAO layer oriented data.
Chapter 8
Astrometry with MAD
8.1
Observations
The observations analyzed here were conducted with the MAD instrument and the
multi pyramid sensor in the LO mode. I analyzed data from two globular clusters,
47 Tucanae (NGC 104) and NGC 6388, which were observed during the demonstration
run in September 2007 by our colleagues from the INAF Osservatorio Astronomico di
Padova, Italy (Arcidiacono et al., 2008; Ragazzoni et al., 2008; Falomo et al., 2009).
Due to bad luck with the observing conditions, these two sets are the best data obtained
during the only layer oriented run carried out with MAD. In the case of the data of
47 Tuc I analyzed the data of the central 5700 × 5700 FoV in the context of this work.
In the case of the NGC 6388 cluster I analyzed a data set observed under good initial
seeing conditions (0.4600 ), which lies at an outer part of the cluster. Another field in
the center of the cluster was also observed, but with an initial seeing of 1.7600 , which I
did not analyze in this work. Unfortunately there will be no more data to compare my
results with, as MAD is no longer offered.
I want to emphasize that the MAD instrument, as its name already says, was built to
demonstrate that MCAO correction over a big field of view is possible. It did this with
great success! But one also has to keep in mind, when interpreting the results shown
in the next chapters, that this data is test data with all the possible problems during
first observations. Not the full performance of this new AO correction technique can
be expected, for this one has to wait for the next generation of MCAO instruments,
which will be fully optimized. Nevertheless, this is the first attempt to analyze and
characterize a layer oriented MCAO system with respect to its astrometric performance.
8.1.1
GLAO - 47 Tuc
The observations of the globular cluster 47 Tuc were obtained on September 22nd 2007
using only the Ground Layer Adaptive Optics (GLAO) approach. The center of the
cluster, RA(J2000)=00:24:05.6, DEC(J2000)=-72:04:49.4, was observed in the Br-γ filter. The camera was not moved during the exposures to scan the full FoV but instead
97
98
CHAPTER 8. ASTROMETRY WITH MAD
Figure 8.1: MAD image of the core of the globular cluster 47 Tuc. The triangles mark
the positions of the AO guide stars relative to center of the observed FoV. The
numbers close to the stars correspond to their F606W (visual) magnitude.
stayed at the same position, observing a 5700 × 5700 field. Four guide stars with magnitudes V = 11.9, 11.9, 12.4 and 12.5 mag1 , corresponding to an integrated magnitude
of 10.63 (Arcidiacono et al., 2008), positioned around this field with one guide star in
the lower left corner of this field were used to sense the wavefront distortions due to
the atmosphere, see Fig. 8.1. Altogether I have 19 frames with NDIT = 15 exposures
of an integration time of DIT = 2 seconds. Each frame is averaged to correspond to
a 2 seconds exposure. Seven sky frames were obtained before the cluster observation
with the same values for DIT and NDIT and also averaged to frames with an exposure
time of 2 seconds each. In Table 8.1.1 the observation is summarized together with the
atmospheric conditions during the observations, indicated by the seeing value and the
performance of the system indicated by the value of the fitted FWHM.
1
HST F606W photometry data
8.1 OBSERVATIONS
Object
47 Tuc
frame
99
DIT
NDIT
rel. jitter
x
y
AO correct.
seeing [00 ]
FWHM [00 ]
in V
in Brγ
1
2
15
0
0
GLAO
1.09
0.178
2
2
15
0
0
GLAO
1.15
0.186
3
2
15
0
0
GLAO
1.13
0.206
4
2
15
0
0
GLAO
1.08
0.169
5
2
15
0
0
GLAO
1.09
0.178
6
2
15
0
0
GLAO
1.08
0.147
7
2
15
0
0
GLAO
1.15
0.157
8
2
15
0
0
GLAO
1.17
0.193
9
2
15
0
0
GLAO
1.17
0.173
10
2
15
0
0
GLAO
1.15
0.178
11
2
15
0
0
GLAO
1.14
0.145
12
2
15
0
0
GLAO
1.15
0.148
13
2
15
0
0
GLAO
1.11
0.144
14
2
15
0
0
GLAO
1.15
0.166
15
2
15
0
0
GLAO
1.14
0.183
16
2
15
0
0
GLAO
1.15
0.183
17
2
15
0
0
GLAO
1.19
0.187
18
2
15
0
0
GLAO
1.13
0.200
19
2
15
0
0
GLAO
1.11
0.174
Table 8.1: Summary of the observations of the cluster 47 Tuc. The seeing value is measured
by the DIMM seeing monitor in V band and the FWHM value corresponds to the
one measured in the data (see Chap. 8.5)
8.1.2
MCAO - NGC 6388
The data of the globular cluster NGC 6388 was obtained on September 27th 2007
using the full MCAO capability of MAD. The observations are in the Ks filter using 5
guide stars with V = 15.0, 15.0, 15.6, 15.7 and 16.3 mag, corresponding to an integrated
magnitude of 13.67 (Arcidiacono et al., 2008). The field together with the guide stars
is shown in Fig. 8.2. The observed field lies at the lower left corner of the cluster
at RA(J2000)=17:36:22.86, DEC(J2000)=-44:45:35.53. All together 30 frames were
obtained, the first five in GLAO and the last 25 in full MCAO. A jitter pattern of five
positions was used, repeated six times with three slightly different central points, to
scan part of the 20 × 20 FoV (Fig. 8.3). The first 10 frames were obtained with DIT
= 10 seconds and NDIT = 24, and in the last 20 frames the number of exposures was
reduced to NDIT = 12. All frames are averaged to correspond to a 10 second exposure.
100
CHAPTER 8. ASTROMETRY WITH MAD
Figure 8.2: MAD image of the globular cluster NGC 6388. The triangles mark the positions
of the AO guide stars relative to center of the observed FoV. The numbers close
to the stars correspond to their F606W (visual) magnitude.
After the science frames, five sky frames were taken outside the cluster using the same
jitter pattern as in the science frame. At each position 24 frames with 10 seconds
exposure time, averaged to one frame of 10 seconds each, were obtained.
In Table 8.1.2 the observations are summarized together with the atmospheric conditions during the observations, indicated by the seeing value.
8.2
Data Reduction
NGC 6388
Each science frame of the NGC 6388 cluster data was flatfielded by the flatfield image
obtained from sky flats at the beginning of the night and badpixel corrected, by replacing the marked pixels with the median of the pixels themselves and their 8 nearest
8.2 DATA REDUCTION
Object
frame
NGC 6388
DIT
101
NDIT
rel. jitter
x
y
AO correct.
seeing [00 ]
FWHM [00 ]
in V
in Ks
1
10
24
0
7
GLAO
0.43
0.098
2
10
24
5
12
GLAO
0.41
0.094
3
10
24
5
2
GLAO
0.49
0.099
4
10
24
-5
12
GLAO
0.55
0.094
5
10
24
-5
2
GLAO
0.51
0.090
6
10
24
0
7
MCAO
0.41
0.095
7
10
24
5
12
MCAO
0.38
0.097
8
10
24
5
2
MCAO
0.37
0.098
9
10
24
-5
12
MCAO
0.39
0.103
10
10
24
-5
2
MCAO
0.40
0.106
11
10
12
2
5
MCAO
0.45
0.126
12
10
12
2
5
MCAO
0.43
0.117
13
10
12
7
10
MCAO
0.45
0.130
14
10
12
7
10
MCAO
0.50
0.158
15
10
12
7
0
MCAO
0.51
0.130
16
10
12
7
0
MCAO
0.49
0.155
17
10
12
-3
10
MCAO
0.48
0.170
18
10
12
-3
10
MCAO
0.49
0.140
19
10
12
-3
0
MCAO
0.45
0.119
20
10
12
-3
0
MCAO
0.41
0.133
21
10
12
-3
6
MCAO
0.42
0.120
22
10
12
-3
6
MCAO
0.44
0.131
23
10
12
2
11
MCAO
0.54
0.139
24
10
12
2
11
MCAO
0.56
0.158
25
10
12
2
1
MCAO
0.43
0.176
26
10
12
2
1
MCAO
0.50
0.153
27
10
12
-8
11
MCAO
0.46
0.143
28
10
12
-8
11
MCAO
0.48
0.135
29
10
12
-8
1
MCAO
0.47
0.120
30
10
12
-8
1
MCAO
0.47
0.134
Table 8.2: Summary of the observations of the cluster NGC 6388. The seeing value is measured by the DIMM seeing monitor in V band and the FWHM value corresponds
to the one measured in the data (see Chap. 8.5)
102
CHAPTER 8. ASTROMETRY WITH MAD
Camera Positions rel. to Center of FoV
10
Dd ["]
4,9
17,18
0
21,22
-5
29,30
-10
-10
5,10
19,20
-5
23,24
2,7
13,14
45.4’’
27,28
5
1,6
11,12
25,26
0
Da ["]
2’
42.5’’
3,8
15,16
5
10
2’
Figure 8.3: Left: Jitter pattern of the observations of NGC 6388. The offsets are in seconds
of arc and relative to the center of the FoV. The numbers indicate the frame
which was taken at that position. Note the change of taking two images at one
position for frames 11-30 and then moving to the next jitter position instead of
taking one frame per position and execute the jitter pattern twice for frames 110. Right: Frames taken in the 20 × 20 FoV and the finally cut-out area common
to all frames drawn with thick lines. This common area was used to investigate
the astrometric precision.
neighbors, with a badpixel mask obtained from the same flat-field images. I did not
create the flatfield and badpixel mask by my own in this case, but used those made by
our colleagues from Italy. A description of how these frames were created can be found
in (Moretti et al., 2009). Briefly, the obtained sky flats were median combined to one
flatfield and also used to create the badpixel mask.
Sky subtraction was done by median combining all sky and science frames to get one
single sky frame. Using only the sky frames obtained right after the science frames to
create the sky frame, did not work satisfactorily. As only five sky-frames were available,
the remaining stars in the frames did not average out perfectly, leaving small holes in
the science frames after subtracting the sky frame. Also κ − σ clipping did not yield
a satisfactory result. Using all science frames together with the sky frames to create a
median image for the sky estimation yielded the best result. Therefore, this resulting
image was used for sky subtraction. This sky frame was then normalized to the median
counts of the science frame before subtraction.
Additionally, NaN and infinite values which were still in the frames after the data reduction, were substituted by the median of their 8 nearest neighbors, as the routines
for the following analysis had problems with such pixel values.
In the first demonstration runs with MAD a problem occurred. Unfiltered light was
reflected on the instrument bench and could enter the science camera, producing a
banana like arc on the lower left side of the images. The intensity and position of this
arc depends on the position of the camera while jittering to scan the FoV. This reflection could be avoided in the later science demonstration runs by shielding the camera
entrance with a tube. Nevertheless, I see this arc in the images, see Fig. 8.4. As it
is position dependent, the arc cannot be removed with the flatfield correction or sky
subtraction. The distribution of the arc is always close to the border of the detector
8.2 DATA REDUCTION
103
and only affecting a small portion of the stars. While this extra, unshielded light is for
sure a problem for photometry, its influence on the astrometric positions of the stars
may be small. Nevertheless, I tried to avoid stars, affected by the arc in my analysis.
In the case of the NGC 6388 data, jittering was used during the observations and I cut
the images to the common FoV which was part of all images after the data reduction.
That left us with a slightly smaller field of the size of 1517 px×1623 px (42.500 ×45.400 ),
see Fig. 8.3, right panel. Only the stars in this common field are taken into account for
the following astrometric analysis.
Figure 8.4: Arc in the MAD frames produced by unshielded and unfiltered reflected light.
Several examples from different jitter positions show the intensity dependence
of this arc on the jitter position. Some frames have a prominent arc, whereas it
is not visible in others.
104
CHAPTER 8. ASTROMETRY WITH MAD
47 Tuc
In the case of 47 Tuc no flat-field images were taken in Brγ (central wavelength
= 2.116µm) in that night. As I had no other choice I used the same flatfield image for
correction as for the NGC 6388 data observed in Ks (central wavelength = 2.15µm),
assuming that the pixel to pixel variations are the same or very similar in these two
filters, whose central wavelength is not too different. As the data I got from our collaborators for the cluster 47 Tuc was already sky subtracted and corrected for the reflecting
arc, I only corrected for the flatfield, badpixel and NaN and infinite pixel values in the
same way as for the NGC 6388 data. The data of 47 Tuc was obtained without moving
the camera. All frames contain the same field and I did not cut the single frames.
8.3
Strehl Maps
As a check of the AO performance I calculated Strehl-maps for each frame2 . A theoretical diffraction limited PSF for MAD was computed and normalized to a flux of one.
With the Source Extractor package (Bertin and Arnouts, 1996) the stars in the frame
were detected and aperture photometry was used to calculate the flux of the stars.
After normalizing the flux to one, a comparison of the theoretical peak value and the
maximum value of the star yields the Strehl ratio. After interpolating values for areas
where no stars were found a smooth surface was fitted to the data, leading to a two
dimensional Strehl-map for each frame. In Fig 8.5 and 8.6 some of these Strehl-maps
are shown, the first, middle and last frame. The Strehl-maps for all frames are attached
in Appendix A.
In the case of NGC 6388 a slight degradation in performance with time can be seen,
following roughly the change in the seeing. The first five frames obtained with GLAO
have the highest Strehl ratio and do not change with the changing seeing conditions.
The Strehl is fairly even over the field of view of the camera with a small drop-off to
the edges of the field. This shows how well and uniformly the layer oriented MCAO
approach corrects wavefront distortions. The drop-off to the edges of the FoV can
partly be explained by the MCAO and the atmospheric tomography approach. The
light coming from the different directions of the guide stars is optically co-added and
a correction is computed based on this light distribution. But the footprints of the
columns above the telescope in the direction of the guide stars overlap more in the
middle of the field than at the edges in the higher layer. If the control software is not
optimized to correct over the whole FoV very evenly, the middle of the field will be
corrected better. As the data I am looking at is the first data of MCAO and layer
oriented mode, I am not surprised to see such an effect. A performance evaluation of
these data can be found in Arcidiacono et al. (2008). In the case of 47 Tuc the Strehl is
smaller than in the case of the NGC 6388 data. The main reason for that is not that this
data is taken with correcting only the ground layer, but more due to the circumstance
that already the initial conditions for the AO correction was very poor with a seeing of
2
Program provided by Felix Hormuth
8.3 STREHL MAPS
105
Figure 8.5: Strehl-maps for some of the mad frames of the cluster NGC 6388. Shown are
the first (GLAO), middle and last exposure (both MCAO). The degradation in
performance with time, as described in the text, can be seen. A collection of the
Strehl-maps for all frames can be found in Appendix A.
106
CHAPTER 8. ASTROMETRY WITH MAD
Figure 8.6: Strehl-maps for some of the mad frames of the cluster 47 Tuc obtained with
GLAO. Shown are the first, middle and last exposure. A collection of the Strehlmaps for all frames can be found in Appendix A.
8.4 PSF TESTS
107
1.09 - 1.19 arcseconds. An AO system can only enhance the performance significantly,
if the initial atmospheric conditions are fairly good. Nevertheless, an even Strehl ratio
of ∼ 15 % over a 10 × 10 FoV is already an enhancement compared to the seeing limited
case.
8.4
PSF Tests
Before I decided what kind of model or empiric PSF I should use to fit the positions of
the stars in the frames, I conducted several tests of the PSFs in the frames. I analyzed
the distributions of shape and orientation of the PSF over the field of view, to check if I
can fit the same PSF to all stars or have to use different PSFs for different parts of the
FoV, which is mostly the case in classic single guide star adaptive optics imaging. As I
first only had the data of the cluster NGC 6388, I conducted all these PSF tests with
this data set and based the following decisions and steps on the consequential results.
Later, after the analysis of the NGC 6388 data set was finished, I got the opportunity
to also analyze GLAO data of the cluster 47 Tuc for which I conducted the same tests.
Though they show certain differences, I nevertheless analyzed this data set in the same
way as the NGC 6388 data, to conserve the possibility of a direct comparison of the
results of the two data sets.
For the PSF tests I first used the StarFinder program (see next section for details)
to measure the positions of the stars in the field. I then cut a quadratic box around
each star and fitted a two dimensional, rotatable Moffat function (see Equ. 4.1) to
each star, using the non-linear least square fitting package MPFIT2DPEAK, provided
by Craig Markwardt (Markwardt, 2009). After flagging those stars which were too
close to the frame edges or other stars, as well as those where the fit with the Moffat
function obviously gave wrong parameters, I worked with the remaining 780 - 800 stars
in both data sets. The fitting routine has some problems if more than one star is
present in the box cut around the star which one wants to fit. If the two stars have
similar brightness, the valley between them is fitted and if the second star is brighter
than the one gained for, always the brighter one is fitted. I therefore checked for large
changes in the position and brightness of the fitted Moffat function to the values given
by StarFinder and flagged those stars where the fit did not went well.
Looking at the distribution of the rotation angle Θ of the fitted Moffat function, a small
dependency of the orientation of the PSFs with the position of the stars can be seen
in the case of the NGC 6388 data. The angle is measured clockwise from the detector
x-axis to the major axis and is in the range [− π2 , π2 ]. In Fig. 8.7, left panels, an example
of the distribution of the orientation of the PSF is shown for one frame of the MCAO
frames of the NGC 6388 data. The upper panel shows the distribution as function
of the distance of the stars from the center of the detector. The middle panel the
distribution as function of the distance in x-direction and the lower panel as function
of the y-distance. Although it seems, that fewer orientations near Θ = 0◦ are fitted,
this effect is not dependent on the distance of the stars from the center. I calculated
correlation coefficients for these plot, yielding values of 0.019, 0.438 and -0.093 for the
distance from the center, the x-direction and the y-direction respectively. Whereas the
108
CHAPTER 8. ASTROMETRY WITH MAD
Orientation of PSF
Eccentricity of PSF
100
1.0
0.8
Eccentricity
Θ [deg]
50
0
0.6
0.4
−50
0.2
−100
0
10
20
30
Distance to center of detector ["]
0.0
0
40
Orientation of PSF
40
Eccentricity of PSF
100
1.0
0.8
Eccentricity
50
Θ [deg]
10
20
30
Distance to center of detector ["]
0
0.6
0.4
−50
0.2
−100
−30
−20
−10
0
10
20
x−Distance to center of detector ["]
0.0
−30
30
Orientation of PSF
30
Eccentricity of PSF
100
1.0
0.8
Eccentricity
50
Θ [deg]
−20
−10
0
10
20
x−Distance to center of detector ["]
0
0.6
0.4
−50
0.2
−100
−30
−20
−10
0
10
20
y−Distance to center of detector ["]
30
0.0
−30
−20
−10
0
10
20
y−Distance to center of detector ["]
30
Figure 8.7: Distribution of the orientation and eccentricity of the PSF in one of the MCAO
frames of the NGC 6388 data as an example. The upper panels show the distribution as function of the distance of the stars from the center of the detector.
The middle panels the distribution as function of the distance in x-direction and
the lower panels as function of the y-distance.
8.4 PSF TESTS
109
1.0
Eccentricity
0.8
0.6
0.4
0.2
0.0
−100
−50
0
Θ [deg]
50
100
Figure 8.8: Correlation of the eccentricity and orientation of the PSF in one of the MCAO
frames of the NGC 6388 data.
coefficients for the center distance and the y-direction are not significant, the one for
the x direction shows a moderate correlation. I plotted the fitted Moffat functions as
enlarged ellipses overimposed on the positions of the stars. Fig. 8.9 shows the distribution in the right panel. One can see the different orientation of the stars in the lower
right corner compared to the ones in the upper left corner. This behavior changes
only slightly over the different frames. In the case of plotting angles, one has to be
careful with the interpretation of the correlation coefficient, because of the periodicity
of the angle. Plotting the distribution with different cut values, as for example [0, π] or
[ π2 , 23 π], can yield different correlation coefficients. I therefor calculated the coefficient
for different cut values. The correlation coefficients changes slightly, but yielded all to
the same interpretation as above.
I also looked at the shape, i.e. the eccentricity, of the fitted√ Moffat functions. In
2
2
Fig. 8.7, right panels, the distribution of the eccentricity, e = a a−b as a function of
distance to the detector center is shown. Here a is the half width at half maximum
(HWHM) of the larger axis and b the HWHM of the smaller axis. Although the mean
eccentricity is ∼ 0.4, no dependency of the shape of the PSF with the position of the
stars on the detector is observable. The calculated correlation coefficients are 0.125,
-0.216 and 0.115 for the distance from the center, the x direction and the y direction
respectively. All these values show a very weak correlation. I also looked at the distribution of the eccentricity in the four quadrants of the detector, to check if there is
any correlation, as for example one of the guide stars is located close to the upper left
quadrant for the NGC 6388 observations (see Fig. 8.2). Each quadrant shows a similar
distribution of eccentricities.
As a last check I looked for a correlation between the orientation of the PSF and its ec-
110
CHAPTER 8. ASTROMETRY WITH MAD
Orientation and Shape of PSFs, NGC 6388
2000
1500
1500
y−Direction [pixel]
y−Direction [pixel]
Orientation and Shape of PSFs, 47 Tuc
2000
1000
500
1000
500
0
0
0
500
1000
1500
x−Direction [pixel]
2000
0
500
1000
1500
x−Direction [pixel]
2000
Figure 8.9: Fitted Moffat functions, displayed as enlarged ellipses at the positions of the
stars, showing the orientation and shape of the PSFs in the 47 Tuc data in the
left panel. The different sizes of the ellipses reflect the different fitted FWHM
of the Moffat functions. The red ellipse marks the guide star contained in the
detector FoV. For comparison the same plot is shown for a MCAO frame of the
NGC 6388 data in the right panel with the same enlargement factor.
centricity. As one can see in Fig. 8.8 there is no preferred eccentricity for a specific angle.
Compared to the single guide star AO correction, where the stars are elongated in the
direction to the guide star and the effect is stronger the further away the star is from
the guide star, these results already show the advantage and improvement of observing
with MCAO.
And although I see some small spatial correlations, I decided to use one PSF model
to fit all stars in the field in the way as described in the next section (8.5). Using the
Moffat fit in all frames was also not an option, because of the above mentioned fitting
problems with close stars pairs.
When I later analyzed the data of the cluster 47 Tuc, I performed the same tests. As the
initial observing conditions were worse than in the case of the NGC 6388 data, also the
overall performance was a bit worse (compare Tables 8.1.1 and 8.1.2 and Figs. 8.5 and
8.6). Already in the image one can see an elongation of the stars in one direction. The
analysis of the orientation of the fitted Moffat function confirms this impression, see
Fig. 8.10. A clear preference of orientation angles between Θ = −30◦ and Θ = −50◦ ,
with an average of Θ = −43.5◦ , is recognizable, with a small dependency on the ypositions of the stars in the field. Stars in the upper part of the detector seem to be
oriented a bit more to the y-direction than the stars in the lower part of the field, see
also Fig. 8.9. The eccentricity distribution, with a mean eccentricity of 0.55, also does
only show a very weak dependency on the position on the detector. But the mean
8.4 PSF TESTS
111
Orientation of PSF
Eccentricity of PSF
100
1.0
0.8
Eccentricity
Θ [deg]
50
0
0.6
0.4
−50
0.2
−100
0
10
20
30
Distance to center of detector ["]
0.0
0
40
Orientation of PSF
40
Eccentricity of PSF
100
1.0
0.8
Eccentricity
50
Θ [deg]
10
20
30
Distance to center of detector ["]
0
0.6
0.4
−50
0.2
−100
−30
−20
−10
0
10
20
x−Distance to center of detector ["]
0.0
−30
30
Orientation of PSF
30
Eccentricity of PSF
100
1.0
0.8
Eccentricity
50
Θ [deg]
−20
−10
0
10
20
x−Distance to center of detector ["]
0
0.6
0.4
−50
0.2
−100
−30
−20
−10
0
10
20
y−Distance to center of detector ["]
30
0.0
−30
−20
−10
0
10
20
y−Distance to center of detector ["]
30
Figure 8.10: Distribution of the orientation and eccentricity of the PSF in one frame of the
47 Tuc data as an example. The upper panel shows the distribution as function
of the distance of the star from the center of the detector. The middle panel
the distribution as function of the distance in x-direction and the lower panel
as function of the y-distance.
112
CHAPTER 8. ASTROMETRY WITH MAD
eccentricity is higher than in the case of the NGC 6388 data obtained under good initial conditions. One of the used guide stars is in the lower left corner of the FoV. If
the shape of the stellar PSFs depended on the position of this guide star, Θ should
change over the field depending on the position angle between the stars and the guide
star. Instead the guide star itself is elongated and oriented in the same way as the
stars around it (Θ = 30.25◦ , e = 0.59). In the left panel of Fig. 8.9 the fitted Moffat
functions are displayed as ellipses at the positions of the stars, showing the orientation
and shape of the PSFs. This shows nicely the principle of the layer oriented correction
approach, where the light of all guide stars is added and measured for the correction of
a certain layer and the correction itself is not so much dependent on the single position
of the guide stars, as in the classical, single guide star approach. On the other hand it
also shows, that the correction was not yet perfect in this data set, leaving this residual
elongation and orientation of the stars.
Nevertheless I continued measuring the positions of the stars and their uncertainties
in the same way as in the case for the data of the cluster NGC 6388 to compare the
two sets later under the aspect of GLAO versus MCAO correction. Most of the stars
in the 47 Tuc data show the same orientation and eccentricity, making it again feasible
to work with one PSF model for the full FoV.
8.5
Position Measurements
To measure the positions of the stars in the single images of both clusters I used the
program StarFinder (Diolaiti et al., 2000b,a), which is an IDL based code to analyze
AO images of stellar fields. The following description is for 47 Tuc, but it is the same
for the NGC 6388 data.
I directly extracted the Point Spread Functions (PSF) for the star fits from the images,
by using in each frame the same 30 stars. In Fig. 8.11 the stars used for the PSF
extraction are marked for both fields.
I assumed here that the PSF does not vary strongly over the FoV, as would be the
case in single guide star adaptive optics. But as seen in the analysis before (Sect. 8.4),
the PSF did not show a huge variation over the field and as I do not intend to do
photometry, I assume that the center of the PSF can still be determined accurately
enough from a fit with an averaged PSF. In Fig. 8.12, (left), an example of such an
extracted PSF is shown. The flux is normalized to one. The 30 stars were chosen
to be equally distributed over the entire field and to give a good mixture of brighter
and fainter stars. After marking the stars, StarFinder provides the opportunity to
remove secondary sources such as close stars, which may be in the box cut around the
marked stars. The final PSF is then generated by a median combination of the selected
stars. At the end I cut the square box containing the PSF into a round one of 80 pixel
diameter, which corresponds more to the base of a PSF.
Some of the brighter stars are saturated. In the case of MAD with its IR detector these
8.5 POSITION MEASUREMENTS
113
Figure 8.11: Stars used for PSF extraction in the fields 47 Tuc (left) and NGC 6388 (right).
Note the different scales: The 47 Tuc field is 5700 × 5700 and the NGC 6388 field
is cut to 42.500 × 45.400 .
saturated stars do not have a flat plateau in place of the PSF tip, but instead have a
hole in the middle (Fig. 8.12, right).
StarFinder had some problems, fitting those stars, as the remaining ring of the PSF
is not constant in flux. Several stars were fitted along the ring. In principle it should
be possible to also fit the extracted PSF to the saturated stars by first filling the holes
and then using a function of StarFinder, which repairs the PSF during the fit. But as
it is difficult to evaluate how the position measurement is affected by this procedure
and I had only a few saturated stars in the field, I decided to not use these stars in
the following analysis. After deleting the false detections in all frames, ∼ 1450 stars
remained in the 47 Tuc data and ∼ 1500 stars in the first 10 frames of the NGC 6388
data and ∼ 600 in the last 20 frames. The huge difference in the numbers for the
NGC 6388 data is due to the change in integration time after the first 10 frames (see
Tab. 8.1.2), that led to a lower signal to noise ratio in the averaged frame and therefore
to fewer detections of the faint stars.
The position uncertainties for the stars were computed with photon statistics as an
estimate. The extracted PSF from each frame was cut at the value of half maximum.
Then a rotatable ellipse was fitted to the slice-plane, giving values for the semi-major
and -minor axes of this ellipse, corresponding to the HWHM of the two axes. After parametrization the ellipse equation I calculated the maximum projection values,
xmax , ymax , of the semi-major axis onto the x- and y-axes. The positional errors,
∆x, ∆y, for the single stars of each frame were then computed following photon statistics by:
xmax
∆x = √
n
ymax
∆y = √
n
(8.1)
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CHAPTER 8. ASTROMETRY WITH MAD
Figure 8.12: Left: Example of a with StarFinder extracted PSF. The flux is normalized to
one. Right: Example of a saturated PSF. The PSF has a hole instead of a tip.
The different peaks at the ’top-ring’ can clearly be seen, causing StarFinder to
fit several stars instead of one.
where n is the total number of photons. To derive n one has to take into account
the gain factor of the detector, g = 2.9e− /ADU for the CAMCAO camera, and the
number of exposures, NDIT, averaged to create this frame. In Tab. 8.1.1 and 8.1.2, last
column, the mean FWHM value, calculated as the mean of the FWHM of the minorand major-axis, is noted. This approach is somewhat conservative compared to the
√
classical approach, calculating the error by σ/ n, but I did not want to underestimate
the errors. The positional errors from the fit derived like this range from 0.010 mas
(0.00034 px) to 0.555 mas (0.020 px) for the brightest and faintest star, respectively,
in the 47 Tuc data set and from 0.004 mas to 0.141 mas (0.0001 - 0.005 px) in the
NGC 6388 data set. The difference in these values for the two data sets comes mainly
from the smaller FWHM of the PSFs in the NGC 6388 data, which itself is mainly due
to the better initial observing conditions.
8.6
Ensquared Energy
In adaptive optics observations often the encircled or ensquared energy is taken as a
measure of performance besides the FWHM. The encircled energy in percent is defined
as the flux of a PSF contained in a certain radius from the middle point divided by
the total flux. In the case of an image where the flux is given as flux per pixel, like
any detector image, the ensquared energy is used, which is the flux within a certain
quadratic box with the size of n pixel times n pixel. To correctly calculate the comprised
flux one needs to know how the program used to calculate the ensquared energy, defines
the pixel center. I used IDL in my analysis, where a pixel is defined from -0.5 to 0.5.
For instance, a peak of a PSF given by the coordinates 50, 50 is therefore defined
in the middle of the pixel and not for example at its lower left corner. Knowing
this, I measured the flux in square boxes with 1 × 1, 3 × 3, 5 × 5...79 × 79 pixel in
diameter, covering the full size of the extracted PSFs. In Fig. 8.6 (left), an example
of the development of the ensquared energy with pixel distance from the center pixel
8.7 DISTORTION MAPPING
115
Relation between FWHM and ensqu. energy
Ensquared energy
0.30
1.0
47 Tuc (GLAO)
NGC 6388 (MCAO)
47 Tuc (GLAO)
NGC 6388 (MCAO)
0.25
FWHM [arcsec]
Ensquared energy [%]
0.8
0.6
0.4
0.2
0.0
0
0.20
0.15
0.10
4.67 6.40 10
radius with 50%
ensquared energy
20
radius [pixel]
30
40
0.05
4
5
6
7
radius of 50% ensquared energy [px]
8
Figure 8.13: Left: Example of the development of the ensquared energy with box size for
the MCAO and the GLAO case. The radius containing 50% of the energy is
marked. Right: Relation between measured FWHM and radius containing 50%
of the energy for the MCAO and GLAO correction.
is shown. Here radius denotes the half diameter of the box. Finally I calculated the
radius in pixels within which 50% of the energy of the PSF is contained and used this
radius as a performance indicator. The smaller the radius the better the AO correction,
moving flux from the seeing halo of the PSF into the central peak. As comparison I
show in Fig. 8.6, right panel, the relation between the measured FWHM of the PSF
and the radius of 50% ensquared energy, r50 . Clearly visible is a relation, the larger
the FWHM, the larger r50 . For the MCAO data this relation is sharper and better
defined as for the ground layer data, reflecting the better initial observing conditions
and therefore the better correction in the single frames.
8.7
Distortion Mapping
To investigate the stability of the MCAO and GLAO performance in terms of astrometric precision over time, I first had to correct for distortions of the field. During the
observations problems with the de-rotator occurred due to a software problem, leading
to a bigger rotational error in several frames. If the AO correction is very stable over
time, the relative positions of the stars should be the same after correcting for effects
such as the de-rotator problem.
The first thing I did, was to set up a coordinate reference frame to which I mapped the
single frame coordinates. For this I set up some criteria for the stars chosen to create
this reference frame: After the measurement of the positions of the stars in each field
following the procedure described in Sect. 8.5, I compared the lists of stars found in
the individual frames and identified the subset common to all frames of the NGC 6388
and 47 Tuc data, respectively. The next step was to exclude faint stars from the list,
namely those that had less than 10000 counts, i.e. a peak value around 85 ADU in the
case of 47 Tuc and those with less than 2800 counts, peak ∼ 50 ADU, in the case of
116
CHAPTER 8. ASTROMETRY WITH MAD
NGC 6388. These values were chosen, because in the NGC 6388 data there are fewer
bright stars than in the 47 Tuc data. Additionally, all stars that have a close neighbor
at less than 2 HWHM separation were excluded. This left ∼ 280 stars for the 47 Tuc
field and ∼ 130 stars for the NGC 6388 field, still absolutely enough to calculate the
transformations.
In order to create the reference frame, I started to use the best frame I had, chosen according to the highest mean Strehl in the images, as a first reference frame and mapped
all the positions from each individual frame onto this reference frame. I then calculated the shift and scale in x and y and the rotation between these frames using the
MIDAS3 data reduction software and simple affine transformations. I was not applying
any interpolation directly to the images, but instead just worked with the measured
coordinates. After correcting for the derived rotation, as well as for the shift and scale
in x and y of each stellar position I created a master-coordinate-frame by averaging
the position of each star over all frames. This was my new masterframe for the coming
analysis. I then mapped all coordinates from each frame to this masterframe, leading to
a better transformation. One could think that one can get even better transformations
between the frames by applying this method iteratively, creating once more a mastercoordinate-frame by calculating mean positions and averaging those positions. If the
distortions in the images, left over from the AO or systematic, were homogeneous over
the FoV the transformations should not change or enhance a lot. But if the distortions
are not homogeneous, but depend, for example, on the camera position in the FoV, I
would introduce warpings in the master-coordinate-frame which I cannot map with a
simple shift, scale and rotation anymore. I therefore stopped after one iteration.
I then calculated the residual separations between the positions of the stars in the
master-coordinate-frame to the new positions in every frame calculated with the obtained transformation parameters and analyzed them as a measure of astrometric precision.
3
http://www.eso.org/sci/data-processing/software/esomidas/
Chapter 9
Results
To set my results concerning the astrometric precision achievable with MCAO into a
context, I looked at overall performance indicators, such as the FWHM of the fitted
PSF and seeing. This led to several interesting results. In Fig. 9.1 I have plotted
the FWHM calculated from the extracted PSF over the measured mean DIMM seeing
for each frame. The seeing is measured in the V band, while the observations were
made in Brγ (47 Tuc) and Ks (NGC 6388). Hence the difference between the seeing
and the measured FWHM cannot be directly taken as the correction factor achieved.
Nevertheless, the principal relation between these two values is preserved as Fig. 9.1
demonstrates. In the case of the NGC 6388 data (right panel), where the first 5 frames
were obtained with GLAO correction and the last 25 with full MCAO correction, I
marked the GLAO frames in red. Interestingly, the FWHM does not change significantly in the GLAO frames, although the seeing does, but in the MCAO case, I see a
correlation between seeing and FWHM, as expected. A similar behavior of the FWHM
can be seen in the 47 Tuc data, which is also in ground layer mode. The FWHM gets
only slightly larger, the larger the seeing does. On the one hand the different FWHM
values for nearly the same seeing may be explained by slightly different eccentricities
of the PSFs in the different frames, leading to different mean FWHM values when calculated as the mean of the FWHM values of the minor- and major-axes. On the other
hand, this can be interpreted as a sign of the non-correlation of these two parameters
during GLAO correction. But this has to be seen with caution, as I have no other data
set confirming or disproving this interpretation. A possible reason for the behavior in
the NGC 6388 data is that the AO system was optimized for the ground layer correction in the nights before. The GLAO correction alone worked very well and even better
than expected1 . Switching on the full MCAO correction can in a first attempt lead to
a small degradation in the performance, if the controller of the system has not yet been
optimized for the correction of two layers. Also in the case of MAD, the subsystems
for the correction of the two layers are nested, meaning that one sensor 'sees 'the correction applied to the distortions of the other layer. This could lead to a degradation
of the overall performance, if the system is not yet fully optimized. As this is the first
time the full MCAO approach was tested, this resulting behavior of the performance is
interesting but comprehensible and should not lead to wrong conclusions about the full
1
C. Arcidiacono (private communication)
117
118
CHAPTER 9. RESULTS
Relation between FWHM and seeing
Relation between FWHM and seeing
0.30
0.30
47 Tuc (GLAO)
NGC 6388 (GLAO)
NGC 6388 (MCAO)
0.25
FWHM [arcsec]
FWHM [arcsec]
0.25
0.20
0.15
0.10
0.05
1.06
0.20
0.15
0.10
1.08
1.10 1.12 1.14 1.16
DIMM seeing [arcsec]
1.18
1.20
0.05
0.35
0.40
0.45
0.50
DIMM seeing [arcsec]
0.55
0.60
Figure 9.1: FWHM over seeing plotted for the 47 Tuc (left) and NGC 6388 (right) data. The
five ground layer frames are marked in red in the right panel with the NGC 6388
data.
capacity of the MCAO correction. To analyze this behavior further, one would need
more data and more detailed information about the applied correction parameters of
the system, both of which we do not have. Also our goal is not a performance analysis,
this is done by the group that built the layer oriented part of MAD (Arcidiacono et al.,
2008), but to see how precise astrometric measurements can be conducted under the
given observing circumstances and corrections. Altogether the seeing does not change
a lot during the observing time in both cases, but rather varies between 1.0800 − 1.1900
in the case of the 47 Tuc data and 0.3700 − 0.5600 during the NGC 6388 observations
(see also Tables 8.1.1 and 8.1.2).
9.1
Separation Measurements
One test I performed, was to measure the relative separation between various pairs of
stars all over the FoV. I wanted to derive a time sequence of the separation, to see
how stable it is. If only a steady distortion were present in the single frames, then the
separations should be stable over time or only scatter within a certain range given by
the accuracy of the determination of the position of the stars. If differential distortions
between the single frames are present, but these distortions are random, the scatter of
the separations should increase. A not perfectly corrected defocus for example would
change the absolute separation between two stars, but, to first order, not the relative
one measured in the frames, if this defocus is stable. An uncorrected rotation between
the frames would change the separation of two stars in the x and y direction, but not
their full separation, r.
0
190
191
192
5
10 15 20
frame number
25
--- Mean X-Distance [px] = 191.779
193 -- Std. Dev [px] = 0.219
1005 --- Mean X-Distance [px] = 995.234
-- Std. Dev [px] = 1.352
1000
995
990
985
215
--- Mean X-Distance [px] = 212.391
214 -- Std. Dev [px] = 0.271
213
212
211
210
84
85
86 -- Std. Dev [px] = 0.157
--- Mean X-Distance [px] = 84.881
30
133
--- Mean Y-Distance [px] = 140.106
-- Std. Dev [px] = 1.184
-- Std. Dev [px] = 1.481
0
107
108
5
10 15 20
frame number
-- Std. Dev [px] = 0.131
25
0
219
218
220
221
5
10 15 20
frame number
-- Std. Dev [px] = 0.230
25
1550
222 --- Mean Distance [px] = 220.309
1205
110 --- Mean Y-Distance [px] = 108.429
109
1560
1210
1580
1590 --- Mean Distance [px] =1570.751
252
254
--- Mean Distance [px] = 254.440
-- Std. Dev [px] = 0.274
256
214
215
1570
30
--- Mean Distance [px] = 215.445
217 -- Std. Dev [px] = 0.208
216
222
223
224
Separation between stars [px]
226 --- Mean Distance [px] = 223.739
225 -- Std. Dev [px] = 0.197
1215
1220
1225 --- Mean Y-Distance [px] =1215.224
139
140
141 -- Std. Dev [px] = 0.118
197
198
178
200 --- Mean Y-Distance [px] = 198.019
-- Std. Dev [px] = 0.191
199
179
132
180
--- Mean Y-Distance [px] = 179.473
Y-Separation between stars [px]
181 -- Std. Dev [px] = 0.166
134
-- Std. Dev [px] = 0.191
135 --- Mean X-Distance [px] = 133.599
X-Separation between stars [px]
30
9.1 SEPARATION MEASUREMENTS
119
Figure 9.2: Separation between pairs of stars plotted against frame number. The left panel
shows the separation in x-direction,
p the middle panel in y-direction and the right
panel the full separation r = ∆x2 + ∆y2. The small straight lines mark the
frames after which a new five points sequence of jitter movements was started.
120
CHAPTER 9. RESULTS
Performing this test for several stars with short and large separations and with different
position angles between the stars showed in the case of the NGC 6388 data a recurring pattern in the separation in x, y, r, which is not observable in the 47 Tuc data.
Fig. 9.2 shows the separation in x, y and r over the frame number for five representative pairs of stars in the NGC 6388 data. Looking at the pattern, which repeats after
five frames for the first 10 frames and after 10 frames for the following frames (where
always two images were taken at the same jitter position before moving to the next
position), this change in separation seems to be correlated with the jitter movement
during the observations which also has a five points pattern with an additional change
of center position (see Fig. 8.3). In the case of the MAD instrument the camera itself
is moved in the focal plane to execute the jitter pattern. This can lead to vignetting
effects, if the jitter offset is too large and it seems to introduce distortions dependent
on the position of the camera in the field of view. It is unlikely that this pattern is due
to the problems with the de-rotator, because of the uniform repetition of the pattern.
Also this pattern is not seen in the 47 Tuc data, which was obtained without jitter
movements, but experienced the same de-rotator problems.
I performed this test with the same star pairs after applying the calculated distortion
correction for shift, scale and rotation (see Sect. 8.7). The strong pattern is gone, leaving a more random variation of the separation. Also the calculated standard deviation
is much smaller, ranging from a factor of ∼ 3 up to a factor of ∼ 19 times smaller!
Comparing the single standard deviations shows a smaller scatter among their values
than before the distortion correction. All this yields to the conclusion, that the calculated and applied distortions remove a large amount of the separation scatter, but
not all of it. The remaining scatter of the separations between the stars in the single
frames still ranges from ∼ 1.2 − 2.8 mas, well above the scatter expected from photon
statistics, pointing to uncorrected higher order distortions. But the values here are
only calculated for a small fraction of the stars in the frames. I therefore also had a
look on a more global scale of the positional residuals.
9.2
Residual Mapping
To look at the spatial distribution of the residuals after the distortion correction, where
I corrected for x and y-shift, x and y-scale and rotation relative to the masterframe,
I created contour plots of the residuals by fitting a minimum curvature surface to the
data of each frame. In Fig 9.3, upper panels, an example of such a contour plot is
shown for the two data sets. The maps for the other frames look pretty much the same
with small variations in distribution and size of the residuals. But the main goal of
this test was to check for any strong spatial variation of the residuals over the FoV.
Similar to the Strehl maps and the PSF tests I made in Chap. 8.4, no strong spatial
variation can be seen, such as for example a strong gradient in one direction. The
high residuals in the two left corners of the 47 Tuc data are an artifact of the surface
fit, as there were not enough data points in these areas for a good fit. Additionally
I created arrow diagrams showing not only the strength, but also the direction of the
residuals for each star used to calculate the transformation. Looking at these maps,
Fig. 9.3 lower panels, the orientation of the arrows is random. I corrected for scale
and rotation, so I do not expect any prominent residual due to these parameters. A
9.2 RESIDUAL MAPPING
121
DEC [arcsec]
85
1.4
1
0.
1.41
1.
41
1.
97
20
7
1.9
1
1.41
10
4.2
5.92
1.9
20
30
RA [arcsec]
40
50
0
10
47 Tuc 1 Residuals
20
RA [arcsec]
30
7
41 1.9
1.9
1
10
1.41
0
1.4
0
8
5
0.2 0.8
3
7
1.41
0.85
4
23..510
7
54
2.
0.85
1.
1.97
1.4
1
1.4
10
0
0.85
30
0.85
7
1.9
4.32.10
3
1.4
0.85
85
1.9
1.
4.2397
5.92 3.10
20
1
1.41
1.9
7
1.4
1
30
0.
7
1.97
45.2.92
3
1. 23..10
97 54
4
1.41
9
1.
0.85
DEC [arcsec]
2.5
2.
1.41
7
1.9
40
2.54
40
54
0.8
5
1.4
1
50
NGC 6388 6 Residuals
0.28
2.5
34.
10
47 Tuc 1 Residuals
7
40
NGC 6388 6 Residuals
2000
1500
Y-Direction [pixel]
Y-Direction [pixel]
1500
1000
1000
500
500
0
0
500
1000
1500
X-Direction [pixel]
2000
0
0
200
400
600 800 1000 1200 1400
X-Direction [pixel]
Figure 9.3: Example contour plots of the residuals (in mas) of the positions of the stars
after the distortion correction in the upper panels. In the lower panels, the
corresponding arrow maps are shown. The arrows are extended by a factor of
1000. The left panels show the data of the cluster 47 Tuc and the right panels
the data of the cluster NGC 6388. The empty areas in the 47 Tuc data are due
to the applied selection criterias for the stars.
residual scale would lead to a pattern, where the arrows all point radially away from
one area and a residual rotation would leave arrows arranged on circles, all facing in
the direction of the rotation. No pattern of this kind can be seen.
I plotted the calculated distortion parameters for x-scale, y-scale and rotation over
the frame number, which can be seen as a time-series, in Fig. 9.4 for both data sets.
Whereas the parameter for the rotation correction looks random, but with some high
values, indicating the de-rotator problem, the correction parameters for the scale in x
and y show a pattern in the case of the NGC 6388 data set (right). This pattern repeats
after five (10) frames, as does the pattern for the separation measurement. As these are
the applied correction parameters, they show nicely the existence of the pattern and
122
CHAPTER 9. RESULTS
NGC 6388 Distortion Parameters
x-scale
1.001
1.000
0.999
y-scale
1.001
1.000
0.999
rotation [deg]
rotation [deg]
y-scale
x-scale
47 Tuc Distortion Parameters
0.02
0.00
-0.02
0
5
10
frame number
15
20
1.005
1.000
0.995
1.005
1.000
0.995
0.02
0.00
-0.02
-0.04
-0.06
-0.08
0
5
10
15
20
frame number
25
30
Figure 9.4: Applied distortion parameters over frame number for the 47 Tuc (left) and
NGC 6388 data (right). The panels show from top to bottom the calculated
distortion parameters for x scale, y scale and rotation for each frame. While the
rotation parameter is random, the scale parameter of the NGC 6388 data show
a pattern which is not visible in the 47 Tuc data.
the ability of correcting for these major distortions. In the 47 Tuc data there is also
some scatter, which is expectable, but no repeating pattern can be seen. Additionally,
the values are smaller in the case of the ground layer data which was obtained without
jitter (note the different scaling of the two plots).
Finally, I calculated the mean residuals over the full FoV for both data sets separately
for the x, y and r direction for each frame. The mean values are very close to zero
(∼ 10−5 − 10−6 ), supporting the results from the arrow plots of random orientation,
but looking at the mean of the absolute values of the residuals shows how large they
still are. In Fig. 9.5 the mean of the absolute residuals over the full FoV in the x and
y direction and in the separation are plotted over the radius of 50% ensquared energy
of the corresponding extracted PSF of each frame and each data set. No correlation
of the size of the residuals with the ensquared energy can be seen in the 47 Tuc data
set but a small correlation in the x-direction in the NGC 6388 data. What is visible,
is that the absolute values of the residuals and their scatter are larger in the case
of the NGC 6388 data set compared to the 47 Tuc data set, even though the initial
observing conditions were better and the measured FWHM values are smaller. This
gives a first impression on how precise the astrometry is in these MAD data. The
mean absolute residuals are between 0.025 px and 0.092 px (0.7 - 2.6 mas) in the case
for the ground layer corrected 47 Tuc data set and between 0.028 px and 0.114 px
(0.8 - 3.2 mas) in the MCAO corrected NGC 6388 data set. With photon statistics
alone, the positions should vary in a way smaller range. Taking the positional accuracy
calculated from photon statistics for the faintest stars used in this set, the residuals
should be within 0.005 px (0.14 mas) in the 47 Tuc case and 0.012 (0.33 mas) in the
6.5
7.0
7.5
radius with 50% ensquared energy [px]
0.10
0.08
2.5
2.0
0.06
1.5
0.04
0.02
0.00
6.0
1.0
6.5
7.0
7.5
radius with 50% ensquared energy [px]
0.10
0.08
0.5
0.0
8.0
2.5
2.0
0.06
1.5
0.04
0.02
0.00
6.0
0.5
0.0
8.0
[mas]
0.02
0.00
6.0
1.0
1.0
6.5
7.0
7.5
radius with 50% ensquared energy [px]
0.5
0.0
8.0
2.0
0.06
1.5
0.04
0.02
0.00
4.0
1.0
4.5
5.0
5.5
6.0
radius with 50% ensquared energy [px]
0.10
0.08
2.0
1.5
0.04
1.0
4.5
5.0
5.5
6.0
radius with 50% ensquared energy [px]
0.14
0.12
0.5
0.0
6.5
3.5
3.0
0.10
2.5
0.08
0.06
0.04
4.0
0.5
0.0
6.5
2.5
0.06
0.02
0.00
4.0
[mas]
0.04
2.5
[mas]
1.5
Mean abs. positional residuals, NGC 6388
0.10
0.08
[mas]
0.06
[mas]
2.0
mean Y-Residuals [px]
2.5
mean Residuals [px]
Mean abs. positional residuals, 47 Tuc
0.10
0.08
mean X-Residuals [px]
123
[mas]
mean Residuals [px]
mean Y-Residuals [px]
mean X-Residuals [px]
9.3 MEAN POSITIONS
2.0
1.5
4.5
5.0
5.5
6.0
radius with 50% ensquared energy [px]
6.5
Figure 9.5: Mean absolute positional residuals over the radius of 50% ensquared energy. In
the left panel for the 47 Tuc data set and in the right panel for the NGC 6388
data set. The plot shows from top to bottom the mean of the absolute values
of the residuals to the masterframe in the x- and y-direction and the separation
after the correction of x and y-shift, x and y-scale and rotation.
The overplotted
√
error bars correspond to the error of the mean value (σ/ n), with n equal to
the number of stars used to calculate the mean value and σ being the standard
deviation. The left y-axis shows the residuals in units of pixel and the right one
in units of mas.
NGC 6388 case. The calculated values show a residual positional scatter that cannot
be explained by simple statistical uncertainties. It rather shows that even after a basic
distortion correction, the remaining positional scatter is fairly large for the purpose
of high precision astrometry. This scatter seems to have its origin in higher order
distortions present in the images, as it seems largely independent from the size of the
PSF. Additionally, the residuals and scatter are larger in the case where the camera
was jittering to scan a bigger field of view. This jitter movement introduced distortions,
which I already saw in the separation measurements and in the distortion correction
parameters calculated for scale and rotation. But also the AO correction can introduce
higher order distortions as it dynamically adapts to atmospheric turbulence changes.
9.3
Mean Positions
As a last step I calculated the mean position for each star over all frames. This is
one way to measure the astrometric positions and their uncertainties. In Fig. 9.6 the
achieved astrometric precision is plotted over the K magnitude for each star in the
final lists of both data sets. The given magnitude is not an accurate value, calculated
124
CHAPTER 9. RESULTS
107
Counts [e−]
106
105
0.20
47 Tuc
NGC 6388
5
3
0.10
2
0.15
X RMS [px]
4
α RMS [mas]
0.15
5
0.05
4
3
0.10
2
0.05
1
0.00
1
0
6
107
8
10
K Magnitude
Counts [e−]
106
0.00
10
12
105
12
14
K Magnitude
16
0
18
Counts [e−]
106
105
107
0.20
0.20
NGC 6388
5
0.15
0.15
2
0.05
Y RMS [px]
3
δ RMS [mas]
4
0.10
5
4
3
0.10
2
δ RMS [mas]
47 Tuc
Y RMS [px]
α RMS [mas]
0.20
X RMS [px]
Counts [e−]
106
105
107
0.05
1
0.00
0
6
8
10
K Magnitude
12
1
0.00
10
12
14
K Magnitude
16
0
18
Figure 9.6: MAD positional RMS over Magnitude
with careful photometry, but rather represents the from the measured flux estimated
2MASS K magnitude of the stars. For the conversion from counts to magnitudes I used
the counts of the stars calculated by starfinder. For the brighter isolated stars I took
the corresponding K magnitudes from the 2MASS catalog and calculated with theses
values the zero point of the conversion for each of these bright stars. After taking the
mean of these zero points I could convert the measured counts for all stars into their
corresponding K magnitude (mag = zero − point − 2.5 log(flux)). These values are not
meant to be understood as being exact, but will be good enough to see the principal
relation between precision and intensity. For completeness I indicated the exact counts
at the upper x-axis of the plots. I made this conversion also for the data set of the
globular cluster 47 Tuc even though it was observed in Brγ . Therefore note that the
given numbers for the K magnitudes in the plots are not measured, but are related to
a certain measured flux.
As one can see in the plots for the NGC 6388 data set (right panels), the fainter stars
have less precision in their position than the brighter ones. The mean positional precision of the stars between 14 and 18 mag is ±0.073 pixel or ±2.057 mas, calculated as
a mean value of the x- and y-directions. This is the achievable astrometric precision
9.3 MEAN POSITIONS
NGC 6388
Photon
125
1
25
pixel
47 Tuc
statistics
Photon
statistics
±0.073 px
±0012 px
±0.040 px
±0.051 px
±0.005 px
±2.057 mas
±0.326 mas
±1.127 mas
±1.437 mas
±0.137 mas
Table 9.1: Summary of the expected and achieved astrometric precisions. The first and
fourth columns list the measured mean astrometric precision for the NGC 6388
and 47 Tuc data, respectively. The second and last columns list the expected
precision from photon statistics for the faintest star in the respective magnitude
ranges for the two data sets, second column for the NGC 6388 data and the last
column for the 47 Tuc data. The middle column lists the corresponding pixel and
milli-arcsecond values for a precision of 1/25 pixel.
with the available MAD data in full MCAO mode. Theoretically, as stated above, the
faintest stars in this regime should have a precision of about ±0.012 px (±0.33 mas)
assuming only photon statistics. The measured precision is a factor of 6.2 worse. This
is a quite large discrepancy, although one has to take more than photon statistics into
account for calculating a correct error budget. On the other hand, aiming at a in astrometry reasonable measurement accuracy of 1/25 - 1/50 pixel (1.127 - 0.564 mas) the
achieved mean astrometric precision is still a factor of 1.8 - 3.6 larger than the expected
possible precision.
In the GLAO data set of 47 Tuc the mean astrometric precision for stars between 9
and 12 mag is ±0.051 pixel (±1.437 mas). Although the fainter stars seem to have
slightly larger uncertainties, this correlation is less distinctive than in the MCAO case
(NGC 6388). And again, comparing these values with the theoretically achievable values of ∼ 1/25 pixel shows, by a factor of 1.3, a less precise position measurement than
expected.
In Tab. 9.1 the expected and achieved results for the astrometric precision in the two
data sets are summarized.
Comparing the results for the ground layer correction with those of the MCAO correction shows a higher precision in the GLAO data. One could expect it the other
way round as the initial observing conditions and the average Strehl are better in the
MCAO data. Also the FWHM and the radius of 50% ensquared energy are smaller in
the MCAO data. Additionally, the exposure time was longer for the NGC 6388 cluster.
But this is no advantage in the sense of better signal, as NGC 6388 is further away,
9.5 kpc, than 47 Tuc, 4.0 kpc, and is fainter. In Fig. 9.6 one can see by comparing the
upper x-axes, that the final counts distribution is the same for the two sets of stars.
One of the main differences in the two data sets is the jitter movement. As could
already be seen, this movement introduces distortions. I corrected for shift, scale and
rotation, but still the afterwards achieved precision is smaller, indicating distortions of
higher order, quadratic or even higher.
126
CHAPTER 9. RESULTS
9.4
Discussion and Conclusion
I have analyzed the first multi conjugated adaptive optics data available in the layer
oriented approach with respect to astrometric performance. The data were taken with
the MCAO demonstrator MAD at the VLT. Two sets of data of globular clusters,
observed in two different approaches were analyzed: The globular cluster 47 Tucanae
with only ground layer correction and the globular cluster NGC 6388 in full MCAO
correction.
The first data analyzed was those of the globular cluster NGC 6388 taken with full
MCAO correction. I calculated Strehl maps for each frame. The Strehl is fairly uniform over the FoV with a small degradation to the edges of the FoV and average values
between 11% and 23%. The performance was slightly degrading over the time of the
observation which can be seen in the lower Strehl values and larger FWHM values in the
later frames. The first five frames were obtained with only the ground layer corrected.
The FWHM and Strehl are stable in these frames, even though the seeing measured by
the DIMM seeing monitor, changed slightly.
I created a master frame with positions of bright isolated stars in the field and calculated distortion parameters for a shift and scale in x and y direction and a rotation
for each frame to this master frame. Separation measurements between stars before
and after the distortion correction showed that this correction is indeed reducing the
scatter in the separations measured over all frames. But it also shows a residual scatter,
which is probably due to higher order distortions. A pattern visible in the separation
measurements as well as in the applied distortion parameters is thought to be due to
the jitter movement of the camera during the observations. This movement introduced
additional distortions which could partly be corrected with the distortion correction.
The precision of the positions of the stars, calculated by the scatter of the mean position of the stars over all frames, is ±0.073 pixel or ±2.057 mas in the corresponding
2MASS K magnitude range from 14 to 18 mag. The positions are calculated in the
detector coordinate system.
The positional precision, as well as the scatter in the separation measurements and the
mean residuals of the positions of the stars after the distortion correction to the positions in the master frame, all lie in the same range, showing the astrometric precision
achievable with these data.
I compared my results with the unpublished ones calculated by Alessia Moretti2 . She
analyzed the same data set photometrically with the DAOPHOT reduction package
(Stetson, 1987). DAOPHOT uses a variable Penny function to fit the PSFs of the
stars. The main goal of her work is the photometric analysis of the cluster data, but
she also performed astrometric measurements. Her calculated mean astrometric precision of stars with a K magnitude between 14 and 18 mag is 0.061 pixel. The precision
was calculated in a final image which was created by combining the four best frames,
allowing for translation, scale and rotation of the images before addition. Our results
are conforming, showing that the use of a mean PSF for fitting the positions of the
stars is not the reason for the uncertainty in the astrometric precision, and the scatter
seen in my results is really in the data and not an artefact of the fitting procedure.
2
Private communication
9.4 DISCUSSION AND CONCLUSION
127
The second data set I analyzed was conducted with only correcting the ground layer
turbulences. The calculated Strehl maps yielded smaller but still fairly uniform values
between 9% and 14%. The lower Strehl in the 47 Tuc data set can partly be explained
by the fact that only the distortions due to the ground layer were corrected, but also the
initial atmospheric conditions were worse, which leads to a degradation in the possible
performance of an AO system. To disentangle these two possible causes, one needs to
analyze more data sets.
I also created a master frame and calculated distortion parameters for each frame to
this master frame. The separation measurements and distortion parameters did not
show a pattern as was the case in the NGC 6388 data set.
The precision of the positions of stars in the corresponding 2MASS K magnitude range
between 9 and 12 mag is ±0.051 pixel (±1.437 mas) in this data set.
Astrometric analysis of the core of 47 Tuc were also performed by McLaughlin et al.
(2006), who used several epochs of Data from the Hubble Space Telescope (HST). They
derived positional precisions in the single epoch data, taken with the High Resolution
Camera (HRC) of the Advanced Camera for Surveys (ACS) for most stars in the range
of 0.01-0.05 pixel. With a plate-scale of 0.027 arcsec/pixel this corresponds to 0.27-1.35
mas. The errors were calculated in the same way as in this work, taking the standard
deviation of the positions in all frames as uncertainties. Detailed distortion corrections
were computed for ACS by Anderson (2002), which were applied to the data in the
work of McLaughlin et al.. This shows that the precision derived with MAD is already
comparable to HST astrometry and with a good distortion characterization, future
instruments will yield even higher astrometric precision.
Although the Strehl is smaller and the FWHM is larger in the GLAO data of the cluster 47 Tuc, the achieved astrometric precision is higher. Also the observing conditions
were worse during the GLAO observations compared to the MCAO observations with
a mean seeing of 1.1300 and 0.4600 , respectively. All this leads to the conclusion that
the degradation of the astrometric precision in the MCAO data set is mainly due to
the jitter movement during the observations, which introduced additional distortions.
But also the more complex correction of two layers could have introduced higher order
distortions, which we could not correct for. To verify this one needs to compare all the
results with the applied correction parameters. But going more into detail would be a
distortion characterization, which is indeed a very interesting task, but not goal of this
work. To fully characterize the remaining distortions, one would need more data, taken
under various seeing conditions and observation configurations. As MAD will not be
offered again, a fully satisfactory analysis is not possible.
For future MCAO observations one should try to either build an instrument, where
the camera is not moved to execute the jitter movement or completely avoid jittering.
As the latter one is often not possible in IR observations, one should take great care
of the distortions present in the frames and fully characterize those for high precision
astrometric observations.
All the results presented here are still given in detector coordinates, as I analyzed the
data in matters of the adaptive optics correction and instrumentation stability over the
time of the full length of the observation. Going to celestial coordinates would involve
the correction for effects such as differential aberration and differential refraction to
128
CHAPTER 9. RESULTS
derive the true positions of the stars. As the observed FoV is large, these effects can
reach several milliseconds of arc of displacement between stars at different points on the
detector. These transformations introduce additional position uncertainties, degrading
the astrometric precision further, but need to be done when comparing data from
different epochs (as seen in Chap. 4).
Appendix A
Strehl plots
129
130
CHAPTER A. STREHL PLOTS
131
132
CHAPTER A. STREHL PLOTS
133
134
CHAPTER A. STREHL PLOTS
135
136
CHAPTER A. STREHL PLOTS
Appendix B
Acronyms
AO
Adaptive Optics
AU
Astronomical Unit
DIMM
Differential Image Motion Monitor
DIT
Detector Integration Time
DM
Deformable Mirror
ESO
European Southern Observatory
FCUL
Faculdade de Ciências da Universidade de Lisboa
FoV
Field of View
GLAO
Ground Layer Adaptive Optics
HLAO
High Layer Adaptive Optics
HST
Hubble Space Telescope
LGS
Laser Guide Star
LO
Layer Oriented
MAD
Multi conjugated Adaptive optics Demonstrator
MCAO
Multi Conjugative Adaptive Optics
NACO
NAOS-CONICA
NDIT
Number of Detector Integration Time
NGC
Natural Guide Star
PSF
Point Spread Function
PWS
Pyramid Wavefront Sensor
RV
Radial Velocity
SDI
Spectral Differential Imaging
SHS
Shack-Hartmann Sensor
137
138
CHAPTER B. ACRONYMS
SNR
Signal to Noise Ratio
SO
Star Oriented
SR
Strehl Ratio
TNG
Telescopio Nationale Galileo
VLT
Very Large Telescope
WCS
World Coordinate System
WF
Wavefront
WFS
Wavefront Sensor
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Acknowledgment
At this point I would like to thank all those people who helped me on my way to finally hand
in this thesis.
First of all I would like to thank my supervisor Martin Kürster for his patience and guidance
over the last three years. A big thank you also for proofreading this thesis and all the helpful
comments.
Thank you to Prof. Hans-Walter Rix and Prof. Joachim Wambsganß agreeing to referee the
thesis.
PD Henrik Beuther and Prof. Werner Aeschbach-Hertig for being jury members at the defense.
Rainer Köhler for fitting and fitting and fitting... and for explaining so much and always having
time for me and never get hacked off :)
Emiliano Arcidiacono for helping me so much with all my MAD questions. And for squeezing
me in in his and Jacopo’s office for two weeks to help define the goals of the MAD analysis.
Roberto Ragazzoni and the full MAD team for providing me with the MAD data.
My former and present office mates for all the conversations in- and outside daily science life.
And the great MPIA studends coffee break members for having everyday 30 min to relax and
chat.
Markus for giving me so much support during all my smaller and bigger problems during the
last years and for running from printer to printer to find the best printed version of the thesis.
Thank you for loving me and asking me to marry you *
A big thank you to my family, who is always supporting me. My brother for reading part of
the thesis. My mother for always believing in me. My father, who cannot celebrate this day
with me, for being the best Dad in the world!
Finally a heartily thank you to all the people I have not mentioned specifically here, but who
helped me on my long way.
Thank you!
149
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