Deconvolution Photometry
of QSO 0957+561 A,B
Candidata Scientiarum thesis by
Anna Kathinka Dalland Evans
Institute of Theoretical Astrophysics
University of Oslo
Norway
November 2003
The years teach much which the days never knew.
Ralph Waldo Emerson
In memory
Til Mormor,
den tøffeste av alle
&
To Granny & Grandad,
as promised
Cover picture: North is up, East is left on all four images. Top left: Observed
image of the quasar components. Top right and bottom left: Residual images.
Bottom right: Deconvolved image.
Thanks!
One of the symptoms of an approaching nervous breakdown
is the belief that one’s work is terribly important.
Bertrand Russell
The work with this thesis turned out to be different than expected. It was
a genuine relief when I discovered that the frustrating ‘obstacles’ in my way
were actually part of my thesis work rather than merely annoying hindrances.
As the road to finishing this work has been rather long, many people have
helped me on the way.
The first load of ‘thank yous’ goes to my supervisor Rolf Stabell, for taking
me on as a student in the first place, for sending me off to Belgium and to the
Canary Islands, for letting me go observing at the NOT, and for pestering me
throughout with his exquisite exactness in writing (and speaking). Ingunn
Burud, thanks for all the help and support you have provided since the first
days in Liège, and for becoming my good friend as well as helper. My most
sincere and heartfelt thanks also goes to Frédéric Courbin, for swift and
steady guidance on using the programs during the last crucial stages.
I thank Wesley N. Colley and Rudy Schild for providing me with photometric results for Q0957 for a comparative analysis, as well as Luis Goicoechea
and Aurora Ullán Nieto for sending me their GLITP data for the same purpose.
From the social side of life, I would like to thank ‘the gang’ in room 303
at ITA, for the good old days: Benedicte Selmer, Jan-Erik Ovaldsen, Kjetil
Kjernsmo and Torstein Sæbø. As it turned out, we were the last group
of students to enjoy the luxury of that room before it was confiscated and
turned into a (mostly empty) conference room. I would also like to thank
my new gang in the prison cell on the 1st floor, where we sat ‘temporarily’
while the institute was besieged by renovation workers. We managed to
work practically on top of each other for close to a year without any murders
being committed. Thanks to Eirik Mysen and Martin Ytre-Eide for providing
comic relief, and to Morten Wergeland Hansen for not throwing me out of
the window. The largest thanks of all goes to Mari Anne Killie: You have
ii
followed this work and all my ups and downs on a daily basis, and I could
not have done without you! You’re the kind of girl that fits in with my world.
I also have to mention a few other astro-people. Thanks to Håkon Dahle
and Vidar Aarum-Ulvås, for your enduring patience with my IRAF questions
and for helping me probe my light curves. To Jan-Erik Ovaldsen (again!)
and Jan Teuber of AstroConsult A/S I am extremely grateful for help and
counselling, for useful comments on my written material and for pointing
out many of the seemingly small but oh so crucial details involved in image
processing. Thanks to Øystein Elgarøy, for your always clear and thorough
explanations to my various questions. To Egil Leer, thank you for asking me
‘how I am doing today’, and for being genuinely interested. I would also like
to thank Anlaug Amanda Kaas at the NOT who has helped me with various
technical details via e-mail from La Palma. For fun&games, thanks to the
rest of the institute’s X-mas party organising committee 2001 not mentioned
elsewhere: Astrid Fossum, Eirik Endeve and Torben Leifsen.
I cannot resist thanking Google, the friend in need when you are all alone,
and Cambridge Dictionaries Online.
To my friends outside the astrophysical community, I would like to thank
you all for being there for me and for helping me keep my sanity which would
otherwise have been gravely endangered after spending all this time with
computer nerds. Special thanks to Aina Kristin Seland Nilsen, for remaining
my oldest friend in Oslo and for being a rock whenever I freak out. To
Øyvind Kristiansen, for late evenings and early mornings. To Benedicte
Karen Fasmer Waaler, for encouragement and for everything else, especially
for the year 1995–96. To Henriette Nilsson, for coffee and opera in the
morning. And to Rikke Kristin Gåsholt, for throwing great parties!
I would also like to thank my fantastic mathematics teacher at Bergen
Katedralskole, Anne Berit Holme, and Geir Ole Øverby for helping me trust
my own sense of logic when I returned to the natural sciences.
To my closest family: My mother Tordis Dalland, my father Bernard
John Evans, my brother Peter John Evans, my uncle Arild Jensen and Pus
the Cat, thank you all for your interest and support. Thanks are also due
to my newly acquired extended family: Kjell Myhren, Bozena Elizabeth
Jakubowska, Rosallina and Toby, for taking such good care of my parents.
Finally: Terje Fredvik - my IDL guru, my sparring partner in discussions
on observational astronomy, my playmate and friend. It has been such a
lot of fun sharing this with you, both at work and off, and you know how
invaluable you have been in pushing me on and getting me through. Thanks
for your endless support, for reading the thesis, for holding my hand and
especially for being the one thing during this time that was always right.
I’m on another planet with you!
Contents
Preface
i
1 Introduction
1.1 General outline of the thesis . . . . . . . . . . . . . . . . . . .
2 Studying QSO 0957+561 A,B
2.1 Data sets . . . . . . . . . . . .
2.1.1 The field . . . . . . . . .
2.2 Gravitational lensing of quasars
2.2.1 Time delay . . . . . . .
2.2.2 Microlensing . . . . . . .
2.3 A brief time delay history . . .
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3 Pre-processing the ALFOSC data set
3.1 A look at the equipment: The CCD . . . . . . . . . .
3.1.1 The ALFOSC CCD . . . . . . . . . . . . . . .
3.2 Bias and overscan . . . . . . . . . . . . . . . . . . . .
3.3 Flat-fielding . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Inspecting the flats . . . . . . . . . . . . . . .
3.3.2 The moving blob . . . . . . . . . . . . . . . .
3.4 Other preparations . . . . . . . . . . . . . . . . . . .
3.4.1 Combining images . . . . . . . . . . . . . . .
3.4.2 Scaling, aligning and background subtraction .
3.4.3 Bad Columns . . . . . . . . . . . . . . . . . .
3.4.4 The importance of careful pre-processing . . .
4 The MCS procedure
4.1 Convolution . . . . . . . . . .
4.1.1 The sampling theorem
4.2 Deconvolution . . . . . . . . .
4.2.1 Traditional methods .
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iv
CONTENTS
4.2.2 Deconvolution with correct sampling:
4.3 Preparations for running MCS . . . . . . . .
4.4 The aim of MCS . . . . . . . . . . . . . . .
4.5 General outline of MCS . . . . . . . . . . .
4.5.1 Extracting data frames . . . . . . . .
4.5.2 Constructing the PSF . . . . . . . .
4.5.3 Deconvolution with the profile s . . .
4.5.4 Residual frames . . . . . . . . . . . .
4.6 Simultaneous deconvolution . . . . . . . . .
4.6.1 Final deconvolution results . . . . . .
4.6.2 Obstacles . . . . . . . . . . . . . . .
4.6.3 Testing MCS . . . . . . . . . . . . .
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5 Photometry from the ALFOSC data
5.1 Basic principles of photometric studies . . . . . . .
5.1.1 Standard systems and reference magnitudes
5.1.2 Colours . . . . . . . . . . . . . . . . . . . .
5.1.3 Photometry from MCS . . . . . . . . . . . .
5.2 Photometric results . . . . . . . . . . . . . . . . . .
5.2.1 Error estimations . . . . . . . . . . . . . . .
5.2.2 Light curves from March 2001 . . . . . . . .
5.2.3 Light curves from January 2000 . . . . . . .
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6 Discussion
6.1 Testing the results . . . . . . . . . .
6.1.1 A simple χ2 test . . . . . . . .
6.1.2 Seeing . . . . . . . . . . . . .
6.1.3 Possible influence of the Moon
6.1.4 Colour diagrams . . . . . . .
6.2 Comparing the time shifted data . .
6.3 Zero lag correlation . . . . . . . . . .
6.4 Comparisons with other results . . .
6.4.1 Results from Ovaldsen et. al. .
6.4.2 Results from Colley et. al. . .
6.4.3 Summary . . . . . . . . . . .
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7 Reduction and photometry of
7.1 Pre-processing . . . . . . . .
7.2 Running MCS . . . . . . . .
7.3 Results . . . . . . . . . . . .
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the StanCam
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data
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CONTENTS
v
8 Conclusions
115
8.1 Future studies . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
Bibliography
A The Image Reduction and Analysis Facility (IRAF)
A.1 Using IRAF . . . . . . . . . . . . . . . . . . . . . . .
A.2 Pre-processing with IRAF . . . . . . . . . . . . . . .
A.2.1 Zerocombine . . . . . . . . . . . . . . . . . . .
A.2.2 Ccdproc and Flatcombine . . . . . . . . . . . .
A.3 Other tasks using IRAF . . . . . . . . . . . . . . . .
A.3.1 Image scaling and background subtraction . .
A.3.2 Stacking the images . . . . . . . . . . . . . . .
119
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123
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B Text files associated with the MCS programs
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C Tables of output files from the MCS programs
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D Miscellaneous
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D.1 Integrating the Gaussian function . . . . . . . . . . . . . . . . 139
vi
CONTENTS
Chapter 1
Introduction
“The time has come,” The Walrus said, “To talk of many things”
Lewis Carroll, Through the Looking-Glass and what Alice found there
Observational astronomy is one of the earliest human activities that can be
described as scientific, at least in the sense of making predictions based on
careful studies of natural phenomena. The most famous early examples are
probably the ancient Oriental astronomers, the Greeks and the Aztec civilisation. The impressive knowledge these societies had of the movements of
the stars and planets never stops to fascinate and intrigue us. They not only
gathered information from observations, they used this data to calculate the
distance around the Earth and make predictions about the future movements
of celestial objects.
It was, however, the business of the philosophers of ancient Greece to
discover ‘how the world really works’. The main function of astronomers,
through antiquity and the Middle Ages, was to provide the mathematical
basis for astrology. Not until the period of what is known as The Scientific
Revolution1 did the natural sciences emerge as an autonomous field of research. During this epoch, the foundation for the scientific method was laid.
Speculation and theory about the nature of the universe on the one hand,
and the empirical gathering of information on the other, were joined in a
mutually dependent system.
Another aspect of this period is the technological development that occurred. With the invention of the telescope in the early 1600s came the first
apparatus that enhanced the sensitivity of the organic light detector – the
human eye. Intriguing observations stimulated the development of bigger
and better telescopes. The ensuing astronomical discoveries in turn inspired
1
Classically dated as 1543 – 1687, that is, the years between the death of Copernicus (and the publication of his work De revolutionibus orbium cælestium) and Newton’s
Philosophiae Naturalis Principia Mathematica.
2
Introduction
the building of even better telescopes. This interplay between technology and
science is of course a continuing process, but it is notably marked by sporadic progress following particular inventions opening up hitherto untouched
territories. When Joseph Fraunhofer (1787–1826) invented the prism spectroscope, he initiated the investigation of the chemical composition of the
Sun. When gelatine-based photographic emulsions and the introduction of
silver bromide, which enhanced the light sensitivity of photographic film, became common in the late 19th century, astronomers quickly started applying
this to make better observational records. This trend has continued from the
Renaissance to the present day in an ever-increasing spiral. Each improvement of scientific apparatus helps to provide answers to old questions as well
as raising a whole host of new ones, which in turn drives the quest for better
equipment.
Better equipment is one way of obtaining better observations. Another
approach, which has accelerated in importance in later years with advancing
computer technology, is the development of new and better image processing
routines. With this approach, images are improved in quality after the observation itself has been conducted. Here we see again the effect technological
progress has on scientific observations. Developing computer programs that
reduce noise and enhance the resolution of images, effectively increases and
hopefully improves the science we can get out of our observations. Many
devices to this effect are created today.
1.1
General outline of the thesis
Accurate photometry is crucial to obtaining reliable measurements of, e.g.,
the time delay between images of a multiply lensed quasar. Precision photometry of QSO 0957+561A,B is the main objective of this thesis work.
We will be using a relatively new method of improving image quality. The
MCS routine, named after its developers Pierre Magain, Frédéric Courbin
and Sandrine Sohy, uses deconvolution with correct sampling to improve the
resolution of astronomical images, without the drawbacks traditional deconvolution routines have suffered from, see Magain et al. (1998). The method
is described in Chapter 4. We will utilise MCS to obtain photometric results
from our data sets of the gravitationally lensed quasar QSO 0957+561 and
some adjoining field stars. In Chapter 2 we describe the field of our observations and the quasar we are studying, and we briefly discuss the principles of
gravitational lensing in general as well as some of its applications. The data
sets we use were obtained by us and other groups during several observing
sessions, from the Nordic Optical Telescope situated on the Roque de los
1.1 General outline of the thesis
3
Muchachos on La Palma. The main part of this thesis consists of analysing
two data sets from observations with the ALFOSC instrument. Before starting to extract photometry from the images, we must correct, as well as can be
achieved, for effects introduced by the passage of light through the detector.
This pre-processing is described in Chapter 3. Results from the photometry itself follow immediately after the description of the MCS programs,
in Chapter 5. A detailed discussion and testing of the results is presented
in Chapter 6. In addition, we have analysed some images provided by the
GLITP cooperation from the instrument StanCam at the NOT. The results
from this data set are presented in Chapter 7. Finally, a short summary of
our main conclusions is provided in Chapter 8.
Hoping that it can be of assistance to others who want to start penetrating
the world of IRAF reductions, we include in Appendix A a brief description
of the tasks and parameters we have used during the course of this work.
This is not in any way intended as a general introduction to the many reduction options available in IRAF, but more as a guideline for students and
others seeking to do similar work as what we have done here. Since few such
overviews exist (see, however, Appendix A in Dahle, 1994), we found this
relevant to include.
4
Introduction
Chapter 2
Studying QSO 0957+561 A,B
The truth is rarely pure and never simple.
Oscar Wilde, The Importance of Being Earnest, Act I
Since its discovery and identification as a gravitationally lensed double object
by Walsh et al. (1979), the quasi-stellar object scientifically named QSO
0957+561 and familiarly dubbed the ‘Twin Quasar’ has been the target of
much study and debate. Shortly after its discovery, changes in brightness of
the quasar images were discovered and the system became the first for which
an attempt to measure the time delay was reported, see the introduction in
Kundic et al. (1995) for a list of references. Accuracy of the photometric
results is crucial when using light curves to measure the time delay and
eventually e.g. calculate the Hubble parameter H0 (see Section 2.2). This
thesis work is a detailed study of optical monitoring of the quasar, from three
sets of observations. As our data were unsuited for time delay measurements,
we focus on precision photometry.
In this chapter, we first describe our data sets and discuss some basic
elements from the field of gravitational lensing and its uses as a cosmological
probe. In the last section, we recapture briefly the history of the Twin Quasar
and recent efforts to measure the time delay.
2.1
Data sets
In this section, we briefly describe the data sets we have reduced and which
form the observational basis for this thesis work. All observations were obtained at the Nordic Optical Telescope (NOT) situated at the Observatorio
del Roque de los Muchachos on the island of La Palma in the Canary Islands.
The data and the CCDs are described in further detail in Section 3.1.1 and
Chapter 7.
6
Studying QSO 0957+561 A,B
We have analysed two sets of observations from the ALFOSC instrument
(The Andalucia Faint Object Spectrograph and Camera).1 Our main set of
observations consisted of 997 images, 806 in the R filter and 191 in V . We
obtained these images during 4 nights of photometric observing conditions
from March 14th–18th 2001.2 This constitutes the main data set for this
thesis work.
Our second set of observations from ALFOSC were from an observing run
in January 2000.3 This set consisted of data from a single night, January
25th. We had a total of 171 R and 39 V images from this night.
The data from these two observing sessions was obtained as part of the
‘QuOC Around the Clock’4 campaign 2000/2001, a collaborating network
involving 10 participating observatories (12 in 2001). The campaign was initiated by Wesley Colley and Rudy Schild, with the aim of closely monitoring
QSO 0957+561 in order to determine more accurately the time delay of the
system. The quasar was observed continuously for a total of 10 nights in
January 2000 and again in March 2001. Colley et al. (2002) and Colley et al.
(2003) describe the official QuOC results.
The observations were, as requested from the QuOC team, run in cycles
of 60 minutes of imaging in the R band, 15 minutes in V, 60 minutes R, 15
minutes V, ... 60 minutes R. The finding chart in Figure 2.1 identifies five
standard stars that the QuOC team asked to make sure were unsaturated;
the brightest is star G. The ideal exposure would bring this star to about
90% of the saturation limit.
We have also reduced a third data set, which was obtained from the
GLITP5 collaboration. Luis Goicoechea and Aurora Ullán Nieto kindly provided us with a set of data from the StanCam instrument at the NOT, a
total of 75 R and 53 V images. The dates on these observations ranged from
February 4th to March 30th 2000. With some modifications, the data from
each night consisted of two R frames and one V frames, see Chapter 7 for
details.
1
ALFOSC is owned by the Instituto de Astrofisica de Andalucia (IAA) and operated
at the Nordic Optical Telescope under agreement between IAA and the NBIfAFG of the
Astronomical Observatory of Copenhagen.
2
Observers were Kjetil Kjernsmo, Øyvind Saanum and Anna Kathinka Dalland Evans.
3
Observers were Kjetil Kjernsmo and Øyvind Saanum.
4
Quasar Observing Consortium for Around the Clock Monitoring.
5
The Gravitational Lenses International Time Project.
2.2 Gravitational lensing of quasars
2.1.1
7
The field
Figure 2.1 shows the central field of all our image frames from the ALFOSC
data. We see that there are several stars in the field, the brightest are for
historical reasons marked with letters.
There are two images of the quasar, labeled A and B, with a separation
of 6.2 00 . The main lensing galaxy, close to the centre of a galaxy cluster, is
an elliptical (cD) galaxy, at redshift zL = 0.36. The redshift of the quasar is
zQSO = 1.41. The centre of the galaxy is located only ∼1 00 from the centre
of the B image, slightly to the North–East of B.
Figure 2.1: The central part of our image frames, showing the quasar components
A and B of QSO 0957+561 and the ‘classic’ field stars F, G, H, E, D as well as the
extra stars labeled X, R, P and Q. This image is 3.8 0 × 3.0 0 . North is up, East
is left.
2.2
Gravitational lensing of quasars
According to Einstein’s General Theory of Relativity, matter bends light.
This prediction was first verified during the total solar eclipse of 1919, when
the apparent position of stars just outside the solar limb were found to have
temporarily changed due to the gravitational field of the sun.
8
Studying QSO 0957+561 A,B
Light rays travelling towards us from distant celestial sources, can pass
through large concentrations of matter such as galaxies or galaxy clusters.
The trajectories of photons passing through gravity fields, are curved as the
photons follow the geodesics of curved space-time. From our point of view,
the intervening mass between us and the source may be seen to act as a lens,
splitting and distorting the image of the distant object. The manner of this
distortion will, in addition to the amount of intervening mass and the length
of the distances involved, depend on the exact geometrical configuration of
the triumvirate observer – lensing mass – lensed object.
The nature of Quasi Stellar Objects (‘quasars’ or QSOs) is one of the
great enigmas in modern cosmology. Quasars are very remote and extremely
luminous, apparently starlike objects with strong emission lines. They are
thought to be the central parts of active galaxies. The intense radiation emitted from quasars is often variable in time. Their high redshifts, along with
their intense luminosity, makes them excellent candidates for gravitational
lensing. That they are far away increases the probability that their light will
pass through large mass concentrations on the way to our detectors. That
they are so luminous is a prerequisite to us observing them at all at redshifts
up to (so far) 6.4.6
2.2.1
Time delay
The geometrical difference between the lensed and unlensed photon paths
introduces a time-lag between the arrival times of the lensed and unlensed
photons at the position of the observer. This time-lag is called the geometrical time delay. While passing in the immediate vicinity of the gravity field
of the lens, the light is also affected by a gravitational time delay. A ‘lensed’
photon will be seen by an observer with a total time delay which is the sum
of these two effects, with respect to the observation of the same photon if it
were not lensed. When two or more images of the source are observed, it is
possible to compare the arrival times of the lensed images and to determine
a ‘relative time delay’. As the unlensed source is never visible, this is in
fact the only truly measurable quantity, rather than the actual time delay
between the lensed and unlensed paths.
Refsdal (1964) proposed an original method to use gravitational lensing as
a tool to measure the Hubble parameter H0 . This application of gravitational
lensing is one of the main motivating factors for the detailed photometric
monitoring of lensed quasars like the Twin Quasar.
The ‘Refsdal method’, or ‘wavefront method’ of determining H0 through
6
Sloan Digital Sky Survey: http://www.sdss.org/news/releases/20030109.quasar.html
2.2 Gravitational lensing of quasars
9
the time delay, is based on the deformation of wavefronts as light waves move
from the source, past the deflector to the observer. This deformation is caused
by curvature effects and time retardation described above. Each passage
of the wavefront at the observer corresponds to an image perpendicular to
the wavefront, forming multiple images. All points located on a wavefront
have identical propagation times. The distance between the sections of the
wavefront at the observer gives the time delay between the corresponding
images.
We use a double imaged system as an example, and assume redshifts
small enough so that Hubble’s law is valid and an axially symmetric lens. In
Figure 2.2 we show a wavefront diagram for this case. Note that the angle
between wavefront and line-of-sight should be 90 degrees. The wavefronts
cross each other at the symmetry point E, where they represent the same
light propagation time. For an observer O located a distance x from the
symmetry axis, the time delay must be equal to the distance between the
wavefronts at the observer, divided by the speed of light (c).
An expression for H0 or, equivalently, for the Hubble age τ0 of our universe, can be derived from Figure 2.2 and the assumption of a deflecting law
of the type
α̂ ∝ |ξ|(−1) ,
where α̂ is the deflection angle of a light ray passing within a distance ξ of
the lens.
The expression for H0 derived in Refsdal and Surdej (1994), is given by
τ0 = H0−1 =
2(zs − zd )∆t
.
zd zs θAB (θA + θB )(2 − )
(2.1)
Note that this equation is solely in terms of observable quantities. zs and
zd are, respectively, the redshifts of the source and the lens. The relative
time delay is ∆t, θA and θB are the angular distances from image to lens,
and θAB is the angular separation between the images A and B. For large
redshifts, the modification of Hubble’s law makes it necessary to introduce
a correction factor in Equation 2.1. Kayser and Refsdal (1983) showed that
the correction factor is less than 10–20%.
The main difficulty in calculating H0 from QSO 0957+561, is the determination of . This is very difficult for the Twin Quasar, since the surrounding
galaxy cluster, as well as the central galaxy, acts as a lens.
Conversely, Equation 2.1 can be used to constrain the mass distribution
in a given lens, once H0 has been determined form other methods, or from
other QSO systems.
10
B
:9:9 :9:9
!"
Studying QSO 0957+561 A,B
;<; ;<;
87< 87 <
S
#$ %&
(' *)*) 0
0//
34 65 12 34 65 ,
+,+
.- .-
ξ
ξ
M
B
A
A
θS
θB
θA
~
~θ A B
θA B
WAVE FRONT
E
~
~θ A B
c ∆t
O
x
Figure 2.2: The wavefront method applied to two selected light rays for the
determination of the time delay. Reproduced from a figure in Refsdal and Surdej
(1994). See text for details.
2.2.2
Microlensing
In Figure 2.2 we showed how light from a distant source was lensed by an axially symmetric lens, represented as a spherical point (M ). As a result of the
deflection, the observer sees two images of the source. In reality, the lensing
2.2 Gravitational lensing of quasars
11
galaxy is an extended object containing billions of stars. These stars, and
other compact objects (MACHOs), may act as microlenses, perturbing the
light bundles so that additional images appear. However, as the typical image
separation is proportional to the mass of the lens, these micro-images have
a separation of the order of micro-arcseconds, hence the term microlensing.
These additional images are well below the resolution of present optical telescopes, so they cannot be seen directly. However, the micro-images
change in strength and number with the relative transverse motion between
the observer, lens and source. This may cause brightness variations between
the macro-images. What we observe, therefore, is a ‘twinkling’ of the quasar
components.
The detection of microlensing can be useful in several respects. It can be
used to extract information about the mass of the microlenses in the lensing
galaxy, as well as the size of the continuum and the line emitting regions in
the quasar.
We see why lensed objects are so valuable a tool. For a single quasar, we
do not know the intrinsic variability, as we do not have a complete theory
of the nature of quasars. At present, our only chance to prove microlensing
is to make use of macrolensed multiple quasars. Intrinsic variability in the
quasar will appear (time shifted) in all lensed images. Microlensing, on the
other hand, appears independently in the different images of the quasar. In
order to use a macrolensed object to prove the existence of microlensing, we
have to know the time delay so that we can separate the recurring variations
from the independent ones. The usual procedure is to time shift the light
curves of the quasar components and subtract one from the other.7 This
produces a difference light curve which contains the microlensing signal. A
difficult issue is that the time delay itself may be extra difficult to measure
if the light curves are a mixture of intrinsic variations and microlensing.
Gravitational lensing itself is achromatic: the strength of the deflection
of the light rays is not dependent on wavelength. As the macro-images we
observe are images of the same quasar, the magnitude difference in different
bands should therefore be equal. If they are observed to be different, the
cause could be differential absorption along the different light paths, or it
could be caused by microlensing. Microlensing can produce an apparent
chromatic effect. This is because the average amplification factor may depend
on source size, which in turn depends on wavelength. This can cause different
parts of the quasar to be magnified differently, which we may observe as a
difference in flux ratio at different wavelengths between the components. We
return to this issue in Section 6.2.
7
One must also shift the curves in magnitude to correct for the different magnifications.
12
2.3
Studying QSO 0957+561 A,B
A brief time delay history
Measuring the time delay between the images of lensed quasars is not a trivial
task. In spite of extensive monitoring and investigations by several groups,
the time delay between the two components of QSO 0957+561 has proved
difficult to determine. Regular monitoring over time periods longer than the
time delay is made difficult by the long time delay of the system, which is
over a year. As long as microlensing signals are not clearly distinguished from
intrinsic variations, microlensing complicates time delay measurements.
More than 15 years after its discovery, there were two favoured time delay
values: ∼540 days and ∼415 days. Kundic et al. (1997) convincingly rejected
the higher value, finding a best-fit delay of 417 ± 3 days. Burud et al. (2001),
using the public data set published by Serra-Ricart et al. (1999), found a
value of 423 ± 9 days. In recent years, reports of time delays between 416–
425 days have been reported. Again, two leading candidates seem to emerge:
417 days (e.g. Kundic et al., 1997, Colley and Schild, 2000) and 425 days
(e.g. Serra-Ricart et al., 1999, Ovaldsen et al., 2003a).
Chapter 3
Pre-processing the ALFOSC
data set
Facts do not cease to exist just because they are ignored.
Aldous Huxley, Proper Studies
A key feature of observational astronomy has been the keeping of records:
the preservation of data archives for future reference. In the earliest of times,
buildings and monuments were designed that had special functions on solstices and equinoxes, thus passing on information about the movements of the
heavenly bodies. Observers started drawing what they saw, either through
their own eyes or, later, by looking through the first telescopes. The inventions of the camera and photographic techniques spurred the development of
record-keeping further. In this chapter, we will take a look at the modern
way: Images are obtained at the telescope through electronic devices and
converted to digital numbers which can then be processed, or reduced, by
computer programs. The first step of data reduction is to minimise noise
and enhance image quality, before attempting to extract any scientific information. We will describe how we performed this pre-processing on our
set of data, as well as discuss some other necessary preparations. Our main
focus is the pre-processing of the main ALFOSC data set, from March 2001.
Pre-processing the set from January 2000 followed essentially the same lines.
3.1
A look at the equipment: The CCD
The great advantage of the photographic plate is that it can build up a picture
of a faint object by accumulating light on its emulsion for a long period of
time. Charge-coupled devices, or CCDs, have further revolutionised modern
14
Pre-processing the ALFOSC data set
astronomy. The first astronomical image from a CCD was produced in 1976,1
and today the CCD is in use at most professional observatories as well as in
many amateur telescopes. The CCD has a supremely high quantum efficiency
(QE), that is, ability to detect and store light signals. The QE is formally
defined as the ratio of incoming photons to those actually detected. The
CCD has a broad spectral range where it is sensitive to light (the bandpass).
Typical CCD QE curves today peak near 90%, while maintaining a 60% or
more efficiency over two thirds of their total spectral range.
A CCD is an array of a finite number of individual, regularly spaced,
‘picture elements’, or pixels, made of silicon. These pixels detect the light
reaching them by converting photons to electrons or ADUs via the photoelectric effect.2 Present day CCDs in use in astrophysical contexts normally
come in sizes of equal length and width, ranging from 512 × 512 up to 4096
× 4096 pixels. The basic physical principle behind the CCD is the photoelectric effect: A photon with the target energy3 is absorbed by the silicon,
which is forced to give up a valence electron. This photo-generated electron then migrates into the conduction band, is collected by a gate structure
within the pixel and held in a potential well until the end of the exposure,
a process known as charge storage. After the exposure, a change of voltage
potentials transfers the charge collected within each pixel row by row to an
output register where the charge is measured as a voltage and converted into
an output digital number. This number is referred to as counts or ADUs.
The number of collected electrons (or received photons) needed to produce
1 ADU is termed the gain of the CCD. For a CCD, there exists a linear relation between input (detected photons) and output (digital number), where
the gain of the CCD is the proportionality constant.
Because of the properties of silicon, early CCDs were less sensitive in
blue wavelengths. Since then, various CCD coatings have been developed to
remedy this, usually consisting of organic phosphors that convert incident
light into longer wavelengths.
3.1.1
The ALFOSC CCD
Two of the data sets discussed in this thesis were obtained using the instrument ALFOSC.4 We record here a few technical details about the CCD and
1
Scientists from the Jet Propulsion Laboratory imaged the planet Uranus at a wavelength of 890 nm (Janesick and Blouke, 1987).
2
ADU = analog-to-digital unit.
3
Silicon has a useful photoelectric effect range of 1.1 to about 10 eV, covering the
near-IR to the soft X-ray region (Howell, 2000 and references therein).
4
The Andalucia Faint Object Spectrograph and Camera
3.1 A look at the equipment: The CCD
15
the images that were obtained. Our third set of data, obtained from the
StanCam instrument, is described in Chapter 7.
The CCD detector consists of 2048 × 2048 pixels. Such a relatively large
device gives a rather long read-out time, which is the time it takes for the
CCD to register the image. As we did not need such a large field-of-view,
the object frames were cut in the y direction in order to save read-out time.
The dimension of our object frames was thus reduced to 2048 × 1200 pixels
and our read-out time to ∼50 seconds.
2048
pixels
1200
pixels
2048 pixels
Figure 3.1: Schematic illustration of how a total of 848 rows on the CCD was ‘cut
off’ in order to save read-out time. The entire field shown in Fig. 2.1 is contained
within the 2048 × 1200 pixels shown here.
The CCD has a gain of 1 electron per ADU, a read-out noise of 6 electrons
per pixel and a pixel scale of 0.188 00 per pixel.
As mentioned, the ALFOSC data was obtained as part of the ‘QuOC
Around the Clock’ campaign. To accommodate the requested observation
cycles in R and V , see Section 2.1, our frames therefore had varying exposure
times, ranging from 32 to 240 seconds in the R band and from 36 to 175
seconds in the V band, but with an approximate mean for each night of 60
seconds in R and 80 seconds in V. Indeed, even after cutting part of the CCD
as shown in Figure 3.1, our exposure time was quite often shorter than the
read-out time.
The telescope was stepped slightly (a few pixels) in the x and y direction
between successive exposures to allow us to average out any small-scale nonuniformities. This also allowed us to remove unwanted stars from the sky
flats, see Section 3.3.
16
Pre-processing the ALFOSC data set
3.2
Bias and overscan
To avoid negative values in the output image, CCDs are set up to add a
positive pedestal level of a few hundred ADUs to each image. The output
signal is thus ‘biased’ by this level. The first step in pre-processing data is
to subtract this level from the image frames, both flats (see Section 3.3) and
object frames. We obtained sets of 10 bias frames each night.5 A bias frame
is simply a read-out of unexposed CCD pixels: It has an exposure time of
zero seconds, and is taken with the shutter closed. Individual bias frames
are averaged so as to eliminate cosmic rays,6 read-out noise variations and
random fluctuations.
An averaged bias frame with no overall structure in the image would
show a Gaussian distribution in a histogram, see for example Howell (2000).
The top panel of Figure 3.2 shows a histogram of how our pixel values were
distributed for a typical, averaged bias. We have also fitted a Gaussian
distribution with three coefficients; height, width and centre of the function,
to the data. The fit is not very good in the top panel of Figure 3.2. There
is a surplus of values below the average value, compared to the Gaussian
fit. This is because there is a column-wise structure in our (averaged) bias.
The left side of the bias frame has a higher value, by approximately 5 ADU,
than the right side. This structure can be seen in Figure 3.3, where we have
plotted the average of each column of a bias frame. To illustrate that the bad
Gaussian fit is a result of the structure in the bias, we removed the structure
on a typical averaged bias frame by subtracting a polynomial of degree 2, and
then we fitted a Gaussian again. The result is shown in the bottom panel of
Figure 3.2. We have searched in the literature for theories about what causes
the structure in the bias frames, but have been unsuccessful in coming up
with an answer.
Overscan strips are columns of ‘pseudo-pixels’ added to and stored with
each image frame. This region, lying outside the physical area of the CCD, is
generated by sending additional clock cycles to the CCD output electronics
after reading out the real CCD. Like a bias frame, the overscan region gives
a measure of the zero offset level and the uncertainty of this level. Figure
3.4 shows that the variation across the overscan region of a given frame is
typically only a few ADUs, but that the level can vary considerably through
one night. In Figure 3.4, we see that the level has increased through the night
5
The exception was the night of March 14th–15th when we obtained only 5 bias frames.
‘Cosmic’ rays can sometimes be caused by weakly radioactive materials used in the
construction of CCD dewars (Florentin-Nielsen, Anderson & Nielsen, 1995). The elimination of these ‘hot’ pixels in the combination of individual bias frames are due to the
rejection option minmax in IRAF, see Appendix A
6
3.2 Bias and overscan
17
Figure 3.2: A typical pixel value distribution of an averaged bias frame, from the
night of March 15th–16th. We have plotted the number of pixels vs. pixel ADU
value. The mean bias level offset of the data is 363.55 with standard deviation
14.26. Overplotted is a Gaussian fit to the data. Top: The Gaussian has its centre
at a pixel value of 364.57, and the standard deviation of the Gaussian is 2.77 ADU
(the larger standard deviation of the actual data is caused by ∼500 pixels having
values below 350, which do not appear in the figure). Bottom: The structure in
the bias frame has been removed by subtracting a polynomial of degree 2. We
notice a good Gaussian fit. The mean of the Gaussian is 364.34, the standard
deviation is 1.79.
by ∼15 ADU. From the image headers, we checked the CCD temperature
throughout the nights, but found no systematic changes, so the increase in
overscan level remains a puzzle. When performing bias subtraction and flat
field division in IRAF (see appendix A), we can monitor any changes in
pedestal level from graphs of the overscan region such as Figure 3.4.
A bias frame includes more information than the overscan region, as it
represents any two-dimensional structure that may exist in the CCD bias
level. If this variation across the frame is very low, it may be sufficient to
subtract only the mean bias level provided by the overscan region. However,
as we had some structure in our bias frames, see Figure 3.3, we used subtrac-
18
Pre-processing the ALFOSC data set
Figure 3.3: Variation across an (averaged) bias frame. The average of each column
is plotted. The high values in the rightmost part is the overscan region, see also
Figure 3.4
tion of the bias frame on our data. To perform bias frame subtraction for an
object frame, IRAF will first subtract an average of the overscan region from
the given frame before subtracting the variation, corrected for each overscan
level, across the frame. The variation across the frame is provided by the
(averaged) bias frame.
Dark frames are images taken with the shutter closed, but for some time
period, usually equal to that of the object frames. These frames are used to
measure the thermal noise in a CCD, which is the production of electrons
caused by the random motion of atoms. However, the efficient cooling of
CCDs using liquid nitrogen makes the dark current practically negligible,
especially for relatively short exposure times like ours.
3.3
Flat-fielding
Across a CCD, there are variations in the individual pixels’ response to light.
The quantum efficiency and gain of each pixel is slightly different from that of
its neighbours. In order to correct for these pixel-to-pixel variations, we obtain flat field images. Flat fields are exposures with uniform illumination and
a high signal-to-noise ratio,7 providing a read-out of the internal variation of
the pixels within the CCD. Dome flats are exposures of the illuminated inside
7
See Section 3.4.
3.3 Flat-fielding
19
Figure 3.4: The overscan region from two different frames. The lower line is from
a frame taken early at night, while the upper is from a frame taken early next
morning, at the end of the observing run. The average value of each column is
plotted.
of the telescope dome, while sky flats are exposures of the bright twilight sky
in the morning or (preferably) evening.
A flat-fielded object image is obtained by first subtracting the bias level
and then dividing by the bias-subtracted flat field image. In equational form,
this is:
Final Reduced Object Frame =
Raw Object Frame − Bias Frame
Flat Field Frame − Bias Frame
The flat field frame is normalised to a mean value of 1 before division. A
pixel in the flat frame with less than average value will thus have a normalised
value < 1. Division by this value will cause the corresponding pixel value in
the object frame to increase, which evens out pixel variations, as this pixel
had, from the outset, a lesser value than it would have had if all pixels reacted
alike.
A dust grain on the filter or on the window of the CCD dewar will cause
a shadow on the image frames. Flat field images usually show a collection
of dark ‘doughnut’ shapes on a brighter background. The doughnut shape is
caused by an out-of-focus negative image of the telescope main mirror, with
the central obstruction caused by the secondary mirror.
20
Pre-processing the ALFOSC data set
While the main role of flat field images is to remove pixel-to-pixel variations, they will also compensate for any vignetting, i.e. unintended darkening
of corners, and for time-varying dust accumulation that may occur on the
dewar window and/or filters within the optical path. Flat field calibration
frames are needed for each wavelength region (R and V filter in our case) and
for each observing night. This is because the variation in the pixels’ response
itself varies with wavelength, and because conditions may change from night
to night (which in fact proved to be a bit of a challenge, see section 3.3.2).
3.3.1
Inspecting the flats
The image pre-processing was done using the IRAF (Image Reduction and
Analysis Facility) environment. See appendix A for a short discussion on
IRAF, including details on what parameters we used for the different tasks.
We obtained both dome and sky flats at the observatory. However, the
dome flats turned out to be very uneven: Parts of each doughnut were much
darker than other parts. We are not sure what caused this, but we could
see from inspection of some of the brighter ‘raw’ object frames that the
doughnuts were even in appearance, and we suspected that something was
probably wrong with our dome flats. We could only speculate as to what
caused this, perhaps the lamp used to illuminate the telescope dome was
badly placed. Dome flats from the StanCam observations also showed this
effect, see Section 7.1.
A ‘master flat’ is obtained by combining several individual flats. To
further test the dome flats, we constructed one master flat consisting entirely
of sky flats, one entirely of dome flats and one of a combination of the two
flat types. After examining the results we decided that the sky flats were
superior and discarded our dome flats. This of course reduced our number
of flats considerably. However, in all cases except one we managed to retain
five or more good sky flats for each filter and each night, which suffices for
a decent flat-fielding. As a rule of thumb, 5 flats or more is acceptable
although it is far better to have a few good flats than many bad ones. On
the one occasion (night of the 14th–15th, filter V ) that we had only three
good flat frames left, we adjusted the combination parameters accordingly,
see appendix A for details.
When combining images, one can form a type of average value of each
pixel in all frames. Not all pixel values need to be taken into account when
combining images, indeed not all of them should be. Cosmic rays, stars
appearing on the sky flats etc. result in some pixel values deviating noticeably
and erroneously from the rest. Reckoning with this value could wrongly
offset the statistics of the combined frame. Therefore, a rejection criterion
3.3 Flat-fielding
21
of some sort produces a more realistic combination. In combining n images
the routine must decide at each pixel which of the n data values, if any,
to reject in forming the average. In IRAF, there are a number of different
rejection operations to choose from. Depending on what frames that are to
be combined, different criteria for rejecting ‘deviating’ pixels can be chosen.
The up side of using sky flats is that the colour of the night sky is usually
closer to the colour of the actual object frames. The down side is that stars
may be present on sky flats, which can render them unusable. Since we used
only sky flats, we wanted to be sure to use the most appropriate combination
parameters for eliminating stars from the flat frames. We examined all sky
flats individually, looking for prominent stars located in sensitive parts of
the images (i.e. around the areas where the quasar components and the field
stars shown in Figure 2.1 would be positioned) that could not be eliminated.
We did have to discard a couple of flats with too many stars. On those nights
when the telescope had been stepped between exposures, faint stars could
be removed by using the median instead of the average when combining
individual flats. For the nights when the telescope had not been stepped
between flat field exposures, we had to discard sky flats containing stars.
The flat field and the bias images utilised the full extent of the CCD, i.e.
they were not cut in the y direction during the observing run. We therefore
had to cut these frames accordingly, to extract the correct section of the
flat field that would correspond exactly to the (pre-cut) object frame. We
made a short IDL program that cut the flat fields appropriately. Figure 3.1
shows how the full CCD detector was cut when obtaining the data frames
and subsequently how we had to cut the flat fields afterwards, to make them
correspond to the object frames.
Figure 3.5 shows the variation across a typical flat frame. Here, we have
plotted the average of each column in the frame. We see some edge effects,
but this is a trimmed flat frame so the severe edge effects have been trimmed
off. The dips and troughs are discussed in section 3.4.3.
3.3.2
The moving blob
There were a lot of out-of-focus dust grains causing doughnut shapes on our
flat frames. After close inspection of the flats for our third night, the night
of the 16th–17th, we found that while the 5 sky flats from the evening were
internally consistent in structure, the flats from the early morning were not:
A strangely shaped, quadruple doughnut had shifted quite considerably in
position.
This apparently sudden shift in position of the blob was apparently not
affected by filter changes (R or V ), and it occurred twice: once for the night
22
Pre-processing the ALFOSC data set
Figure 3.5: An example of variation across a typical flat frame. The graph shows
the average ADU value of all columns. The two sharp declines at x = 1232 and
x ∼ 1370, as well as the apparent vertical line at x ∼ 1786, will be discussed in
Section 3.4.3. This figure shows a combined flat, so the extreme low values at the
far edges have been trimmed off. The ordinate on the left show the counts on the
images (ADU), while the ordinate on the right and the abscissa show the y and x
axes of the image frame.
16th–17th and then again for the night 17th–18th. We discussed several
theories as to what this ‘object’ might be. We now think the blob may
be a conglomeration of 3–4 grains of dust, giving rise to several doughnuts
situated very close together. This would make it heavier, and it is thus a
possibility that this ‘giant dust grain’ has been blown by a chance wind and,
the individual dust grains sticking together, has flipped and rolled over, while
the other, lighter ones stayed in place. Although we have not investigated
this in more detail, this is the most likely hypothesis concerning the nature
of the moving blob that we came up with. We decided to leave the matter of
the nature of the blob here, and concentrate on how to avoid any problems
it might cause.
On the night of the 16th–17th, the shift in position of the blob occured
while we were taking our object frames. The logical solution was to regroup
3.3 Flat-fielding
23
Figure 3.6: The moving blob. Top: Master flat from the evening of the 16th.
The ‘blob’ is in the bottom right-hand quadrant. Bottom: Master flat from the
morning of the 18th. Notice how the blob has moved in relation to the other,
stable doughnuts. Also visible on these frames are some of the bad columns and
the circular area of reduced sensitivity at x ∼ 1370, y ∼ 170 discussed in Section
3.4.3.
the flats, producing two master flats with the blob in the two different positions. We proceeded to flat the object frames with their respective flat field
images.
The night of the 17th–18th provided more of a challenge. On this night,
the blob moved very early in the evening, while we were taking our first flat
frames. This meant that our only flats with the blob in the same place as
on (all of) the object frames were from the morning of the 18th. Also, many
24
Pre-processing the ALFOSC data set
of the flats from the morning seemed to contain quite a few stars. To get
enough good flats (i.e. not containing stars), we first considered using flats
from both the evening and the morning, which would mean combining flats
with the blob in different positions. As the object frames were stepped in
position between each exposure, the coordinates of the stars were different on
all 301 frames. To observe the blob on all these frames, we made a film that
showed all object frames in succession, with a rectangular box indicating the
two positions of the blob. We concluded that the stars F, G, H, X and R, see
Figure 2.1, were never positioned in the areas affected by the blob. Stars D
and E, however, were affected on several frames. As we wanted to use these
stars in the photometry, we ended up using only flats from the morning. This
meant we only had 4 flat frames in R and 5 in V , but we considered this the
best option as we did not want to jeopardise the D and E stars.
3.4
Other preparations
The signal of an image is produced by the object being imaged. The signalto-noise ratio (S/N), measures the amount of signal in relation to the amount
of noise, which is important for the degree of detail that can be seen in the
image.
For performing deconvolution of images, a high S/N ratio is needed. We
therefore decided to make a summation of 10 images at a time, which corresponds to increasing the exposure time in our final image, which again
increases the S/N ratio. With an exposure time of 60 seconds for each individual image, this would mean a ‘new’ exposure time of 600 seconds, or 10
minutes.
3.4.1
Combining images
Combining images, either by taking the sum, the mean or the median of
several individual exposures, can be useful for several reasons, especially for
increasing the S/N ratio.
The standard deviation is the simplest measure of the noise in the image.
The standard deviation (σ) of a stochastic variable x, or rather its probability
distribution, is defined by:
σ=
p
hx2 i − hxi2 ,
where ‘hxi’ denotes the mean of the variable x. The standard deviation per
3.4 Other preparations
25
pixel,8 when summing N images, is given by:
q
2
σper pixel = σ12 + σ22 + . . . + σN
(3.1)
In terms of describing electromagnetic radiation as photons, the rate of
arrival of individual photons is a random process. Independently of the
nature of the detector there is an irreducible level of fluctuations, known as
Poisson noise. There are several ways of computing the signal (S) divided
by the noise (N ). For sources that behave according to √
Poisson statistics, a
signal level of N∗ photons has an associated 1σ error of N∗ . That is, if the
noise is dominated by the Poisson noise from the signal itself, we have that:
p
S
N∗
'√
= N∗ .
N
N∗
Additional noise comes from sources such as background sky level, light pollution, heat in the camera, the process of reading the data out of the camera
etc.
For images dominated by Poisson noise, summing several images will
increase the noise (see the expression for σper pixel above), but will increase
the signal more, causing the signal-to-noise ratio (S/N) to increase.
An additional motivation for combining images was the relative shortness
of our exposure times. Also, there are several steps in the MCS procedure
that has to be performed manually for each image that is to be deconvolved.
Reducing our number of images from close to 1000 to less than 100 (93), was
therefore very helpful.
We should also mention that the number of images for each filter and
each night did not always add up to 10. The rejection operation we used
in the combination procedure, see appendix A, favoured a large number of
images, so instead of having, e.g., 3 images in the last batch and 10 in the
second last, we evened them out, giving 7+6 instead of 10+3. If we had 10
images in the second last batch and only one or two in the last, we combined
the 11 or 12 individual images to one image.
3.4.2
Scaling, aligning and background subtraction
Turbulence in the atmosphere gives rise to a broadening of the PSF of an
image, known as seeing. As conditions change through the night, so does the
8
By standard deviation per pixel, we mean an estimate of what the standard deviation
(σ) would be in this one pixel, if we were able to repeat the observation an infinite number
of times.
26
Pre-processing the ALFOSC data set
PSF. Seeing is easily measured by the FWHM of stars. In our images, the
seeing varied from 0.7 00 to 1.5 00 .
We can think of an astronomical image as being a composite of a background level with a signal (i.e. a star) on top. The background, or ‘sky’ level,
depends on the exposure time and also on the observing conditions: the time
of night, whether the moon is present, clouds and dust in the atmosphere
etc. The signal, on the other hand, increases linearly with the exposure time
and does not (ideally) depend on when we made the observation.9
Variations in background level and in ellipticity, varying seeing conditions
and different exposure times can produce unwanted effects when combining
images, especially in the wings of stellar profiles. We found that combining
images without background subtraction, could lead to ‘holes’ in the outer
regions of stars. The σ rejection method we used when combining images
(see Appendix A), would also be affected if we combined images with different background levels. Higher signal means higher noise, so an image with
higher level would contribute with an ‘artificially’ high σ in Equation 3.1,
affecting the number of pixels that are rejected. We therefore subtracted the
background from all images before combining. As a measure of the background level in an image, we used the median of the entire image, as this will
be completely dominated by the background.
In Section 3.1, we mentioned that the average exposure time for the R and
V frames were, respectively, approximately 60 and 80 seconds. We decided
to scale the images to these exposure times. This also proved useful later,
when we plotted magnitudes versus sky level, see Section 6.1.3.
Aligning
As mentioned, the object frames were stepped a few pixels in the x and y
direction. Before combining the images we therefore had to align them so
that the stars had the same coordinates before combining. We aligned the
frames so that 10 and 10 frames had identical coordinates.
To perform the alignment, we needed coordinates for as many stars as
possible from all frames, in standardised text files to use as input to the IRAF
aligning routine. This was supplied by the first part of Jan-Erik Ovaldsen and
Jan Teuber’s photometry program (Ovaldsen, 2002), giving high-precision coordinates for those of the stars F, G, H, E, D, X, R that were unsaturated
and not too close to bad columns (see Section 3.4.3) in each image. The
procedure performs an initial centring by fitting a paraboloid to the nine
9
In reality of course, the signal will be affected by factors such as an increase of air
mass as the star sinks towards the horizon.
3.4 Other preparations
27
central pixels, i.e. the pixel with maximum intensity and the eight surrounding pixels. Then, an iterative and more robust centring algorithm improves
the first centroid by correlating the PSF with several synthetic images of the
star in question – each offset by sub-pixel distances.
The scaling, aligning and background subtraction were performed in IRAF.
We note that the alignment procedure, using bilinear interpolation, will tend
to smooth the images somewhat. The automatic alignment procedure would
often result in large residual values, that is, differences between the reference
and the transformed coordinates of up to ∼0.5 pixels. We therefore used an
interactive feature in order to remove from the transformation the stars on
these images that caused higher residual values, possibly due to bad pixels or
cosmic rays, see Appendix A for details. This procedure effectively reduced
the residual values from the alignment to around ∼0.03 pixels.
3.4.3
Bad Columns
According to the online specification of the ALFOSC detector,10 there are a
total of 16 charge traps and bad columns on the CCD, as well as many dark
‘specks’ scattered across the imaging area. The specks are areas a few pixels
wide with reduced sensitivity, of diverse origin. Some of them are probably
results of defects in the coating of the CCD. All the specks are removed by
flat-fielding.
Bad columns are columns of pixels that for some reason do not work
properly. There were several bad columns on the CCD we used. Two of
the bad columns11 were a particular nuisance, as they were close to our
target objects. Although we tried to avoid these particular bad columns
when observing, we did not always succeed in doing so, as the bad columns
depended on illumination and are not always clearly visible on the frames.
As we discovered after the observations had been completed, the images of
the quasar components and some of the field stars came close to and even
directly hit the bad columns on ∼10% of the images.
We wanted to discard images where a bad column came too close to the
point sources. Tests performed by Jan-Erik Ovaldsen, replacing the bad column by an average of the two immediately surrounding columns, placed a
limit on how close to a field star or quasar component a bad column could
come before the interpolation yield an unsatisfactory result, i.e. the flux of
the star was severely affected by the interpolation. Near a star, the gradients
10
Note that the ALFOSC CCD has recently been replaced (September 24th 2003). The
specifications for the CCD we used for our observations are still located at
http://www.not.iac.es/technical/astronomical/detectors/ CCD7/PreCommissioningReport/
11
Coordinates x = 1232 and x = 1409 on our (pre-processed) frames.
28
Pre-processing the ALFOSC data set
across columns are so great that simple interpolation will not give a satisfactory result. A total of 86 frames, 78 R and 8 V , were rejected from our data
set because a bad column came too close to an object of interest. It might
have been possible to try and figure out a complex interpolation scheme to
deal with bad columns this close to stars, but we chose not to as we had an
abundance of images in the first place. Rejection seemed the safest choice.
When running the MCS deconvolution, we extract smaller images of 64
× 64 pixels around the objects of interest. These ‘sub-images’ should be
free of bad columns. When combining 10 successive images, we made use
of procedures that reject ‘out of bounds’ pixels. An average of the values
of each pixel is computed and values deviating from the average by some
factor are rejected, see Appendix A for details. As we aligned the images
before combining them, the bad columns had different coordinates on every
frame. We therefore initially trusted the rejection method to eliminate the
bad columns.
However, the bad columns did not all disappear on the combined frames.
With hindsight, we realized that we should have interpolated the bad columns
before combination, while they still had the same coordinates on every frame.
The explanation for why the rejection method did not work as expected,
could be that we used the ‘σ clipping’ rejection algorithm mentioned in Section 3.4.2, which rejects pixels with values higher or lower than a specified
σ value (usually 2 or 3σ). When a bad pixel is taken into the computations,
the σ computed will be artificially large because of this deviant value. This
leads to a threshold value which is too high and therefore too few or none
of the pixels will be rejected. As it involved quite a large amount of work
to do the interactive aligning over again, we decided not to. However, when
deconvolving, we had to inspect the (93) combined images, and could then
check for any remaining bad columns in the extracted sub-images. We could
then replace these pixels by a relatively high value in the ‘noise map’ produced in MCS, which would cause the deconvolution routine to disregard
these particular pixels, see Section 4.5.1.
Referring back to Figure 3.5, we notice some bad columns and also an
area several pixels wide of reduced sensitivity. What appears to be a vertical
line at x = 1785 to x = 1787, is actually a steep incline followed by an equally
steep decline. See Fig. 3.7 for a view of a magnified portion of Fig. 3.5. There
are two bright bad columns followed by one dark. These columns did not
affect our observations, as they were never in the vicinity of any of the objects
we studied.
In Figure 3.5 we also see a trough in the graph around x ∼ 1370. This
corresponds to a circular spot of low QE on the CCD, approximately 9 pixels
in radius. The spot can also be seen in Figure 3.6. The lowest value of this
3.4 Other preparations
29
Figure 3.7: Bad columns. Closeup of Figure 3.5.
spot, in the centre, has a value of ∼20% of the surroundings. By comparison,
the doughnuts have only 2-4% lower value. The spot does not flat-field away
properly on all images, and we suspect it might be a defect in the chip.
However, no objects of importance to our investigations come near this spot
on any of the frames. The sharp decline at x = 1232, one of the bad columns
that did give us some problems, can also be seen in Figure 3.5.
3.4.4
The importance of careful pre-processing
In concluding this chapter, we want to stress that careful pre-processing of
data is a crucial step towards obtaining accurate photometry. Simple flaws
in the CCD, or a diversity of other conditions, can seriously affect the results
if not noted and, where possible, corrected for. Careful inspection of the
bias, flats (darks where applicable) and object frames is important, as well
as choosing the particular pre-processing methods that fit our particular
observations. Understanding the principles of the noise affecting the images
is paramount to obtaining reliable results.
30
Pre-processing the ALFOSC data set
Chapter 4
The MCS procedure
Everything you say to me
Takes me one step closer to the edge...
Linkin Park, One step closer
In this chapter, we describe the basic ideas behind the MCS deconvolution
procedure, named after its developers Pierre Magain, Frédéric Courbin and
Sandrine Sohy. A few elements from Fourier theory and digital image processing theory are mentioned in Section 4.1. To illustrate the general principles
of the MCS procedure, we first discuss deconvolving one individual image,
and in Section 4.6 we expand our discussion to the deconvolution of several
images simultaneously. Simultaneous deconvolution has the advantage of
utilising the combined S/N ratio of all the images. We also describe in some
detail the process of running the programs. As a certain amount of experience is needed in order to run MCS in the most efficient manner, we spent
quite some time familiarising ourselves with the different routines. This part
of the chapter is therefore partly intended as a guide for future users of MCS.
4.1
Convolution
As light passes through our atmosphere and our detectors, it is distorted
and blurred. This is a result of atmospheric turbulence, the finite resolving
power of our detectors and instrumental noise. The light distribution of point
sources such as stars, is therefore broader and the resolution of the images
we obtain is poorer than it would have been if we were observing outside the
Earth’s atmosphere with an ‘ideal’ telescope (i.e. a telescope with infinite
resolving power).
The mathematical representation of this broadening involves perceiving
the original light distribution as convolved with a function representing the
32
The MCS procedure
total instrumental profile. This function is termed the point spread function
(PSF). The imaging equation describing the broadening is:
d(~x) = t(~x) ∗ f (~x) + n(~x) ,
(4.1)
where t(~x) is the total PSF for the image, f (~x) and d(~x) are, respectively,
the original and observed light distributions and n(~x) is the noise affecting
the data. The symbol ‘ * ’ signifies convolution.
Convolution is a process which ‘folds’ functions together. In the onedimensional, continuous case it is expressed mathematically as:
Z ∞
y(t) = f (t) ∗ g(t) =
f (α)g(t − α) d(α) ,
−∞
where α is an integration variable. To find y(t) at a specified t, the integrand
f (α)g(t − α) is first computed as a function of α. Then, integration with
respect to α is performed, resulting in y(t).
These mathematical operations have simple graphical interpretations, see
Figure 4.1 for illustration. First, plot f (α) and the flipped and shifted g(t−α)
on the α axis, where t is fixed. Second, multiply the two signals and integrate
the resulting function of α from −∞ to ∞. These operations can be repeated
for every value of t of interest.
In essence, the Fourier transform decomposes a function into sinusoids of
different frequency which sum to the original function. The Fourier transform
F (ω) of a continuous function f (t), where t ∈ R, is given by:
Z ∞
1
f (t) e−iωt dt .
F (ω) = √
2π −∞
The transform is here expressed using the angular frequency ω(= 2πν) and
the time t. We have restricted ourselves to the one-dimensional case since it
is sufficient for our illustrative purposes. Inversely, we can obtain f (t) from
F (ω):
Z ∞
1
F (ω) eiωt dω .
f (t) = √
2π −∞
The expressions for F (ω) and f (t) are called a Fourier transform pair, and
they translate between the Fourier (or frequency) domain and the ‘time’
domain.
One of the theorems that constitute a basic link between the Fourier
and the spatial domain is called the convolution theorem. It simply states
that the convolution of two functions in the ‘time’ domain corresponds to
4.1 Convolution
33
g( α )
f( α )
1
1
α
1
α
1
-1
t
α
1
1
t
α
y(t)
t
Figure 4.1: Convolution of f and g. Top: The two functions f (α) and g(α).
Middle: The flipped g for t = 12 and t = 52 . Imagine g sliding towards the right,
across f . The marked area shows the multiplication of f (α) with g(t − α) for these
specific values of t. Bottom: Integrating the products for all values of t of interest
produces the function y(t) which equals the convolution of f (t) and g(t). The two
points originating from integration of the marked areas in the middle panel, are
shown as filled dots on y(t).
ordinary multiplication in the Fourier domain. In other words, f (t) ∗ g(t)
and F (ω)G(ω) constitute a Fourier transform pair. We have that:
f (t) ∗ g(t) ⇔ F (ω)G(ω) ,
34
The MCS procedure
and vice versa:
f (t)g(t) ⇔ F (ω) ∗ G(ω) .
The task of deconvolving Equation (4.1) thus seems in principle quite
an easy task when using Fourier transforms, as the equation in the Fourier
domain is simply:
D(ω) = T (ω)F (ω) + N (ω) ,
where the functions D, T , F and N are the Fourier transforms of d, t, f and
n from Eq. 4.1 respectively. To recover F we then simply put:
F (ω) =
D(ω) − N (ω)
.
T (ω)
(4.2)
When one has obtained F (ω), one can simply transform back to the spatial
domain to find f (x). However, on practically all modern light detectors used
in astrophysics the observed light distribution is not continuous but sampled and this introduces an important factor to be considered when solving
Equation (4.2). Also, the noise N (ω) is unknown and has to be modelled.
4.1.1
The sampling theorem
Most astronomical detectors in use today are electronic charge-coupled devices (CCDs), see Section 3.1. This means that the light we observe with our
instruments is sampled, i.e. known only at regularly spaced intervals, while
the original light distribution from the star is continuous.
The sampling theorem was first described by Shannon (1949). It is an
important theorem, determining the maximum sampling interval allowed if
a continuous function is to be regained completely from discrete samples.
The theorem states that if there exists a cutoff frequency ω0 above which
the Fourier transform of the function is zero, then the function can be fully
specified by sampling values spaced at equal intervals not greater than 1/2ω0 .
Imagine a continuous function f (x), whose Fourier transform is bandlimited as described above. That is, the Fourier transform F (ω) vanishes for
values of ω outside the interval [−ω0 , ω0 ]. To obtain a sampled version of this
function, multiply f (x) by a ‘sampling function’ g(x) consisting of a train of
impulses (δ-functions) an arbitrary ∆x units apart. By the convolution theorem described in the previous section, multiplication in the spatial domain
corresponds to convolution in the Fourier domain. The Fourier transform
of the product f (x)g(x) is periodic, with a period (∆x)−1 . If the individual
repetitions of F (ω) overlap, this will contaminate the recovery of f (x). The
centre of the overlapped region in the first period occurs at ω = (2∆x)−1
4.2 Deconvolution
35
if this quantity is less than ω0 . To be able to isolate F (ω), the following
requirement has to be satisfied:
∆x ≤
1
.
2ω0
(4.3)
The minimum sampling frequency ω0 is called the Nyquist frequency. This
is the sparsest sampling allowed if one is to avoid overlapping in the Fourier
domain.
4.2
Deconvolution
The problem of improving the resolution of astronomical images by regaining (some of the) precision that has been lost, is the inverse problem of
deconvolving the images. This corresponds to recovering the function f (~x)
in Equation 4.1. Deconvolution tries to recover high frequencies, which will
tend to increase the noise.
We now turn to a description of the MCS programs, and how we used
them on our data of QSO 0957+561.
4.2.1
Traditional methods
Being an inverse problem, deconvolution is ill-posed, meaning that there are
a large number of possible solutions compatible with the original light distribution. Some additional constraint must be applied in order to chose between
the plausible solutions. Several methods have been developed to apply constraints to this problem. A typical method consists of minimising the χ2 of
the differences between model and data, in addition to applying a constraint
imposing smoothness of the solution. The original light distribution fi is
then the solution that minimises the function
N
N
2
X
1 X
S1 =
+ λH(f1 , ..., fN ) .
t
f
−
d
ij
i
i
σ 2 j=1
i=1 i
(4.4)
The first part of S1 is the sum of the χ2 with σi the standard deviation at
the ith sampling point. N is the number of sampling points, dj , fj and nj
are vector components giving the sampled values of d(~x), f (~x) and n(~x) in
Equation (4.1) respectively, and tij is the value at point j of the PSF centred
on point i. The χ2 represents sampled deconvolved model, reconvolved with
the PSF, minus the original data points. In the second part of S1 , H is a
smoothing function and λ is a Lagrange parameter that is determined so that
the reconstructed model is statistically compatible with the data (χ2 ≈ N ).
36
The MCS procedure
It is also very useful to consider any prior knowledge in order to choose
the correct solution. For example, the maximum entropy method automatically rejects all solutions with negative values as no negative flux can be
detected. This is also the case for another method, the Richardson-Lucy
iterative algorithm (Richardson, 1972; Lucy, 1974).
Traditional deconvolution methods tend to introduce artifacts in the images, as a result of violating the sampling theorem. For example, ‘ringing’
artifacts can be produced around point sources in the image. The relative
intensities of different parts of the image are not conserved either, which
precludes quantitative photometry from the image.
The main problem with classical deconvolution algorithms is that they
try to deconvolve to infinite resolution. However, if the observed signal is
sampled so that the sampling theorem is obeyed, the deconvolved data will
generally violate that same theorem. Increasing the resolution means recovering higher Fourier frequencies, thus increasing the cutoff frequency, which
by Equation 4.3 makes the correct sampling denser.
4.2.2
Deconvolution with correct sampling: a preview
The MCS algorithm is based on the principle that sampled data cannot be
fully deconvolved without violating the sampling theorem. It differs from
other deconvolution methods in that it does not deconvolve with the total
PSF, but with a narrower PSF chosen to accommodate the sampling theorem.
In other words, since the data are sampled we should not try to deconvolve the
image to achieve a resolution from an ideal telescope in perfect surroundings
(i.e. a space telescope located above the atmosphere with infinite resolving
power), but only a resolution as would be obtained with a better telescope.
Along with the technique for smoothing the background, see Section 4.4 and
4.5.4, this removes the ringing artifacts. The method also allows for accurate
photometric and astrometric measurements on the deconvolved frame.
Figure 4.2 shows an example of a deconvolved image of the components
of QSO 0957+561. The left panel shows the original image, and the right
panel the deconvolved image.
4.3
Preparations for running MCS
Running MCS on such a relatively large number of images requires some
preparation. The background must be the same in all images, so they must
be background subtracted. We also need the approximate positions of the
objects of interest, and the standard deviation in the background of each
4.3 Preparations for running MCS
37
Figure 4.2: Left: The extracted image from the original, combined (R) frame.
North is up, East is left. Exposure time for the combined image is 600 seconds, seeing 0.75 00 , plate scale 0.188 00 /pixel. Right: The final deconvolved frame. FWHM
is 2 pixels. The deconvolved image is subsampled by a factor 2. See text for
details.
image, which we measured with IRAF. When simultaneously deconvolving
many images, see Section 4.6, we aligned all the images to the same reference
image, as the simultaneous deconvolution requires all objects to have equal
coordinates within one pixel.
Starlight reaching us is scattered and absorbed by molecules in the Earth’s
atmosphere. As the star moves across the sky during each night, it is affected
by varying amounts of air mass that the light has to pass through in order to
reach us, and the apparent brightness will decrease as the air mass increases.
The air mass depends upon the zenith distance of the object, as shown in
Figure 4.3. As a first approximation, the air mass varies with sec(z). At
angles more than about 60 degrees from the zenith, this approximation is less
accurate due to the curvature of the atmosphere and increased refraction.
Changes in air mass or cirrus affect the quasar and reference stars in the
same way. These effects cannot be easily modelled, but they can be accounted
for by normalising by a star in the field. By normalising, we correct for the
effects of any changes in cirrus or air mass that may occur with time. We
also normalised the other field stars we were interested in deconvolving, by
the same reference star. We chose the star D in Figure 2.1 as our normalising
star.
In addition to choosing a normalising star, we must choose a star to create
the PSF from. In case of PSF variations across the field, the star chosen for
constructing the PSF should be close to the objects we are interested in. We
considered using the two relatively weak stars P and Q as PSF stars, but in
the end decided to use the star E, as P and Q are considerably dimmer than
the quasar components. This star is bright, so it has a high S/N ratio, but
it is not too close to the saturation limit. It is also relatively close to the
quasar components and the other field stars. We performed some tests with
different PSF stars to see if the residuals varied much from star to star, but
38
The MCS procedure
Zenith
*
Star
path length
~ sec(z)
z
Horizon
Figure 4.3: Changes in air mass are proportional to sec(z). This figure shows a
simplified, plane-parallel atmosphere.
found that they were quite similar. So the PSF does not appear to change
much across the CCD.
In the MCS procedure, one may also choose to construct the PSF from
several stars. If several stars are used, the different residuals after Moffat
subtraction, see Section 4.5.2, should be approximately equal.
We deconvolve to a FWHM of 2 pixels, which makes the construction of
the PSF, s, easier compared to deconvolving to a broader FWHM, see Section
4.5.2. Our deconvolutions will thus try to recover the sharpest possible light
distribution allowed by the sampling theorem.
4.4
The aim of MCS
The MCS deconvolution method does not use the total PSF t(~x) of an image, but a narrower function s(~x) chosen so as not to violate the sampling
theorem. The two PSFs are related by the convolution
t(~x) = r(~x) ∗ s(~x) .
(4.5)
Here, t(~x) is the total PSF of the image, and s(~x) is the narrower function that
we want to use for deconvolving our images. To create the function s(~x) we
use the relation above and first deconvolve t(~x) with a pre-chosen function
r(~x). This analytical function r(~x) must have a certain width, i.e. not be
point-like. In the MCS algorithm r(~x) is set to be a Gaussian distribution,
4.5 General outline of MCS
39
whose FWHM must be at least 2 pixels in order not to violate the sampling
theorem. In the final deconvolution of the image, see Eq. 4.7, the narrower
function s(~x) is used instead of the total PSF t(~x). Instead of using t(~x) to
deconvolve the observed image, thereby aiming to achieve a ‘perfect’ pointlike resolution, the narrower function s(~x) is used to deconvolve the observed
image to the resolution of r(~x). See Figure 4.4 for a comparison of the total
(t) and the narrower (s) point spread functions. The MCS method does not
aim at obtaining the point-like distribution f , but instead a distribution, r,
with a certain width. For this purpose we must deconvolve the data with
the PSF, s, which we have constructed, instead of t, which is the original
PSF. The deconvolved image will thus have an improved but not an infinite
resolution.
A method like the maximum entropy deconvolution looks for the solution
where the entropy is maximum, which corresponds to the flattest solution
compatible with the image. Understandably, this would tend to underestimate the intensity of point sources. In the MCS method, the knowledge of
the shape of the point sources in the deconvolved image is a very strong prior
knowledge that can be used to constrain the solution f (~x). Since the profile
r is known and chosen by the user, the deconvolved image f can be decomposed into a sum of point sources plus a diffuse background, see Figures 4.4
and 4.7. The program returns positions ~ck and intensities ak for the M point
sources, while the ‘background’ consists of a deconvolved numerical image
for all extended sources h(~x). The solution f (~x) can thus be written
f (~x) = h(~x) +
M
X
k=1
4.5
ak r(~x − ~ck ) .
(4.6)
General outline of MCS
The MCS procedure consists of 4 FORTRAN 77 programs that are run individually and consecutively. From these programs we obtain as output 5 image
files per PSF star, as well as 14 other files, both images and text files. The
PSF stars are the stars chosen for the construction of the PSF, s. Thus, if we
run the program on one image using 3 PSF stars, we end up with a total of
14 + 5 ∗ 3 = 29 output files. This is the ‘grand total’ number of output files,
where the output from one program is generally used as input in the next.
Some of these output files are relics from older versions of the programs and
are not used but still given as output, see the overview in Appendix C.
The MCS programs are, in running order:
40
The MCS procedure
d
=
t
*
f
d
=
s
*
r
Figure 4.4: Above: d is the observed distribution, t is the total PSF and f is the
original (point-like) distribution. Below: s is the narrower PSF that we want to use
in the deconvolution in order to obtain our chosen solution r from our observations,
r being the Gaussian function with a FWHM that we choose (≥ 2 pixels). d and
t have the same width and can be interchanged for practical purposes.
1. extract.f
2. psfm prive.f
3. psf prive.f
4. dec prive.f
These programs use a total of 4 input text files, to be filled in by the user,
at various stages (see examples of input files in Appendix B). In addition, 2
sub-routines used for setting the image sizes and the number of point sources
in the image to be deconvolved must be present and filled in correctly. We
will now go through each stage of the MCS deconvolution process, and for
illustrative purposes first describe how to deconvolve a single image. The
simultaneous deconvolution of several frames is one of the main advantages
of the MCS method, and will be described in later sections.
4.5 General outline of MCS
4.5.1
41
Extracting data frames
The program extract.f extracts frames of optional size from the original
image. We have set the frame size to 64 × 64 pixels, which encompasses
both the individual field stars and the quasar images. The program takes
as input the text file extract.txt, where we enter the coordinates of the
centres of the objects we want to extract. This means that we have to
specify approximate coordinates of the stars we want to use for constructing
the PSF, s, and of the object image itself (the quasar components or a star).
The program extract.f gives as output the extracted images psfnn.fits,
that is, in numerical order psf01.fits, psf02.fits . . . , of the PSF stars,
normalised to a peak intensity of 1, and the extracted image g.fits of the
object.
The routine extract.f also calculates an estimate of the photon noise in
each pixel of the normalised PSF frames, and returns this as a ‘noise map’
image, termed psfsignn.fits and sig.fits for the PSF and object frames
respectively. For a sky subtracted image, the noise in one pixel expressed as
a standard deviation σimage is given by:
q
2
σimage = I/g + σsky
,
where σsky is the standard deviation in the background, which we have to
measure on all image frames. I is the flux of the pixel and g is the gain of
the detector.
In extract.txt, the user can also choose to use ‘small pixels’ which
means that the pixel size in the deconvolved image will be half that of the
original frame. This ‘subsampling’ allows for a finer resolution in the deconvolved image. The plate scale of the detector is 0.188 00 per pixel. The pixel
size of the deconvolved image is therefore 0.188 00 /2 = 0.094 00 .
4.5.2
Constructing the PSF
The profiles of astronomical point sources, typically imaged on two-dimensional
arrays, are referred to as point spread functions or PSFs. A star image is
inherently complex and, in practise, the telescope and detectors themselves
impose their own signatures on the stellar images recorded. There are therefore many ways to encode the detailed morphology of star images, and the
subject has been much debated, e.g. in Moffat (1969), King (1971) and Stetson et al. (1990). To obtain a good stellar profile, it is customary to construct
the PSF as a combination of an analytical function and an empirical lookup table of residuals from the analytic fit. The assumption is that this will
provide good representations of the data itself.
42
The MCS procedure
In this section, we describe the two programs in the MCS procedure that
construct the PSF, points 2 and 3 in the list in Section 4.5.1. The analytical
part is a Moffat function. The numerical part consists of the residuals from
the analytic fit. Both parts are obtained by a deconvolution combined with
a χ2 -minimisation. After a short discussion of the type of analytical function
used, we will describe how the two parts of the PSF are constructed.
The Moffat Function
The analytical part of stellar profiles can be approximated by several mathematical functions. Three functions in particular have been found to be
useful descriptions of at least parts of a stellar brightness profile: the Gaussian G(r), the modified Lorentzian L(r) and the Moffat M (r) function. In
their simplest form, these functions can be written as 1
r2
G(r) ∝ e( 2a2 )
1
L(r) ∝
2
1 + ( ar 2 )β
1
M (r) ∝
2
(1 + ar 2 )β
In these equations, r is the radius of the distribution and a and β are fitting
parameters. All of these functions have been tried in various combinations. A
typical stellar image has a sharp peak and rather broad wings. The Gaussian
is a good representation of a stellar profile in the central parts, but not so
good in the wings as these are low for a Gaussian.
When an analytic function is used to represent a star image, either in
its entirety or in combination with a look-up table of residuals, it is useful to generalise that function with additional parameters which allow for
the inclusion of ellipticity in the PSF, i.e. non-circular images produced by
e.g. telescope aberrations.
The MCS programs utilise such a Moffat function as the analytical part
of the PSF. In generalised form, this can be written
M∝
1
(1 +
x2
α2x
+
y2
α2y
+
xy β
)
αxy
,
for a profile centred in origo. In this equation, the parameters αx , αy and
αxy can be adjusted to account for ellipticity and rotation of the profile. By
1
For a more detailed treatment of the Gaussian distribution, see appendix D.
4.5 General outline of MCS
43
rotation, we mean that the elliptical Moffat is allowed to be inclined with
respect to the rows and columns of the detector. The Moffat function used
in the MCS programming is explicitly described by
a m φβ ,
where
φ = 1 + b1 (x − cmx )2 + b2 (y − cmy )2 + b3 (x − cmx )(y − cmy ) .
The centre of the function is (cmx , cmy ). The intensity is am , while b1 , b2 , b3
and β are shape parameters. Comparing the two expressions for a generalised
Moffat function, we see that b1 and b2 represent the relation between the
semi-major and -minor axes in an ellipse, with b1 and b2 corresponding to
1/αx2 and 1/αy2 respectively, while b3 = 1/αxy represents any rotation of the
ellipse. The exponent β adjusts the width of the distribution in relation to the
wings. In contrast to the Gaussian function, where the σ alone determines
this relationship, so that the relation between centre and wings are ‘locked’,
the Moffat can have a very sharp peak and still significant wings according
to β. Figure 4.5 shows a Moffat function with typical values for b1 , b2 , b3 and
β returned from the program.
1
0.8
0.6
0.4
0.2
0
–15
15
–10
10
5
–5
0
x
0
5
–5
10
15
–10
y
Figure 4.5: A normalised Moffat function with typical values for the shape parameters b1 , b2 , b3 and β returned from the program.
44
The MCS procedure
Figure 4.6: A typical image of the Moffat, mofc.fits, returned from the program
psfm prive.f. Some ellipticity and rotation can be seen.
The analytical part of the PSF
This program takes as input the text file psfmof.txt. The program fits an
analytical Moffat distribution to the PSF stars extracted in the preceding
program. By χ2 -minimisation of the difference between the extracted star
and the Moffat (convolved with the function r), an image mofc.fits of the
Moffat with a resolution of s is obtained and returned as output from the
program. This is the analytical part of the point spread function. Figure 4.6
shows what a typical Moffat image looks like.
The Moffat profile, reconvolved2 with the profile r to a resolution of t, is
subtracted from the PSF stars psfnn.fits and the residuals are returned as
images difcnn.fits. That is, difc01.fits = psf01.fits − mofc.fits,
etc. After running this program, we examine the data residuals difcnn.fits.
If more than one star is used, we check that the residuals are approximately
equal in structure and values. If one residual image differs noticeably from
the rest, this star is discarded as PSF star and only stars with approximately
equal residuals are retained.
In psfmof.txt we enter the starting value for the rotation parameter b 3
and the starting β value. The β is closely correlated to the b-parameters,3 and
the program will perform iterations until it finds a satisfactory fit for all shape
2
To subtract the Moffat from the data residuals, these must have the same resolution,
namely that of t.
3
See Stetson et al. (1990).
4.5 General outline of MCS
45
parameters. We also enter the number of stars that are used in constructing
the PSF, the FWHM value in the observed image and the FWHM value
that we wish to obtain after deconvolution, i.e. the width of r. Both FWHM
values must be entered in the coordinate system of the deconvolved image.
That is, if we are using small pixels we must multiply the observed FWHM
value by 2. Finally, we enter the maximum intensity (=1) and coordinates
of the PSF stars in the extracted images.
The numerical part of the PSF
The next program also takes the text file psfmof.txt as input. The final aim
of this program is to add the residuals from the Moffat fit to the analytical
function, in order to construct the final PSF, s, see Figure 4.7.
In order to do this, the residuals and the Moffat must have the same
resolution. The Moffat already has the resolution of s. The data residuals
however, have the resolution of t. Referring to Figure 4.4, we see that to
obtain a resolution of s we have to deconvolve the residuals difcnn.fits,
which have the resolution of t, with the Gaussian r. This has to be done
numerically, since we are deconvolving data residuals and not an analytical
function. This deconvolution is easier when r is sharper, which is why we
chose to deconvolve to 2 pixels FWHM, as we found some correlation in the
noise in the modelled residuals when using 3 pixels FWHM as the width of
r.
The program psf prive.f performs this deconvolution and, in this process, has to interpolate and smooth the data residuals. Another χ2 -minimisation
is performed, minimising the difference between the data residuals and the
modelled residuals (convolved with r).
The program will ask the user for values of ‘inner’ (λ1) and ‘outer’ (λ2)
smoothing parameters that must be specified, as well as the radius of the
circle that separates the areas where these smoothings apply. This radius
should be 3–4 times the FWHM before deconvolution. Typical smoothing
values are ∼ 10 for low smoothing and up to 100 000 for heavy smoothing,
but can in principle vary from close to 0 to 106 .
The modelled residuals, that is, the numerical part of the PSF with a
resolution of s, is returned from the program as an image psff.fits. For
comparison with the actual data residuals, an image of the modelled residuals
with resolution t is also made, by reconvolving psff.fits with r. This image
is returned as output psfr.fits, which can be compared directly to the
actual data residuals difcnn.fits as these now have the same resolution
(t).
We want to smooth the model of the residuals to avoid noise. However,
46
The MCS procedure
the residuals must also be comparable to the data residuals that we are trying to model. That is, psfr.fits must have approximately the same values
as difcnn.fits. That they are comparable in value must be checked manually after the program has been run. The program is re-run with different
values of λ1 and λ2 until the values match. One has to find the highest
smoothing possible that will still give a psfr.fits sharp enough compared
to difcnn.fits (and a psff.fits that is not too noisy).
Next, the program masks the outer parts of the modelled residuals as
these consist mostly of noise. The signal outside the region defined by a
mask radius is deleted and replaced by zero. The masking radius is supplied
by the user who investigates the images and decides where to set the mask.
PSF
=
Moffat
+
residuals
s.fits
=
mofc.fits
+
psffond.fits
Figure 4.7: psf prive.f: Constructing the point spread function (freehand drawing for illustrative purposes only).
The masked model of the residuals is returned as the image psffond.fits.
The final part of the program psf prive.f performs this addition: s.fits
(final PSF) = mofc.fits (analytical part) + psffond.fits (numerical part)
– see Figure 4.7. We now have the PSF, s, that we will use in the final deconvolution. It is normalised to a total flux of one, and with the pixels
re-ordered so that the peak intensity of the PSF falls on the lower left corner of the image, with coordinates (1,1). This re-ordering is simply done
for technical reasons, to facilitate the programming which uses FFT (Fast
Fourier Transform).
4.5 General outline of MCS
4.5.3
47
Deconvolution with the profile s
The last MCS program, dec prive.f, performs the final deconvolution of
the data image g.fits. This program takes as input the image g.fits
and its associated noise map sig.fits, as well as the PSF s.fits created
during the two previous stages. The program tries to minimise a complicated
function of many variables using two χ2 -minimising algorithms: minimi and
conjugate gradient. By minimising this function, see Eq. 4.7, the program
tries to minimise the differences between the deconvolved image, reconvolved
with the PSF, and the original data, to make as close a fit as possible.
The minimisation method assumes that all variables are independent and
minimises an N dimensional function along N independent directions. Minimisation is done along the N directions until each individual derivative
changes sign. This is done iteratively. When the derivatives of the function relative to one variable changes sign, the step is divided by two and the
minimisations continue in the opposite direction. The step is increased by
10% if the derivative does not change sign between the next iterations. The
iterations stop when the minimum of the function does not decrease significantly or even increases instead of decreasing. The algorithm minimi is the
main algorithm, while conjugate gradient fine-tunes the result.
There are two input text files to fill in before running this final program:
deconv.txt for the deconvolution parameters and in.txt for the photometry. In these text files the user has to enter the initial parameters, the
step values and the maximum step size for the minimising algorithms, the
smoothing strength to be applied to the extended component of the image
(the background), the peak intensity of the point sources after deconvolution
as well as the coordinates of these point sources.
The function that is minimised is, in one dimension:
N h
N
N
N
M
i2
i2
X
X
X
X
1 hX
hi −
rij hj (4.7)
S2 =
sij (hj +
ak r(xj −ck ) )−di +λ
σ2
i=1
j=1
i=1 i j=1
k=1
The function S2 is minimised with respect to the unknowns hj (j = 1 . . . N ),
ak and ck (k = 1 . . . M ). The first term is the χ2 fit, and the second term
is the smoothing function for the background. We can see from S2 that the
image is decomposed into a sum of point sources and a diffuse background,
see also Equation 4.6.
We also have that:
N is the number of pixels (sampling points),
M is the number of point sources,
48
The MCS procedure
σi is the standard deviation of the image intensity measured at the ith sampling point,
k indicates a point source with intensity ak and position ck ,
h is the background,
λ is a Lagrange parameter for the background which is determined so that
the reconstructed model is statistically compatible with the data (λ can also
be a varying function across the image frame).
Note that in the first term of Equation 4.7, the deconvolution is performed
using the calculated PSF, s, instead of the total PSF t. Comparing Equations
4.4 and 4.7, we also see that the solution f has been replaced by Equation
4.6.
We assume that what is not point sources is a smooth background. In the
second term of Equation 4.7, λ gives the intensity of the smoothing of the
background. The value of λ is entered into deconv.txt. In contrast to our
previous smoothing parameter in the program psf prive.f, higher values of
λ will here decrease the smoothing.
Since the width of the Gaussian profile of the point sources in the deconvolved image is chosen by the user, and thus known a priori, the peak
intensity is uniquely determined. An approximate value for the peak intensity after deconvolution is entered into in.txt. This can easily be calculated,
as the deconvolution preserves fluxes. Approximating the light distribution
with a Gaussian, the flux F of a point source is given by
F = 2πσ 2 Ipeak .
(4.8)
This expression is derived from integrating a Gaussian function, see Appendix
D for the calculation. Figure 4.8 is an illustration of the relationship between
flux and peak intensity before and after deconvolution. Since F is conserved,
we get
σ1
σ12 Ipeak,1 = σ22 Ipeak,2 ⇒ Ipeak,2 = Ipeak,1 ( )2 ,
(4.9)
σ2
where the indices 1 and 2 refer to values before and after deconvolution,
respectively. We also have that σ relates linearly to the FWHM by the
relation 4
√
(4.10)
FWHM = 2σ 2ln2 .
Thus, we can calculate the intensity after deconvolution, Ipeak,2 , by using
Equations 4.9 and 4.10. To obtain σ1 , we measure the FWHM of the
point sources in the original image (before deconvolution), whereas σ2 is
4
See Appendix D for details.
4.5 General outline of MCS
49
the FWHM after deconvolution, the FWHM we have chosen: the profile r
whose width has to be ≥ 2 pixels at FWHM. We measure the peak intensities
for the point sources, multiply by the ratio of the σs and enter this in the
input text file in.txt.
I (peak)
I (peak)
deconvolution
FWHM
FWHM
Figure 4.8: The FWHM decreases and the peak intensity increases after deconvolution, but the flux (the area under the graph) is preserved.
The output from the program dec prive.f is given as two text files,
out.txt and in2.txt, as well as the following images: dec01.fits, back.fits,
resi01.fits and resi sm01.fits. The image dec01.fits shows the two
point sources (+ the background) after deconvolution. The image back.fits
contains only the background. In our case of studying the QSO 0957+561
system, the galaxy should appear here, especially if we deconvolve several
frames simultaneously to obtain more information on the background. The
text file out.txt contains the output information from the program (source
intensity and coordinates), as well as some information on the χ2 reached
and the number of iterations used to reach it. The text file in2.txt contains
the output photometry but in the same FORTRAN format as the input text file
in.txt. This is to facilitate re-running of the program. The deconvolution
may be run several times, once with relatively large steps to improve too
rough estimates of the initial parameters, and then with smaller steps, using
the results obtained in the first run as new input.
There are some additional parameters to set in in.txt and deconv.txt
if several images are to be deconvolved simultaneously, see Section 4.6.
We obtain a numerical model of the galaxy, which we could improve by
using an option in the program to introduce a variable λ map.
4.5.4
Residual frames
The quality of the results from the MCS deconvolution can be checked from
‘residual maps’: images showing the residual in each pixel, i.e. the match
50
The MCS procedure
between the deconvolved model image (reconvolved with the PSF, s) and
the data.
Examples of two residual maps, from our final deconvolution results, can
be seen in the top right and bottom left panels of Figure 4.9. The image
resi01.fits is simply the difference between the data g.fits and model
dec01.fits, in units of the photon noise (that is, divided by sig.fits).
The image resi sm01.fits is resi01.fits squared: it is the χ 2 value
in each pixel, smoothed and reconvolved with the PSF, s. The smoothing is
done with an appropriate function so that any value is replaced by a weighted
mean on a neighbourhood containing a few dozen pixels. This smoothing and
reconvolution makes it easier for the eye to see the structure in the fit and
to recognise objects that may be fitted less well, e.g. the galaxy. We see that
there are some residuals where the point sources are located.
The parameter λ in deconv.txt determines the smoothing strength.
This value is adjusted by the user and the deconvolution is re–run until
resi sm01.fits is as flat as possible with a mean value of 1. If we have
smoothed too much (low λ value in deconv.txt), this image will have a
mean value significantly above 1, and if we smooth too little (high λ value),
the mean value will be lower than 1. We try to smooth so as to obtain a
value in this image of approximately 1. In the point sources the value will
be slightly higher, but we try to obtain as good a fit as possible by checking
this image.
As deconvolution tries to recover high frequencies, it will increase the
noise in the images. The last term in Equation 4.7 tries to minimise this
increased noise. The background h is smoothed with the profile r, revealing
the frequencies in the background that are too high for the chosen sampling.
These higher frequencies are then subtracted from the background, thus ridding the image of artifacts resulting from too high frequencies.
4.6
Simultaneous deconvolution
In addition to deconvolving a single frame, the MCS deconvolution can be
used to simultaneously deconvolve several individual frames. Indeed this is
one of the main points of the MCS method. When deconvolving simultaneously, one takes advantage of the S/N of all the images to construct a
background, hence obtaining a numerical model of the galaxy. Each data
frame has its own PSF s1 , s2 , . . . sn . The extended sources and the positions
of the point sources are forced to be equal in all images.
We find the deconvolved image that is the best fit between each data
image and the deconvolved model image (reconvolved with the PSF), using
4.6 Simultaneous deconvolution
51
the respective point spread functions and sigma images (one for each image).
In this way, the solution is compatible with all the images. The intensities
of the point sources are allowed to vary, while the positions are constrained
by all the images used. We obtain n deconvolved images that are essentially
one image, the only difference between them being the peak intensities.
When running the simultaneous deconvolution, initial intensities for the
point sources are entered for each frame. In order to detect intrinsic variations
in the quasar components, the intensities of the point sources are allowed to
converge to different values from one frame to another.
One set of initial positions are entered for all the frames together, which is
the reason we had to align all g.fits images to within one pixel of each other.
We also enter one guess peak intensity for all images in the set, calculated
from an image with representative seeing.
The minimisation performed during simultaneous deconvolution can be
expressed by the following:
N
N
M
i2
X
X
λ1 h X
~
a1,k r(~xj − ~ck + ~δ1 ) − d1,i
S3 =
s1,ij α1 h(~x + δ1 ) + β1 +
i
σ
i=1 1,i j=1
k=1
+
..
.
N
N
M
i2
X
X
λ2 h X
s2,ij α2 h(~x + ~δ2 ) + β2 +
a2,k r(~xj − ~ck + ~δ2 ) − d2,i
σ
i
i=1 2,i j=1
k=1
N
M
N
i2
X
X
λn h X
~
~
s2,ij αn h(~x + δn ) + βn +
an,k r(~xj − ~ck + δn ) − dn,i
+
σ
i
i=1 n,i j=1
k=1
+
N h
X
i=1
hi −
N
X
j=1
rij hj
i2
(4.11)
The function S3 that is minimised, consists of a deconvolution part and a
smoothing term (the last term in Equation 4.11). Three new parameters are
included when deconvolving several frames: α, β and δ.
The background h can be adjusted for each image by a multiplicative
factor α and an additive term β. The factor α can correct for effects due to
different exposure times and varying transparency of the atmosphere, while
β corrects for errors in the sky–subtraction.
To account for small shifts in position from frame to frame, a δx and δy
parameter are entered in in.txt. These affect the positions of the centres of
the point sources during the deconvolution in the following way.
xn = x 1 + δ x
52
The MCS procedure
yn = y 1 + δ y
where n indicates the frame being deconvolved, and the subscript 1 refers to
the first image in the list which is chosen as reference image.
The reason we need these shift parameters, is that the centres of the point
sources, after aligning the frames to within one pixel of each other, will be
located in different places within this one pixel. The dithering in x and y
improves the sampling by combining several offset samples of the same light
distribution, similar to a technique known as variable-pixel linear reconstruction, or ‘drizzling’, which is used on WFPC2 images from the Hubble Space
Telescope to optimise the information obtained from undersampled images,
see Fruchter and Hook (2002). This technique is implemented in the MCS
deconvolution method.
4.6.1
Final deconvolution results
For our two final, simultaneous deconvolutions of the March 2001 data, we
used our 73 R and 15 V combined image frames. We experienced some
problems when deconvolving both the quasar components and the stars. Our
results show relatively large residuals, although this varied considerably from
frame to frame. In general, the amount of residuals did not seem to affect
the photometry, although there were exceptions to this in extreme cases, see
Section 5.2.2. The deconvolved image also shows some artifacts, which we
comment on in this section.
Figure 4.9 shows our final deconvolution results of the quasar components,
in the R band. The top left panel shows the original, extracted image of the
two quasar components A and B. The top right and bottom left panels show
the two residual images discussed in Section 4.5.4, and the bottom right panel
shows the final, deconvolved image. The galaxy can be seen to the North–
East of the B component. In the deconvolved image, the galaxy is separated
from the B component.
There is some noise in the deconvolved image, as well as some ‘holes’ or
rings around the point sources. This may be a result of deconvolving to a
FWHM value of 2 pixels, the maximum deconvolution allowed. However,
we chose this FWHM in order to facilitate the construction of the PSF, see
Section 4.5.2.
One of the main problems we encountered in the deconvolution process,
apart from the residuals, was to get enough flux into the galaxy. When
examining preliminary deconvolution results, we would often find that the
galaxy was flat in the central parts. This indicated that the program, for some
reason, had stopped iterating before completely building up the galaxy. Our
4.6 Simultaneous deconvolution
53
solution to this was to construct the sharp centre of the galaxy in the first
run, using low smoothing and relatively large steps for the background, see
Section 4.5.3. In the second deconvolution run, using the output solution
from the first run as input, we used smaller steps to fine-tune the result and
applied more smoothing.
The point sources in the deconvolved image, lower right panel in Figure
4.9, does not seem perfectly symmetric. Examining the image back.fits, we
see that the background is not completely zero where the points sources are
located. There are values of maximum ∼3% of the maximum peak intensity
after deconvolution, at the positions of the point sources in the background
image. This causes the point sources in the deconvolved image to appear
non-circular.
We attribute the residuals mainly to errors in the PSF fit, although we
did not find evidence of a large PSF variation across the CCD, see Section
4.3. It may be that the PSF star was too bright compared to the quasar
components, and that a PSF of a more comparable brightness to these objects
would have produced less residuals. When analysing different objects, there
have been some indications by users of MCS that the PSF star chosen in the
MCS programs should not be too bright compared to the object that is being
deconvolved (Ingunn Burud, private communication). In our case, there were
no ‘ideal’ PSF star candidates. Our PSF star, E, is approximately 4 times
as bright in flux as the QSO A component. The stars P and Q, which we
considered for PSF stars, are about a quarter of the brightness of A. On the
other hand, we had even larger residuals when deconvolving the star H, up to
10 times larger values in the χ2 image, but this star is about twice as bright
as E. When deconvolving the bright field stars we needed to apply more
smoothing. Residuals from the deconvolution of star R were comparable to
those from the deconvolution of the quasar.
4.6.2
Obstacles
When deconvolving the March 2001 data in the V filter, the δx and δy increased inexplicably a little above 1 on two frames. We tested several combinations of step sizes without any improvement. There did not seem to be
any problems with the rms values of the aligning from these images, so in the
end we simply discarded them, regarding this as the safest choice. After discarding the images, MCS ran smoothly on the rest of the images. The same
thing happened in R, but only affecting one frame, which was also discarded
from the final deconvolution.
Sometimes the program psfm prive.f would stop iterating. Changing
the start value for β helped.
54
The MCS procedure
Figure 4.9: North is up, East is left on all four images. Top left: Observed,
extracted image of the quasar components. Top right: Difference between data
and model, divided by the photon noise. Bottom left: The squared, reconvolved
and smoothed residual image. Bottom right: Final, deconvolved image. Maximum
value in the background, i.e. in the galaxy, is ∼5 ADU. For the observed and
deconvolved images, the colour table used produces the colour black for values
above 5, and white for values below zero. For the lower left residual image the
levels were set to 0 and 10, and for the upper right residual image the levels are
–10 and 10.
There is a certain amount of ‘art’ involved in running MCS efficiently,
in the sense that the user has to familiarise him- or herself with how things
should ‘look’, which takes some experience and time.
We found the interplay between smoothing strength and step size to be
very important when deconvolving. The effect of a given step on the function
to be minimised is stronger with high smoothing than with low smoothing.
This implies that, in general, heavier smoothing requires smaller step sizes.
In our initial runs of MCS, we found strong correlation between quasar
4.6 Simultaneous deconvolution
55
components and the stars, at the level of several tenths of magnitudes. We
found that we had cut too much of the PSF, so we re-made the PSFs using
almost no mask. As a consequence, we produced noisier PSFs. However, the
residuals from the Moffat fit were so extensive that we had to make the mask
very large. Essentially no mask was used unless to cut off a cosmic. Changing
the PSFs in this manner, as well as running more images simultaneously and
running the deconvolution program for a long time with many iterations,
effectively reduced the correlation to the millimagnitude level.
In about 20% of the images from the main data set, we experienced some
problems when running psf prive.f. The modelled residuals that were
produced were far too weak compared to the actual data residuals, with
values at least 10 times too weak. On these images, we used P and Q as
PSF stars instead. It is possible that large gradients in the bright PSF star
E caused the problem. We noted the images, and examined all light curves
without finding any deviant behaviour for these particular points.
4.6.3
Testing MCS
As we start the deconvolution with equal peak intensities for all point sources,
it may be argued that this is the reason for the small standard deviation
in the results presented in Chapter 5. To test whether the starting values
we enter in the photometry parameter file for the deconvolution had any
significance for our results, i.e. for the magnitude levels and the dispersion of
the data points, we performed two tests. In the following, what we refer to as
our ‘reference run’ is not the final deconvolution results discussed in Section
4.6.1 but a very similar result that we obtained earlier. First, we switched
the guess peak intensities after deconvolution for the two quasar components
A and B. For this test, the values for A and B were still the same for all
images, but their internal ratio was changed, i.e. B brighter than A as input
instead of vice versa.
For the second test, we selected a wide range of guess peak intensities for
A and B. The output values from the deconvolution programs, two numbers
representing the peak intensities of the quasar components, for technical
reasons relative to some normalisation factors, had a standard deviation of
approximately 0.7% of their mean value. The range of guess peak intensities
we entered for this test, varied from 4 to 44% of the mean output value
from the reference run of MCS. In other words, the variations we introduced
artificially were much larger than the variations we had obtained from the
reference run. We now wanted to see if MCS could return the relatively
constant magnitudes for A and B, and whether the ratio between A and B
would be affected by introducing these variations.
56
The MCS procedure
Figure 4.10: Results of testing MCS for stability with different entries for the
guess peak intensity after deconvolution. Left: The change in magnitude level of
the A component is very small, both for test 1 (dashed line) and for test 2 (dotted
line). Right: The change in magnitude level for the B component is very small for
test 1 but larger for test 2.
The results, displaying the change in magnitudes for the two tests, are
shown in Figure 4.10. For the reference run, the mean magnitude difference
between the two components was 0.10 mag, with A the brightest component.
For the first test, MCS quickly switched back to A being brighter than B. The
components ended up being just slightly closer (A brighter than B by 0.09
mag) than for the reference run. The magnitude levels were approximately
the same as before, for both components. For the second test, A ended up
being brighter than B by a mean value of 0.06, mostly due to the somewhat
larger change in magnitude of the B component, see Figure 4.10.
Table 4.1: Standard deviation of the A and B magnitudes for the reference run
and for each test.
Reference
Test 1
Test 2
A component
0.0077
0.0078
0.0077
B component
0.0095
0.0093
0.0090
We see from Table 4.1 that the low dispersion in magnitudes from the
reference run was regained very well in both tests. We note with some surprise
that the biggest change in standard deviation is that component B shows a
4.6 Simultaneous deconvolution
57
lower standard deviation in test 2 than in the reference run. All in all, we note
that the program is very robust against ‘artificially introduced’ variations,
consistently producing magnitudes with low standard deviation. Note that
our tests were performed with the rest of the input text files being equal,
i.e. with the same step sizes for the different parameters.
58
The MCS procedure
Chapter 5
Photometry from the ALFOSC
data
May I borrow some of the light that you possess?
The 3rd and the Mortal, Oceana
In this chapter, we first discuss a few basic principles of photometry studies.
We explain how we obtained magnitudes from the results of MCS, and how
we have estimated the errors in these magnitudes. In the main part of the
chapter we present our photometric results from the ALFOSC data, from
our main data set from March 2001, as well as from the one night in January
2000. We display light curves showing our results. A discussion of the results,
as well as a comparative analysis with the findings of Ovaldsen et al. (2003b)
and Colley et al. (2003), follows in Chapter 6.
5.1
Basic principles of photometric studies
Photometry, or the study of brightness measurements, is one of the fundamental branches of astronomy. The flux F of an astronomical object relates
directly to what we measure with a telescope. The monochromatic flux of
energy through a surface, or into a detector, is the amount of energy per unit
time passing through a unit area of the surface per unit frequency interval.
When performing photometric investigations, the amount of flux emitted
from an object at different wavelengths is studied, as well as how this flux
changes in time. Photometry is important for objects as various as planets,
stars and galaxies, and the time scales of interest range from single observations to careful monitoring over months and years. For the purposes of
this work, we are solely interested in studying stars and other objects that
appear to us as point sources, i.e. the twin quasar.
60
Photometry from the ALFOSC data
Apparent magnitude, that is, the magnitude we observe here on Earth, is
defined by
F m = −2.5 log10
,
(5.1)
F0
where F0 is an arbitrary ‘standard’ flux. The magnitudes of any two stars
are then related by
F 1
m1 − m2 = −2.5 log10
,
(5.2)
F2
where m1 and m2 are the magnitudes of the two stars and F1 and F2 are
their respective fluxes. Note that m1 > m2 when F2 > F1 , that is, brighter
objects have numerically smaller magnitudes.
Because the magnitude scale is a logarithmic scale, the mean value of
a series of magnitudes is not equal to the mean value of the corresponding
fluxes. To calculate the mean value of magnitudes corresponding to the
fluxes, we should first convert the magnitudes into fluxes, using F = 10(−0.4m) ,
calculate the mean of these fluxes and then convert this number back into
magnitudes. In this thesis, we specify when we calculate directly the mean
value of the magnitudes. The results we display in the tables in this chapter,
would not be affected by how we calculate the mean, as the difference in our
case comes only at the 5th decimal point. When calculating the magnification
ratio in Section 6.2, we also checked that the two methods provided equal
results to the given accuracy.
There are two main types of photometric techniques, aperture and pointspread function (PSF) photometry. The aperture technique makes no assumption about the actual shape of the source PSF. The observed counts
within a chosen aperture, centred on the source, are collected and summed
together. The contribution from the background level is estimated by nearby
pixels and subtracted, and the net result constitutes the signal from the
source. Although sounding deceptively simple, there are many intricate steps
involved, including determining the image centre and the size of the aperture, correcting for partial pixels that are intersected by the (usually circular)
aperture boundary and deciding how and where to measure the background
level. Aperture photometry works best with a group of stars that are not
too crowded nor too faint. If this prerequisite is not present, PSF photometry is a more appropriate technique. As the name implies, this technique
assumes that all point sources on the CCD frame have the same shape, barring distortions introduced by camera optics. This shape differs from a point
source δ–function because the light from the point source is blurred when
passing through the Earth’s atmosphere and the telescope. Performing PSF
photometry consists of constructing the point-spread function and scaling
5.1 Basic principles of photometric studies
61
it to fit the different point sources on the image. There are many ways of
constructing a PSF, some of which were mentioned in Section 4.5.2.
There are of course a myriad of photometric techniques derived in some
way from these two main types. Described in Ovaldsen (2002), is a hybrid
method consisting of ‘PSF cleaning’ of a nearby source followed by aperture
flux measurement of the other. The deconvolution method we have adopted
in this thesis work, utilises a PSF constructed from one or several stars on
the frame and increases the resolution of the images by deconvolving them
as well as exploiting the S/N ratio of several images simultaneously.
5.1.1
Standard systems and reference magnitudes
In modern usage, a photometric system 1 consists of a set of discreet wavelength bands and a set of primary standard stars. Each wavelength band has
a known sensitivity to incident radiation, defined by the detectors and filters
used. The standard stars define the magnitude scale of the system.
A catalogue of primary standard stars for a given photometric system is
usually published when the system is defined. Sometimes, for widely used
systems, further catalogues of ‘secondary’ standards will be compiled by making observations calibrated with the original primary standards. Many catalogues of photometric standard stars exist, e.g. the widely used Landolt
catalogue.
Reference magnitudes encountered in the literature are derived, step by
step, from the first standards. Since this process can result in the reference
magnitudes differing from each other, the system one has chosen must be
specified.
The magnitudes we produce are instrumental, i.e. they are dependent
on the combination of the bandpass of the filters and the response of the
CCD. This means that the magnitudes are specific to our data set. In order
to compare results to that of others, it is usually necessary to transform
the instrumental magnitudes into a standard photometric system. This is
done by re-observing the standard stars in the system and comparing the
instrumental and standard magnitudes.
In this thesis, we are mostly interested in measuring variations and obtaining relative photometric measurements. Ovaldsen et al. (2003b) have
used the same ALFOSC data set that we have. In chapter 6, we wish to
compare our magnitudes with those obtained by these authors and by Colley
et al. (2003), as well as comparing our relative photometry, e.g. the internal
ratio between A and B, and any variations in the light curves. We therefore
1
For example the Johnson-Morgan UBV system, the Strömgren system etc.
62
Photometry from the ALFOSC data
calibrate our magnitudes using the same values for the reference magnitudes
as these authors. This does not produce absolute photometry in the strict
sense, but it places our data on a common scale so we can compare results
directly.
Our reference magnitudes are listed in Tables 5.1 and 5.2. The R references were obtained by Ovaldsen et al. (2003b), by a bootstrapping procedure
of magnitudes originally supplied by Rudy Schild (private communication).
The reference V magnitudes were obtained by the same authors using V − R
colour terms from Serra-Ricart et al. (1999). In our combined images, the
brightest stars (F, G and X, see Figure 2.1) were often saturated on at least
one of the 10 individual images we stacked. As a consequence of this, we did
not use these stars in our photometric study, but their magnitudes are still
listed here as reference.
Table 5.1: Reference magnitudes in the R filter for 7 field stars, see Figure 2.1 for
location of the stars.
Star
Ref. mag
F
13.758
G
13.731
H
13.972
E
14.778
D
14.513
X
13.425
R
16.329
Table 5.2: Reference magnitudes in the V filter for 7 field stars, see Figure 2.1 for
location of the stars.
Star
Ref. mag
5.1.2
F
14.093
G
14.054
H
14.497
E
15.247
D
14.948
X
13.770
R
17.086
Colours
The intensity of the light emitted by stars and other astronomical objects
varies strongly with wavelength. The apparent magnitude observed for a
given star by a detector therefore depends on the shape of the filter response
function, i.e. the width of the observed wavelength band and the transmittance as a function of wavelength. A quantitative measure of the colour of a
star is given by its colour index (CI). This is defined as the difference between
magnitudes at different wavelengths:
CI = m(λ1 ) − m(λ2 ) .
5.1 Basic principles of photometric studies
63
Because the colour index is a magnitude difference, it corresponds by Equation 5.2 to a flux ratio. The star Vega, or α Lyrae, is defined to have zero
magnitude at all wavelengths.
We note that the star we used for normalisation (star D in Figure 2.1)
has a different colour than the quasar. In fact, all the field stars are redder
than the quasar components. In Section 6.1.4 we perform some simple tests
based on colour diagrams.
5.1.3
Photometry from MCS
The MCS procedure was described in Chapter 4. We have used MCS as a
means of obtaining photometry from the QSO 0957+561 system. In this section we describe how we calculate object magnitudes from the MCS output.
The photometric output information from MCS consists of a list of the
maximum intensities of the point sources. In Section 4.5.3 we mentioned the
relationship between the flux and peak intensity of a star, where the star is
approximated by a Gaussian distribution. We show in Appendix D that this
is given by
F = 2πσ 2 Ipeak .
When using subsampled pixels, we effectively divide each pixel into four
smaller pixels. We therefore divide the expression above by four, giving
1
F = πσ 2 Ipeak .
2
Inserting this expression for the flux into Equation 5.2, we obtain, by comparison with a reference star with a known standard magnitude:
mobject − mstar = −2.5log10
F
object
Fstar
= −2.5log10
I
object
Istar
,
(5.3)
where the subscript ‘object’ denotes the object we are investigating, whether
a quasar component or another star. The σs cancel each other in the fraction,
as they are the same for the object and the reference star. This σ is related
by Eq. 4.10 to the FWHM we choose in the MCS programs. In Figure 4.4,
this is the width r of the Gaussian distribution. Since we use the same PSF
to deconvolve the quasar components and the reference star, this σ is the
same for Fobject and Fstar .
The MCS procedure requires the background to be the same in all images. To deconvolve the quasar, which has a galaxy in the ‘background’, we
therefore normalise the images by a ‘normalising star’, which is assumed to
be constant. This also has the effect of removing air mass and cirrus effects,
64
Photometry from the ALFOSC data
as described in Section 4.3. The reference star we chose for normalisation
was star D. The direct output from MCS thus consisted of a list of peak
intensities representing the ratio Iobject /ID . We therefore merely had to use
Equation 5.3 directly, giving:
mobject = −2.5log10
I
object
ID
+ mD .
(5.4)
We considered using the mean of several reference stars for calibration
purposes, which is a good idea if we have several reference stars available
which are known to be constant. If one of them should vary, however, we
would introduce systematic effects in the calibration. A detailed study of
the behaviour of the field stars in Figure 2.1 with special focus on variability
is still to be done. As mentioned earlier, we also have a reduced number
of available stars because of saturation of the brightest field stars. We finally chose one reference star (star D), to use for both normalisation and
calibration as described. We deconvolved other stars in the field to check for
correlated temporal intensity variations of the stars and the quasar components, which would indicate that the chosen reference star was variable. We
first suspected this was the case, see Section 4.6.2 for a description of the
strong correlation from our preliminary runs. In our final runs, we did not
find any indications of variability in the reference star.
5.2
Photometric results
In this section, we present our photometric results from the ALFOSC data
sets, from March 2001 and January 2000. The results are discussed and
analysed in more detail in Chapter 6. All results are from the simultaneous
deconvolution.
5.2.1
Error estimations
We calculated our errors for the quasar component magnitudes as the sum
of a systematic part (attributed mainly to PSF errors) and a random part
(the photon noise).
2
2
2
σtot
= σsyst
+ σphoton
(5.5)
The star R is of similar brightness as the quasar components. Assuming
that this star is not variable, we used it to compute the systematic part
of the errors. These systematics are an a posteriori measure of how well
the PSF fits the object we study (in this case the star R). The underlying
5.2 Photometric results
65
assumption is that the systematic errors are the same for the star and the
quasar components.
We used the standard deviation around the mean value of the flux of R
from each night as a measure of σtot for the star. See Table 5.7 for a list of the
standard deviations of the magnitudes. The standard deviation of the star
was calculated after variations due to air mass etc. had been removed through
normalisation. For the night of March 14th–15th, the last two data points in
the top panel (R band) of Figure 5.5, showing the star’s light curve, differed
significantly from the rest. We thought it unlikely that these points were
representative of this star’s assumed constant behaviour. When calculating
the errors for the quasar components for the remaining images this night,
we therefore calculated the systematic errors from the star without these
two points. When calculating the error bars for the two ‘deviating’ points
themselves, we used the standard deviation of all points that night. This
produces, appropriately, larger error bars for the last two points.
We calculated σphoton for the star R (in ADUs) by using the following
formula:
s
Flux(star R)
2
σphoton (star R) =
+ npix σsky
.
(5.6)
g
In this formula, for a sky subtracted image, g is the gain of the CCD
and σsky is the standard deviation in the background. When squared, σsky
approximates the original background level. The first term underneath the
square root represents the contribution to the total photon noise from the
source itself, in this case the star R. The second term is the contribution
from the background. In order to estimate the background contribution from
inside an aperture of approximately the size of the object, we multiply σsky
by the number of pixels inside this aperture, npix . The size of the aperture
was set in the following way: We measured the distance from the centre to
the point where the intensity of the object was ∼10% of the peak intensity,
and used this as radius in the circular aperture.
For the systematic errors in star R, we therefore have, from Equation 5.5:
2
2
2
σsyst
(star R) = σtot
(star R) − σphoton
(star R) ,
(5.7)
2
2
where σtot
is measured from our results and σphoton
is calculated from Equation 5.6. Since we did not use star R in constructing the PSF, the systematic errors calculated in this way will include the errors introduced by the
difference between PSF and the object we are deconvolving. We add these
systematics, in quadrature, to the errors from the photon noise for the quasar
components:
66
Photometry from the ALFOSC data
2
2
2
σtot
(QSO) = σsyst
(star R) + σphoton
(QSO)
(5.8)
Using an equation like Eq. 5.6 to calculate the photon noise for the quasar
2
components, we can now estimate the total errors σtot
(QSO). When calculating these errors, we found that they were dominated by systematics, i.e. the
systematic part of Equation 5.8 was much larger than the random part. This
we attribute to the rather large residuals from the deconvolution. Error bars
for the other field stars were calculated in the same manner.
To convert the total error bars from flux to magnitude, we ‘differentiate’
Equation 5.1:
d(mag) = −2.5 d(log10 (Flux)) .
Using the following relation to differentiate the logarithm:
log10 (x) =
ln(x)
,
ln(10)
where ln(x) is the natural logarithm, we obtain
d(mag) = −2.5
5.2.2
dFlux
1
.
Flux ln(10)
(5.9)
Light curves from March 2001
In this section, we present our results from our main data set; the four nights
in March 2001. Tables 5.3 and 5.4 display the mean of the R magnitudes
each night for the quasar components A and B, respectively. Also given are
the standard deviations of the magnitudes each night. The individual error
estimates were calculated in the manner discussed in Section 5.2.1. The mean
of these individual errors for each night, is also given in the tables.
Table 5.3: Mean values of the R magnitudes for the A component of QSO
0957+561 for the four nights, the standard deviation of the magnitudes σ mag and
the mean error (average of individual estimated errors) for each night.
2001
2001
2001
2001
Night
March 14–15
March 15–16
March 16–17
March 17–18
Mean R mag
16.422
16.428
16.432
16.436
σmag
0.0026
0.0023
0.0031
0.0032
Mean error
0.0034
0.0033
0.0030
0.0042
5.2 Photometric results
67
Table 5.4: Mean value of the R magnitudes for the B component of QSO 0957+561
for the four nights, the standard deviation of the magnitudes σ mag and the mean
error (average of individual estimated errors) for each night.
2001
2001
2001
2001
Night
March 14–15
March 15–16
March 16–17
March 17–18
Mean R mag
16.523
16.529
16.534
16.540
σmag
0.0041
0.0033
0.0046
0.0033
Mean error
0.0037
0.0036
0.0033
0.0046
Figure 5.1 displays the light curves for the A and B quasar components
for the four nights in March 2001. The horizontal dashed line is the mean of
all data points for all nights, and is meant solely as a guide to the eye. Error
bars are omitted for clarity, and are shown in Figure 5.2. The most striking
feature of Figure 5.1 is the near constant magnitudes, for both A and B. As
the B component and the galaxy are partially overlapping, it is harder to
separate this component from the background. This is probably reflected in
the slightly larger standard deviation each night for component B than for
the A component.
A slight, systematic decrease in brightness from night to night, in both
components, can be seen. Subtracting the mean R magnitude of the night
14th–15th, from the mean magnitude of the night 17th–18th, yields 0.014
mag for the A component, and 0.017 mag for the B component. Comparing
this decrease between the nights, to the standard deviation within each night,
we see that this change in magnitudes is more than 3σmag , which we consider
a significant decrease. Whether the decrease is ‘real’ in the sense of being
intrinsic to the quasar or could be caused by observational effects, will be
discussed further in Chapter 6. We will here mention the main points of
interest.
That the decrease is so similar in both A and B does not particularly
favour the theory that the variation is intrinsic. The two components are
shown from observations of the same night, making the curves theoretically
unrelated. The light rays from the two components have not travelled the
same distance and represent different times in the quasar’s ‘emission history’.
It is of course a possibility that both components happen to exhibit the same
variation at the same time. However, the decreasing magnitude from night
to night is not reflected in the behaviour within each night. Although it is on
a very small scale, there seems to be a variation within each night, similar in
68
Photometry from the ALFOSC data
Figure 5.1: Light curves in the R band for the quasar components A and B during
the four nights. The horizontal dashed line represents the mean of all data points
and is meant as a guide to the eye. Notice the decrease in brightness from night
to night, in both A and B.
A and B. In Figure 5.1 we see a variation which resembles a ‘U’ shape each
night.
In Figure 5.2 we have plotted light curves for each night individually,
including the individual error bars for each point. The variations within
each night, though on a small scale, could still be significant compared to
the (1σtot ) error bars. Since the ‘U’ shape each night is discernible every
night, the probability that it is a significant variation increases, although the
variations themselves are of a scale comparable to the error bars.
We see from Figure 5.1 that the maximum magnitude value each night occurs around midnight. We therefore performed a quick calculation intended
to probe the scale of the nightly variations. Subtracting the mean of the
three data points in the middle of each night, from the mean of the three
points at the start of that night, we obtain values ranging from 0.0005 to
0.0068. Performing the same calculation, but replacing the first data points
each night with the last three points of the night, we find changes in magni-
5.2 Photometric results
69
Figure 5.2: Light curves in the R band for the quasar components A and B for
each night, and with individual error bars. As in Figure 5.1, data points for the
A component are represented by a cross (×) and points for B by a diamond shape
().
tude ranging from 0.0008 to 0.0103. Compared to the average error bars in
Tables 5.3 and 5.4, we see that both computations show some variations on
a larger scale than the error estimations: exceeding 2 and even 3σtot for the
B component. This exercise was not intended to be an exact measure of the
possible variations within each night, but is included as a rough estimate.
We find that the variations within each night, although possibly consistent
with a straight line fit, are on a significant enough scale, especially as they
occur in all nights, to warrant further investigations. These we will pursue
in Chapter 6.
In the V filter, we had fewer images, approximately 50 each night, leading
to 4–5 data points in the corresponding light curves, see Figures 5.3 and 5.4.
70
Photometry from the ALFOSC data
We also had to discard two images because of aligning problems, as mentioned
in Section 4.6.2. Results for the quasar V images are listed in Tables 5.5 and
5.6. The standard deviation of the values for each night is given in the tables,
although we had very few points. We note that the quasar components are
weaker in V than in R.
Table 5.5: Mean values of the V magnitudes for the A component of QSO
0957+561 for the four nights, the standard deviation of the magnitudes σ mag and
the mean error (average of individual estimated errors) for each night.
2001
2001
2001
2001
Night
March 14–15
March 15–16
March 16–17
March 17–18
Mean V mag
16.833
16.833
16.839
16.843
σmag
0.0098
0.0102
0.0033
0.0033
Mean error
0.0068
0.0058
0.0034
0.0060
Table 5.6: Mean values of the V magnitudes for the B component of QSO
0957+561 for the four nights, the standard deviation of the magnitudes σ mag and
the mean error (average of individual estimated errors) for each night.
2001
2001
2001
2001
Night
March 14–15
March 15–16
March 16–17
March 17–18
Mean V mag
16.921
16.918
16.927
16.932
σmag
0.0098
0.0099
0.0029
0.0038
Mean error
0.0073
0.0062
0.0037
0.0066
The temporal sampling in the V filter is not good enough to conclude
anything about magnitude variations internal to each night. However, we
do notice a decrease in brightness from the first night to the last, the same
trend as in the R band. The overall decrease in V is 0.010 and 0.011 mag
for the A and B component, respectively. The tendency of a decline in the
V brightness is thus slightly less pronounced than for the R filter. In both
R and V we have a slightly larger decline in the B component than in A.
The average colour index V − R of all four nights is 0.407 ± 0.010 for
component A and 0.393 ± 0.012 for B. Changes in colour from night to night
will be discussed in Section 6.1.4. Future studies should include more images
in the V filter in order to discuss colour changes within each night.
5.2 Photometric results
71
We note that both quasar components seem to exhibit similar behaviour,
in both R and V . This ‘zero lag correlation’ will be discussed in Section 6.3.
Figure 5.3: Light curves in the V band for the quasar components A and B during
the four nights. The horizontal dashed line represents the mean of all data points
and is meant as a guide to the eye. Notice the decrease in brightness from the first
night to the last, for both A and B.
Results from the field stars
We have also deconvolved and produced light curves from four stars in the
field: the stars R, H, P and Q, see Figures 5.5, 5.6 and 5.7. This is done in
order to have – hopefully stable – reference objects. An important issue when
studying the reference stars, is to see whether they exhibit similar variations
to the ones we observe in the quasar components. An additional reason for
deconvolving the star R was to estimate the systematic errors, as explained
in Section 5.2.1. In this section, we first discuss our results for the star R,
and then the results from the other stars.
72
Photometry from the ALFOSC data
Figure 5.4: Light curves in the V band, for the quasar components A and B for
each night individually. As in previous figures, data points for the A component
are represented by a cross (×) and points for B by a diamond shape ().
In Table 5.7 we present the results from star R. As we estimated the
errors for the first night using the standard deviation of R excluding the last
two points, see Section 5.2.1, this is the value we have entered in the table.
Including these points produces a larger standard deviation, of 0.0049 mag,
while the mean level of the night is unchanged.
In the R filter, for both A and B components, the error bars are largest
on the night of March 17th–18th, which can be seen in Figure 5.2. As our
errors are dominated by systematics, this seems to be the result of the larger
standard deviation, in R, of star R on this night. Similarly, for V , the smaller
standard deviation on the night of March 16th–17th, has resulted in smaller
error bars this night for the quasar components in V , see Figure 5.4.
The magnitude of star R seems fairly stable from one night to the next.
5.2 Photometric results
73
The distribution of points within each night seem more random than for
the quasar. We can discern no ‘U’ shape, but in some cases the brightness
seems to be increasing during the night, see e.g. the last three nights in the
R band. That we do not find the same variations for the star R as for the
quasar components, indicates that the quasar variations are not caused by a
varying normalisation star, as this would probably have caused R to vary in
a similar fashion. Again, there are too few points in the V band to say much
about variations internal to each night.
Table 5.7: Mean values of the R and V magnitudes for the star R for the four
nights, and the standard deviation of the magnitudes. This star, assumed constant
in time, was used to estimate the errors for the quasar components and other stars,
see Section 5.2.1 for details.
2001
2001
2001
2001
Night
March 14–15
March 15–16
March 16–17
March 17–18
Mean R mag
16.389
16.393
16.392
16.388
σmag (R)
0.0029
0.0032
0.0030
0.0039
Mean V mag
17.092
17.092
17.098
17.097
σmag (V )
0.0087
0.0072
0.0043
0.0074
We compare the magnitude levels for the star R, in R and V , with the
reference magnitudes listed in Tables 5.1 and 5.2. Taking the mean of all four
nights as our value obtained for star R, we get a difference in magnitude of
m(star R) − mref (star R) = 0.0615 mag in the R band and, similarly, 0.0088
mag in V . So, our magnitudes for the reference star are slightly higher
(i.e. less bright) than the reference values.
For our further studies of the variations in the light curves of the quasar
components, both the nightly ‘U’ shape and the decrease from night to night,
the lack of similar variations in star R tells us something about what the
variations are probably not be caused by. For example, if the variations
in the quasar light curves are to be interpreted as the product of changes
in observing conditions, these changes must affect the star and the quasar
components differently, as we do not observe any of the same variations in
Figure 5.5. This motivates our discussion of the differences in colour between
the stars and the quasar, in Chapter 6.
Light curves in the R band for the field stars H, P and Q are shown in
Figure 5.6, and Table 5.8 provides the mean magnitudes of these stars for
each of the four nights. For the star H, there is no systematic decrease in
magnitudes from night to night comparable to that observed for the quasar
components. The star appears very constant until the last night, when we
74
Photometry from the ALFOSC data
Figure 5.5: Light curve in the R and V bands for the star R during the four
nights. The horizontal dashed line represents the mean of all data points and is
meant as a guide to the eye. Notice there is no systematic decrease in the level for
each night, as opposed to the systematic decrease seen in the quasar components,
see Figures 5.1 and 5.3.
see an inexplicable drop in brightness of 0.01 mag. The stars P and Q
show a total decrease in brightness of 0.012 mag and 0.014 mag, respectively.
This decrease is systematic, similar to that of the quasar components. The
error bars for P and Q are larger than for the quasar components, which is
appropriate as these stars are much weaker. The bright star H on the other
hand, has error bars of the order of ∼0.4 mmag, which is too small to be
seen in the figure.
Figure 5.7 displays light curves for the field stars H, P and Q in the V
band, and Table 5.9 shows the mean V magnitudes of the stars for each of
the four nights. As for the star R, there are too few points in V to conclude
5.2 Photometric results
75
Table 5.8: Mean values of the R magnitudes for the stars H, P and Q for the four
nights. Numbers in parenthesis are the standard deviation of the magnitudes in
mmag.
2001
2001
2001
2001
Night
March 14–15
March 15–16
March 16–17
March 17–18
H (R mag)
13.966 (0.0031)
13.968 (0.0018)
13.969 (0.0023)
13.976 (0.0020)
P (R mag)
17.960 (0.0060)
17.965 (0.0042)
17.969 (0.0068)
17.972 (0.0061)
Q (R mag)
18.238 (0.0072)
18.243 (0.0057)
18.248 (0.0064)
18.252 (0.0071)
anything about variations internal to each night. Taking the mean value of
each night shows a decrease in brightness from the first night to the last of
0.008 mag for P and 0.015 mag for Q. The light curve for star H contains
even fewer points, so we do not comment further on the variations here,
but include the graph for completeness. In both R and V , the star H was
saturated on some images, and the corresponding points have been discarded
from the light curve. In addition, the deconvolution of three of the extracted
images of H in the V band, resulted in extreme χ2 residuals, of values ∼30
times that of the other images. The corresponding points on the light curve
deviated from the rest by ∼0.5 mag, so the points were discarded from the
final light curve of the star. Again, the error bars for star H are too small to
be seen, they are of the order ∼0.8 mmag.
Table 5.9: Mean values of the V magnitudes for the stars H, P and Q for the four
nights. Numbers in parenthesis are the standard deviation of the magnitudes in
mmag.
2001
2001
2001
2001
Night
March 14–15
March 15–16
March 16–17
March 17–18
H (V mag)
14.642 (0.0032)
14.663 (—)
14.669 (0.0199)
14.658 (0.0125)
P (V mag)
18.836 (0.0088)
18.827 (0.0107)
18.841 (0.0070)
18.844 (0.0080)
Q (V mag)
19.261 (0.0078)
19.265 (0.0146)
19.277 (0.0062)
19.276 (0.0117)
The stars are generally redder than the quasar components, which could
be a cause of dissimilar variations in magnitude. A discussion of colour
indices of the quasar components and the reference stars is presented in
Section 6.1.4.
76
5.2.3
Photometry from the ALFOSC data
Light curves from January 2000
We now turn to our second set (chronologically the first) of observations from
the NOT using ALFOSC, originally 171 R and 39 V images from January
25th 2000. We combined the individual images in stacks of 10, in the same
manner as for the March 2001 data, to 15 R and 4 V images.
The time separation between the two sets is 416 days, from January 25th
2000 to March 16th 2001, which is within the range of the currently held time
delay. Unfortunately, weather conditions proved a severe hindrance for the
observers. Although ten days of NOT observing time was scheduled for this
session, nine nights contributed either no data or very few images of poor
quality. We have thus restrained our analysis to the one night with good
observing conditions, January 25th–26th.
Unfortunately, a determination of time delay from this data set is excluded, due to several facts. One is the weak magnitude variations we observe, the other is the bad observing conditions in January 2000. If more
data from the January 2000 session had been obtained, we could have investigated the light curves for short time scale variations that could possibly
lead to a determination of the time delay. We would have looked for similarities in the behaviour of the (first arriving) A component in January 2000,
compared to the B component in March 2001. The small variations within
each night seem to depend heavily on observing conditions and/or reduction
errors, so to determine the time delay we would probably require variations
in magnitude on the time scale of several days. If we could have discerned a
correlation in the behaviour of the two components, after time shifting one of
them, we could have entered our findings into the still current debate about
which value of the time delay is most appropriate. As it is, we could not hope
to find any events in the one day available to us, that would be reflected in
our data from March 2001.
We have, however, analysed the existing images from the one night in
January 2000. This leads to our discussion of the colour indices and difference
in magnitude of the quasar components, in Section 6.2.
For both R and V bands, we find that the components’ relative brightness
has shifted between the two observing sessions, see Table 5.10 and Figure 5.8.
In January 2000, we find that B is brighter than A, which is the opposite
situation of that in March 2001.
In the R band the B component is dimmer by ∼0.02 mag in January
2000 than in March 2001, which is less than the change in R for the star
R, which is assumed to be constant. Component A, on the other hand, is
dimmer by all of ∼0.2 mag in January 2000. So, chronologically, component
A has brightened considerably more than B from January 2000 to March
5.2 Photometric results
77
Table 5.10: Mean values of the R and V magnitudes for the A and B components
of QSO 0957+561 for the one night in January 2000, the standard deviation of the
magnitudes σmag and the mean errors (average of individual estimated errors) for
each night.
Comp
A
B
Mean R mag
16.632
16.555
σmag
0.0038
0.0051
Mean error
0.0043
0.0040
Comp
A
B
Mean V mag
16.989
16.864
σmag
0.0045
0.0041
Mean error
0.0037
0.0033
Table 5.11: Mean values of the R and V magnitudes for the star R for the one
night in January.
Night
2000 January 25
Mean R mag
16.354
σmag (R)
0.0034
Mean V mag
17.078
σmag (V )
0.0040
2001. B is more or less stable. In the V filter, the components seem to have
interchanged magnitude values, although the A component has changed most
in this band also. A is ∼0.15 mag dimmer in January 2000, while B is ∼0.06
mag brighter.
In Table 5.11 and Figure 5.9 we show our results for the star R on January
25th 2000.
78
Photometry from the ALFOSC data
Figure 5.6: Light curve in the R band for the stars H (top), P (middle) and
Q (bottom) during the four nights in March 2001. The horizontal dashed line
represents the mean of all data points and is meant as a guide to the eye. Some
points for star H have been discarded, see text for details. Notice the systematic
decrease in the mean level of each night for the stars P and Q, similar to the
systematic decrease in the quasar components, see Figure 5.1. The star H shows
no such systematic decrease, but a sudden, unexplained ‘drop’ on the last night.
5.2 Photometric results
79
Figure 5.7: Light curve in the V band for the stars H (top), P (middle) and
Q (bottom) during the four nights in March 2001. The horizontal dashed line
represents the mean of all data points and is meant as a guide to the eye. Some
points for star H have been discarded, see text for details. For stars P and Q, a
decrease in the mean level from the first to the last night can be seen, comparable
to that of the quasar components.
80
Photometry from the ALFOSC data
Figure 5.8: Light curves for the quasar components A and B during January
25th–26th 2000, for the R (left panel) and V band (right panel). Notice that the
relative strengths of the A and B components are shifted compared to the results
from March 2001.
Figure 5.9: Light curves for the star R in R and V during January 25th 2000.
Chapter 6
Discussion
Man: An argument is an intellectual process.
Contradiction is just the automatic gainsaying
of any statement the other person makes.
Arguer: No it isn’t
Monty Python, The Argument Clinic
In this chapter we discuss in more detail the results presented in Chapter 5,
with primary focus on our main data set, from March 2001. We also compare
our results to those of Ovaldsen et al. (2003b) and Colley et al. (2003).
6.1
Testing the results
In this section, we discuss the significance and possible causes of the variations in our light curves, concentrating on the data in the R band from March
2001. Two different variations seen in the light curves are discussed: The
possible ‘U’ shaped variation within each night, and the decrease in brightness from night to night. We performed several tests to see if we could find
any connection between these variations and changes in observing conditions,
such as seeing and moonlight.
6.1.1
A simple χ2 test
The variations seen within each night in the light curve from March 2001
(Figure 5.1) seem to be at least comparable to the estimated errors, see Section 5.2.1. To investigate these variations more quantitatively, we performed
a reduced χ2 test for goodness of fit to each night separately. In this way,
we wanted to examine whether the nightly ‘U’ shapes are consistent with a
horizontal, straight line fit.
82
Discussion
The χ2 is a statistical quantity that characterises the dispersion of the
observed frequencies from the expected frequencies. The χ2 is defined by
χ2 ≡
N
X
(yi − y(xi ))2
n=0
σi2
,
(6.1)
where N is the number of data points and σi are the associated errors for
each point i. The basis for this test is the hypothesis that the optimum
description of a set of data is the one that minimises the weighted sum of the
squares of the deviation of the data yi from the fitting function y(xi ). The
numerator in Equation 6.1 is a measure of the spread of the observations,
while the denominator is a measure of the expected spread.
The expectation value for χ2 is hχ2 i = ν = n − nc , where ν is the number
of degrees of freedom which is equal to the number of data points n minus the
number of parameters describing y(xi ). For our simple model of a horizontal
straight line, we have nc = 1.
The reduced χ2 is conveniently defined as χ2ν ≡ χ2 /ν, and has expectation
value hχ2ν i = 1. Values of χ2ν much larger than 1 may indicate an underestimation of the errors, or it may be the result of real deviations from the
assumed distribution. Values significantly below 1 indicates overestimated
errors.
We calculated the reduced χ2 :
χ2ν
≡
N
X
n=0
(yi − y(xi ))2
1
,
N −1
σi2
(6.2)
where the normalising factor is equal to the number of data points each night
minus one, ν = N − 1. The data points yi are the magnitudes for each night,
and y(xi ) = C, where C is constant, is the constant magnitude level that
we tried fitting to the data. We calculated χ2ν for many values of the free
parameter describing the straight line model. The resulting graphs are shown
in Figs. 6.1 and 6.2 for the A and B component, respectively.
Looking in statistical tables, e.g. in Bevington and Robinson (2003), we
find the probability of exceeding the minimum values of χ2ν given in the
figures, for the corresponding values of ν. For example, the first night a minimum value of χ2ν = 0.6696 for ν = 18 degrees of freedom gives a probability
between 0.80 and 0.90 of obtaining χ2ν ≥ 0.6696. In other words, the chance
of obtaining such a low value of χ2ν by chance, is only between 10 and 20%.
We note that the values of χ2ν are almost all well below or very close to
1, with the exception of the bottom left panel in Figure 6.2. Unfortunately,
this means that our test is inconclusive with regard to the question of variability within each night. In the example mentioned, the probability that
6.1 Testing the results
83
Figure 6.1: Reduced χ2 for the A component for the four nights in March 2001.
Top left: March 14th–15th. Top right: March 15th–16th. Bottom left: March
16th–17th. Bottom right: March 17th–18th.
the straight line would give such a good fit to the data, even if it were not
the correct model, is between 10 and 20%, which means the model might be
considered an unlikely good fit and we can conclude that our errors seem to
be overestimated. From Eqs. 5.7 and 5.8, we have that
2
2
2
2
σtot
(QSO) = [σtot
(star R) − σphoton
(star R)] + σphoton
(QSO) .
(6.3)
The photon noise of star R and the quasar components were approximately
equal, as was expected since they have very similar flux. From Equation 6.3
we see that in this case the photon noise from star R and from the quasar
components will approximately cancel each other. As our errors were dominated by systematics, the most likely cause for the possible overestimation
2
is that σtot
(star R) is too big. However, this quantity is measured directly
from our results for the star, and we corrected for the two points we had any
reason to consider atypical for the assumed constant star, see Section 5.2.1.
In the third panel of Figure 6.2 the value of χ2ν corresponds to a probability
of 0.02 of exceeding χ2ν = 1.8263. By itself this would seem to indicate a
84
Discussion
Figure 6.2: Reduced χ2 for the B component for the four nights in March 2001.
Top left: March 14th–15th. Top right: March 15th–16th. Bottom left: March
16th–17th. Bottom right: March 17th–18th.
model very inadequate at representing the physical situation, and a very good
chance that we have significant variability this night. However, as the rest
of the values for χ2ν are so low, the high value in this particular case seems
slightly suspicious as the estimated errors for all nights are approximately
equal. We note that the highest value of χ2ν in Figure 6.1 is from the same
night.
6.1.2
Seeing
The seeing of the March 2001 observations varied from roughly 0.700 to 1.500 ,
with a median value of 0.800 , see Figure 6.3. There are few points with FWHM
over 1.200 . We calculated the seeing using the FWHM value of a selection of
stars in the field, measured with IRAF. These were rather rough estimates
of the FWHM that we entered into psfmof.txt when constructing the PSFs
for each image. We measured the seeing on our combined images, which were
sums of 10 individual images, each with exposure time scaled to 60 seconds.
6.1 Testing the results
85
Figure 6.3: Changing seeing conditions within each night.
To check if any systematic effects were introduced when seeing conditions
varied, we have in Figures 6.4 and 6.5 plotted magnitudes versus seeing for
all four nights, for both quasar components and the star R. The figures show
no marked effects, neither in R nor in V . Fitting a straight line,
mag = a ∗ seeing + b
to the points, yields practically straight lines, as shown in the figure. Values
for a are −2.5 mmag/00 and −6.7 mmag/00 for the A and B components in
the R filter, see the top panel in Figure 6.4. Corresponding values in the V
filter are a = 7.4 mmag/00 and a = −0.5 mmag/00 . For the star R, the values
of the straight line fit to the data are a = 2.2 mmag/00 and a = 3.8 mmag/00
for the R and V filter, respectively.
To further investigate any relationships between magnitude and seeing,
we have in Figure 6.6 plotted the R magnitude of the quasar components
versus seeing, for each of the four nights individually. We fitted straight
lines to the data points, and obtained values for the gradient a ranging from
−11.0 mmag/00 to 3.0 mmag/00 for the A component, and −13.1 mmag/00
to 1.9 mmag/00 for the B component. We find no evidence of any general
86
Discussion
Figure 6.4: Top: R magnitude for the A (×) and B () quasar components versus
seeing (FWHM) for all four nights in March 2001. Bottom: V magnitude versus
seeing. A straight line is fitted to the data.
Figure 6.5: Top: R magnitude for the R star versus seeing (FWHM). Bottom: V
magnitude versus seeing. A straight line is fitted to the data.
6.1 Testing the results
87
Figure 6.6: R magnitude for component A (top four panels) and component B
(bottom four panels), versus seeing, for each night individually. A straight line is
fitted to the data. We see no general trend of changing magnitude with seeing.
tendency for the magnitude of either quasar component to vary with seeing
within each night. Although seeing generally increases towards morning, see
Figure 6.3, it does not seem that this could have caused the observed increase
in brightness at the end of each night.
88
Discussion
6.1.3
Possible influence of the Moon
Systematic effects introduced by insufficient removal of the background can
depend upon the colour of the object. The moon was present during our
observations in March 2001. We wanted to investigate whether the presence
of the moon had any effect on our results. Table 6.1 shows the time of Moon
rise, how much of the Moon that was illuminated each night seen from Earth
and the altitude of the Moon at 6 a.m., the approximate maximum altitude
the Moon reached before morning twilight.
Table 6.1: Presence of the Moon during the March 2001 observations.
Night
March 14–15
March 15–16
March 16–17
March 17–18
Moon rise (UT)
00:22
01:18
02:13
03:04
% illumination
67.5
57.8
48.0
38.6
Alt. at 6 a. m. (degrees)
42
38
33
27
When the Moon is present, the background level is higher, especially at
blue wavelengths, than when the Moon is absent. In Figure 6.7 we have
plotted the background level for each night, against time. As mentioned
in Section 3.4.2, the background level of each image was subtracted before
adding the images in stacks of 10, each individual image being scaled to the
same exposure time. By adding the levels that we originally subtracted, for
each stack, we obtained the background levels of the combined images. The
images that were not stacks of 10, see Section 3.4.2, were corrected for, by
taking the mean of the background levels in the stack and multiplying by
10. The vertical dotted line in Figure 6.7 represents the time (UT) of Moon
rise each night. The rise in sky level immediately after Moon rise is easily
seen. Note that the range on the y axis is reduced for the bottom panels.
On the first night, the Moon rises to a maximum altitude of 42 degrees and
the sky level is over 5000 ADU. On the last night, the moon only rises to
27 degrees above the horizon, producing a sky level of approximately 3000
ADU. Comparing this to around 3 a.m. on the first night, when the Moon
was also 27 degrees above the horizon, we see that the background level again
has a value of ∼3000 ADU.
To determine whether the moonlight had any effect on our magnitudes,
we constructed a scatter plot showing R and V magnitude versus background
(sky) level for the March 2001 data, see Figure 6.8.
6.1 Testing the results
89
Figure 6.7: Sky level variation for the four nights in March 2001. The dotted line
represents Moon rise.
The Earth’s atmosphere emits photons by fluorescence, a process sometimes referred to as airglow, see Léna (1988). Daily photochemical dissociation causes electrons to recombine with ions which produces fluorescence
photons. The main sources of this line emission are OH molecules, oxygen
(O), nitrogen (N) and sodium (Na). After sunset, there is still some emission
in the atmosphere, especially at red wavelengths. This emission probably affects our data in a different way than the moonlight. We therefore removed
points from images in R taken less than 2 hours after astronomical evening
twilight, from Figure 6.8. The last data point from the night March 14th–
15th was also removed, as the background level in this image was noticeably
higher than for all other points (level at 22400 ADU). This extremely high
background probably originates from morning twilight rather than the Moon.
In Figure 6.8, we have plotted magnitude (both R and V ) against background level for the two quasar components and for the reference star R. A
straight line was fitted to the points, which is also shown in the figure. As
the V plots contain quite few points, we will mainly discuss the results in the
90
Discussion
Figure 6.8: Scatter plots showing the relationship between magnitude and background level. Top: R magnitude for the A (left) and B (right) component. A
marked decrease in magnitude, i.e. increasing brightness, with increasing sky level
is illustrated by the fitted line. Middle: V magnitude for the A (left) and B (right)
component. The fitted line shows increasing magnitude with increasing sky level,
but there are very few points. Bottom: R (left) and V (right) magnitude for the
R star. The straight line fitted to the R plot shows a decreasing magnitude trend
with increasing sky level, but the gradient is clearly smaller than for the quasar
images. Again, the V plot contains too few points to conclude much.
R band. We merely remark that, in V , the fitted line shows a decrease in
brightness with increasing sky level, for the quasar components, and the opposite situation for the star R. However, we see from the figure that only two
points contribute to the ‘lowering’ of the fitted line, so we cannot conclude
whether this is coincidental or a real trend.
In the R band, we see a clear difference in the gradients of the fitted lines
for the quasar components and the star, although the fits have considerable
spread. For the A component, the gradient of the fitted line is 2.2 times
larger than the gradient of the line fitted to star R. For component B, the
gradient is 3.1 times larger than that of the star.
6.1 Testing the results
91
The increase in background level apparently influences the quasar components in a different way than it does the star. This may come from the fact
that the star is redder than the quasar. Assuming the Moon is causing the
trends, we can say that the moonlight affects the quasar and the star very
differently, possibly because of their different colours. It might also be that
insufficient galaxy modelling is causing this effect for the quasar components.
However, as the B component is much closer to the galaxy than A, one would
in that case perhaps expect to find a more marked difference in the two top
panels in Figure 6.8.
It would be very interesting to see what tendencies we could find in V ,
from both quasar and star, with more images in this band. A future study
of this effect should include better temporal sampling in V so as to obtain
more data points for this filter.
Figure 6.9: A model light curve showing the distribution of points with the relation between magnitude and background level shown in Figure 6.8 removed.
Symbols used: × and for the A and B component, respectively, for the points
not included in the scatter plots in Figure 6.8. The symbols + and are used for
the modelled points for the A and B component, respectively.
In order to better see how the light curves in Figure 5.1 would have looked
92
Discussion
without the connection between background level and magnitude we have
just shown, we constructed a model light curve for the quasar components.
This is shown in Figure 6.9. The data points originating from the images
that were removed from the moonlight analysis, as they were obtained less
than two hours after astronomical twilight, are plotted with the same symbol
conventions for the A and B components used in Chapter 5. The new points
that have been modelled, have corresponding symbols rotated 45 degrees
(i.e. a plus sign for the A component and a square for the B component).
To construct the model curve, we subtracted the effect of the straight line
fit from the original magnitudes, using the following equation:
mi = moriginal − [f (xi ) − f (xfix )] .
Here, mi is the magnitude of the model curve and moriginal is the actually
observed magnitude. The straight line we fitted is f (x). We assume this is
the correct relation between magnitude and background level, and calculate
f (xi ), the magnitude corresponding to background xi from the equation of
the straight line, and f (xfix ), which is the magnitude corresponding to the
lowest background level in Figure 6.8. We thus obtain the magnitude mi that
we would have had, if the Moon had not been present, assuming that the
Moon is causing the increase in background level and that the straight line
fit is the correct relation between increasing background level and increasing
magnitude.
Figure 6.9 also shows that the straight line fit is not very exact. We see
from Figure 6.8 that there is considerable spread in the points. The drop we
see in brightness at the end of the first night, is an example of the mismatch
of the model. It corresponds to the fact that there are a few points in Figure
5.1 on this night that actually decrease in brightness towards the end of
the night. The residuals from the deconvolution for these images were not
particularly large compared to other images.
However, disregarding these last points on the first night, we see that we
have with some success ‘straightened’ the curve, at least for the first two
nights. The last two nights are a bit messier, but we recall from Figure 6.7
that the background levels here are lower as the Moon rises later in the night.
The conclusion seems to be that the ‘U’ shape each night can be a result
of background effects caused by the Moon. However, we can see in Figure 6.9
that our model is not in perfect agreement with the data. There might be
other aspects, e.g. atmospheric changes, which introduces other effects that
mix with that of the moonlight to form a more complex situation than we
have modelled.
6.1 Testing the results
6.1.4
93
Colour diagrams
We wanted to study whether the variations we see in quasar brightness from
March 2001 correlate at all with changes in colour. This could indicate
that the observing conditions changed. We were also interested in studying
the colours of the reference stars, and compare this with the light curves
presented in Section 5.2.2.
As our temporal sampling in V was not good enough, we did not investigate variations with colour within each night. Only the possible colour
dependency of the decline in brightness from night to night was studied. We
calculated V − R for the four nights in March 2001, using the average magnitude of each night for each filter. We checked that the results of calculating
the mean directly from the magnitudes and via the flux, gave identical results
to the given accuracy. The colour indices of the quasar components and the
star R are plotted in Figure 6.10. Our symbol conventions still holds: The
A component is represented by a cross (×), the B component by a diamond
shape () and the R star by a triangle (M).
Figure 6.10: V − R plotted against time for the quasar components (top panel)
and the star R (bottom panel). The A component is represented by a cross (×)
and the B component by a diamond shape (). The star R is represented by a
triangle (M).
94
Discussion
When subtracting two magnitudes, the errors increase. The error bars in
Figure 6.10 were calculated in the following manner:
q
σV −R = σV2 + σR2 ,
where σ is the standard deviation of the magnitudes within each night.
For all four nights, the mean value of V − R is 0.407 ± 0.010 for the
A component and 0.393 ± 0.012 for B. Differences in colour between the
A and B component could indicate differential extinction along the light
paths, or possibly microlensing. Extinction should influence the light from B
most, because light from this component travels through the lensing galaxy.
Scattering by material in the galaxy affects blue wavelengths most,1 so we
would expect this to cause the light from B to be redder than the light from
A. We find the opposite: A is slightly redder than B. However, the difference
in V and R magnitudes is very small.
The V −R values for the A component range from 0.405±0.010 to 0.410±
0.010 for the four nights. For the B component, the range is 0.392 ± 0.005 to
0.400 ± 0.010. In other words, the changes we see of colour versus time are
within the error bars. The ‘dip’ from the first to the second night in Figure
6.10, constitutes 0.005 magnitudes for (V − R)A and 0.008 for (V − R)B .
The corresponding dip for the R star is (V − R)R = 0.004. This suggests
a possible, observational influence that produces the same colour change,
towards the blue, in all three objects between the two nights. Changes in
colour might be due to atmospheric conditions, perhaps in combination with
moonlight. However, the variations in colour are not significant compared to
the error bars. We also note that we do not find any correlating change in
colour index with the stepwise decrease in brightness from night to night.
As the light curves for the stars P and Q show a similar systematic decrease from night to night as that of the quasar components, we wanted to
examine the colours of these stars. The field stars R and H were known to
be redder than the quasar components, but we did not have any reference
magnitudes for P and Q. If the colour of the objects influence the variations
from night to night, we might expect the stars P and Q to be bluer than
R and H, more similar in colour, perhaps, to the quasar components. In
Table 6.2, we show the mean V − R value, for all four nights, of the quasar
components and all field stars we have studied.
Our results show that the stars P and Q were in fact significantly redder
than the stars R and H. We therefore did not find any evidence that the
1
The amount of scattering that occurs to a beam of light is dependent upon the size
of the particles and the wavelength of the light. Blue light has a shorter wavelength than
red light, and is therefore scattered more.
6.2 Comparing the time shifted data
95
Table 6.2: Mean values of the quasar components A and B, and the field stars
R, H, P and Q, for the four nights in March 2001. The standard deviation of the
V − R values are also given, computed in the same manner as the error bars in
Figure 6.10.
Object
V −R
σV −R
A
0.407
0.010
B
0.393
0.012
R
0.705
0.008
H
0.748
0.016
P
0.870
0.013
Q
1.024
0.015
decrease in brightness from night to night is correlated with the colour of the
object. Pondering as to what else could cause P and Q to behave so similar
to the quasar components, we wondered whether the decreasing tendency
has something to do with the brightness of the object. The A component
is approximately four times brighter in flux than P and five times brighter
than Q. The star H, on the other hand, is almost ten times brighter than A.
Conceivably, the nightly decrease is seen only in weak objects. However, the
star R is only slightly brighter than A, so in that case we would expect to
see the decreasing tendency in this star also, which we do not.
In Figure 6.11, we show a colour magnitude diagram of V −R versus R for
the quasar components and the reference star R. The groups of four points
are results from the four nights in March 2001, while the isolated points are
from the one night in January 2000.
6.2
Comparing the time shifted data
Gravitational lenses are achromatic. The angle that light rays are bent by
the intervening mass is not dependent on wavelength. From this it follows that the magnitude difference between the images of a gravitationally
lensed quasar should be the same in different bands, at the same point in
the quasar’s emission history. As our data sets (March 2001 and January
2000) are separated in time by the approximate time delay, the time shifted
difference in magnitude between the two components of QSO 0SO 957+561
provides the optical continuum magnification ratio. A is the first arriving
component, followed by B a number of days equal to the time delay later.
We therefore calculate the magnitude difference between the components,
using the magnitude of A from the chronologically first observing session,
January 2000, and the magnitude of B from March 2001. We calculate the
96
Discussion
Figure 6.11: V − R plotted against R magnitude for the quasar components
(top panel) and the star R (bottom panel). The symbol conventions for the QSO
components and the star are the same as in Figure 6.10. The groups of four
points close together are data from March 2001, while the isolated points are from
January 2000.
magnitude difference in both R and V :
mB, 2001 (R) − mA, 2000 (R)
mB, 2001 (V ) − mA, 2000 (V )
In these calculations, we have used the mean level of all four nights for the
2001 data, although we observed a decrease in brightness from night to night
of 0.014 and 0.017 magnitudes for A and B, respectively. For the 2000 data we
used the mean value of the one available night. Converting these magnitude
6.2 Comparing the time shifted data
97
differences into B/A flux magnification ratios, we get 1.096 ± 0.022 in R and
1.061 ± 0.024 in V . To calculate the errors for the magnification ratio, we
inverted and differentiated Equation 5.1.
As mentioned in Section 2.2.2, a discrepancy between the magnifications
in R and V can be caused by either differential absorption along the different
light paths, or by microlensing. However, the difference between the magnification ratios is within the error estimations, so it is not very significant.
We also calculate the colour indices of A and B and find:
mA, 2000 (V ) − mA,
mB, 2001 (V ) − mB,
2000 (R)
= 0.357 ± 0.006
2001 (R) = 0.393 ± 0.012
where the errors were calculated in the same manner as in Section 6.1.4. We
see that, for the time shifted data, B is slightly redder than A. The lensing
galaxy is significantly redder than the QSO. Therefore, one would expect to
find that B is redder than A if no galaxy correction has been done. If the
galaxy was perfectly corrected for, one might still expect B to be redder than
A, as the blue light has probably been scattered more as described in Section
6.1.4.
Compared to our V − R calculations in Section 6.1.4, the situation has
changed. Without time shifting, we found A slightly redder than B, although
within the errors. We do not know of any reason why the quasar itself should
change colour. It is probable that something else has caused the change.
Microlensing by compact objects in the lensing galaxy or galaxy cluster,
can cause the quasar components to become brighter, or weaker. We see
from Figure 6.11 that component A becomes brighter in R by ∼0.2 mag
from January 2000 to March 2001. Component B has approximately the
same magnitude in March 2001 as in January 2000, it brightens by only
∼0.02 mag. If the changes in brightness were the result of microlensing, we
would expect that as the quasar brightens, it would also change in colour
towards the blue. This is because microlensing would mainly amplify the
smaller, central parts of the quasar, which are also the bluest. From Figure
6.11 and Table 6.2, we find that V − R for the A component in January
2000 was 0.357 ± 0.006, while in March 2001 the value was 0.407 ± 0.010.
In other words, as the A component brightens, it also becomes more red.
This makes it unlikely that the change in brightness of component A is the
result of microlensing, and more probable that this is an intrinsic brightness
variation in the quasar. As the brightness change for the A and B component
are so dissimilar, it is probably not a result of different observing conditions
in January and March. We do not have any explanation for the change
in colour, aside from speculations involving quasar models with radiating
‘spots’.
98
6.3
Discussion
Zero lag correlation
Because of the time delay, the light curves for the two quasar images are in
theory uncorrelated, and should exhibit independent behaviour. The appearance of correlation without correcting for the time delay, is therefore noted
with some apprehension.
There have been several reports on a ‘zero lag correlation’ between the
A and B images, for time-scales of weeks and months (Kundic et al., 1995;
Ovaldsen, 2002; Ovaldsen et al., 2003a) as well as for shorter time-scales
(Ovaldsen et al., 2003b; Colley et al., 2003).
In Ovaldsen et al. (2003b), the authors find a small brightness decrease
in the mean magnitude level from night to night, using the same ALFOSC
data set that we use in this thesis. Their decrease is similar to what we have
found, although our decrease is about twice as large in terms of magnitudes.
From March 14th to March 18th, Ovaldsen et al. found a decrease of 0.005
R mag for A and 0.008 mag for B. For the V filter, their decreases are 0.006
mag for both A and B. This small decrease is suggested by the authors as a
possible manifestation of the correlation at zero lag.
Colley et al. (2003), in addition to the same ALFOSC data, use data
from 11 other observatories. From their results, ranging from March 12th to
March 21st, they report a surprising zero lag agreement of the curves of A
and B. They interpret this as evidence of an as yet uncorrected photometric
problem at the few millimagnitude level.
Referring to Figure 5.1 and Figure 5.2, we see that the nightly variations
in both components show a tendency to correlate at zero lag. For example,
the magnitudes increases from around midnight towards the morning. As it
has been argued for by e.g. the moonlight test, this is probably caused by
some or more observational and/or photometric effects. Our discussion in
Section 6.1.3 has shown that the presence of moonlight is a possible explanation for the small increase in brightness at the end of each night. Only five
nights of optical monitoring does not suffice to draw any firm conclusions
about this effect. However, the correlation between the components at zero
lag within each night, as well as its possible connection with moonlight, constitutes an indication that our variations within each night may be caused
by observational effects rather than intrinsic variations in the quasar. The
zero lag correlation within each night can also be seen in the V band, see
Figures 5.3 and 5.4.
The decrease in brightness from night to night could also be a variety
of this unexplained correlation at zero lag, as suggested by Ovaldsen et al.
(2003b). We also tentatively note, that if our model of quasar magnitude
versus sky level is correct, it could be possible that the variation from night
6.4 Comparisons with other results
99
to night really is reflected in the nightly observations. If so, that could
mean either i) evidence of zero lag correlation on a time scale of the four
observation nights, or ii) both quasar components happen to exhibit very
similar decreases in flux during these four nights. A more sophisticated model
than our straight line fit should probably be constructed before making any
definite claims.
6.4
Comparisons with other results
In this section, we directly compare some of our results to those of Ovaldsen
et al. (2003b), and to results kindly provided by Wesley Colley (private communication). The latter constitute the NOT part of the official QuOC results
(Colley et al., 2002, 2003). Tables 6.3 and 6.4 show results from the three
groups in the R band for the quasar components. Our results, presented in
the second column in the table, are the same as in Tables 5.3 and 5.4. We
only compare our results in R, as the QuOC team has not published any
results using the V images.
Table 6.3: Our mean values of the R magnitudes for the A component of QSO
0957+561 for the four nights, the results of Ovaldsen et al. (2003b) and results from
the data provided by Wesley Colley for the same nights. Numbers in parenthesis
are the standard deviation of the magnitudes in mmag.
2001
2001
2001
2001
Night
March 14–15
March 15–16
March 16–17
March 17–18
Mean R mag
16.422 (2.6)
16.428 (2.3)
16.432 (3.1)
16.436 (3.2)
Ovaldsen
16.394 (5.7)
16.397 (4.4)
16.399 (4.0)
16.399 (5.1)
Colley
16.374 (7.6)
16.377 (4.8)
16.374 (5.0)
16.378 (5.9)
A note on the calibrations: As mentioned in Section 5.1.1, we have used
the exact same reference magnitudes as those of Ovaldsen et al., see Tables
5.1 and 5.2. The latter authors use a mean of several reference stars for
the calibration, whereas we use one star (D). The reference magnitudes were
supplied, to two decimal points, by Rudy Schild, but have been slightly modified by the bootstrapping applied by Ovaldsen et al. Of the stars supplied
by Schild, only H differ at all from the reference magnitudes in Table 5.1, by
∼0.01 mag. We therefore note that there is a possibility that a small systematic difference between our own results and those of Colley et al. has been
100
Discussion
Table 6.4: Our mean values of the R magnitudes for the B component of QSO
0957+561 for the four nights, the results of Ovaldsen et al. (2003b) and results from
the data provided by Wesley Colley for the same nights. Numbers in parenthesis
are the standard deviation of the magnitudes in mmag.
2001
2001
2001
2001
Night
March 14–15
March 15–16
March 16–17
March 17–18
Mean R mag
16.523 (4.1)
16.529 (3.3)
16.534 (4.6)
16.540 (3.3)
Ovaldsen
16.486 (6.2)
16.491 (4.0)
16.495 (4.4)
16.494 (5.7)
Colley
16.296 (5.9)
16.299 (4.2)
16.297 (4.5)
16.300 (5.5)
introduced by the difference in one of the reference magnitudes we believe
was used by these authors.
6.4.1
Results from Ovaldsen et. al.
Ovaldsen et al. (2003b) discuss the detailed variations in brightness for QSO
0957+561, employing the newly developed aperture photometry scheme from
Ovaldsen (2002). The method involves subtracting a galaxy model from
each image frame. Then, one of the quasar components is ‘cleaned’ by subtracting a PSF, followed by aperture photometry of the other component,
and vice versa. The PSF subtraction is done in order to correct for crosscontamination between the quasar components. In Figure 6.12 we show their
results for the A and B component, for R in 2001. In order to facilitate comparisons, this figure is presented in the same format as our light curves in
Section 5.2. The figure shows all 673 individual data points, without binning.
Error bars are omitted for clarity. The mean value of individual formal error
bars for A is 5.6 mmag, and for B 6.2 mmag. These were typical errors. Mean
values of magnitudes from each night as well as the standard deviation, are
shown in Tables 6.3 and 6.4.
We have several findings that agree with those of Ovaldsen et al. This
is interesting and reassuring, as these authors have used the exact same
ALFOSC data sets as we have done, but a very different method. Both
groups find that, for March 2001, A is brighter than B in both R and V
by approximately 0.1 mag. We also find the opposite situation, B brighter
than A, in January 2000, both groups finding that B is brighter than A by
∼0.07 mag in R and ∼0.12 mag in V . Table 6.5 shows a comparison between
the magnitude differences of the A and B components for the two observing
6.4 Comparisons with other results
101
sessions.
Another similarity between our results is the systematic night–to–night
decrease in brightness, seen in March 2001 in both quasar components, see
Figure 6.12 and Tables 6.3 and 6.4. The overall decrease in brightness, from
the first to the last night, is 0.005 mag for A and 0.008 mag for B. Compared
to our own total decreases of 0.014 mag for A and 0.017 mag for B, we see
that the trend is the same but our decrease is considerably more than that
of Ovaldsen et al. It is interesting that both methods yield a systematic
decrease.
Table 6.5: Differences in magnitudes between the two QSO components in the
R and V filters, for our results and the results of Ovaldsen et al. The mean
magnitudes of each night of component A have been subtracted from the mean of
each night of component B. Values from our results, subscripted EV, are shown
in the second and third columns for R and V respectively, while the results from
Ovaldsen et al. (2003b), with subscripts OV, are displayed in the fourth and fifth
columns, also for R and V .
Night
2000 January 25–26
2001 March 14–15
2001 March 15–16
2001 March 16–17
2001 March 17–18
(R)EV
–0.077
0.101
0.101
0.102
0.104
(V )EV
–0.125
0.088
0.085
0.088
0.089
(R)OV
–0.069
0.092
0.094
0.096
0.095
(V )OV
–0.115
0.088
0.087
0.087
0.088
From Tables 6.3 and 6.4 we see that the magnitudes in Ovaldsen et al. are
between 0.03–0.04 mag brighter than our results. Table 6.5 shows that the
magnitude difference between the two quasar components remain stable for
the four nights, both for our results and those in Ovaldsen et al. The differences in magnitudes between the components are also very similar for the
two methods, compare columns two and four, and columns three and five, in
Table 6.5. For the star R, there is a difference in brightness between our results and Ovaldsen et al. of ∼0.06 mag. In this case also, Ovaldsen et al. have
the brightest values, so this seems to be a systematic difference introduced
by the two photometric reduction methods.
We find a discrepancy in the colour indices for A and B between the two
groups. For the time shifted results, Ovaldsen et al. find that A is redder
than B, while we have the opposite: B is redder than A. V − R magnitudes
for A and B from Ovaldsen et al., are 0.357 ± 0.007 and 0.318 ± 0.009,
102
Discussion
Figure 6.12: Light curve from the results of Ovaldsen et al. (2003b) for the quasar
components A and B in R during the four nights in March 2001. The horizontal
dashed line represents the mean of all data points and is meant as a guide to the
eye. Notice the systematic decrease in brightness from night to night, similar to
our results, although less pronounced.
respectively. In Section 6.2 we stated our V − R values for A and B: 0.357 ±
0.006 and 0.393 ± 0.012. In other words, the colour indices for component A
are practically identical for us and Ovaldsen et al., while for B they differ by
∼0.08 mag, our results showing a higher V − R value, thus a B component
more red in colour. Since the values for A are so similar, it is probably the
different galaxy corrections that makes our results differ, as the galaxy affects
the B component the most. Ovaldsen et al. subtract a galaxy model, while
we produce a numerical model of the galaxy through deconvolution.
6.4.2
Results from Colley et. al.
In the paper reporting the results from the QuOC campaign’s second session,
Colley et al. (2003) mention the data from the NOT, i.e. our main data set
in this thesis, as being part of the ‘back-bone’ data, providing many frames
6.4 Comparisons with other results
103
with excellent statistics. On request, Wesley Colley kindly provided us with
results from their reduction of the NOT data in the R band.
Colley et al. applied aperture photometry with a single correction for
aperture cross-talk, i.e. the contribution of light from component A to the B
aperture and vice versa, and galaxy contamination, as a function of seeing.
We have plotted the results of Colley et al. in Figure 6.13, choosing to plot the
743 individual data points as this is how we showed the results of Ovaldsen
et al. in Section 6.4.1. In Colley et al., hourly means of the magnitudes
are used. As in Figure 6.12, error bars are omitted for clarity. Mean of
individual errors in magnitude, provided by Wesley Colley, is 4.1 mmag for
the A component and 3.9 mmag for B. We note that the number of data
points are 70 more than those of Ovaldsen et al., probably due to different
rejection criteria.
Figure 6.13: Light curves for the quasar components A and B in R during the
four nights in March 2001, from the results supplied to us by Wesley Colley. The
horizontal dashed line represents the mean of all data points and is meant as a
guide to the eye. Notice there is no systematic decrease in magnitude from night
to night, and that the B component is stronger than A.
Comparing the brightness ratio between A and B in Figure 6.13, we
104
Discussion
see that, in this case, we have the opposite situation of that in our own
results and the results from Ovaldsen et al. The B component – with a
mean magnitude for all four nights of 16.298 mag – is here brighter than A,
which has a mean of 16.375 mag. This discrepancy is due to the fact that
the method used by Colley et al. involves a correction for the galaxy as a
function of seeing, but not a galaxy subtraction. As the galaxy contributes
∼18% to the aperture photometry of component B (Rudy Schild, private
communication), the result is that B appears brighter than A. Converting the
uncorrected B magnitudes into fluxes, subtracting 18% and then averaging
over all points, yields a mean value of mB, corrected = 16.513 mag, a value in
good agreement with our own B magnitude. With this value for the corrected
B magnitude, the internal ratio between the components A and B also agree
with our results. The difference in magnitude between the two components
for the QuOC results in March 2001, become mB, corrected − mA = 0.138
mag. This value is a little higher than the value we have obtained, see
Table 6.5. However, the internal brightness ratio between A and B are now
approximately the same for all three groups, with A the brightest component.
For January 2000, the mean magnitudes from the QuOC results, without
subtracting the galaxy contribution, are 16.46 mag for the A component and
16.29 mag for B (supplied by Rudy Schild to two decimal points, private
communication). That is, B is very similar to the (uncorrected) March 2001
value. The A component is brighter in 2001 by ∼0.1 mag, while B is less
bright in 2001 by merely ∼0.01 mag. In fact, all three groups find that the
A component has changed much more than B during the ∼416 days between
the two observing sessions. As mentioned in Section 6.2, taking the mean of
all four nights for the results in March 2001, we find that the A component is
∼0.2 mag brighter in 2001 than in 2000, while B is only ∼0.02 mag brighter
in 2001, see Tables 5.3, 5.4 and 5.10. For Ovaldsen et al. (2003b), A brightens
by ∼0.1 mag from 2000 to 2001, while B has become ∼0.04 mag less bright.
Assuming the contribution of the galaxy to the B aperture in January
2000 is still 18% in the results supplied by Schild, we subtracted the galaxy
contribution and note that, for January 2001, mB, corrected − mA = 0.05. In
other words, the A and B components are closer in magnitude than in 2001,
but A is still brighter than B. Both we and Ovaldsen et al. found B to be
brighter than A in January 2000, see Section 6.4.1.
We also note from Figure 6.13 that there is no evidence of the systematic
decrease from night to night comparable to that observed by us, see Section
5.2, and to a somewhat lesser extent by Ovaldsen et al., see Section 6.4.1.
Although there is a difference in mean magnitude of ∼0.04 mag between the
first and last nights in both components, in Figure 6.13, the mean of each
night shows a decrease in brightness between the first and second nights,
6.4 Comparisons with other results
105
then an increase from the second to the third night, followed by another
decrease from the third to the fourth night. This pattern is discernible for
both components, see Tables 6.3 and 6.4.
6.4.3
Summary
In general, our results support several of the findings of Ovaldsen et al.
(2003b), as well as those of Colley et al. (2003) after subtracting the galaxy
contamination.
As regards the decrease in brightness from night to night, our results
agree with the findings of Ovaldsen et al. (2003b). It also supports these
authors’ values for the brightness ratio of the quasar images for the March
2001 data: A is stronger than B with approximately 0.1 mag in both R and
V . For the R band, this is also the result we found in the QuOC data after
subtracting the galaxy influence in the B aperture. Furthermore, our analysis
also supports the shift in relative brightness reported by Ovaldsen et al.
The night–to–night decrease in magnitudes seen in the observations from
March 2001 are puzzling, because the decrease is not reflected in the variations internal to each night and because the decrease is so similar in both
components (although we did find a decrease that was 0.003 mag larger in
the B component). We have shown that moonlight may be causing the ‘U’
shape in each night. There is thus a possibility that the night to night decrease is evidence of intrinsic behaviour, and the fact that we do not see
this decrease during the night could be due to some observational conditions. For example, moonlight and changing atmospheric conditions could
prevent us from observing it. However, the fact that the decrease from night
to night is so similar in both components, suggests the possibility that it is
an observational effect.
It remains a puzzle why no such systematic decrease in brightness from
night to night is observed in the results from Colley et al. (2003). The trend
of increase/decrease in brightness from one night to next are the same in
both components for Colley et al., which could indicate that the night–to–
night changes are manifestations of the correlation at zero lag. However, we
note that our results and those of Ovaldsen et al. both show the systematic
decrease from night to night, and that our decreases are larger in magnitude
and more significant compared to the error bars. We have also, in Section
5.2.2, found the same trend for the reference stars P and Q, while stars R
and H did not show this trend.
As regards the time shifted colour indices, our results show that component B is redder than A, while Ovaldsen et al. have the opposite result. This
is probably due to differences the galaxy corrections.
106
Discussion
Chapter 7
Reduction and photometry of
the StanCam data
The most exciting phrase to hear in science, the one that heralds
new discoveries, is not ‘Eureka!’ (I found it!) but ‘That’s funny ...’
Isaac Asimov
The Gravitational Lenses International Time Project (GLITP), is a cooperative venture between lensing communities in several European countries as
well as the Harvard–Smithsonian Center for Astrophysics, the Anglo Australian Observatory and the Canada–France–Hawaii Telescope Corporation.
Members monitor and analyse gravitationally lensed objects. In the field of
optical monitoring, the group has focused particularly on the systems QSO
0957+561 and QSO 2237+0305, the so-called Einstein Cross. Ullán et al.
(2003) discuss the daily, weekly and monthly variability of QSO 0957 based
on GLITP observations from the instrument StanCam at the NOT from
February 3rd to March 31st 2000, in the R and V bands. On request, Luis
Goicoechea and Aurora Ullàn Nieto kindly provided us with this data set.
In this chapter, we present our reduction of their R images, as an example of using the MCS procedure on a very different data set than our main
observations.
As we reduced the StanCam data in a similar way as that described previously, we will here mainly mention the differences between the two reduction
processes.
7.1
Pre-processing
The StanCam CCD is 1024 × 1024 pixels, with a gain of 1.68 e− /ADU and
a read-out noise of 6.5 e− /pixel. An overscan section of 50 columns is added
to each side of the frames.
108
Reduction and photometry of the StanCam data
There were a total of 75 R images in the data set, but we discarded one
image due to a bad line of ∼80% lower pixel value crossing one of the quasar
components. As a rule, there were two R images from each observing night,
although this number varied between one and four. The images had been
obtained approximately every second or third night through February and
March 2000. Exposure times were 150 seconds for most of the images. A
total of 17 images had exposure times of 60, 100, 180 or 300 seconds. Seeing
conditions varied from approximately 0.500 to 2.800 , measured by the FWHM
of point sources.
The field did not show all the stars present in our ALFOSC images, see
Figure 2.1. However, most of the StanCam images included stars H, E, D, P
and R.
As explained in Chapter 3 and Appendix A, we have used IRAF for the
flat-fielding of the object frames.
Most of the bias frames had a lot of structure. Many of them were filled
with a wide variety of ‘stripes’ of different widths, with somewhat different
ADU counts from the rest of the frames. As there was an overscan section
on all frames, we decided in this case to discard the bias frames and subtract
only the overscan level. Using the interactive overscan fitting in IRAF on a
selection of the images, we determined an appropriate overscan section that
did not exhibit any gradient.
There were sky flats on 2 out of 39 nights, so we were forced to use
mainly dome flats. The doughnuts on the dome flats were uneven, as we also
found for the ALFOSC flat frames in Chapter 3. The rotator angle had been
shifted 90 degrees at irregular intervals when obtaining the dome flats, which
resulted in a rotation of the irregular doughnuts on the frames. Assuming
that this rotation was done in order to correct for the uneven doughnuts,
we tried combining dome flats with different rotation angles. This resulted
in more even doughnuts, so we decided to combine dome flats with different
rotation angles. When executing the task flatcombine in IRAF, we used the
average combination choice with rejection method avsigclip, see Appendix
A.
Whenever possible, we constructed one master flat for each filter each
night. However, not all nights contained flats, so we were sometimes forced
to use dome flats from the night immediately before or after.
Examining all flat frames carefully, we noted spurious artifacts occurring
at irregular intervals: bright or dark columns, or even broad, black stripes
down the middle of the frames. We discarded all flats containing these artifacts.
After flat-fielding the images, we subtracted the sky background in each
image, taking the median of each image as a measure of the background level.
7.2 Running MCS
7.2
109
Running MCS
Using the same techniques as explained in Chapter 3 and Appendix A, we
aligned all images to the same coordinates as one reference image. This was
done in order to run the simultaneous deconvolution.
The PSF star and the normalisation star were chosen to be E and D,
respectively, the same as in our previous analysis. The residuals from the
Moffat fit were similar to those from the ALFOSC data, i.e. quite extensive
residuals, so we again utilised a wide mask of ∼60 subsampled pixels radius.
When running the program that constructs the analytical Moffat adjustment to the PSF star, we again experienced that the program was sensitive
to the β value in the parameter input text file psfmof.txt, as mentioned in
Section 4.6.2. However, it was not easy to distinguish a direct relationship
between the input and the final β value that would make the program run
smoothly. For example, the program would crash when given a starting β
value of 3, but would run if β was changed to 1.3, producing a final β value
of 2.7.
We did not experience any problems with modelling the residuals from
the Moffat fit, as mentioned for the ALFOSC images in Section 4.6.2.
7.3
Results
Light curves for the quasar components A and B, with error bars, are shown
in Figure 7.1. As in our previous work, we used the normalisation star D for
calibrating the magnitudes.
The error bars shown in the figure were computed as described in Section
5.2.1. The star R was present on only 37 out of 74 frames. When computing
the error bars for the frames where R was missing, we ‘simulated’ star R
using the quasar component A. Taking the average of the fluxes of R on
the frames where the star was present, we found that star R was 1.28 times
brighter than component A. Star R was then simulated on the frames where
it was missing, by multiplying the flux of A by 1.28.
Two points were discarded when computing the standard deviation of
R for the error estimations. The flux of star R on these images varied by
more than 25% from the flux on the other images. As the star is assumed
constant, we did not consider these two frames to show a correct value for
the star. When calculating error bars for the quasar components on these
same frames, the standard deviation of all R fluxes was used, producing very
large error bars for the two frames. In fact, the error bars were too large to
be included in Figure 7.1. The two points, one from February 4th and the
110
Reduction and photometry of the StanCam data
other from March 2nd, are plotted without error bars in the figure.
The standard deviation of star R, with the two deviant points removed
and the simulated stars included, was 25% larger for the StanCam data than
for the ALFOSC data. The stacked ALFOSC data were sums of 10 individual
images, each with exposure time ∼60 seconds, so they had longer ‘combined’
exposure times than the StanCam images. This has probably resulted in
the rather sizeable error bars for the A and B components for the StanCam
results, typically ∼0.03 mag, approximately 10 times larger in mag than the
error bars for the quasar components in Chapter 5.
Noting that the start of this observing run is only a few days after our
one night of ALFOSC observations in January 2000, we compare the magnitudes of A and B obtained here with our previous results. Our mean R
values for January 25th were 16.632 for A and 16.555 for B, see Table 5.10.
Corresponding mean A and B magnitudes from Figure 7.1, are 16.582±0.035
mag and 16.515 ± 0.025 mag, where the errors are the standard deviation of
the magnitudes of all points. Our result from January 25th are in reasonable
good agreement with these values, although not within the errors expressed
as the standard deviation of the mean. Our A and B magnitudes from January are ∼0.05 mag dimmer than the mean of the StanCam data points in
Figure 7.1. They also agree that B is brighter than A by approximately
∼0.07 mag.
Ullán et al. (2003) obtain their final light curves using the psfphot technique (e.g. McLeod et al., 1998). The method uses PSF fitting to extract the
fluxes, and assumes a de Vaucouleurs profile,1 convolved with a PSF, as the
lens galaxy model.
Our light curves look qualitatively similar to the results of Ullán et al.
Over the entire observation period, component A seems to increase somewhat
in brightness, by ∼0.04 mag, while there is a possible decrease in component
B of ∼0.02 mag. Ullán et al. concluded they did not find evidence of any
daily variability of the quasar. The daily variations were very similar in both
quasar components, and they conclude that these correlated variations are
due to some ‘observational noise’. From Figure 7.1, we see that our results
also show evidence of zero lag correlation between the two components, especially in the first part of the period. However, as Ullán et al. also find,
there seems to be uncorrelated gradients in both A and B with respect to
time scales of several weeks.
As Ullán et al. performed a straight line fit to their data points, we have
also attempted this, in order to compare the two results. We first note a few
1
The de Vaucouleurs profile is an empirical luminosity profile of elliptical galaxies, given
by I(r) ∝ Ie exp[−(r/re )], where Ie is the intensity at the effective radius re .
7.3 Results
111
differences between the two data sets. Ullán et al. combined images for each
night, while we have included all individual points. They also rejected points
from 10 nights, based on criteria inherent in their photometry method. The
χ2 residuals from the deconvolution did not provide us with reasons to discard
any images, so we included them all in Figure 7.1. On request, Aurora Ullàn
provided us with a list of the images they had discarded, which we compared
to our own results. The points of conspicuously large magnitude in Figure
7.1 from February 7th were on the list of discarded images. These points
differ noticeably from all other points, for both components, indicating some
observational cause for the deviation. As we perform the straight line fit
solely in order to compare our results with those of Ullán et al., we found it
reasonable to reject these two points when fitting the line.
The straight line fits are shown in Figure 7.2. The gradient for component
A is –0.63 mmag/day, and for component B +0.57 mmag/day. Gradients
found by Ullán et al. are –0.8 mmag/day for A and +0.3 mmag/day for B.
Our results show similar gradients, although a slightly smaller absolute value
for the A component and a relatively larger value for B. Discrepancies in our
results probably originate from the different photometry methods, as well as
the difference in data sets.
In conclusion of this chapter, we note that the components seem to be
moving closer in terms of brightness values. The gradient in component A
is towards brighter values, while B seems to become weaker. If this trend
continued, it could point to the shift that, according to our results in Chapter
5, has occurred in March 2001, when component A is brighter than B. It
would indeed be interesting to analyse more data from the year 2000 and
the beginning of 2001, to see if this shift in the brightness ratio constitutes
a continuous trend, or an abrupt change.
Preliminary results from aperture and PSF photometry using data obtained at the Spanish Calar Alto 1.5 meter telescope between February and
June 2003, show that B is again brighter than A in both R and V . Results
for the R band is ∼16.58 mag for the A component and ∼16.47 mag for B. In
the V band the results are ∼16.95 mag for A and ∼16.70 mag for B. These
results are obtained from Jan-Erik Ovaldsen and Aurora Ullán Nieto (private
communication), using the photometry methods of Ovaldsen et al. (2003b).
112
Reduction and photometry of the StanCam data
Figure 7.1: Light curves in the R band for the quasar components A and B from
the StanCam data. As in Chapter 5, the A component is represented by a cross
(×) and B by a diamond ().
7.3 Results
113
Figure 7.2: Straight line fits to the data points in the R band for the quasar
components A and B from the StanCam data. Two points seen in Figure 7.1 are
rejected in the line fits, see text for details. The gradients for the two lines are
–0.63 mmag/day for component A, and +0.57 mmag/day for component B.
114
Reduction and photometry of the StanCam data
Chapter 8
Conclusions
I could be bounded in a nutshell
and count myself a king of infinite space
were it not that I have bad dreams.
William Shakespeare, Hamlet, Prince of Denmark, Act II, Scene II
In this thesis, we have presented photometric results from three data sets,
all obtained at the Nordic Optical Telescope. Two of the data sets contained
observations from January–March 2000, and one set was from four nights in
March 2001.
Optical monitoring of QSO 0957+561 has proved a fascinating study in
itself, especially when comparing our results to that of others who have used
the same data but different methods of analysis. Applications of time delay measurements and studies of microlensing are diverse and important, but
they require firm observational ground. This thesis work has been a contribution to that platform from which one can probe the distribution of matter in
lensing galaxies, constrain the different regions of quasars and even calculate
the Hubble parameter.
The main part of this work has been a detailed analysis of the light
curves of QSO 0957+561 during a relatively short time period, four nights
in March 2001. Comparison with the one night of observations in January
2000 provided an opportunity to calculate the time shifted colour indices and
magnification ratio of the quasar components, which we compared to results
in Ovaldsen et al. (2003b).
Analysis of the StanCam data provided us with an opportunity to study
variations of the quasar on a time scale of two months.
For our main data set, from March 2001, we studied the light curves
in detail. Our investigations of the influence of the Moon in Section 6.1.3
showed that moonlight is a possible explanation for the increase in brightness
116
Conclusions
seen in both components towards the end of each night. We have no simple
explanation for why moonlight should have this effect. Chromatic effects
seem to be important, as the star R is not influenced in the same degree as
the quasar components. The effect may be caused by the fact that the quasar
is bluer than star R. It may also be an effect arising from our normalisation
of the quasar with star D, which is also more red than the quasar.
If the moonlight increases the brightness of the quasar components towards the end of each night, it could explain the slight increase in the last
part of each night, which starts approximately at the time of moon-rise. Referring to the model light curve in Figure 6.9, we note that the last points on,
e.g., the first night, that have low brightness (large magnitude value) might
be caused by inadequacies in the model. However, disregarding some of these
points, it is tempting to suggest that the sinking tendency throughout the
four nights could be reflected in the observations within each night. Other,
unknown observational effects might also be influencing the data, which further studies will hopefully shed some light on. This could mean that the
trend seen from night to night is evidence of intrinsic variation in the quasar,
and that both quasar components happen to decrease during this period.
The aber of interpreting the light curves as sinking steadily, is of course
the fact that the tendencies are so similar in the theoretically unrelated A
and B components. Although we acknowledge that this might be interpreted
as too much of a coincidence, we also remark that it is not impossible. After
all, both components must exhibit some behaviour at all times, e.g. a nightto-night decrease in brightness, and the observed decreasing tendency is a
bit larger (0.003 mag) in the B component than in A.
On the other hand, we note that two of the field stars, P and Q, exhibited
a very similar decrease in brightness from night to night as that of the quasar
components. This could be a further indication that the decrease has origin
in something other than the quasar itself. What could cause such an effect
in both the quasar components and P and Q is uncertain. The two stars
showing the same effect as the (bluer) quasar components, are the reddest
stars we have studied. The stars H and R, that are closer in colour to the
quasar components, do not show this systematic decrease. This indicates
that the cause of the decrease, if based on colour differences, is complex. It
could also indicate that the explanation has to be found elsewhere.
Apart from the possible decrease in brightness for both components, we
have not found any evidence of intrinsic night-to-night variability in the ALFOSC data. For the StanCam data there is a possible gradient on the time
scale of several weeks, towards increasing brightness for component A and
decreasing for B.
The most conspicuous brightness variation seen in our results, however, is
8.1 Future studies
117
the change from 2000 to 2001. Within this period, the two components have
shifted in relative brightness, from B brighter than A in 2000 to A brighter
than B in 2001.
8.1
Future studies
First of all, it would be intriguing to study further the colour variations of
the quasar components and field stars. Both the variations we have studied,
the ‘U’ shape within each night and the decreasing brightness from night to
night, could be further analysed with more information on colour variations.
For example, it could be interesting to study the spectra of the quasar components. As this thesis work has shown that stars of different colour seem
to behave differently, possibly because of reacting differently to changes in
observing conditions such as moonlight, it would be informative to obtain
specific information about variations in the continuum and line emitting regions of the quasar, how the intensities in these regions change in time and,
possibly, how they respond to changes in observing conditions.
The observations we used for the main part of this thesis work, i.e. the
ALFOSC data, were obtained from directions from the QuOC team, see
Section 2.1, designed for their time delay analysis. We find that the studies we
ended up performing could have had a more optimally designed observation
schedule. As the observations were done on an hourly basis (60 minutes R,
15 minutes V etc.), there was a small and irregular number of V images. One
15 minute session with V could yield 9 frames while most gave us 3–6, when
images with bad columns had been subtracted from the data set, see Section
3.4.3. When summing 10 and 10 images, we thus ended up combining V
frames separated by time intervals of ∼1 hour. As we had few data points in
V , and they were separated by large intervals, we could not study detailed
variations in V − R within each night.
If planning the observations for this thesis work from scratch, it would be
better to observe, e.g. 10 R frames followed by 10 V frames etc. in cycles. This
procedure could in principle have provided us with a better time resolution
for studying colour variations, especially if we consistently avoided the bad
column so we did not have to reject so many images. Also, on the technical
side, we would have stepped all the sky flats to remove stars more easily, and
we would have checked the dome flats in the beginning of the observation
run and if they seemed bad, we would take only sky flats.
As regards the brightness variations seen in the course of one year, more
observations from April 2000 and onwards, if available, could reveal how the
shift in relative brightness between the two components occured, whether a
118
Conclusions
slow trend – perhaps starting with the trend we see in the StanCam data
– or a sudden change. The preliminary results we mentioned in Section 7.3
indicate that the relative brightness of the components has shifted again,
which would also be interesting to study further.
As a last remark, we note that if we had obtained more data from the
January 2000 session, we could possibly have used our data sets to make a
time delay analysis, as data from the two observing sessions were separated
by ∼416 days. Since different groups report time delays ranging from ∼417
to ∼425 days, see Section 2.3, observing sessions lasting for several days,
seperated by an approximate time delay, might be necessary to resolve the
issue.
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Appendix A
The Image Reduction and
Analysis Facility (IRAF)
IRAF is a general purpose software system for the reduction and analysis of
astronomical data, developed by the National Optical Astronomy Observatories (NOAO) in Tucson, Arizona. The following is not in any way meant
to be an exhaustive discussion of the reduction possibilities in IRAF. It is
a description of the tasks we have used in the course of this work, in more
detail than included in previous chapters.
A.1
Using IRAF
IRAF tasks are commands that perform certain operations. The parameters
for the tasks determine how these operations will be done. There are two
types of IRAF parameters: required (‘query’) and hidden parameters, see
figure A.1. Practically speaking, using IRAF consists of deciding upon and
filling in different parameters for the tasks you want to execute. The epar1
[taskname] command opens the editing environment. There are certain default values for each task. In the following, we will primarily discuss the
parameters we used when they differ from the default values.
An IRAF task does not have to be repeated manually for each image.
One way of repeating the same task for several images, is to provide a list
of images as the input parameter. This enables IRAF to perform the same
task on all images in the list. A list is indicated by the prescript @. We
made use of lists for the tasks zerocombine, ccdproc and flatcombine.
Another method of dealing with many images, is to make a small script,
which performs the same task for many images while allowing the user to
1
‘epar’ = ‘edit parameters’.
124
The Image Reduction and Analysis Facility (IRAF)
Figure A.1: Example of environment for editing parameters in IRAF, for the
task imarith. The first four parameters are query parameters: They must be
specified in order to execute the task. The parameters in parentheses are the
hidden parameters: If they are not specified, the current default values will be
used.
vary parameters if necessary. We made use of scripts when running the tasks
geomap, gregister and imcombine.
A.2
A.2.1
Pre-processing with IRAF
Zerocombine
As described in Section 3.2, we need to combine our bias frames to avoid
random fluctuations and ‘cosmics’ present in the individual frames. For this
purpose we used the IRAF task zerocombine (the image type code for bias
frames in IRAF is ‘zero’). For this task, the rejection type minmax was used
to reject any pixels with deviating value, before combinaion. This method
rejects the n low and n high pixels, where n is specified by the user. We used
the default value of nhigh=1, nlow=0. Thus, the bias frames for each night
were averaged, but with the highest value being ignored when forming the
average for each given pixel.
A.2.2
Ccdproc and Flatcombine
The task ccdproc processes CCD images to correct and calibrate for detector
defects, bias level, dark counts and fringing. We have used the task for two
purposes: preparing the flat frames for the task flatcombine and the flat
field calibration of the object frames themselves.
In the editing environment of ccdproc, we enter the ‘trim section’, which
is the part of the image that will be saved, and the ‘bias section’, which is the
A.2 Pre-processing with IRAF
125
section of the raw image that will be used to fit the bias level. For the March
2001 data, we used a trim section of [50:2000, 5:1195] and a bias section of
[2053:2088, 5:1195].
The task flatcombine performs the combination of a list of individual
flat frames. We used the combine option average when combining most of
the flats, but median on the night of March 17th–18th. On this night, we
combined sky flats that had been shifted in x and y, and used the median
option to remove the effects of a few stars appearing on the flat frames.
The reject parameter selects a type of rejection operation to be applied
to the pixels when combining the flats. A rejection algorithm is useful when
forming the average or median of n pixels, as one or several of the pixels
may be affected by a cosmic ray, or for some other reason have an unrealistic deviant value. We compared two rejection options for flatcombine:
avsigclip and crreject.
Both rejection methods assume that the image values are proportional to
the number of photons recorded. Pixels above or below the median or average
by N σ are rejected, where N has a default value of 3 but can be modified
by the user. We chose the median here, to avoid that extreme-valued pixels
should dominate the calculation. Values deviating by more than the specified
σ threshold factors are rejected. These steps are repeated, until no further
pixels are rejected or there are fewer than three pixels.
The avsigclip method determines σ empirically, by making an estimate
of the expected σ value based upon the data itself. The crreject method on
the other hand, determines σ from the CCD noise characteristics (i.e. readout noise and gain) that we provide a priori.
For every night, we made one image with each reject parameter, and
subtracted these images from each other. The residuals we obtained were
random fluctuations above and below zero. Since no distinguishable pattern
was apparent, we concluded that the routines were more or less equivalent
for our data.
We used avsigclip when combining most of our flats. On the one occasion when we had only 3 flats (night of 14th–15th in the V filter), we used
crreject as this option uses the noise characteristics we provide instead of
calculating them from the image itself. Assuming that this is the best option
when few images are available, we used it on the one occasion when we had
only 3 individual flats.
126
A.3
The Image Reduction and Analysis Facility (IRAF)
Other tasks using IRAF
Apart from the flat-fielding, there were several other operations that we
performed on the images using different tasks in IRAF. One of them was
‘stacking’ the images to obtain a better S/N ratio. In addition, we used
IRAF to perform arithmetical operations on the images like subtracting the
background, normalisation, multiplication etc.
A.3.1
Image scaling and background subtraction
To perform arithmetical operations on the images, we used the task imarith,
see Figure A.1. We made a short IDL routine that read exposure times from
the image headers to a text file, and we then made lists of the image names
and the exposure times, so we could use imarith on all images (of each night
and each filter) simultaneously.
The background subtraction, described in Section 3.4.2, was also done
using imarith, and by making lists of the image names and of the background
level in each image. In this way the background subtraction could be done
efficiently on all ∼1000 images.
A.3.2
Stacking the images
As described in Section 3.4.2, we wanted to sum the images in batches of 10.
The way we proceeded to do this was to first form an average frame from 10
images, and then multiply this image by 10. This provided us with the S/N
ratio of a much longer exposure time than that of our individual frames.
Aligning the images using geomap and gregister
There are many ways of aligning, or registering, images in IRAF, ranging from
simple x and y transformations to more complex coordinate transformations.
We have used the tasks geomap and gregister for both our alignments.
First, we aligned images using simple shifts in x and y, to construct our
‘combined’ image frame from 10 individual frames. Second, we aligned all
combined images, in preparation for the simultaneous deconvolution, using a
parameter that computed the rotation as well as shifts in x and y. Aligning
the images twice like this is not ideal, but circumstances forced us to do this.
The task geomap computes the spatial transformations required to map
input coordinates to a reference coordinate system. The format of the input
file to geomap includes one object per line, with the x and y coordinates of
the reference image followed by the x and y coordinates of the image to be
A.3 Other tasks using IRAF
127
registered. These input files were supplied to us by Jan-Erik Ovaldsen, as
described in Section 3.4.2.
The transformations have the form
xin = f (xref , yref )
yin = g(xref , yref ) ,
and are stored in a text database file. The functions f and g are by default
a power series polynomial surface of order 2. For our simple case of a shift
in x and y (fitgeom = shift, see below), this becomes
xin = a + xref
yin = b + yref ,
where a and b give the shift in the x and y direction respectively.
Geomap may be run interactively by switching on the interactive
parameter. In interactive mode, we have the option of changing the fit parameters and displaying the data graphically, using keystroke commands,
until a satisfactory fit has been achieved. We checked manually each shift
and rejected stars that had large residual values, that is, differences between
the reference and the transformed coordinates. As remarked in Section 3.4.2,
this resulted in satisfactory residuals.
We performed a few tests to see how the peak value and the flux of
stars changed after the transformation, and found that the peak value could
change by ∼5% while the total flux only changed by ∼0.3%. The peak value
changes more than the flux, as the values of the pixels are distributed a bit
differently after the transformation.
For the second alignment, we used rxyscale as the fitgeom parameter
(‘fitting geometry’), thus taking any rotation of the images into account when
aligning images. This second alignment was done because the images had
to be aligned so that the centre of the point sources fell within the same
pixel on each frame, for performing the simultaneous deconvolution. We
chose 6 unsaturated stars for the transformation, choosing stars that were
well ‘spread out’, i.e. located at different positions in x and y coordinates on
the frame.
On the first night, we noticed a sudden rotation from R image number 195
to 196. The rest of the images this night, about 20, were also rotated in the
same way with respect to all previous. We avoided this problem in the first
alignment by only combining images not rotated in relation to each other.
However, this (unexplained) rotation needed to be taken into account when
aligning images for the simultaneous deconvolution, if we were to utilise the
128
The Image Reduction and Analysis Facility (IRAF)
last two stacked images. This was achieved by using the rxyscale option for
fitgeom. In retrospect, we realise we should have first aligned all images to
the same coordinates, using rxyscale, and then combined 10 and 10 images.
In this way, we would only have needed to align the images once. However,
tests showed that even with the rxyscale option, the automatic alignment
produced relatively large residuals, and we did not have time to perform the
interactive alignment on all ∼1000 individual images over again.
The output file created by geomap, containing the transformation parameters, is used as input to the task gregister, which performs the actual
transformation. The transformation is stored as the coefficients of a polynomial surface. The transformed coordinates at intermediate pixel values are
determined by bilinear interpolation.
The parameter geometry determines the type of geometry to be applied
by geomap. We found that with the default option geometric, which computes the ‘full transformation’, the task had a tendency to ‘overfit’, i.e. introduce unnecessary effects, so we changed this to the option linear which
computes shifts and rotations, which sufficed for our purposes.
We ran geomap and gregister using simple IRAF scripts. A script is a
text file containing a sequence of IRAF commands, and are well suited for
performing the same task on a large number of images. The script was edited
in emacs, and executed by typing ‘cl < (script name)’.
Combining the images using imcombine
After aligning the images to their reference coordinates, we used the task
imcombine to construct an average image frame from the 10 individual
images.
As regards the reject parameter, the same options as for flatcombine
apply. This time, however, we could not use avsigclip or crreject as these
methods assume a linear relation between ADU and photons, and we had
already subtracted the background from the image frames before forming the
average, see Section 3.4.2. When we subtracted the background we violated
the linearity criterium. Instead, we used the reject option sigclip. This
method computes the median (or average) at each output pixel, excluding
the highest and lowest value, and then computes the σ about this estimate.
For this method to work well, it is recommended that at least 10 images are
used, which is just what we were using. In the last sets for each night and
filter, we therefore distributed the images evenly in the two last sets. That
is, 10 + 4 number of images in the last two sets was changed to 7 + 7.
We again used a simple IRAF script to combine the images efficiently.
The input images to be combined were spectified by a list. We made one list
A.3 Other tasks using IRAF
129
for each set of 10 images to be combined.
After combining, we used imarith to multiply all the (combined) images
by 10.
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The Image Reduction and Analysis Facility (IRAF)
Appendix B
Text files associated with the
MCS programs
To illustrate the discussion of the MCS deconvolution programs in Chapter
4, we include examples of the input text files that must be filled in by the
user of the programs. To illustrate the format, we also include an example
of one of the the output text files.
Figure B.1: extract.txt: The input text file for extract.f
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Text files associated with the MCS programs
Figure B.1 displays the input text file for the MCS program that extracts
sub-images of optional size (la taille des images a extraire) from the image.
See Section 4.5.1 for a description of the program.
Figure B.2: psfmof.txt: The input text file for psfm prive.f and psf prive.f
Figure B.2 displays the input text file for the two MCS programs that
are used for constructing the PSF, see Section 4.5.2.
Figures B.3 and B.4 show the two input text files for the program that
performs the final deconvolution, see Section 4.5.3. Figure B.3 shows the text
file for the photometry parameters and Figure B.4 the input deconvolution
parameters.
Figure B.5 shows an example of an output text file, with information
on the positions and peak intensities, in units specific to the normalisations
applied and to this particular image, as well as information on the χ2 reached
and specifications on the minimising routines. The output text file in2.txt
contains the same information on the intensities and positions reached, but
in the format of Figure B.3.
133
Figure B.3: in.txt: Input text file for dec prive.f
134
Text files associated with the MCS programs
Figure B.4: deconv.txt: Input text file for dec prive.f
135
Figure B.5: out.txt: An example of an output text file from dec prive.f
136
Text files associated with the MCS programs
Appendix C
Tables of output files from the
MCS programs
We include an overview of the output files from the programs. Files to the
right of the double line in Tables C.2 and C.3 were used in earlier versions
of the program and are still given as output but not used by us. The left
column shows the name of the FORTRAN 77 program in bold text.
The first program extracts images of the objects of interest: the object to
be deconvolved and the stars that are to be used for constructing the PSF.
Table C.1: The program extract.f and its output files. The extracted images
are located in the second column, and their associated noise maps in the third
column, see Section 4.5.1
extract.f
g.fits
psfnn.fits
sig.fits
psfsignn.fits
The next two programs aim at constructing the PSF, s, with an improved,
but not infinite, resolution.
Table C.2: The program psfm prive.f and its output files: The residuals
difcnn.fits are the residuals remaining after subtracting the moffat mofc.fits from
the stars psfnn.fits, see Table C.1 and Section 4.5.2.
psfm prive.f
difcnn.fits
mofc.fits
sigcnn.fits
138
Tables of output files from the MCS programs
Table C.3: The program psf prive.f and its output files. The constructed PSF
is s.fits. The other files in the second and third column are the modeled residuals,
see Section 4.5.2 for details.
psf prive.f
psfr.fits
psff.fits
psffond.fits
s.fits
xixinn.fits
t.fits
The last program performs the actual deconvolution with the PSF that
was constructed in the preceeding two programs.
Table C.4: The program dec prive.f and its output files. The deconvolved
images are decnn.fits. The intensity and position of the point sources are listed
in the text file out.txt. The residual frames are resinn.fits and resi smnn.fits. The
background of the deconvolved image is returned as back.fits. See Section 4.5.3
for details.
dec prive.f
decnn.fits
resinn.fits.fits
back.fits
resi smnn.fits
out.txt
in2.txt
Appendix D
Miscellaneous
D.1
Integrating the Gaussian function
Point sources in the deconvolved images from the MCS procedure are represented as Gaussian distributions. In this appendix, we show how we can
obtain Eq. D.3, the relationship between flux and peak intensity of a Gaussian that we used in Section 4.5.3.
In two dimensions, the Gaussian function is given by
I(x, y) = Ipeak e−[(x−µx )
2 +(y−µ
y)
2 ]/2σ 2
.
(D.1)
The centre of the function is (µx , µy ) and the standard deviation is σ. We
have assumed that the function is circular, i.e. σ = σx = σy . The intensity of
the distribution is I. When the function is centred in origo, i.e. µx = µy = 0,
we easily see that the peak value of I is given by
x = y = 0 ⇒ I = Ipeak .
The flux is obtained by intergrating Equation D.1. We can therefore write
Z ∞Z ∞
2
2
2
F = Ipeak
e−(x +y )/2σ dx dy ,
(D.2)
−∞
−∞
where F is the flux of the star. Considering the double integral, we see
that it can be written as the multiplication of two terms, since x and y are
uncorrelated. That is, we can write
Z ∞
Z ∞
Z ∞Z ∞
2
y2
− x2
−(x2 +y 2 )/2σ 2
e− 2σ2 dy ,
dx dy =
e 2σ dx
e
−∞
−∞
−∞
−∞
Equation D.2 can thus be written
F = Ipeak
Z
∞
e
−∞
−
x2
2σ 2
dx
2
.
(D.3)
140
Miscellaneous
From Rottmann (1995), this integral is given by
Z ∞
√
x2
e− 2σ2 dx = 2π σ .
−∞
This brings us from Eq. D.3 to the following relation between flux and peak
intensity:
F = 2πσ 2 Ipeak .
To obtain Eq. 4.10, the relation between the FWHM and standard deviation, also used in Section 4.5.3, we use the one-dimensional Gaussian. Setting
Ipeak = 1, we solve
−(x0 −µ)2
1
e 2σ2 = f (xmax ) ,
2
where x0 is the x position at FWHM. We have that f (xmax ) = 1 at x0 = µ.
From this, we get
−(x0 −µ)2
1
1
e 2σ2 = f (µ) = .
2
2
Taking the natural logarithm on both sides of the equation we obtain
and thus
1
(x0 − µ)2
= ln
= −ln2 ,
−
2σ 2
2
√
x0 = ±σ 2ln2 + µ .
The full width at half maximum is therefore given by
√
FWHM = x+ − x− = 2σ 2ln2 ≈ 2.3548σ.