Extrasolar Moons as Gravitational Microlenses Christine Liebig

Extrasolar Moons
as
Gravitational Microlenses
Christine Liebig

Faculty of Physics and Astronomy
University of Heidelberg
Diploma Thesis in Physics
submitted
13 February 2009
by
Christine Liebig
born in Schwerte

Extrasolar Moons
as Gravitational Microlenses
This thesis has been carried out
at the
Astronomisches Rechen-Institut
Zentrum für Astronomie der Universität Heidelberg
under the supervision of
Prof. Dr. Joachim Wambsganß

Abstract
Extrasolare Monde als Mikrogravitationslinsen: Durch den Mikrolinseneffekt
wurden bereits acht Exoplaneten detektiert. Die Methode ist prinzipiell in der Lage,
Körper mit einer Erdmasse oder sogar geringerer Masse zu entdecken. Daher erscheint
es naheliegend, die möglichen Gravitationslinseneffekte extrasolarer Monde
auf beobachtbare Lichtkurven zu untersuchen. Die einfachste Gravitationslinsenkonstellation
stellt ein extrasolares System mit Mond als Dreikörperlinsensystem
dar, bestehend aus den drei Massen Stern, Planet und Mond. Da der Mond sich
im Orbit um den Planeten befindet, wird der Abstand zwischen Mond und Planet
stets klein gegenüber dem Abstand zwischen Stern und Planet sein. Diese Vorgabe
kann zu komplexen Interferenzen der Planetenkaustik mit der Mondkaustik führen.
Wir quantifizieren Detektierbarkeit und Detektionsgrenzen, indem wir Lichtkurven
der Dreikörperlinse mit angepassten Binärlichtkurven, resultierend aus dem System
Stern und Planet ohne Mond, vergleichen. Durch Anwendung von inversem Ray-
Shooting simulieren wir Verstärkungskarten mit realistischen Massen- und Winkelabstandsgrößen.
Von diesen Karten wird jeweils eine große Anzahl von Lichtkurven
extrahiert und für jede Lichtkurve die Unterscheidbarkeit zum Binärfall analysiert.
Wir benutzen ein Chi-Quadrat-Kriterium um die Detektierbarkeit des Mondes in
einigen ausgewählten Dreifachlinsenszenarios zu bestimmen.
Extrasolar Moons as Gravitational Microlenses: Up to now eight exoplanets
have been detected with microlensing, and the method is sensitive to masses as low
as an Earth mass or even a fraction of it. Hence it seems natural to theoretically
investigate the microlensing effects of moons around extrasolar planets. From a microlensing
point of view, an extrasolar system consisting of a star, a planet and a
moon can be modelled as a triple-lens system with mass ratios very different from
unity. As the moon orbits the planet, the planet-moon separation will be small
compared to the distance between planet and star. Such a configuration can lead to
a complex interference of caustics. We quantify detectability and detection limits by
comparing triple-lens light curves to best-fit binary light curves as caused by host
star and planet only – without moon. We simulate magnification patterns of realistic
mass and separation parameters using the inverse ray-shooting technique. These
patterns are processed by analysing a large number of light curves and fitting a
binary case to each of them. A chi-squared criterion is used to indicate detectability
of the moon in a number of selected triple-lens scenarios.

Never know anything about it. Waste of
time. Gasballs spinning about, crossing
each other, passing. Same old dingdong
always. Gas: then solid: then
world: then cold: then dead shell
drifting around, frozen rock, like that
pineapple rock. The moon. Must be a
new moon out, she said. I believe there is.
— James Joyce, Ulysses

Contents
1 Introduction 3
2 Gravitational Lensing 7
2.1 Historical Development . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 Triple-Lens Equation . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Search for Planets via Galactic Microlensing . . . . . . . . . . . . . . 13
3 Method 15
3.1 Magnification Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.1 The microlens code . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.2 The moonlens code . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Light Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2.1 Light curve extraction . . . . . . . . . . . . . . . . . . . . . . 18
3.2.2 Light curve fitting . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3.1 Comparing data and a theoretical model . . . . . . . . . . . . 24
3.3.2 Comparing two theoretical models . . . . . . . . . . . . . . . . 25
4 Choice of Scenarios 31
4.1 Parameters Relevant for Magnification Patterns . . . . . . . . . . . . 31
4.1.1 Mass ratio of planet and star . . . . . . . . . . . . . . . . . . 32
4.1.2 Mass ratio of moon and planet . . . . . . . . . . . . . . . . . . 32
4.1.3 Angular separation of planet and star . . . . . . . . . . . . . . 33
4.1.4 Angular separation of moon and planet . . . . . . . . . . . . . 35
4.1.5 Position angle of moon with respect to planet-star axis . . . . 37
4.2 Parameters Relevant for Light Curve Analysis . . . . . . . . . . . . . 37
4.2.1 Distance to source plane . . . . . . . . . . . . . . . . . . . . . 38
4.2.2 Distance to lens plane . . . . . . . . . . . . . . . . . . . . . . 38
4.2.3 Mass of lensing star . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2.4 Source size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2.5 Sampling rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2.6 Photometric uncertainty of observed data . . . . . . . . . . . 42
4.3 The Standard Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . 42
1

2 CONTENTS
5 Results: Detectability of Extrasolar Moons in Microlensing 45
5.1 The moonlight Output for a Single Magnification Pattern . . . . . . 45
5.2 Results for Selected Scenarios . . . . . . . . . . . . . . . . . . . . . . 46
5.2.1 Standard scenario . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.2.2 Changing the angular separation of planet and moon . . . . . 47
5.2.3 Alteration of the mass of the moon . . . . . . . . . . . . . . . 47
5.2.4 Changing the planetary mass ratio . . . . . . . . . . . . . . . 48
5.2.5 Increased separation between star and planet . . . . . . . . . . 49
5.2.6 Different sampling rates . . . . . . . . . . . . . . . . . . . . . 49
5.2.7 Source size variations . . . . . . . . . . . . . . . . . . . . . . . 49
5.3 Possible Limitations and Solutions . . . . . . . . . . . . . . . . . . . 50
6 Conclusion 53
A Analysed Magnification Patterns 55
B Numerical Results 65
Bibliography 69
Acknowledgements 77

Chapter 1
Introduction
Astronomy, and in particular the search for planets outside our solar system, begins
to touch what has long been a question reserved for philosophers to ponder about:
Do other worlds exist?
After centuries of speculation and research, it was less than two decades ago
when it was confirmed for the first time that planets exist in the universe that circle
stars other than our sun. By now hundreds have been detected with five different
techniques that are described in the overview below. For all we know, none of
these extrasolar planets offer physical conditions permitting any form of life. But
the search for planets potentially harbouring life and the search for indicators of
habitability is ongoing. One of these indicators might be the presence of a large
natural satellite – a moon – which stabilises the rotation axis of the planet and
thereby the surface climate.
This work sets out to simulate and evaluate the success rate of one of the most
promising techniques for detecting extrasolar moons. This technique is gravitational
lensing. The deflection of light by massive bodies is a consequence of the theory of
general relativity and has been experimentally verified since 1919. It is now the
basis for diverse areas of research in astronomy, because of its ability to work as a
magnifying glass to make distant or small objects visible that could not be detected
otherwise. Even small-mass objects like planets can leave a detectable mark in
Galactic gravitational lensing events. While a lensing star passing in front of a
background source star can be detected as a transient brightening of the source,
the gravitational influence of a planet in orbit around the star can be detected as
an irregularity in this brightening. The method is sensitive to masses as low as an
Earth mass. From theoretical works about planet formation we know that massive
moons can exist in stable orbits around giant planets. Therefore it is promising to
apply the technique of gravitational lensing to the search for extrasolar moons.
Overview of different methods for finding extrasolar planets. . .
The following list provides a brief overview of the various techniques that have
successfully discovered extrasolar planets in the last years, with a reference to the
3

4
first confirmed detection by the respective method.
• Pulsar timing (Wolszczan and Frail, 1992)
The regular emission of radio pulses of a pulsar is perturbed by the presence of
orbiting secondary masses. Pulsars are relatively rare, therefore not many extrasolar
planets will be detected with this method. The lowest mass extrasolar
planet up-to-now was detected with this method with a mass of 0.02 M⊙.
• Radial velocity measurements or “Doppler wobble” (Mayor and Queloz, 1995)
The motion of star and planet around their common barycentre can be detected
as a shifting in the spectral emission of the star. The method has
detected more than 300 planets. It is most sensitive to massive close-orbit
planets. In is only possible to determine a minimum mass, because the true
mass depends on the orbital inclination with respect to the observer.
• Transit observations (Henry et al., 2000)
A planet temporarily occults its host star when passing in front of it. The
orbit inclination has to be favourable for this to happen and planets with
larger radii are easier to detect. More than 50 planets have been detected
with this method.
• Gravitational microlensing (Bond et al., 2004)
The steady and symmetric curve of a single-source, single-lens microlensing
event can show anomalies due the presence of further lensing masses. Discoveries
are favoured by large masses and an orbit of several AU. Eight planet
detections were reported up-to-now.
• Direct imaging (Marois et al., 2008; Kalas et al., 2008)
Thermal emission and reflected light from the planet can be imaged by using
a coronagraph, a mask to blend out the dominating light from the host star.
Large orbits are easier to capture. Two independent detections of planetary
systems were announced last November.
We refer to Cassen et al. (2006) for a comprehensive introduction to the study
of extrasolar planets. For up-to-date listings of discovered exoplanets and their
properties, we refer to the Extrasolar Planets Encyclopaedia 1 .
. . . and searching for moons
The majority of discovered planets has been confirmed through radial velocity measurements.
Unfortunately, this method does not have sensitivity to satellites of those
planets.
As early as 1999 it has been suggested that extrasolar moons might be detected
through transit observations (Sartoretti and Schneider, 1999). Later Han
and Han (2002) performed a feasibility study whether microlensing might be the
1 exoplanet.eu

CHAPTER 1. INTRODUCTION 5
tool to discover an Earth-Moon analogue, but concluded that finite source effects
would probably be too severe in realistic observing situations. Williams and Knacke
(2004) published the quite original suggestion to look for spectral signatures of
Earth-sized moons in the spectra of Jupiter-sized planets. Cabrera and Schneider
(2007) proposed a sophisticated transit approach to detect “binary planets”,
i.e. planet-companion systems with a mass ratio close to unity. Lewis et al. (2008)
analysed pulsar time-of-arrival signals for lunar signatures. None of these methods
has achieved successful detections yet.
Han (2008) undertook a new qualitative study of a number of microlensing constellations
finding non-negligible light curve signals occur in the case of an Earthmass
companion orbiting a 10 Earth-mass planet at the right angular separation.
For planets with masses in the mass range between Uranus and Jupiter, he defines
regions of possible satellite detections as the region of angular separations between
the lower limit of one planetary Einstein radius and an upper limit at the projected
Hill-radius average. We do not quite agree with this proposition, as will become
clear during the presentation of our results.
Outline
In chapter 2, the basic concepts of gravitational lensing are outlined and in particular
the application to planetary lensing is briefly summarised. Chapter 3 gives an
extensive overview over how we approached the given problem of identifying light
curve perturbations caused by a moon, and which methods are used to solve it.
Because we had to restrict our research to a subset of possible scenarios involving
a moon, these choices are motivated in chapter 4 by discussing the astrophysical
context that we are dealing with. Chapter 5 presents the results of our simulations
and a first interpretation. This thesis ends with concluding remarks in chapter 6.
The appendix chapters A and B contain additional documentation in a visual and
numerical form, respectively.

6
Figure 1.1: The solar system seen as a gravitational lens.
A magnification map of the solar system. Celestial bodies are at the positions marked
with crosses. Along the lines connecting the planets with the sun, one finds the
planetary caustics. With a larger separation of the planet, the caustic moves towards
the planet. Separations in this image correspond to the semi-major axes, if this was
a lensing system illuminated by a background source at a distance of 2 kpc seen from
an observer on the other side at a distance of 6 kpc. In this setting Jupiter might
well be detectable from outside the solar system.

Chapter 2
Gravitational Lensing
The phenomenon of gravitational lensing occurs when light rays of a background
source are deflected by an intervening massive body. The light of the source appears
magnified, if more light rays are deflected towards the observing point, than without
the lens. Accordingly, the light can also be demagnified, if light rays are deflected
away from the observer.
This chapter aims at providing the reader with a short introduction to the history
(section 2.1), the basic equations (section 2.2) and the application to planet searches
(section 2.3) of gravitational lensing. Better and more comprehensive reviews are
plentiful in literature, e.g. Meylan et al. (2006).
2.1 Historical Development
The deflection of light by massive bodies must be concluded from Einstein’s general
theory of relativity, which was first summarised in 1916 (Einstein, 1916). In this
work Einstein first published the correct derivation of the deflection angle of a light
ray passing the sun close to the surface. He had thought “ Über den Einfluß der
Schwerkraft auf die Ausbreitung des Lichtes” 1 before, such was the title of a paper
published in the “Annalen der Physik” (Einstein, 1911), where he also references an
earlier publication (Einstein, 1908). Even this is preceded by more than a century
in form of a paper by Soldner (1801), who applied Newtonian gravity to compute a
deflection angle of 0.84 arcsec for the sun, half of the correct value.
Einstein had called for a test of his calculations in 19112 and, indeed, an expedition
set out to measure precise star positions close to the limb of the sun during solar
eclipse in 1914, but was hindered in their task by the outbreak of World War I. This
was very fortunate for Einstein, as it gave him time to develop general relativity
1 “About the influence of gravity on the propagation of light”
2 “Es wäre dringend zu wünschen, daß sich Astronomen der hier aufgerollten Frage annähmen,
auch wenn die im vorigen gegebenen Überlegungen ungenügend fundiert oder gar abenteuerlich
erscheinen sollten. Denn abgesehen von jeder Theorie muß man sich fragen, ob mit den heutigen
Mitteln ein Einfluß der Gravitationsfelder auf die Ausbreitung des Lichtes sich konstantieren
laesst.”
7

8 2.2. BASIC EQUATIONS
and publish the value of 1.7 arcsec, which was confirmed to within 20% in a 1919
expedition by Dyson et al. (1920). This result was recognised worldwide and was
the basis of Einstein’s fame.
In the following decade a number of people published their ideas concerning
gravitational lensing. Chwolson (1924) investigated the geometry of star-on-star
lensing, finding that the observer would see two images of the source star, one of
which is inverted like a mirror image, except for the case of perfect alignment of
source, lens and observer, when a ring around the lens star would appear (now
usually referred to as Einstein ring). Then Einstein himself had worked out the
magnification of the apparent brightness of a background source star, if a lensing
star and the observer are appropriately positioned (Einstein, 1936). Zwicky (1937a)
pointed out the great possibilities of lensing galaxies acting as a magnifying glass
on the distant universe 3 and added later that it is “practically a certainty” to find
galaxies acting as lenses (Zwicky, 1937b).
The first time that the effects of a gravitational lens other than our sun were
observed, was not until 40 years later, when Walsh et al. (1979) announced discovery
of gravitationally lensed quasar QSO 0957+561.
Paczyński (1986) was first to point out the possibility of gravitational lensing
due to masses as small as 10 −11 M⊙ in the halo of the Milky Way with background
stars in the nearby galaxies. If found, those masses would provide an explanation of
the dark matter in the halo. The idea was eventually picked up and several searches
for microlensing events were called into life in the early 1990s. 4 Alcock et al. (1993)
and Aubourg et al. (1993) reported discoveries towards the Large Magellanic Cloud.
Udalski et al. (1993) found events towards the Galactic bulge. As a general study
of lensing behaviour Schneider and Weiß (1986) had investigated the case of binary
point mass lenses. Paczyński (1991) provided details of lensing by Galactic bulge
stars. On this foundation, Mao and Paczyński (1991) proposed to look for lensing
signatures of binary stars and also, mentioned here for the first time in the context
of gravitational lensing, extrasolar planets.
2.2 Basic Equations
From the theory of general relativity we know that light is deflected by gravity. More
precisely, light always follows the geodesics of spacetime, but spacetime is bent in
the vicinity of a massive body. The mass acts as a lens, that focusses light rays
that are passing near-by. We can work with the thin lens approximation for almost
all practical applications, assuming that the action of deflection takes place in an
imaginary plane at the location of the lensing mass, the so called lens plane. This
simplification finds its justification in the fact, that in all astronomical settings the
3 “Any such extension of the known parts of the universe promises to throw very welcome new
light on a number of cosmological problems.”
4 There is an interesting footnote in Zwicky (1937b), stating that as early as 1923 plans were
made “for the search of such lens effects among stars”.

CHAPTER 2. GRAVITATIONAL LENSING 9
distances covered by the light path are very large in comparison to the extent of
mass distribution of the lens.
DS
I2
DLS
DL
L
O
β
θ
η S I1
Figure 2.1: Sketch of a gravitational
lensing system. Notation
is explained in the text.
ξ
α
˜α
In figures 2.1 and 2.2 the relevant parameters
of point-mass lensing are sketched. I1 and I2
denote the apparent locations of the two images
of the source S. There are always images for a
point mass lens L and a single source S (compare
equation (2.2) and its derivation below), except
for the singular case of perfect alignment, when
an Einstein ring is visible. The observer O sits in
the observer plane. The hyperbolic light paths
are approximated by two straight rays with a
sharp bend at the lens plane in figure 2.1.
General relativity predicts that a light ray,
which passes by a point mass M at a minimum
distance ξ, is deflected by an angle
ˆα = 4GM
c2 . (2.1)
ξ
G denotes the Gravitational constant, c the speed
of light. The point mass is a safe approximation
for a Galactic lensing situation, where two stars
align to form a gravitational lens system. It is
justified to approximate the lensing body as a
point mass or point lens under the assumption
that all light rays pass at a distance from the centre of mass, which is larger than
the radius of the lensing star.
Considering simple geometry in figure 2.1 we find the lens equation
βDS = θDS − ˜αDLS
for θ, β, ˜α

10 2.2. BASIC EQUATIONS
We can interpret this equation to state that a source with true position β will have
the apparent position θ for the observer O. The number of solutions for equation
(2.2) gives us the number of images produced by the lensing system. The lens
equation describes a mapping θ −→ β from the lens plane to the source plane and is,
in principle, solvable for any given mass distribution in the lens plane. Problematic,
and not in all cases analytically solvable, is the inversion of the lens equation, which
is necessary to determine image positions θ for a given source position β.
If lens and source lie on the same line of sight from the observer, β equals zero
and we can see an Einstein ring. The angular radius of this ring is called Einstein
radius:
4GM DLS
θEinstein =
. (2.3)
c 2
DLDS
This quantity defines the angular scale of the system. We can reformulate the lens
equation, to obtain
β = θ − θ2 E
θ .
For a point lens and a single source, there will always be two images of the background
source, one inside and one outside the Einstein ring radius, except for perfect
alignment, when we see an Einstein ring. The positions of the images are the two
solutions of the lens equation,
Figure 2.2: The view of the
lens plane from the observing
point O. Figure reproduced
from Paczyński (1996).
L
S
I 2
I 1
θ1,2 = 1
β ± β
2
2 + 4θ2
E . (2.4)
The magnification µ of an image is defined
by the ratio between the solid angles of the magnified
image and the original one. Here, that
translates to the ratio between the solid angle
of the image and the solid angle of the source,
since the surface brightness is conserved during
the lensing process,
µ = θ dθ
. (2.5)
β dβ
In the discussed spherically symmetric case, the
image magnification can be written as
µ1,2 =
1 −
θE
θ1,2
4 −1
= 1
2 ± u2 + 2
2u √ u 2 + 4 ,

CHAPTER 2. GRAVITATIONAL LENSING 11
where the impact parameter
u = β
is used, the angular separation of the source from the point mass in units of the
Einstein angle. The magnification of the image inside the Einstein ring is formally
negative, because it has negative parity. It is a mirror-inverted image of the source.
The sum of the two magnifications is unity,
θE
µ1 + µ2 = 1,
but the net magnification of the source flux in the two images is obtained by adding
the absolute magnifications
µ = |µ1| + |µ2| = u2 + 2
u √ u 2 + 4 .
If the source is located at an angular separation of one Einstein radius from the lens,
the impact parameter will become u = 1 and the total magnification
µ = 3
√ 5 = 1.34.
If the source moves closer towards the lens, the magnification will increase (figure
2.3). For β → 0 it reaches infinity. In reality, this is prevented by the finite extent
of the source. The magnification is converted into units of magnitudes via
mag = −2.5 log 10 µ. (2.6)
According to the astrophysical convention, more negative values denote brighter
luminosity.
2.2.1 Triple-Lens Equation
Planetary systems have the characteristic trait of being multiple lens systems. After
having described the single-lens case in the last section, we ought to have a look
at how multiple lenses manifest themselves in light curves and how they can be
accounted for in the lens equation.
Using complex coordinates, let us express the projected source position as η and
the image position as ξ. For N point masses the lens equation then becomes
η = ξ +
N qi
, (2.7)
ξi − ξ
i=1
following Witt (1990) and also referring to Gaudi et al. (1998). ξi denotes the
position of lens i. qi is the mass ratio between lens i and the primary lens (qi = Mi
M1 ).

12 2.2. BASIC EQUATIONS
−∆mag
Figure 2.3: If a source star passes behind a lensing star along the source trajectory
depicted in the magnification map on the left, this gives rise to a point-lens, pointsource
light curve, also called Paczyński curve.
All lengths are in units of Einstein radii of the primary lens. The magnification
µI of image I, which has been derived for a single lens in an earlier section 2.2, is
now obtained as the inverse of the determinant of the Jacobian of the the mapping
equation (2.7),
µI =
1
det J(ξI)
time
∂η ∂η
with det J(ξ) = 1 −
∂ξ ∂ξ .
The Jacobian J can be equal to zero, it that case a hypothetical point source would
be infinitely magnified. The mapping of all image positions satisfying J = 0 to
the source plane forms the continuous caustics. These are lines of formally infinite
magnification that arise due to the lens astigmatism, compare figure 2.4.
Choosing complex coordinates is of course just a question of preference, the same
calculation can be performed with Euclidean coordinates (see e.g. Bozza, 1999), but
one has to be aware that the Jacobian J becomes much more complex to explicitly
express.
In this work, we focus on the triple lens case with the three bodies of the host
star, the planet and the moon in orbit of the planet. The equation gets the following
form, if we place the primary lens in the origin of the lens plane,
η = ξ − 1
ξ
qPlanet
− −
ξ − ξPlanet
qMoon
. (2.8)
ξ − ξMoon
As pointed out in Rhie (1997), and explicitly calculated in Rhie (2002), the triplelens
equation is a tenth-order polynomial equation in ξ. Equation (2.8) is therefore
analytically solvable.

CHAPTER 2. GRAVITATIONAL LENSING 13
−∆mag
Figure 2.4: Caustics form when there are multiple lenses. The magnification map
on the left corresponds to a star-planet binary, with a mass ratio of 10 −3 . If the source
trajectory crosses caustic lines this gives rise to a characteristic deviation from the
Paczyński curve, a planetary signature.
2.3 Search for Planets via Galactic Microlensing
When we observe a Galactic lensing event, the parameters that we deal with will
have typical orders of magnitude. Imagine, that we observe an alignment of two solar
sized stars in our Milky Way. The background star will most probably lie in the
centre of the Galactic bulge at a distance of DS = 8 kpc for there we find the highest
density of stars, and indeed, that is where all Galactic lensing survey programs direct
their telescopes. The lensing star will be somewhere in the foreground, DL = 6 kpc
should be a safe assumption. We can already calculate the Einstein angle, to find
θE = 0.6 milliarcseconds.
This is far beyond the resolution abilities of real telescopes. However, even if the
individual images cannot be resolved, the magnification can still be detected if lens
and source are in relative motion to each other. This passing of the lens in front
of the source gives rise to a transient brightening as shown in figure 2.3. The
light curve that results if a point lens moves in front of a point source is often called
Paczyński curve, as it was depicted in Paczyński (1986). A magnification of µ = 1.34
corresponds to a difference in magnitude of ∆mag = 0.32 mag, which is easily
detectable. Events with a magnification of µ = 100 or more are not uncommon.
The search for massive compact halo object as potential dark matter, mentioned
it section 2.1 was not very fruitful (see e.g. Afonso et al., 2003). Microlensing
experiments carried out towards the Galactic bulge had originally been intended
as test experiments for the halo surveys, but Mao and Paczyński (1991) showed
that roughly 10% of the lensing events must have signature of a binary companion.
It was realised, that through constantly monitoring a very large number of stars
one would surely detect binary systems and possibly planets. Gould and Loeb
time

14 2.3. SEARCH FOR PLANETS VIA GALACTIC MICROLENSING
(1992) qualitatively estimated the fraction of light curves that would show signs of
companions of Jupiter or Saturn mass. Bennett and Rhie (1996) find that Earthmass
planets are principally detectable. Finally, Wambsganß (1997) provided a
detailed study of possible planetary light curve perturbations with different massratio
and angular-separation settings, concluding with the identification of the socalled
“lensing zone” between angular star-planet separations of 0.6 to 1.6 θE that
favours detections, because the planetary anomalies will settle on the slope of the
Paczyński curve – on top of the ongoing primary lens magnification.
Steady monitoring of the Galactic bulge was realised by different groups, since
1992. As of 2009, OGLE (The Optical Gravitational Lens Experiment, see Udalski
et al. (1992)) and MOA (Microlensing Observations in Astronomy, see Muraki et al.
(1999)) are active survey collaborations that complement each other by operating
wide-field telescopes in Chile and New Zealand, respectively. Every season these
groups detect microlensing events in their hundreds. In 2004 they jointly discovered
a planet of 1.5 Jupiter masses with an orbit of ∼ 3 AU, which made history as the
first microlensing detected planet (Bond et al., 2004).
Already 1995, a cooperative follow-up strategy had been realised by the PLANET
network (Albrow et al., 1998). The idea behind it was, and still is, to find a compromise
between the field-of-view, the sampling rate, the limiting magnitude and
the resolution of the targets. To maximise the gain, the follow-up community only
react to event alerts by the survey groups. These are sent out by OGLE and MOA,
if a microlensing event promises to be interesting - because it is evolving towards a
very high magnification or because it already shows deviations from the Paczyński
curve. Several groups are active in the field of follow-up observations, 5 the common
goal is to get dense time resolution on interesting light curve features to be able to
constrain a planetary model (or a binary star model, or a source star atmosphere
model etc.) for that light curve. Crucial to the success of these operations is the
continuous exchange of life data during the observing season 6 that enables fitting of
preliminary models and thereby tentative predictions about the development of the
light curve, to decide about the priority of observing the given event. This approach
has led to seven further reported planets. 7
For an up-to-date status report of the past, present and future of planet searching
via Galactic microlensing the two ESA white papers of last year are a good source
of reference (Beaulieu et al., 2008; Dominik et al., 2008).
5 In the year 2008 these follow-up groups were namely, PLANET (planet.iap.fr), Micro-
FUN (www.astronomy.ohio-state.edu/~microfun), RoboNet-II/LCOGT (www.lcogt.com) and
MiNDSTEp (www.mindstep-science.org).
6 The Galactic microlensing season lasts roughly from April to September, the time of year when
the Galactic bulge is well visible from the southern hemisphere, at least for part of the night.
7 To list them individually, they were “a Jovian-mass planet” in event OGLE-2005-BLG-071
(Udalski et al., 2005), “ a cool planet of 5.5 Earth masses” in OGLE 2005-BLG-390 (Beaulieu et al.,
2006), a “cool Neptune-like planet” in OGLE-2005-BLG-169 (Gould et al., 2006), “a Jupiter/Saturn
analog” in OGLE-2006-BLG-109 (Gaudi et al., 2008), “a low-mass planet” in MOA-2007-BLG-192
(Bennett et al., 2008) and “a cool Jovian-mass planet” in MOA-2007-BLG-400 (Dong et al., 2008).

Chapter 3
Method
The search for extrasolar planets via microlensing is well established today. See
Gould (2008b) for a recent summary of successful (and some not so successful)
planet findings. Microlensing probes deeply into the low-mass range, even Earthmass
bodies are within the range of possible discoveries. Hence it seems natural to
expand theoretical investigations to the effects that satellites of extrasolar planets
have on microlensing light curves. The goal of this work is to evaluate and quantify
under which conditions an extrasolar moon will be detectable in the observational
situation. We propose that lunar effects will first show up as noticeable irregularities
in light curves that have been initially observed and classified as light curves with
planetary signatures. The simplest model is that of a binary lens 1 , consisting of the
lensing star and a planetary companion.
The simplest system incorporating an extrasolar moon is a triple-lens system of
the lensing star, a planet and a moon in orbit around that planet (sketched in figure
4.1. To measure the detectability of triple-lens systems among binary lenses, we
have to determine whether the resulting triple-lens light curves differ significantly
from light curves from a corresponding binary-lens system.
To obtain many independent light curves of a selected lens system with moon, we
first produce a magnification pattern for the given parameter configuration. This will
be explained in detail in section 3.1. Light curves are obtained as one dimensional
cuts through this pattern (section 3.2). Analogously, light curves of a corresponding
binary-lens system will be produced. We fit the latter to each examined triplelens
light curve. For the best-fit binary light curve we employ χ 2 -statistics to see
whether the triple-lens light curve could be explained as a normal fluctuation within
the error boundaries of the binary case (section 3.3). If this is not the case, the
moon is considered detectable.
1 Throughout this text, the term binary lens will always refer to a system consisting of a host
star and a planet, i.e. a binary with a mass-ratio very different from unity. In this text it never
denotes a binary of stars.
15

16 3.1. MAGNIFICATION PATTERNS
3.1 Magnification Patterns
The three-mass lens equation, a tenth-order polynomial, can in principle be solved
analytically. For a more detailed discussion refer to section 2.2.1. Han and Han
(2002) reported that numerical noise in the polynomial coefficients caused by limited
computer precision was too high (∼ 10 −15 ) when solving the polynomial for the
very small mass ratios of satellite and star (∼ 10 −5 ). Therefore, we employ the
inverse ray-shooting technique, which has the further advantages of being able to
account for finite source sizes and non-uniform source brightness profiles more easily.
It also gives us the option of incorporating additional lenses (further planets or
moons) without a rise in complexity. This technique was developed by Schneider
and Weiß (1986) and Kayser et al. (1986) and enhanced by Wambsganß (1990) and
Wambsganß (1999). My starting basis was a January 2008 version of the Wambsganß
code, subsequently referred to as microlens.
3.1.1 The microlens code
Inverse ray-shooting means that light rays are traced from the observer back to the
source plane. This is equivalent to tracing rays from the background source star
to the observer plane. The influence of all individual masses in the lens plane on
the path of light is calculated. In the thin lens approximation, the deflection angle
is just the sum of the deflection angles of every single lens. After deflection, all
light rays are collected in pixels of the source plane. Lengths in this map can be
translated to angular separations or – assuming a constant relative velocity between
source and lens – to time intervals. Any common plotting programme that is capable
of handling two-dimensional data can be used to visualise the result, for example
Gnuplot. Thus a magnification pattern (cf. figure 3.1) is produced. The number of
collected light rays per pixel N is correlated to the magnification µ of a background
source at the respective position in the source plane. It is scaled to the usual measure
of luminosity in magnitudes as follows:
magpixel = −2.5 log 10 µ = −2.5 log 10
N
Naverage
(3.1)
To get the magnification of a certain source, the magnification pattern is convolved
with the brightness profile of the source. A light curve is extracted as a
one-dimensional cut through the convolved magnification pattern along the source
path.
microlens goes far beyond being a ray-shooting programme. It makes use of a
two-dimensional, hierarchical tree code and a multipole expansion to decide which
lenses have to be included exactly to determine the deflection angle for a given light
ray, and which can be bundled and evaluated together as a single mass, because they
are too distant from the light ray to add significantly to the total deflection angle.
Thanks to this algorithm it is possible to account for a very large number of lenses
within reasonable computing time. Another feature of microlens is its ability to

CHAPTER 3. METHOD 17
approximate a three-dimensional mass distribution as many successive lens planes.
Combining all this, the programme is very well equipped for simulating quasar
microlensing (where the background light of a quasar is microlensed by individual
stars of the strongly lensing foreground galaxy), as well as simulating strong lensing
on the cosmological scale. Wambsganß (1999) provides a discussion of the algorithm
and further references to its applications.
3.1.2 The moonlens code
The original microlens code is optimised for a huge number (∼ 10 6 ) of lenses
(Wambsganß, 1999). Features like the underlying cell structure of the hierarchical
tree code and the random lens generator are redundant for the simulation of planetary
systems. On the other hand, it is fundamental to be able to choose the lens
positions and masses to match realistic systems.
These modification have been implemented and from then on this simplified
version was used, referred to as moonlens from now on. moonlens is capable of
handling a chosen number of lenses (currently up to 10 2 ) at fixed positions. The
primary lens has unit mass, whereas all other lens masses are given in mass ratios
relative to this. By assigning a physical mass to the primary lens, the physical
mass scale is determined for the set of lenses. The lenses are described in three
parameters (relative mass, x- and y-position in units of θE of the primary lens). So
it is easily possible to simulate a given planetary system, for example our own (see
figure 1.1). But we concentrate on the ternary lens case with extreme mass ratios
of 0.3 M⊙ : 10 −3 : 10 −5 , i.e. a subsolar mass star, a Saturn-mass companion and a
satellite with 10 −2 Saturn masses.
moonlens is written in Fortran77 and can be compiled with any standard Fortran
compiler, for example g77 and gfortran. It consists of one main routine and several
subroutines. Parameters are read in from an input file, so that no recompiling is
necessary for changing the physical parameters. Processing time is of the order of a
few CPU hours on a PC for a reasonable resolution (1000 × 1000 pixels, 2000 rays
shot per pixel), when zoomed in on a planetary caustic. Output is currently given in
two formats: FITS files are printed using the CFITSIO library and standard routines
from the cookbook 2 . For alternative processing (and useful for control purposes) an
ASCII matrix is written out. We used it with Gnuplot to create the colour plots in
this thesis.
3.2 Light Curves
To investigate the effects that satellites of extrasolar planets have on microlensing,
we must examine how the changed gravitational potential influences the light curves,
because this is what we will eventually detect through observations. We analyse a
2 heasarc.gsfc.nasa.gov/fitsio/

18 3.2. LIGHT CURVES
Figure 3.1: A magnification map of the planetary caustic of a lensing system consisting
of an M-Dwarf, a Saturn-mass planet and an Earth-mass satellite of that planet,
i.e. mass ratios of 0.3 M⊙ : 10 −3 : 10 −5 . The side length of the pattern is given in
stellar Einstein radii. Difference in colour correspond to units of magnitudes. Note
that brighter colour means higher magnification at that source location. A light curve
(figure 3.2) can be extracted from the magnification pattern as a one-dimensional cut.
number of physically different lensing systems (see also chapter 5) that are represented
by a set of magnification patterns. We have written a programme (called
moonlight from now on) to extract (section 3.2.1) a large number of light curves
from each of the patterns, perform a binary case fit on each of them (section 3.2.2)
and then to statistically analyse the detectability of the moon in each triple lens
light curve (section 3.3).
3.2.1 Light curve extraction
In physical terms, the magnification pattern is a virtual map of the magnification a
point light source would experience at any position of the source plane covered by the
map. While it does not have a physical counterpart, it still serves a useful purpose

CHAPTER 3. METHOD 19
by visualising the lensing system. Even more important is the fact that it carries
the information about observable light curves. A light curve is obtained as a onedimensional
cut through the magnification pattern. Only a few more assumptions
are necessary to simulate realistic, in principle observable, light curves – angular
source size, relative motion of lens and source, lens mass. Angular separations get
a physical meaning once the distances to the source plane and to the lens plane are
fixed.
To get the magnification of a certain source, the brightness profile of the source is
convolved with the magnification pattern. As a first approximation to the physical
brightness profiles of stars (ignoring e.g. limb-darkening effects), we use a flat surface
brightness profile. The extent of the profile is given by the radius of the source
RSource.
−∆mag
3
2
1
0
Triple Lens Light Curve
0 0.05 0.1 0.15 0.2 0.25 0.3
t/tE
Figure 3.2: The light curve resulting from the source trajectory shown in figure 3.1.
The caustic-crossing close to the cusp shows up as a distinct planetary signature.
The path of the source is described in map coordinates as a combination of a point
and a directional step vector. For each light curve, the start and the end point are
determined, while taking into account the size of the source by being limited to the
area RSource away from the map boundaries. The light curve is sampled at equidistant
intervals, the length of which is given as an input parameter in units of pixels.
See also section 4.2.5 for physical implications of the sampling parameter. At each
sampling point the brightness profile is convolved with the surrounding magnification
values and an integration over all pixels up to RSource away is performed. This is the
most computationally intensive operation in our light curve analysis programme,
the computing time scaling with R 2 Source .

20 3.2. LIGHT CURVES
3.2.2 Light curve fitting
In our method we presume that the ternary mass (that of the moon) will only be
detectable in a triple-lens light curve before a background of binary-lens light curves.
In other words, the planet will have been detected first (through a caustic-crossing
signature in the light curve), but there will be cases where we achieve a better fit
employing a triple-lens model. In order to make any statements of detectability, we
have to quantify the difference between a given triple-lens light curve and its best-fit
binary counterpart. For this comparison we produce an additional magnification
(a) (b)
Figure 3.3: Is the difference detectable? The caustic of a Saturn-mass planet around
an M-Dwarf with (a) and without (b) moon. The depicted source trajectories result
in light curves as shown in figure 3.4.
pattern of the corresponding binary-lens system, where the mass of the moon is
added to the planetary mass. As a first approximation, one can compare two light
curves with identical source track parameters (see figures 3.3 and 3.4). But one
has to act cautiously, numerically big differences can occur without a significant
topological difference. This has to be avoided. The light curves in figure 3.4 show
a large discrepancy that is clearly not dominated by the central lunar peak of the
triple-lens light curve, but by the fact that the triple-lens caustic is slightly deformed
by the lunar perturbation. The binary light curve and the triple light curve are
simply not very well matched. To get a better fitting, we could modify every one
of the many parameters involved in the computation of the light curve (see also
section 4.2). The physical configuration of the binary lensing system is determined
by the planetary mass ratio and the angular separation between planet and star.
There is also the velocity of the relative movement of source and lens, for which we
could assume a different value to stretch or shorten the light curve. Furthermore,
the source track can be changed, adopting a new location and direction. We could
try to search the whole parameter space to find the binary-lens light curve with

CHAPTER 3. METHOD 21
−∆mag
3
2
1
Triple Lens Binary Lens Difference
0
0.12 0.13 0.14 0.15 0.16 0.17 0.18
t/tE
Figure 3.4: The two light curves with identical source track parameters obtained
from 3.3(a) and 3.3(b) and their absolute difference in magnitudes. The plot is
zoomed in on the caustic-crossing features, to clearly show the signature of the moon,
i.e. the third peak in the centre of the red curve. The absolute difference of the two
curves is plotted in blue. Obviously, the large peaks on the left and right are not
caused by the moon, but by a mismatch of the two light curves that can be avoided
with a better choice of the binary-lens light curve.
the absolute least-square difference compared to the triple-lens light curve. As it is,
an alteration of the lensing system parameters would require us to generate a new
magnification pattern, which would be computationally costly.
Since the perturbation introduced by the moon is small however, we argue that
we have excellent starting conditions using the binary magnification pattern that
corresponds to the triple-lens magnification pattern. Keeping the planet-star separation
fixed and changing the planetary mass to the sum of the masses of planet and
moon, we cannot be far from the absolute best-fit values. In our simplified approach,
we only open up the parameter space of the source trajectory and search there for
the best-fitting binary-lens light curve, which already yields very convincing results.
For the fitting we use a least-square method. We search for the minimum of the
sum of the absolute difference between triple and binary-lens light curve squared.
S :=
(xtriple − xbinary) 2
light curve points
In moonlight this is done by taking the parameters (coordinate position, direction
vector) of the original light curve as the starting set of parameters. Parallel source
tracks are drawn on both sides, more light curves are taken by rotating the tracks.
Of the order of 10 2 binary light curves in the immediate environment around the

22 3.3. STATISTICAL ANALYSIS
original source track are compared to the triple-lens light curve. When the sum is
calculated, the light curves are also allowed a longitudinal translation, which gives
a further factor to the number of binary light curves. For every pairing the S value
is being calculated and from all parameter sets, the curve with the minimal S is
selected as the best-fit. If it happens that one or more of the parameters of the
best-fit curve hit a boundary of the original parameter range, then the programme
redefines the current best-fit light curve as the new starting point and the search
continues until the first minimum is found that does not touch the boundaries of
the parameter space for any of the three parameters. The result of this procedure
is our best-fit binary-lens light curve.
−∆mag
3
2
1
Triple Lens Binary Lens Difference
0
0.12 0.13 0.14 0.15 0.16 0.17 0.18
t/tE
Figure 3.5: The triple-lens light curve is unchanged. But while the binary lens
light curve is again obtained from 3.3(b), we get a much better fit to the triple-lens
light curve after having allowed the source trajectory to move freely. All residing
differences can be attributed to the moon.
3.3 Statistical Analysis
Grid of light curves
In order to be able to make robust statistical statements about the detectability of
a moon in our chosen setting, we analyse a grid of source trajectories that delivers
about two hundred light curves, as shown in figure 3.6. To have an unbiased sample,
the grid is chosen independently of lunar caustic features, but all trajectories are
required to pass through or close to the planetary caustic. This is realised by
generating a grid of source trajectories that is oriented at a manually given central

CHAPTER 3. METHOD 23
point of the planetary caustic through which a first horizontal track is cut, then 9
tracks are drawn in parallel on either side, after a rotation of 15 ◦ again 19 parallel
tracks are drawn, and so on, until the magnification map is covered with the evenly
spaced-grid.
Figure 3.6: 228 individual light curves are extracted from each triple lens magnification
pattern. To have an unbiased statistical sample, the source trajectories
are chosen independently of the lunar caustic features, though all light curves are required
to pass through or very close to the planetary caustic. This is realised through
generating a grid of source trajectories that is only oriented at of the planetary caustic.
In the standard case we use solar sized source, the size of which is indicated in
the lower left corner of the pattern.
Curve comparison
We want to use the properties of the χ 2 -distribution for a test of significance of
deviation between two simulated curves. We start in section 3.3.1 with describing
the standard procedure for quantifying the goodness-of-fit for any fit of a theoretical

24 3.3. STATISTICAL ANALYSIS
curve to experimental data. In section 3.3.2 a method for directly comparing two
simulated (light) curves is presented.
3.3.1 Using the χ 2 -distribution as a measure for significance
of deviation between data and a theoretical curve
After observing a microlensing event, we are eventually left with light curve data.
Each of the data points has an associated photometric error and in general the
data are randomly scattered with a standard deviation of σi, including error sources
such as the photon noise √ photon count of data i or the scintillator noise. For
comparing the data with a physical explanation, one has to fix a model and then
test the goodness-of-fit of the model light curve to the data light curve to determine
the optimal parameter values of the model, or to reject the whole model if no
satisfying fit can be found. To test the goodness-of-fit of some theoretical light
curve, the cumulative distribution function of the χ 2 -distribution is used to quantify
the agreement between observations and theory.
Let us recall the general usage of the χ 2 -test. Let Xi (with i = 1, . . . , n) be n
independent, normally distributed random variables that have the mean µi = 0 and
the standard deviation σi = 1 for all i. The sum over all X 2 i
χ 2 ∼ Q 2 =
is again a random variable and follows the χ 2 -distribution and can be called χ 2 -
distributed with n degrees of freedom. See any standard book with an introduction
to calculus of probability, for example Bronstein et al. (2000) or Bevington (1969).
To compare data xi that has an associated error, or standard deviation, of σi
with a theoretical curve µi, we can create a new random variable
Q 2 µ =
i=1
n
i=1
σi
X 2 i
n
2 xi − µi
.
Q 2 µ will follow the χ 2 -distribution, if the assumed theory is correct (⇔ 〈xi − µi〉 = 0)
and each summand is weighted with the standard deviation σi of the distribution
function of the data spread. As in all science, it is not possible to derive positive
affirmation of a theory and ultimately verify it. We can only try to find a discrepancy
and falsify it.
With a given probability density function, it is possible to calculate the probability
that a random variable X lies in an interval [a; b]
P (a ≤ X < b) =
b
a
f(x)dx.

CHAPTER 3. METHOD 25
Now, we can look up the probability density function of the χ2-distribution with n
degrees of freedom and find
⎧
⎪⎨ x
fn(x) =
⎪⎩
n
2 −1 x
− e 2
2 n
2 Γ x > 0
n
,
2
0 x ≤ 0
where Γ(r) is the gamma function. Γ
n can be evaluated via the formulae
2
1
Γ =
2
√ π , Γ(1) = 1
Γ(r + 1) = r · Γ(r) with r ∈ R + .
The probability P (χ 2 ≥ Q 2 µ) that any random set of n data points compared with
the correct(!) theory µ would yield a value of χ 2 as large as or larger than Q 2 µ is
P (χ 2 ≥ Q 2 µ) =
∞
Q 2 µ
fn(x)dx = 1 − Fn(Q 2 µ),
with Fn(z) = z
f(x)dx as the cumulative distribution function. If it is very
−∞
small, Q2 µ is outside the expected range for χ2-distributed random variables and we
conclude that our theory is probably wrong.
For the standard χ2-test one compares Q2 µ with the expectation value of χ2 ,
which is equal to the number of degrees of freedom.
Q 2 µ = 〈χ 2 〉 ± √ 2n = n ± √ 2n
is a rough criterion for an acceptable goodness-of-fit, where too small a value of
Q 2 µ is indicative of possible hidden correlations between the data points or an overestimation
of the standard errors, whereas too high a value can be caused by an
underestimation of the standard errors or, in fact, by using a wrong model.
3.3.2 Using the χ 2 -distribution as a measure for significance
of deviation between two theoretical curves
We are faced with the task of comparing two simulated light curves, one of a ternary
lens system, the other of a related binary lens system. One common approach is
to random-generate artificial data around one of them and then consecutively fit
the two theoretical curves (theoretical meaning the non-random-scattered simulated
curve) to the data with some free parameters. Two χ2-values for the goodness-of-fit
will result. With these the so-called ∆χ2 = χ2 1 − χ2 2 is calculated. A threshold
value ∆χ2 thresh is, more or less arbitrarily, chosen. By keeping it rather large one
can try to account for systematic errors. ∆χ2 > ∆χ2 thresh then is the condition
for reported non-negligible deviation and, thus, detectability of the deviation. A

26 3.3. STATISTICAL ANALYSIS
detailed description and of the algorithm and its application has been presented by
Gaudi and Sackett (2000).
For two reasons we were led to consider other possible techniques. The first
being the unsatisfaction evoked by “feeding” our knowledge of the data distribution
to a random number generator and thereby “loosing” it in order to get an unbiased
χ2-sample, when at the same time we have all necessary information to calculate
a much more representative χ2-value. The second reason is the comparably high
computational effort required to produce artificial data and then re-fit. We have
developed a method to quantify significance of deviations between two theoretical
functions that avoids the steps of data simulation and subsequent fitting, by making
use of the definitions of statistic quantities and their well-known relations.
Assume, we had n data points on an observed triple-lens light curve. We want
to know, whether this set of data could be described by the binary-lens model.
The question is, could it be just random scatter of binary-lens data points, well
within the range of normally expected fluctuations? If the answer that we get, is
“No, it is very improbable that this is a random fluctuation of binary-lens data.”,
then we feel justified to say the deviation between triple lens data and binarylens
model is significant. Now, we do not have observational data, but we have
a simulated triple lens light curve. We will directly use the intrinsic information
about the data distribution that we would have had to put into the random data
generator otherwise. So we hypothesise, every simulated point on our triple-lens
light curve is in fact the mean µ t i of the distribution of a random variable Xi that
is distributed according to the Gaussian probability density function fi(xi) with a
standard deviation of σt i. The variance is then (σt i) 2 . We recall the definitions of
mean and variance, which we will need later on. For the random variable X with
probability density function f(x), they are in general
µ =
σ 2 =
∞
−∞
∞
−∞
x f(x) dx = 〈X〉 and
(x − µ) 2 f(x) dx = 〈(X − µ) 2 〉,
where we also introduced the notation 〈X〉 for the expectation value of X. To be
able to use the properties of the χ2-distribution we can create new random variables
Zi with mean µZi = 0 and σ2 = 1, Zi
Zi = Xi − µ t i
σt .
i
Then the sum of all Z2 i will be χ2-distributed. χ 2 n
∼
i=1
Returning to our original question, we now examine whether the Xi could be described
equally well with a binary-lens model µ b i. To be able to employ the χ 2 -test
Z 2 i

CHAPTER 3. METHOD 27
method, we introduce the new random variable
Q 2 n
Xi − µ
=
b i
i=1
This is the step, where we could simulate data in order to find a somewhat representative
value of Q2 , but instead we simply calculate what the mean value of all
possible Q2 would be. We use the now familiar definitions, to find
〈Q 2
n
Xi − µ
〉 =
b
2
i
=
i=1
n
i=1
1
(σ b i )2
σ b i
σ b i
2
.
(Xi − µ b i) 2
Here, we keep in mind that, in general, 〈x 2 〉 = 〈x〉 2 . Using the parameters µ t i and
σ t i of the distribution fi(xi) that belongs to our Xi, we reduce the equation by
calculating
〈Q 2 〉 =
=
=
=
=
=
=
n
i=1
n
i=1
n
i=1
n
i=1
n
i=1
n
i=1
n
i=1
1
∞
(σb i )2
∞
1
(σb i )2
∞
1
(σb i )2 −∞
∞
1
(σ b i )2
1
(σ b i )2
1
(σ b i )2
1
(σ b i )2
(xi − µ
−∞
b i) 2 fi(xi)dxi.
(xi − µ
−∞
t i + µ t i − µ b i) 2 fi(xi)dxi
(xi − µ t i) 2 + 2(xi − µ t i)(µ t i − µ b i) + (µ t i − µ b i) 2 fi(xi)dxi
(xi − µ
−∞
t i) 2 fi(xi)dxi + 2(µ t i − µ b i)
+(µ t i − µ b i) 2
∞
(σ t i) 2 + 2(µ t i − µ b i)
+(µ t i − µ b i) 2
fi(xi)dxi
−∞
∞
∞
xifi(xi)dxi − µ
−∞
t i
t
(σi) 2 + 2(µ t i − µ b i) µ t i − µ t t
i + (µ i − µ b i) 2
(σ t i) 2 + (µ t i − µ b i) 2 .
(xi − µ
−∞
t i)fi(xi)dxi
∞
fi(xi)dxi
−∞

28 3.3. STATISTICAL ANALYSIS
We used the definitions of µ t i and (σ t i) 2 and the property of the probability density
function fi(xi)dxi = 1. In our case σ t i = σ b i holds true without loss of accuracy,
and indeed, we simplify σi = σ for all i and argue that this does not pose a problem
as long as σ is chosen to be rather large. So we can further reduce to
〈Q 2 〉 =
n
i=1
(1 + 1
σ 2 (µt i − µ b i) 2 ) = n +
n
i=1
t µ i − µ b 2 i
.
σ
We now have calculated the mean value of Q 2 , which is the mean of all Q 2 possibly
resulting, when comparing the simulated binary-lens light curve to randomly scattered
triple-lens light curve points. It shall be our measure of deviation. Fortunately,
we can provide all remaining parameters:
• n is the number of degrees of freedom, equal to the number of compared data
points.
• σ is the assumed standard error of observations.
• (µ t i − µ b i) equals the difference between the two compared light curves squared,
which we already use for our least square fit.
probability density →
χ 2 probability density function
〈χ 2 〉 = degrees of freedom
X →
〈Q 2 〉
P ≤ 0.01
Figure 3.7: The χ 2 probability density function, plotted here only for illustration
with n = 20 degrees of freedom and an arbitrary 〈Q 2 〉. For n = 400, which is
closer to the number of the degrees of freedom in our comparison, the central limit
theorem takes effect and the curve will very closely resemble a Gaussian distribution.
If P (X ≥ 〈Q 2 〉) ≤ 1%, we consider the deviation between the two compared curves
to be significant.

CHAPTER 3. METHOD 29
Significance of deviation
Similarly to the procedure described in section 3.3.1, we are looking for a probability
P to decide if the deviation between our two examined light curves is significant. By
taking 〈Q 2 〉 as χ 2 we can evaluate the cumulative distribution function - either by
looking into a χ 2 -table or, more comfortably, by using an already existing Fortran
code (such as the one published by Davies (1980)) - to find the corresponding probability
for any given χ 2 -distributed random variable X to be as large as or larger
than 〈Q 2 〉
P (X ≥ 〈Q 2 〉) = 1 − Fn(〈Q 2 〉)
If this probability is very small, 〈Q 2 〉 is outside the expected range for a χ 2 -distributed
random variable Q 2 . In that case Q 2 is obviously not χ 2 -distributed with n degrees
of freedom. We must conclude that there is a significant deviation between the triple
lens model light curve and the binary-lens model light curve, which we wanted to
compare.
Choosing a threshold probability, we say we have a significant deviation between
our two curves, if and only if the mean value 〈Q 2 〉 is so high that the probability P
for any random variable X being larger or equally large is less than 1%. We interpret
this to say, only if the probability for a given triple lens light curve with independent,
normally distributed data points to be random fluctuation of the compared binarylens
light curve is less than 1%, we consider it to be principally detectable.
Known limitations
There are two known limitations to our method.
• Systematic errors are not accounted for.
A possible solution could be to add a further term to 〈Q 2 〉 as follows
〈Q 2 new〉 = 〈Q 2 〉 − n
σsys
2
, (3.2)
σ
where σsys can be assumed to lie in the few percent region for realistic data.
• Since we avoid refitting the binary-lens model light curve and always compare
to the one we got as best-fit to the exact (simulated) triple-lens model light
curve, we are over-estimating 〈Q 2 〉. This effect vanishes for n → ∞ and
is negligible for a sufficiently large n. The high sampling frequency we use
ensures that n is always large enough (n > 250 in all cases).

30 3.3. STATISTICAL ANALYSIS

Chapter 4
Choice of Scenarios
This chapter presents the assumptions that we used for our simulations. We open
with a detailed discussion of the astrophysical parameter space that is available
for simulations of a microlensing system consisting of star, planet and moon. We
separate the parameters into those that are relevant for the creation of magnification
patterns (section 4.1), and those necessary for the physical interpretation of the
microlensing light curves (section 4.2). By choosing the most probable or in some
other way most reasonable value for each of the parameters (the reasons for our
choices are explained within the first two sections of this chapter), we create a
standard scenario (section 4.3) that all other parameters are weighted against during
the analysis.
host star
qP S = MPlanet
MStar
dP S
qMP = MMoon
MPlanet
φ
planet
moon
Figure 4.1: For creating the magnification maps, five parameters have to be fixed.
They are the relative masses qP S and qMP as well as the angular separations in the
lens plane dP S and dMP and the position angle of the moon φ. These parameters
determine the relative projected positions of the three bodies.
4.1 Parameters Relevant for Magnification Patterns
There are five parameters describing the lens configuration (see figure 4.1). They
are the mass ratios qP S = MPlanet
MStar and qMP = MMoon , then the angular separations
MPlanet
31
dMP

32 4.1. PARAMETERS RELEVANT FOR MAGNIFICATION PATTERNS
dP S, between planet and star, and dMP , between moon and planet, and as the fifth
parameter φ, the position angle of the moon with respect to the planet-star axis.
These five parameters are barely constrained by the physics of a three-body system,
even if we do require mass ratios very different from unity and separations that allow
for stable orbits. Therefore it is quite impossible for us to cover the whole zoo of
possible magnification patterns.
4.1.1 Mass ratio of planet and star
The mass ratio of planet and star qP S = MPlanet is chosen to match realistic settings.
MStar
Since we have declared an interest in planets (and not binary stars), we choose
a small value of qP S. Our standard value will be 10−3 which is the mass ratio
of Jupiter and the Sun, or could be a Saturn-mass planet around an M-Dwarf of
0.3 M⊙. Please refer also to the discussion of the stellar mass, section 4.2.3. At a
given projected separation between star and planet, this mass ratio determines the
size of the planetary caustic, with a larger qP S leading to a larger caustic. This is
illustrated in figure 4.2. We will also vary qP S to examine scenarios with mass ratios
corresponding to a Jovian mass around an M-dwarf and to a Saturn mass around
the Sun, respectively.
(a) qP S = 3.3 × 10 −3 (b) qP S = 10 −3 (c) qP S = 3.0 × 10 −4
Figure 4.2: The size of the planetary caustic depends on the mass ratio qP S of
planet and star. Shown are wide-angle magnification patterns, with a side length of
1.6 θE, that include the central caustic of the star as well as the planetary caustic.
From left to right the value of the planetary mass ratio decreases, and the size of the
planetary caustic decreases accordingly. All other parameters are frozen.
4.1.2 Mass ratio of moon and planet
To be classified as a moon, the tertiary body must have a mass considerably smaller
than the secondary. The standard case in our examination corresponds to the
Moon/Earth mass ratio of qMP = MLuna
MEarth = 10−2 . We are generous towards the
higher mass end, and include a mass ratio of 10 −1 in our analysis, corresponding
to the Charon/Pluto system, i.e. a double planet (or double plutoid, if you want).

CHAPTER 4. CHOICE OF SCENARIOS 33
Both examples are singular in the solar system, but we argue that more massive
moons are the more interesting ones anyway. A more massive moon can effectively
stabilise the obliquity of a planet, which is thought to be favouring the habitability
of the planet (Benn, 2001). The next highest mass ratio found in the solar system
is two orders of magnitude smaller: qMP = MTriton
MNeptune = 2 × 10−4 . At the low mass
end of our analysis we examine qMP = 10 −3 . See figure 4.3 for the three mass ratios
that we adopt.
There are analogies to the planet-star mass ratio. A higher value of the lunar
mass ratio leads to a higher deflection of light rays by the moon, and will produce a
larger sized lunar caustic. Consequently, the lunar caustic will cover a larger fraction
of the planetary caustic and be noticeable in a larger fraction of source trajectories.
One of our requirements is that the planet has already been detected with a caustic
crossing. It is just spelling out common sense to state that the probability to discover
a massive moon next to an already discovered planet is higher than to find a less
massive natural satellite of the planet.
(a) qMP = 10 −3 (b) qMP = 10 −2 (c) qMP = 10 −1
Figure 4.3: The moon mass determines the size and strength of the lunar caustic,
on which we have zoomed in. From left to right the lunar mass ratio is increased
from qMP = 10 −3 to 10 −1 , all other parameters remain constant. Distinctly different
caustic topologies are induced by the interference of lunar and planetary caustic.
4.1.3 Angular separation of planet and star
The angular separation dP S of a binary of star and planet will evoke a certain
topology of caustics, see figure 4.4. They gradually evolve from the close separation
case with two small triangular caustics on the far side of the star and a small central
caustic at the star position, to a large central caustic for the intermediate case, when
the planet is situated near the stellar Einstein ring, dP S = θE. If the planet is moved
further out, one obtains a small central caustic and a larger isolated, diamond-shaped
planetary caustic (figure 4.5(b)) that is very asymmetric and elongated towards the
primary mass in the beginning, but becomes more and more symmetric if the planet
is placed further outwards (figure 4.5(c)).

34 4.1. PARAMETERS RELEVANT FOR MAGNIFICATION PATTERNS
Figure 4.4: The topology of binary-lens
caustics (red) changes with the angular
separation d. The close, intermediate and
wide-separation case are depicted. The
black lines indicate the limits of each
regime. In our simulations we focussed
on the wide separation case. Figure taken
from Cassan (2008) with kind permission
of the author.
If there is a moon in a not too large separation from the planet, it will show its
influence in any of the caustics. But we have reason to focus our research on the
wide-separation case. Regarding the close-separation triangular caustics, we argue
that the probability to cross one of them is vanishingly small and the magnification
decreases as they move outwards from the star. 1 The intermediate caustic, on the
other hand, is well “visible” because it is always located close to the peak of the
Paczyński curve. There is a whole follow-up network (MicroFUN, Gould (2008a))
dedicated to so-called high-magnification events, meaning those with a source trajectory
passing very close to the star. But regarding the central caustic, we see the
problem that all massive bodies of a given planetary system affect the central caustic
with minor or major perturbations and deformations (Gaudi et al., 1998). There is
no general reason to expect that extrasolar systems are less densely populated by
planets or moons than the solar system. Since we are looking for very small deviations,
we would face the messy task of distinguishing the signature of the moon
from multiple other ones left by low-mass planets and possibly more moons.
The planetary caustic, on the other hand, is practically almost always evoked
by a single planet and interactions due to the close presence of other planets are
unlikely (Bozza, 1999, 2000). Planets with only one dominant moon are at least
common in the solar system (Saturn & Titan, Neptune & Triton, Earth & Luna).
Such a system would be the most straightforward to analyse in real data.
In our view, all of this makes a strong case for concentrating on the wideseparation
planetary caustic. For our standard case we choose a separation in units
of stellar Einstein ring radii of dP S = 1.3 θE.
1 Nonetheless, it could be potentially interesting to examine the influence of a satellite on them.
In principle, our method should work just as well.

CHAPTER 4. CHOICE OF SCENARIOS 35
(a) dP S = 1.1 θE (b) dP S = 1.2 θE (c) dP S = 1.3 θE
Figure 4.5: The caustic shape varies with the angular separation of star and planet
dP S. From left to right, the images illustrate the gradual change in topology from
a (highly asymmetric) central caustic to a smaller central caustic and a diamond
shaped planetary caustic. qP S is equal to 10 −3 .
4.1.4 Angular separation of moon and planet
As the moon by definition orbits the planet, there is an upper limit to the distance
between the two bodies. It must not exceed the distance between the planet and its
inner Lagrange point. We recall that in the gravitational potential formed by the
masses of star and planet, the inner Lagrange point is the saddle point between the
two bodies (see figure 4.6). The distance between the centre of the planet and the
inner Lagrange point is called Hill radius and is calculated as
rHill = aP S
MP
3MS
for circular orbits, with aP S as the semi-major axis of the planetary orbit. This can
be translated to “lensing parameters”, but the nature of the equation changes into
an inequality, because the semi-major axis aP S cannot be directly inferred from the
projected angular separation dP S that we can measure in a caustic model. Therefore,
we need to make assumptions for the position in orbit at which we are seeing the
planet. Fortunately, the angular separation at least gives us information about the
minimal orbital radius of the planet. The projected distance in the lens plane cannot
be larger than the physical distance. Thus we get
rHill = aP S
1
3 qP
1
3
S
1
3
with aP S ≥ dP SDL
(4.1)
and the Hill radius is not a strict constraint anymore.
Bodies in prograde motion with an orbit below 0.5 rHill can have long term stability.
For retrograde motion, the limit is somewhat higher at 0.75 rHill. For these
numbers, we refer to Domingos et al. (2006) and references therein, particularly
Hunter (1967).

36 4.1. PARAMETERS RELEVANT FOR MAGNIFICATION PATTERNS
Figure 4.6: Sketch of the gravitational potential
of a system of star and planet. A moon in stable
orbit is also shown. The five Lagrange points are
marked in green. “L1” marks the saddle point of
the potential between primary mass and secondary
mass and is called the inner Lagrange point.
Image Credits: NASA
A Saturn around an M-dwarf (MS = 0.3M⊙, qP S = 1.0 × 10 −3 ) has a planetary
Einstein radius θ P E = √ qP S θE of roughly 3 % of the stellar Einstein radius θE .
At a lens plane distance of DL = 6 kpc (and with a distance to the source plane
of DS = 8 kpc) that amounts to rP E = 0.06 AU. The Hill radius for this mass
configuration is 7 % of the semi-major axis aP S.
If the planet is located at an angular separation to the star of at least one stellar
Einstein ring radius, all prograde orbits with a radius of less than 3.5 % of θE will be
stable. For our standard case of dP S = 1.3 θE we get an increased (minimal) region
of stability with a 4.5 % of θE radius corresponding to 1.53 θEP or 0.09 AU, where
satellites can safely be assumed to be gravitationally bound by the planet.
(a) dMP = 0.8 θ P E (b) dMP = 1.0 θ P E (c) dMP = 1.2 θ P E
Figure 4.7: The images are zoomed in on the lunar perturbation of the planetary
caustic, with qP S = 10 −3 and qMP = 10 −2 . The topology of interaction varies with
the angular separation of moon and planet dMP , all other parameters remain fixed.
The side length of all three images is 0.055 θE, while the separation is given in units
of planetary Einstein ring radii, θ P E = √ qP S θE.
Finally, our choice of dMP is also motivated by a request to have caustic interaction
between lunar and planetary caustic, because only under this condition the
moon will give rise to light curve features that are distinctly different from those of
a low-mass planet. Microlensing of low-mass planets has been discussed elaborately

CHAPTER 4. CHOICE OF SCENARIOS 37
by previous authors, e.g. Bennett and Rhie (1996). Our standard case will therefore
have an angular separation of moon and planet of one planetary Einstein ring radius,
dMP = 1.0 θP E . We have to refer back to figure 4.3, to stress again that the shape of
the resulting interference depends on the lunar mass as well as the separation.
4.1.5 Position angle of moon with respect to planet-star axis
The last physical parameter necessary for determining the magnification patterns is
the position angle of the moon φ, as in figure 4.1. It is the only parameter we do not
fix for our standard scenario. Instead, we vary it in steps of 30 ◦ to complete a full
circular orbit of the moon around the planet. By doing this, we are aiming at getting
complete coverage of a selected mass/separation scenario. Admittedly, it is not to be
expected that we will ever have an exactly head-on view of a perfectly circular orbit,
but it serves well as a first approximation and we will not loose much generality with
this assumption. Furthermore, an elliptical orbit would be constructed as the sum of
angular separations that vary depending on the position angle. We have simulated
variations in separation, but not yet constructed elliptical orbits.
Caustic interferences of the two caustics of moon and planet can be complex and
we have not yet succeeded in working out a thorough classification. We are considering
only orbits of the moon around its host planet which have a circular projection
centred at the planet. These orbits are symmetric along the planet-star axis, and
if we mirror the position of the moon along this axis, the resulting magnification
pattern will also be mirrored. For the axis perpendicular to the aforementioned
one and crossing the centre of the planetary caustic, this kind of symmetry only
holds approximately. As long as the planet-star separation is sufficiently large, lunar
magnification lines will have a similar appearance on either side of the planet.
Appendix chapter A contains the analysed magnification patterns and visualises
how the position of the moon affects the planetary caustic.
4.2 Parameters Relevant for Light Curve Analysis
In this section further parameters are discussed: those necessary for determining
the physical properties of the inspected lensing system, and also those required for
simulating realistic light curves. The first three parameters discussed, i.e. distance
to the source star DS, distance to the lensing system DL and the mass of the lensing
star MStar set the “big picture” of Galactic microlensing and serve to convert angular
separations (Einstein radii) to lengths (Astronomical Units). Source size RSource and
sampling rate constrain the shape and sampling density of the simulated light curves.
The observational standard error σ is needed for the final analysis of the light curves.

38 4.2. PARAMETERS RELEVANT FOR LIGHT CURVE ANALYSIS
4.2.1 Distance to source plane
All operating planet-searching Galactic microlensing surveys (section 2.3) direct
their telescopes primarily towards the Galactic centre. The highest density of stars
is found in the Galactic bulge, this fact allows high-number statistics thereby raising
the probability for interesting events. We assume (and follow numerous preceding
examples in the microlensing literature) that our source stars will be positioned
around the centre of the Galactic and are using throughout this work a source star
distance of DS = 8.0 kpc. This value is in agreement a recent paper on the Galactic
structure that sets the distance from the Earth to the Galactic centre as 7.9 kpc
(Groenewegen et al., 2008).
4.2.2 Distance to lens plane
The lens plane must surely lie between observer and source. Our knowledge of the
Galactic structure tells us that a lens closer to the region of higher star density, i.e.
the Galactic bulge, is more probable. A typical choice is DL = 6 kpc, and we adopt
this value. One also finds in the literature a value of DL = 0.5 DS = 4 kpc. We point
out that this is relevant for the apparent source size and thus for the resolution of
features in the light curves. For DL = 6 kpc vs. DL = 4 kpc the difference is a factor
of 1.7 in apparent size.
4.2.3 Mass of lensing star
Figure 4.8: Lens star distribution in mass or spectral type: M-dwarfs are the most
abundant type of star towards the Galactic bulge. The probability density function is
plotted with a thin line and corresponds to the scale on the left axis, the cumulative
distribution (thick line) has the scale on the right axis. Figure taken from Dominik
et al. (2008) with kind permission of the author. Look there for a more detailed
discussion of the plot and references to the underlying Galactic model.
The mass of the lens star, i.e. the primary lens, MStar sets the scale for the

CHAPTER 4. CHOICE OF SCENARIOS 39
whole lensing system. All other masses enter the equation in units of the primary
mass. The Einstein radius, see equation (2.3), is the natural measure for angular
separations in our system and is proportional to the square root of this mass. A
given magnification pattern is produced with a unit mass and scales with the mass
of the primary lens (while lengths scale with √ MStar as the Einstein radius).
It is not at all straightforward to obtain the mass of the lensing star through
microlensing observations. It has been achieved in individual cases through a combined
measurement of parallax and the Einstein radius θE , summarised in Gould
(2008b), but lens mass determination is not an easy task. For our simulations we
rely on statistic drawn from Galactic models (figure 4.8). We assume our primary
mass to be an M-dwarf star with a mass of MStar = 0.3M⊙, because that is the
most abundant type of star to be found toward the Galactic bulge. We derive the
corresponding Einstein radius θE(DS, DL, MStar) = 0.32 milliarcseconds.
4.2.4 Source size
−∆mag
3
2
1
0
Triple Lens Binary Lens Figure 4.9: For this pair of
light curves a source size of one
solar radius has been chosen:
Rsource = R⊙. The triple-lens
0 0.1 0.2 0.3 0.4
t/tE
0.5 0.6 0.7
light curve is obtained from the
magnification pattern in figure
4.10. There is a noticeable difference
between the triple-lens light
curve and the binary-lens light
curve and the planetary causticcrossing
features are pronounced
in both.
The source size influences the “time resolution” of a light curve. For every light
curve point we have to integrate over the projected size of the source. Finer caustic
structures can easily get lost in this process. Unfortunately, for our source star
assumptions we have to take into account not only the abundance function of the
Galactic stellar population, but also the luminosity of a given stellar type. M-dwarfs
are in all probability too faint to be seen at a large distance by our telescopes – even
if they are lensed and magnified. Therefore giant stars are more probable as source
stars in our gravitational lensing setting. But the finite source size constitutes
a serious limitation to the discovery of extrasolar moons. In fact, Han and Han
(2002) stated that detecting satellite signals in the lensing light curves will be close
to impossible, because the signals are seriously smeared out by the severe finitesource
effect. They tested various source sizes (and planet-moon separations) for an
Earth-mass planet with a Moon-mass satellite. They find that even for a K0-type
source star any light curve modifications caused by the moon are washed out. Those

40 4.2. PARAMETERS RELEVANT FOR LIGHT CURVE ANALYSIS
effects remain well visible when simulating the event with a hot white dwarf source
- nature’s approximation to a point source.
Last year Han published a new take on microlensing by satellites of extrasolar
planets (Han, 2008). With increased lens masses and using a solar source size
(RSource = R⊙, i.e. a source with the radius of a G2 star), he finds that “nonnegligible
satellite signals occur” in the light curves of planets of 10 to 300 Earth
masses “when the planet-moon separation is similar to or greater than the Einstein
radius of the planet” and the moon has the mass of Earth. After having tested
different source sizes (compare figure 4.11), we also decide to use a solar sized source
(RSource = R⊙) for our standard case, but also present results for RSource = 2 R⊙
and 3 R⊙. In terms of stellar Einstein ring radii this corresponds to 1.8 × 10 −3 θE,
3.6 × 10 −3 θE and 5.4 × 10 −3 θE.
Figure 4.10: Magnification pattern showing a planetary caustic through which a
source trajectory has been cut. Here, a solar sized source is indicated in the lower left
corner. The resulting light curve is shown in figure 4.9. Depending on the assumed
source size different light curves result, as shown in figures 4.11(a) to (d).

CHAPTER 4. CHOICE OF SCENARIOS 41
−∆mag
3
2
1
Triple Lens Binary Lens
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
t/tE
(a) Rsource = 2R⊙. While not as pronounced
as with a solar sized source, the tell-tale signs
of a caustic crossing are still visible.
−∆mag
3
2
1
Triple Lens Binary Lens
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
t/tE
(c) Rsource = 10R⊙. The caustic crossing features
are no longer distinguishable.
−∆mag
3
2
1
Triple Lens Binary Lens
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
t/tE
(b) Rsource = 5R⊙. The planetary double
peak is still noticeable, but the caustic crossing
features are starting to wash out.
−∆mag
3
2
1
Triple Lens Binary Lens
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
t/tE
(d) Rsource = 20R⊙. The planet still reveals
itself as a bump on the slope of the Paczyński
curve.
Figure 4.11: Testing the effect of an increased source radius Rsource. Compare also
figure 4.9.
4.2.5 Sampling rate
When observing a gravitational lensing event, data points are acquired by taking
frames of the object. Typical exposure time has a duration of 30 to 300 seconds.
Higher observing frequency equals better coverage of the resulting light curve. A
constant observing rate facilitates understanding of the planetary population. In real
observations sampling is constrained by numerous factors. Usually, one or two dozen
events must be covered at a reasonable rate during follow-up observations, because
it is never certain which event will evolve to show those interesting anomalies that
we look for. Ground-based observations will always be weather-dependent. Then
there is the Pacific gap; more of a problem, when the nights are getting shorter in
the southern hemisphere.
A normal microlensing event is seen as a transient brightening that lasts for
about a month. A planet will alter the light curve for hours or days; a small planet
or a moon only for minutes or hours. This duration is inversely proportional to the

42 4.3. THE STANDARD SCENARIO
transverse velocity v⊥ with which the lens moves relative to the line of sight or the
relative proper motion µ⊥ between the background source and the lensing system.
We fix this at v⊥ = 200 km/s or µ⊥ = 7.0 mas/year.
In our simulations a realistic observing rate is mimicked by evaluating only a
subset of the available light curve data. When fixing the sampling rate for the light
curve simulations, we are navigating a fine line where the monitoring frequency
still allows for detections of extrasolar moons, but we are not too far advanced
from today’s state of technology. We have to find a good compromise between too
optimistic assumptions and reality. We make concessions to an increase in detections
by assuming a constant rate. We sample our triple lens light curve progressing along
the source trajectory in equal sized steps. Assuming a constant relative velocity
between source star and lens system, this translates to equal-spacing in time. Aiming
for a rate of one frame every 15 minutes for about 50 hours, we choose a step size of
6×10 −4 θE. To see the effect of a decreased sampling rate, we have also examined the
standard scenario (see section 4.3) with a step size of 9 × 10 −4 θE and 12 × 10 −4 θE,
translating to roughly 22 minutes and 30 minutes respectively.
4.2.6 Photometric uncertainty of observed data
The standard error σ of observed data in microlensing can vary significantly. Values
between 0.01 and 0.05 mag seem to be most common after final reduction, compare
with data plots in recent event analyses, e.g. Gaudi et al. (2008) or Dong et al.
(2008). But differently calibrated telescopes under different weather conditions (and
possibly with different data reduction tools) deliver data with different uncertainties.
In our analysis we ignore these differences and choose a fixed σ-value for all sampled
points on the triple-lens light curve.
The photometric uncertainty directly enters the statistical evaluation of each
light curve as described in section 3.3.2. Since it required only negligible computation,
we have drawn results for an (unrealistically) broad σ-range from 0.5 mmag
to 0.5 mag. σ = 0.5 mmag has recently been reached with high-precision photometry
of an exoplanetary transit event at the Danish 1.54 m telescope at ESO La
Silla (Southworth et al., 2009), which is one of the telescopes presently dedicated to
Galactic microlensing, but this low uncertainty is surely not available for standard
microlensing observations.
In our discussion of the results, we regard only the more realistic range from
σ = 5 to 100 mmag. We set σ = 20 mmag to be our standard for the comparison of
different mass and separation scenarios.
4.3 The Standard Scenario
Let us summarise the standard scenario for our analysis. The grand scene is set
with the Galactic distance scale. We expect the background source star to be in
the Galactic bulge at a distance of DS = 8 kpc from an observer on Earth. The

CHAPTER 4. CHOICE OF SCENARIOS 43
lens plane with the three lensing masses of lens star, planet and moon we assume
to lie at a distance of DL = 6 kpc. This planetary system moves with a relative
proper motion µ⊥ = 7 mas/year with respect to the line-of-sight between observer
and source. Our knowledge about the stellar population of our Galaxy motivates
us to choose a mass of MStar = 0.3 M⊙ for the lensing star, corresponding to an
M-dwarf. From these choices, we derive an Einstein radius of θE = 0.32 mas and
the physical Einstein length in the lens plane ξE = 1.96 AU. The so-called Einstein
time, the crossing time of the Einstein ring radius tE then is tE 17 days.
We simulate a scenario where an extrasolar planet of Jovian mass (qP S = 10 −3 )
at an angular separation of 1.3 θE from the host star, i.e. a projected semi-major axis
of aP S = 2.55 AU, has a natural satellite with a moon/planet mass ratio qMP = 10 −2
as that of our Moon and Earth, equivalent in this scenario to the mass of Earth.
The extrasolar moon has a circular orbit around the planet with a radius of one
planetary Einstein ring radius θ P E = √ qP SθE which translates to aMP = 0.06 AU.
The orbit is approximated with twelve different positions of the moon, the position
angle is varied in steps of 30 ◦ .
We analyse the resulting planetary caustics with a grid of source trajectories, see
figures A.2 and 3.6. We choose a source size of one solar radius for the background
star, RSource = R⊙. We sample the triple-lens light curves at constant intervals of
6 × 10 −4 θE, corresponding to time intervals of 15 minutes. For the observational
photometric uncertainty we assume a σ of 20 mmag.
parameter standard value
DS 8 kpc
DL 6 kpc
MStar
θE
0.3 M⊙
0.32 mas
µ⊥ 7 mas/year
tE 17 days
qP S 10−3 θP E 0.01 mas
dP S 1.3 θE
qMP 10 −2
dMP 1.0 θ P E = √ qP SθE
RSource R⊙
sampling 1 frame per 15 minutes
σ 20 mmag
Table 4.1: Table of the standard scenario values.

Chapter 5
Results: Detectability of
Extrasolar Moons in Microlensing
In this chapter, the numerical results of our work are presented and a first interpretation
is made. We begin our analysis by considering the output for a single
magnification pattern in section 5.1. We are then prepared to draw conclusions that
cover complete physical scenarios (section 5.2). We perform a sensitivity analysis,
where we always vary just one parameter to uncover the dependence of the detection
probability on this parameter. We finish the present chapter by analysing the
results for possible limitations of the method and sketching applicable solutions.
5.1 The moonlight Output for a Single Magnification
Pattern
We take the standard case (section 4.3) with the moon at the position φ = 90 ◦
and have moonlight count the number of light curves in which it deems the moon
detectable, out of the total of 228 light curves drawn from the grid of 228 different
source trajectories depicted in figure 3.6. The output is shown in table 5.1. The
σ in mmag significantly deviating
light curves
of a total of 228
detectability
5 211 92.5 %
10 154 67.5 %
20 72 31.6 %
50 23 10.1 %
100 6 2.6 %
Table 5.1: Output for a single magnification pattern (standard scenario A.2, φ =
90 ◦ ), see figure 3.6 for the analysed source trajectories.
45

46 5.2. RESULTS FOR SELECTED SCENARIOS
percentages tell us the detection probability of the moon in light curves that have
crossed the caustic of the host planet, as a function of the photometric uncertainty.
From table 5.1, we can see that the detectability of the moon is 31.6 % provided
we have an observational uncertainty of σ = 20 mmag. A rough thumb-estimate
based on figure 3.6 shows that roughly 30 % of the source tracks pass through or
close to the lunar caustic. So the result seems plausible. This is also valid for the
σ = 50 mmag and σ = 100 mmag results. Obviously, with a higher photometric
error, one is less sensitive to small deviations as caused by the moon. In particular,
σ = 100 mmag ⇒ 2.6 % detectability contains the information that with an error of
0.1 mag it is improbable to detect the moon through microlensing.
Careful interpretation is necessary, though, regarding the lower σ-values. It is
to be expected that the sensitivity to moons rises further as the lunar perturbation
deforms all parts of the planetary caustic, not only in the direct vicinity of the lunar
caustic itself. But we caution that the sensitivity to limitations of the method rises
as well. The percentage of 92.5 % for σ = 5 mmag seems very high. We discuss this
further in section 5.3.
5.2 Results for Selected Scenarios
To make a statement about a physical mass and separation setting, we combine
the results for 12 magnification patterns following a whole lunar orbit with φ =
0 ◦ , 30 ◦ , . . . , 330 ◦ . 5 of these are mirror-symmetric, so are produced and evaluated
only once but counted twice. We assume that we can interpolate for the intermediate
position angles, since all changes in caustic topology occur gradually.
5.2.1 Standard scenario
Percentages of 12 individual magnification maps with varying φ are combined to
cover the standard scenario (compare figure A.2). The scenario is described in
depth in section 4.3.
σ in mmag detectability
5 88.8 %
10 58.8 %
20 29.9 %
50 12.3 %
100 1.5 %
From this table we get to know that 29.9 % of the light curves are deemed to show
detectable signs of the moon. This seems plausible, if you recall our requirements:
• The host planet of the satellite has been detected.
• A small source size is required.

CHAPTER 5. DETECTABILITY OF EXTRASOLAR MOONS 47
• The moon is massive for a satellite. Bodies of Earth-mass are detectable with
state-of-the-art microlensing, be they planets or moons.
• We have assumed data light curves with a constant sampling rate of one frame
per every 15 minutes for about 50 to 70 hours. Constant sampling means
several 24-hour night shifts 1 with no bad-weather gaps.
• No systematic errors are accounted for. 2
Not surprisingly the detection rate increases with higher photometric sensitivity
and decreases with a larger uncertainty. We conclude from our results, that at a
σ-level of 20 mmag or above, the detection of a moon depends on a crossing of the
lunar caustic. With lower observational errors the method becomes sensitive to the
barely noticable deformation of the planetary caustic, but probably also to intrinsic
limitations.
5.2.2 Changing the angular separation of planet and moon
We start changing the angular separation of planet and moon. This following table
lines up the changing detectability for a varying angular separation dMP . All other
parameters as in standard scenario (section 4.3).
dMP in θ P E
detectability
0.8 23.1 %
1.0 29.9 %
1.2 27.7 %
1.4 24.7 %
With the chosen range from dMP = 0.8 θP E to 1.4 θP E , we are still mostly within
the regime of caustic interactions. We find that the percentages of successful “de-
tections” lie close to each other at a 20 − 30 % level. One can make out a peak
arounddMP = 1.0−1.2 θP E which might be an analogy to the star-planet binary case,
but in order to confirm that, further and denser analysis results would be desirable.
5.2.3 Alteration of the mass of the moon
This table lines up the changing detectability for a varying mass ratio qMP of lunar
and planetary mass. All other parameters as in standard scenario (section 4.3).
qMP
detectability
10 −3 1.4 %
10 −2 29.9 %
10 −1 82.7 %
1 Term coined by Penny D. Sackett (2001).
2 It is highly non-trivial to estimate the influence of different sources of systematic errors.

48 5.2. RESULTS FOR SELECTED SCENARIOS
The mass ratio of moon and planet influences the detection probability quite dramatically.
Though, admittedly, the input parameters we chose differ quite dramatically
as well. It is informative to compare the magnification patterns for the different mass
settings, figures A.5 and A.6. We are confident about the plausibility of the results.
The steep rise suggests it would be advisable to analyse intermediate values as well,
in order to further constrain the dependence of detectability on the moon-planet
mass ratio.
5.2.4 Changing the planetary mass ratio
This table lists the changing detectability for a varying ratio of planet mass and star
mass qP S. Since we leave the lunar mass ratio qMP unchanged, the absolute mass of
the moon scales accordingly with the planetary mass ratio. The lunar caustic covers
the same percentage of the planetary caustic in all three cases.
qP S
detectability
3.0 × 10 −4 9.6 %
1.0 × 10 −3 29.9 %
3.0 × 10 −3 57.2 %
The configurations examined here should be regarded as a test of our programme.
The size of the planetary caustic is proportional to the planet-star mass ratio. In
spite of this, we did not adjust the grid parameters of the grid of source trajectories.
With a lower mass ratio, less light curves cut through the planetary caustic. The
ones passing outside should in almost all cases not be influenced by the moon. For a
decreasing mass of the planet, the source size increases relative to the caustic size, so
finer features will be blurred out more in the case of a smaller planet. In figure 5.1,
Detection probability in %
100
80
60
40
20
qP S = 3.0 × 10 −3
qP S = 1.0 × 10 −3
qP S = 3.0 × 10 −4
0
0.001 0.01
σ in mag
0.1
Figure 5.1: The detectabilities
for different planet mass ratios
are plotted as a function
of the photometric uncertainty
σ. They all seem to follow
the same pattern with different
offsets. This was expected,
as we used the same grid parameters
to draw light curves,
with the consequence that for
qP S = 3.0×10 −4 significantly less
light curves cross the planetary
caustic.
the detection probabilities are plotted as a function of the observational standard
error σ. Plotted are the values from table B.3. As we would expect, the detection

CHAPTER 5. DETECTABILITY OF EXTRASOLAR MOONS 49
efficiency at a given σ is proportional to the planetary caustic size. Overall we find
that for all three cases the function follows mostly the same pattern, even if shifted
in absolute value.
5.2.5 Increased separation between star and planet
We have also tested changing the angular separation parameter
dP S in θE detectability
1.3 29.9 %
1.4 33.2 %
There is a slight rise in detectability for a larger planet-star separation dP S. But this
difference in detectability for the two examined values dP S = 1.3 θE and dP S = 1.4 θE
is not pronounced enough to show clear trends. The size of the planetary caustic
decreases with growing separation, but this is counterweighted by the increasing
influence of the lunar mass. As the two bodies are further seperated from the
primary lens, their caustic behaviour more and more resembles a binary case.
5.2.6 Different sampling rates
This table lines up the changing detectability for different sampling rates.
sampling
step size in
θE
corresponding to
time intervals in
minutes
detectability
6 × 10 −4 15 29.9 %
9 × 10 −4 22 27.2 %
12 × 10 −4 30 18.0 %
A lower sampling frequency does lower the detection probability, but we had expected
to see a more pronounced difference. Between the step size values of 6×10 −4
and 9 × 10 −4 θE there is no significant difference, but doubling the time interval for
taking data points decreases the probability to 60 % of the old value.
5.2.7 Source size variations
We have simulated light curves of the standard scenario with three different source
star radii. They correspond to a star the size of our Sun and stars with RSource =
2 R⊙ and 3 R⊙, respectively.

50 5.3. POSSIBLE LIMITATIONS AND SOLUTIONS
RSource in θE corresponding to
RSource in R⊙
detectability
1.8 × 10 −3 1.0 29.9 %
3.6 × 10 −3 2.0 18.9 %
5.4 × 10 −3 3.0 14.0 %
The larger source blurs out the sharp caustic-crossing features of a light curve. On
the other hand, this leads to a broadening of those peaks which can have a positive
effect since it allows for sampling of the peak over a larger time span.
5.3 Possible Limitations and Solutions
It is noteworthy, and in fact needs further attention, that at a σ of only 0.5 mmag
(which has not been used in the presented analyses) all detectability counts closely
approach 100%. While theoretically it is conceivable that the deformation of the
binary-lens magnification map, which is introduced with a third mass, will always be
detectable if the error is chosen small enough, we are led to regard it as a sign that, at
this low error level, intrinsic limitations of the applied method gain dominance. We
suspect that the “brightness” resolution of the magnification patterns is too coarse.
This will have to be examined. The minimal intervall in magnitude between two
pixels could be of the order of 0.1−1 mmag. In that case, the intrinsic imprecision of
the magnification value will prevent low χ 2 -values when fitting two light curves, i.e.
they will be considered to differ significantly because of the non-continuous nature
of the magnification patterns.
Analogously, this problem can arise due to the step size between binary curve
points which might also be too large to enable a fitting at this extremely low σ. We
have tried to account for this by using a step size 8 times smaller than the sampling
of the triple-lens light curve. This should be sufficiently dense, but we actually did
test increased sampling rates of the binary-lens light curve. This has been tested in
order to quantify how the refinement of the fitting influences the final results.
sampling
step size in
θE
corresponding to
time intervals in
minutes
detectability
7.50 × 10 −5 1.9 29.9 %
3.75 × 10 −4 0.9 29.4 %
We notice a very slight improvement in fitting. Less light curves are considered to
have noticeable lunar deviations, but the difference in detectability is not significant.
So we argue that the sampling rate of the binary lens light curve is chosen to be
sufficiently dense to be a good substitute for a continuous curve.
It is absolutely non-trivial to estimate the influence of different sources of systematic
errors. As is their nature, they can work in both ways: raising or lowering

CHAPTER 5. DETECTABILITY OF EXTRASOLAR MOONS 51
the detectability. To be on the safe side, one can globally raise the limit for a
claimed detection, as shown in section 3.3.2, equation (3.2). But we refrained from
adding an arbitrary element to the analysis. Instead we emphasise that our results
are drawn from sound assumptions and offer a realistic, albeit optimistic, view on
the possibilities in the microlensing of extrasolar moons.
While we are not able to exactly quantify the errors introduced by our simplifications,
we know for sure that all our simplifications can only lead to an overestimation
of the detection probabilities. For moderate values of the photometric uncertainty
(i.e. 20 mmag or larger) these overestimations seem to be slight. But in any case
we have calculated definite upper limits on the lunar detectability in microlensing.

52 5.3. POSSIBLE LIMITATIONS AND SOLUTIONS

Chapter 6
Conclusion
We showed that massive extrasolar moons can be detected through the technique of
Galactic microlensing. From our results can be concluded that the detection of an
extrasolar moon – under favourable conditions – is within close reach of available
observing technology. Extrasolar moons have not yet been detected, but based on
studies of planet formation we would hypothesise that Earth-mass moons around
Saturn-mass planets do exist and will be observed. An observing rate of one frame
per 15 minutes is frequent, but not out of question, also the required photometric
uncertainty of 20 mmag can be met with today’s telescopes. But if, for example,
the source is too large to resolve any planetary caustic features, this fact constitutes
a physical limitation to what is detectable and effectively prevents any detection of
a moon. So, in order to find moons it is crucial to be able to monitor small sources,
which equals the need to monitor low-luminosity stars.
This work started out with the character of a feasibility study, but it has been extended
to provide detection probabilities for selected scenarios. Through quantifying
these probabilities we have achieved the stage, where they can be compared against
data or other simulation techniques. Furthermore, we have systematically probed
the available parameter space. We have identified the influence on the detection
rates of almost all individual parameters.
While a complete listing of observational requirements for the detection of exomoons
in general is beyond the scope of this work, we have succeeded in placing
strict upper limits on the detectability of the moons in the different cases studied.
If we assume, for the sake of the argument that systematic errors are negligible, we
know that if the conditions summarised in the standard scenario are met, statistically
we will discover the moon in 30% of the observations at most. This knowledge
would enable us to infer a tentative census of massive extrasolar moons.
Future Prospects
Our simulations can be refined in some ways. The next step would be to perform a
probabilistic inspection of lunar orbits and then simulate inclined orbits accordingly.
Then it is still desirable to enhance the fitting procedure. We can think of two
53

54
further parameters to include in the fitting procedure, without requiring the timeconsuming
generation of additional magnification patterns. One would be to allow
for a relative offset in magnitude (as is reality in observations since most telescopes
do not perform absolute photometry with sufficient precision), the other would be
to free the time scale of the event, i.e. have the fitting procedure dynamically adjust
the sampling step size of one curve to achieve a lower least-square. With improved
fitting, we would be able to lower the upper limits, thereby acquiring an increased
certainty for inferring population statistics.
While possibly not of direct astrophysical interest, it would be advantageous
to enhance the time-performance of our scenario-analysing programme moonlight.
Faster overall processing would enable to examine high-resolution magnification
maps and also more efficient sampling of the whole parameter space. Both of which
would help to work out the observational requirements to discover extrasolar moons
in more detail. For this it is also necessary to test the parameter space, not only
one-dimensionally, as has been done in this work by raising or lowering a single
parameter, but also multi-dimensionally to identify the dominant influences.
It should be possible and might be insightful to apply our method – as it is – to a
different topology of the planetary caustic. However regarding the other two possible
topologies, we reason that the detection probability will be vanishingly small. In
the close-separation case because the caustic size decreases significantly, and in the
intermediate case because real data will always show perturbations of all massive
bodies of the planetary system, so both do not look very promising.
Finally, the application to observational data is required to discover extrasolar
moons. With some parallels to the method we presented for quantifying the
detectability of lunar deviations, modelling of lunar events can be conducted by
searching magnification patterns for best-fit model light curves.
We conclude by stating that it is possible to detect extrasolar moons through
patient and highly precise monitoring of anomalous Galactic microlensing events.

Appendix A
Analysed Magnification Patterns
A note about the following pages: They are meant to serve as a visualisation of
what we analysed, as well as a documentation. On each page, we see one physical
scenario, determined by two mass ratios and two angular separations. The position
angle of the moon with respect to the planet-star axis φ is varied in steps of 30 ◦ ,
starting with φ = 0 ◦ in the upper left corner, and moving the moon clockwise until
φ = 330 ◦ in the lower right. All axis-symmetric magnification patterns were only
produced and analysed once, and the result counted twice for the total.
The shade of grey-scale corresponds to the magnification a point source would
experience at that position. The scaling runs from white equal unit magnification
or ∆mag = 0 to black equal ∆mag = −4. Larger magnifications have been cut-off
to show the best contrast. All maps have side length of 0.1 θE.
Overview
The table offers an overview of the presented magnification maps. For each set of
maps the standard parameters are used, excepting one variable that is lowered or
raised to see the effects this has. The standard scenario itself is shown in A.2.
parameter lower values map standard higher values map map
dMP in θ P E 0.8 A.1 1.0 1.2 A.3 1.4 A.4
qMP 1.0 × 10 −3 A.5 1.0 × 10 −2 1.0 × 10 −1 A.6
qP S 3.0 × 10 −4 A.8 1.0 × 10 −3 3.0 × 10 −3 A.7
dP S in θE 1.3 1.4 A.9
55

56
Figure A.1: Small angular separation of moon: dMP = 0.8 θ P E

APPENDIX A. ANALYSED MAGNIFICATION PATTERNS 57
Figure A.2: Standard: dMP = 1.0 θ P E

58
Figure A.3: Big angular separation of moon: dMP = 1.2 θ P E

APPENDIX A. ANALYSED MAGNIFICATION PATTERNS 59
Figure A.4: Large angular separation of moon: dMP = 1.4 θ P E

60
Figure A.5: Small moon mass. qMP = 10 −3

APPENDIX A. ANALYSED MAGNIFICATION PATTERNS 61
Figure A.6: Large moon mass. qMP = 10 −1

62
Figure A.7: Large planet mass: Jupiter around M-Dwarf. qP S = 3 × 10 −3

APPENDIX A. ANALYSED MAGNIFICATION PATTERNS 63
Figure A.8: Small planet mass: Saturn around the Sun. qP S = 3 × 10 −4

64
Figure A.9: Large angular separation of planet: dP S = 1.4 θE

Appendix B
Numerical Results
In this appendix chapter, the averaged output for every examined scenario is given,
i.e. the detection probabilities of the moon as a function of the photometric uncertainty
σ for the different lens constellations (references to the corresponding maps
in parentheses) averaged over the twelve lunar positions φ = 0 ◦ , 30 ◦ . . . 330 ◦ . For a
discussion of the results see chapter 5.
(a) dMP = 0.8 θ P E (A.1)
σ/mmag detectability
5 77.4 %
10 43.5 %
20 23.1 %
50 4.1 %
100 0.1 %
(c) dMP = 1.2 θ P E (A.3)
σ/mmag detectability
5 83.4 %
10 51.0 %
20 27.7 %
50 10.4 %
100 3.6 %
(b) dMP = 1.0 θ P E (A.2)
σ/mmag detectability
5 88.8 %
10 58.8 %
20 29.9 %
50 12.3 %
100 1.5 %
(d) dMP = 1.4 θ P E (A.4)
σ/mmag detectability
5 80.8 %
10 51.4 %
20 24.7 %
50 8.6 %
100 2.0 %
Table B.1: Results for varying angular separation dMP .
65

66
(a) qMP = 10 −3 (A.5)
σ/mmag detectability
5 23.6 %
10 7.7 %
20 1.4 %
50 0.0 %
100 0.0 %
(b) qMP = 10 −2 (A.2)
σ/mmag detectability
5 88.8 %
10 58.8 %
20 29.9 %
50 12.3 %
100 1.5 %
(c) qMP = 10 −1 (A.6)
σ/mmag detectability
5 99.9 %
10 97.5 %
20 82.7 %
50 57.6 %
100 36.3 %
Table B.2: Results for varying moon-to-planet mass ratio qMP .
(a) qP S = 3 × 10 −4 (A.8)
σ/mmag detectability
5 52.4 %
10 24.1 %
20 9.6 %
50 1.0 %
100 0.0 %
(b) qP S = 1 × 10 −3 (A.2)
σ/mmag detectability
5 88.8 %
10 58.8 %
20 29.9 %
50 12.3 %
100 1.5 %
(c) qP S = 3 × 10 −3 (A.7)
σ/mmag detectability
5 99.0 %
10 84.8 %
20 57.2 %
50 28.8 %
100 13.8 %
Table B.3: Results for varying planet-to-star mass ratio qP S
.
(a) dP S = 1.3 θE (A.2)
σ/mmag detectability
5 88.8 %
10 58.8 %
20 29.9 %
50 12.3 %
100 1.5 %
(b) dP S = 1.4 θE (A.9)
σ/mmag detectability
5 86.5 %
10 61.5 %
20 33.2 %
50 13.5 %
100 2.0 %
Table B.4: Results for varying angular separation dP S.

APPENDIX B. NUMERICAL RESULTS 67
(a) RSource = R⊙
σ/mmag detectability
5 88.8 %
10 58.8 %
20 29.9 %
50 12.3 %
100 1.5 %
(a) sampling step size =
15 minutes
(b) RSource = 2R⊙
σ/mmag detectability
5 67.7 %
10 36.7 %
20 18.9 %
50 4.0 %
100 0.0 %
Table B.5: Results for varying source size RSource. (A.2)
σ/mmag detectability
5 88.8 %
10 58.8 %
20 29.9 %
50 12.3 %
100 1.5 %
(b) sampling step size =
22 minutes
σ/mmag detectability
5 86.1 %
10 53.4 %
20 27.2 %
50 10.8 %
100 1.0 %
(c) RSource = 3R⊙
σ/mmag detectability
5 52.2 %
10 27.0 %
20 14.0 %
50 0.2 %
100 0.0 %
(c) sampling step size =
30 minutes
σ/mmag detectability
5 64.7 %
10 35.3 %
20 18.0 %
50 4.6 %
100 0.1 %
Table B.6: Results for varying sampling of the triple-lens light curve. (A.2)
(a) sampling step size =
1.9 minutes
σ/mmag detectability
5 88.8 %
10 58.8 %
20 29.9 %
50 12.3 %
100 1.5 %
(b) sampling step size =
0.9 minutes
σ/mmag detectability
5 88.2 %
10 57.4 %
20 29.4 %
50 12.1 %
100 1.5 %
Table B.7: Results for normal and high sampling of the binary-lens light curve.
(A.2)

Bibliography
Afonso, Cristina; Albert, J. N.; Andersen, J. et al.
Limits on galactic dark matter with 5 years of EROS SMC data.
Astronomy and Astrophysics, 400: pp. 951, 2003.
Albrow, Michael; Beaulieu, Jean-Philippe; Birch, Peter et al.
The 1995 pilot campaign of PLANET: searching for microlensing anomalies
through precise, rapid, round-the-clock monitoring.
The Astrophysical Journal, 509: pp. 687, 1998.
Alcock, Charles; Akerlof, Carl W; Allsman, Robyn A et al.
Possible gravitational microlensing of a star in the large magellanic cloud.
Nature, 365: p. 621, 1993.
Aubourg, Eric; Bareyre, Pierre; Bréhin, S. et al.
Evidence for gravitational microlensing by dark objects in the galactic halo.
Nature, 365: p. 623, 1993.
Beaulieu, Jean-Philippe; Albrow, Michael; Bennett, David P et al.
Discovery of a cool planet of 5.5 Earth masses through gravitational microlensing.
Nature, 439(7075): pp. 437, 2006.
Beaulieu, Jean-Philippe; Kerins, Eamonn; Mao, Shude et al.
Towards a census of earth-mass exo-planets with gravitational microlensing.
ArXiv e-prints, 2008.
Benn, Chris R.
The moon and the origin of life.
Earth Moon and Planets, 85: pp. 61, 2001.
Bennett, David P.; Bond, Ian A.; Udalski, Andrzej et al.
A low-mass planet with a possible sub-stellar-mass host in microlensing event
moa-2007-blg-192.
Astrophysical Journal, 684: pp. 663, 2008.
Bennett, David P and Rhie, Sun Hong.
Detecting earth-mass planets with gravitational microlensing.
The Astrophysical Journal, 472: pp. 660, 1996.
69

70 BIBLIOGRAPHY
Bevington, Philip R.
Data reduction and error analysis for the physical sciences.
The McGraw-Hill Companies, 1969.
Bond, Ian A; Udalski, Andrzej; Jaroszyński, Micha̷l; and coauthors, 32.
OGLE 2003-BLG-235/MOA 2003-BLG-53: A planetary microlensing event.
The Astrophysical Journal, 606(2): p. L155, 2004.
Bozza, Valerio.
Perturbative analysis in planetary gravitational lensing.
Astronomy and Astrophysics, 348: pp. 311, 1999.
Bozza, Valerio.
Caustics in special multiple lenses.
Astronomy and Astrophysics, 355: pp. 423, 2000.
Bronstein, Ilja Nikolajewitsch; Semendjajew, Konstantin Adolfowitsch; Musiol, Gerhard;
and Mühlig, Heiner.
Taschenbuch der Mathematik.
Verlag Harri Deutsch, Thun und Frankfurt am Main, 5th edn., 2000.
Cabrera, Juan and Schneider, Jean.
Detecting companions to extrasolar planets using mutual events.
Astronomy and Astrophysics, 464: pp. 1133, 2007.
Cassan, Arnaud.
An alternative parameterisation for binary-lens caustic-crossing events.
Astronomy and Astrophysics, 491: pp. 587, 2008.
Cassen, Pat; Guillot, Tristan; Quirrenbach, Andreas et al., editors.
Saas-Fee Advanced Course 31, Saas-Fee Advanced Course 31. 2006.
Chwolson, Orest.
Über eine mögliche Form fiktiver Doppelsterne.
Astronomische Nachrichten, 221: p. 329, 1924.
Davies, Robert B.
The distribution of a linear combination of χ 2 random variables.
Journal of the Royal Statistical Society: Series C (Applied Statistics), 29(3): pp.
323, 1980.
Domingos, R C; Winter, Othon C; and Yokoyama, Takaaki.
Stable satellites around extrasolar giant planets.
Monthly Notices of the Royal Astronomical Society, 373: pp. 1227, 2006.
Dominik, Martin; Jørgensen, Uffe Gr˚ae; Horne, Keith et al.
Inferring statistics of planet populations by means of automated microlensing
searches.

BIBLIOGRAPHY 71
ArXiv e-prints, 2008.
Dong, Subo; Bond, Ian A.; Gould, Andrew et al.
Microlensing event MOA-2007-BLG-400: Exhuming the buried signature of a cool,
Jovian-mass planet.
ArXiv e-prints, 2008.
Dyson, Frank Watson; Eddington, Arthur Stanley; and Davidson, Charles.
A determination of the deflection of light by the sun’s gravitational field, from
observations made at the total eclipse of may 29, 1919.
Royal Society of London Philosophical Transactions Series A, 220: pp. 291, 1920.
Einstein, Albert.
Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen.
Jahrbuch der Radioaktivität und Elektronik, 4: pp. 411, 1908.
Einstein, Albert.
Über den Einfluß der Schwerkraft auf die Ausbreitung des Lichtes.
Annalen der Physik, 340: pp. 898, 1911.
Einstein, Albert.
Die Grundlage der allgemeinen Relativitätstheorie.
Annalen der Physik, 354: pp. 769, 1916.
Einstein, Albert.
Lens-like action of a star by the deviation of light in the gravitational field.
Science, 84: pp. 506, 1936.
Gaudi, B Scott; Bennett, David P; Udalski, Andrzej; Gould, Andrew et al.
Discovery of a Jupiter/Saturn analog with gravitational microlensing.
Science, 319: p. 927, 2008.
Gaudi, B Scott; Naber, Richard M; and Sackett, Penny D.
Microlensing by multiple planets in high-magnification events.
Astrophysical Journal, Letters, 502: pp. , 1998.
Gaudi, B Scott and Sackett, Penny D.
Detection efficiencies of microlensing data sets to stellar and planetary companions.
Astrophysical Journal, 528: pp. 56, 2000.
Gould, Andrew P.
MicroFUN 2007.
In Manchester Microlensing Conference. 2008a.
Gould, Andrew P.
Recent developments in gravitational microlensing.

72 BIBLIOGRAPHY
eprint arXiv:0803.4324, 2008b.
Review presented at ”The Variable Universe: A Celebration of Bohdan Paczynski”,
29 Sept 2007, 24 pages incl. 23 figures.
Gould, Andrew P and Loeb, Abraham.
Discovering planetary systems through gravitational microlenses.
The Astrophysical Journal, 396: p. 104, 1992.
Gould, Andrew P; Udalski, Andrzej; An, Deokkeun et al.
Microlens OGLE-2005-BLG-169 implies that cool neptune-like planets are common.
Astrophysical Journal, Letters, 644: pp. , 2006.
Groenewegen, Martin A T; Udalski, Andrzej; and Bono, Giuseppe.
The distance to the Galactic centre based on Population II Cepheids and RR Lyrae
stars.
Astronomy and Astrophysics, 481(2): pp. 441, 2008.
Han, Cheongho.
Microlensing detection of moons of exoplanets.
submitted to ApJ, 2008.
Han, Cheongho and Han, Wonyong.
On the feasibility of detecting satellites of extrasolar planets via microlensing.
Astrophysical Journal, 580(1): pp. 490, 2002.
Henry, Gregory W; Marcy, Geoffrey W; Butler, R Paul; and Vogt, Steven S.
A transiting “51 Peg-like” planet.
Astrophysical Journal, Letters, 529: pp. , 2000.
Hunter, Rex B.
Motions of satellites and asteroids under the influence of Jupiter and the sun - I.
Stable and unstable orbits.
Monthly Notices of the Royal Astronomical Society, 136: p. 245, 1967.
Kalas, Paul; Graham, James R.; Chiang, Eugene et al.
Optical images of an exosolar planet 25 light-years from earth.
Science, 322: p. 1345, 2008.
Kayser, Rainer; Refsdal, Sjur; and Stabell, Rolf.
Astrophysical applications of gravitational micro-lensing.
Astronomy and Astrophysics, 166(1-2): pp. 36, 1986.
Lewis, Karen M.; Sackett, Penny D.; and Mardling, Rosemary A.
Possibility of detecting moons of pulsar planets through time-of-arrival analysis,
2008.
Submitted to Astrophysical Journal.

BIBLIOGRAPHY 73
Mao, Shude and Paczyński, Bohdan.
Gravitational microlensing by double stars and planetary systems.
The Astrophysical Journal, 374: p. L37, 1991.
Marois, Christian; Macintosh, Bruce; Barman, Travis et al.
Direct imaging of multiple planets orbiting the star HR 8799.
Science, 322: p. 1348, 2008.
Mayor, Michel and Queloz, Didier.
A Jupiter-mass companion to a solar-type star.
Nature, 378: p. 355, 1995.
Meylan, Georges; Jetzer, Philippe; North, Pierre et al., editors.
Saas-Fee Advanced Course 33. 2006.
Muraki, Yasushi; Sumi, Takahiro; Abe, Fumio et al.
Search for MACHOS by the MOA collaboration.
Progress of Theoretical Physics Supplement, 133: pp. 233, 1999.
Paczyński, Bohdan.
Gravitational microlensing by the Galactic halo.
Astrophysical Journal, 304: pp. 1, 1986.
Paczyński, Bohdan.
Gravitational microlensing of the Galactic bulge stars.
Astrophysical Journal, Letters, 371: pp. , 1991.
Paczyński, Bohdan.
Gravitational microlensing in the Local Group.
Annual Review of Astronomy and Astrophysics, 34: pp. 419, 1996.
Rhie, Sun Hong.
Infimum microlensing amplification of the maximum number of images of n-point
lens systems.
The Astrophysical Journal, 484: pp. 63, 1997.
Rhie, Sun Hong.
How cumbersome is a tenth order polynomial?: The case of gravitational triple
lens equation.
2002.
Sackett, Penny D.
Astronomical society of the pacific conference series.
In Gravitational Lensing: Recent Progress and Future Go, edited by T. G. Brainerd
and C. S. Kochanek, vol. 237, p. 227. 2001.

74 BIBLIOGRAPHY
Sartoretti, Paola and Schneider, Jean.
On the detection of satellites of extrasolar planets with the method of transits.
Astronomy and Astrophysics, Supplement, 134: pp. 553, 1999.
Schneider, Peter and Weiß, Achim.
The two-point-mass lens: detailed investigation of a special asymmetric gravitational
lens.
Astronomy and Astrophysics, 164: pp. 237 , 1986.
Soldner, Johann.
Ueber die Ablenkung eines Lichtstrahls von seiner geradlinigen Bewegung, durch
die Attraktion eines Weltkörpers, an welchem er nahe vorbei geht.
Astronomisches Jahrbuch für das Jahr 1804, 29. Jahrgang: pp. 161, 1801.
Southworth, John; Hinse, Tobias C; Jørgensen, Uffe Gr˚ae et al.
High-precision photometry by telescope defocussing. I. the transiting planetary
system WASP-5.
Submitted to Monthly Notices of the Royal Astronomical Society, 2009.
Udalski, Andrzej; Jaroszyński, Micha̷l; Paczyński, Bohdan et al.
A Jovian-mass planet in microlensing event OGLE-2005-BLG-071.
Astrophysical Journal, Letters, 628: pp. , 2005.
Udalski, Andrzej; Szymański, Micha̷l; Ka̷lu˙zny, Janusz; Kubiak, Marcin; and Mateo,
Mario.
The Optical Gravitational Lensing Experiment.
Acta Astronomica, 42: pp. 253, 1992.
Udalski, Andrzej; Szymański, Micha̷l; Ka̷lu˙zny, Janusz et al.
The Optical Gravitational Lensing Experiment. Discovery of the first candidate
microlensing event in the direction of the Galactic bulge.
Acta Astronomica, 43: pp. 289, 1993.
Walsh, Dennis; Carswell, Robert F.; and Weymann, Ray J.
0957 + 561 A, B - twin quasistellar objects or gravitational lens.
Nature, 279: pp. 381, 1979.
Wambsganß, Joachim.
Gravitational Microlensing.
Ph.D. thesis, Ludwig-Maximilians-Universität München, 1990.
Wambsganß, Joachim.
Discovering galactic planets by gravitational microlensing: magnification patterns
and light curves.
Monthly Notices of the Royal Astronomical Society, 284(2): pp. 172, 1997.

BIBLIOGRAPHY 75
Wambsganß, Joachim.
Gravitational lensing: numerical simulations with a hierarchical tree code.
Journal of Computational and Applied Mathematics, 109: pp. 353, 1999.
Williams, Darren M. and Knacke, Roger F.
Looking for planetary moons in the spectra of distant jupiters.
Astrobiology, 4: pp. 400, 2004.
Witt, Hans J.
Investigation of high amplification events in light curves of gravitationally lensed
quasars.
Astronomy and Astrophysics, 236: pp. 311, 1990.
Wolszczan, Aleksander and Frail, Dale A.
A planetary system around the millisecond pulsar PSR1257 + 12.
Nature, 355: pp. 145, 1992.
Zwicky, Fritz.
Nebulae as gravitational lenses.
Physical Review, 51: pp. 290, 1937a.
Zwicky, Fritz.
On the probability of detecting nebulae which act as gravitational lenses.
Physical Review, 51: pp. 679, 1937b.

76 BIBLIOGRAPHY

Acknowledgements
I thank my supervisor Professor Joachim Wambsganß for handing this perfect diploma
thesis topic about the search for extrasolar moons to me, and also for thereby inviting
me to the fantastic Heidelberg gravitational lensing group. For his knowledgeable
guidance that kept me on the right track, while granting me freedom in how to
conduct the details of my work. For giving me the opportunity to participate in
the MiNDSTEp microlensing observations 2008 in Chile and for attending the Microlensing
Workshop 2009 in Paris. This is far more than most diploma students can
hope for and has greatly contributed to the fact that the final year of my diploma
studies has been one of the most enjoyable ones.
I am indebted to Professor Andy Gould for instantly agreeing to act as the second
referee on this thesis. I would like to also thank him for his openness to share
views on topics beyond astronomy and the interesting discussions that have resulted.
I want to thank my parents Winfried and Hildegard, my sister Dorothea, my
brother Martin and my brother-in-law Wolfgang for supporting me throughout my
years of studying. For caring a lot about my successes and for not caring that much
about my less successful phases. It was good to know that I could always rely on
their support, no matter what.
I thank the Evangelisches Studienwerk Villigst for financially supporting my
studies in Heidelberg and at Edinburgh University. Invaluable to me were the unique
opportunities for personal and cultural development offered in all those years. I particularly
want to thank Knut Berner for giving me personal and pastoral advice when
I needed it.
I would like to thank Valerio Bozza for patiently teaching me how to search for
extrasolar planets with the Danish 1.54m telescope on La Silla, Chile, and then
observing OB-08-510 aka “Svenonia-Paolania” with me. It was hugely motivating
to be able to learn from his experience.
I thank the members of the Heidelberg lensing group that I met during my
year at the ARI: Cécile Faure, Jonathan Duke, Janine Fohlmeister, Timo Anguita,
Dominique Sluse, Marta Zub, Arnaud Cassan, Robert Schmidt, Fabian Zimmer,
77

78 BIBLIOGRAPHY
Gabriele Maier.
I thank them all for lots of little and not-so-little help during the daily strains
of scientific work: expert input on my topic, solving my compiling and printing
problems, improving my talks (with technical support and constructive critique),
thesis reading, keeping up the chocolate supplies. . .
I particularly want to thank the ”lunch group”: For the parties, lots of lunchtime
philosophy and all the laughter we shared! It really made a difference.
I thank Sven Marnach for his invaluable and (seemingly) tireless support especially
during the final phases of this thesis. There are no words to express my joy
and my gratitude for having found such a wonderful partner in him.
This work has made use of Fortran77, g77/gfortran, Gnuplot, FITS and the
CFITSIO library, Emacs and L ATEX. The research has relied on NASA’s Astrophysics
Data System and arXiv.

Eigenständigkeitserklärung
Ich versichere, dass ich diese Arbeit selbstständig verfasst habe und keine anderen
als die angegebenen Quellen und Hilfsmittel benutzt habe.
Heidelberg, den 13. Februar 2009
79