Properties of Extrasolar Planets and Their Host Stars the Cases of HAT-P-7 and KIC 8866102

FACULTY OF SCIENCE Department of Physics and Astronomy Institute of Astronomy

Properties of extrasolar planets and their host stars The cases of HAT-P-7 and KIC 8866102

by Vincent VAN EYLEN

Supervisor: prof. dr. H. Kjeldsen∗ Dissertation presented in Co-supervisor: prof. dr. C. Aerts fulﬁllment of the requirements

Mentor: prof. dr. J. Christensen-Dalsgaard∗ for the degree of Master of

(∗Aarhus University, DK) Astronomy and Astrophysics

Academic year 2011-2012 Front picture: Launch of the Kepler spacecraft from Cape Canaveral Air Force Station, Friday March 6, 2009. The telescope was launched into an Earth-trailing heliocentric orbit. Kepler is a NASA mission that has made a large number of signiﬁcant observations for asteroseismology and exoplanet science. It acquired the data used throughout this work. Credit: Ben Cooper/LaunchPhotography.com.

c Copyright by KU Leuven

Without written permission of the promotors and the authors it is forbidden to reproduce or adapt in any form or by any means any part of this publication. Requests for obtaining the right to reproduce or utilize parts of this publication should be addressed to KU Leuven, Facul- teit Wetenschappen, Geel Huis, Kasteelpark Arenberg 11, 3001 Leuven (Heverlee), Telephone +32 16 32 14 01.

A written permission of the promotor is also required to use the methods, products, schematics and programs described in this work for industrial or commercial use, and for submitting this publication in scientiﬁc contests.

No crystal balls, no tarot cards, no horoscopes. Just you, your brain, and your ability to think. Welcome to science. You’re gonna like it here. Phil Plait Preface

This morning, June 6, 2012, I saw Venus move in front of the Sun, an extremely rare astronomical event that will not be seen again for the next 105 years because of the orbital inclination of our Earth’s and Venus’ orbit. We stood there in the physics building, astronomers with eclipse glasses, cameras and telescopes to watch the black dot on the solar disk. As the less scientiﬁc humans of our Earth will start to wake up, even they will be amazed by the many beautiful images that were made of this event from all over the world, by how we have been able to foresee this event, predict it to incredible accuracy and ultimately observe it to a stun- ning level of precision.

The real wonders of astronomy today are literally more exotic. The Kepler satellite is able to detect transits exactly like this one, of the same relative sizes, around stars other than the Sun. These extrasolar planets or exoplanets are discovered by measuring tiny variations in the light of a star when a small part of the surface is blocked by a transiting planet. For distant stars, we cannot resolve the stellar surface to see the planetary trajectory directly, but the transit duration allows us to derive the exact inclination of the planet, and the transit depth (the amount of blocked star light) gives us its relative size. A number of observed transits allows us to measure the planetary period, and variations in the expected transit times can reveal the presence of a gravitationally interacting additional planet hiding in the system. That is truly amazing!

Using the very same Kepler observations, we can measure even smaller variations in the brightness of the star, that are caused by stellar pulsations. As the stellar surface expands and contracts periodically, the slight temperature variations cause tiny brightness variations. We can measure them. Asteroseismology is the field that studies these pulsations and by making models for how different stars will pulsate in different ways, we can make a real image of the inside of the stars. Again: that is truly amazing!

This Master’s thesis combines these two wonderful research ﬁelds and carries out an A to Z analysis of pulsating stars that contain transiting exoplanets. The stellar properties are analysed using the techniques asteroseismology provides us with, while the same time series of observations contains the planetary transits that we use to study the exoplanet’s properties. Ultimately it is the combination of both domains that turns out to be very rewarding.

This work carries out a study on two speciﬁc star-planet systems. HAT-P-7 is a very interesting star that is orbited by a close-in and large planet, and has been extensively studied in the past few years. KIC 8866102 provides us with an entirely diﬀerent perspective: a longer and more distant orbit with a much smaller planet, whose properties have not been published yet. To improve the readability, I have nicknamed KIC 8866102 ‘Sandra’ for this work.

iii iv

The ﬁrst chapter introduces the relevant science to the reader. It gives an overview of the relevant aspects of star pulsations, as well as an introduction to exoplanetary research. The Kepler mission is described, as its state of the art observations have brought us to the golden age of both domains. This satellite has also observed HAT-P-7 and Sandra and we give a short overview of both systems, after which the reader has acquired the level of knowledge to understand the objectives that will be covered in the remainder of this work.

The second chapter deals with stellar pulsations. It is divided into two large parts. The ﬁrst section describes how the observations have been performed and how the time series of observations on the stars can be used to obtain stellar pulsation frequencies. In the other section, I explain how these frequencies can be modeled using the asteroseismic theory, which leads to an accurate determination of a number of stellar parameters (e.g. mass, radius, temperature, age, ...).

The extrasolar planet information enters in chapter three. In a ﬁrst step, the times of the observed planetary transits are determined with high accuracy and this information is used to obtain the planetary period with extraordinary precision. Variations in the transit times are studied carefully to discover potential other bodies (or place upper limits on their possible mass). In a next step, I put together all transits before constructing a transit model. Combined with the stellar parameters derived in the previous chapter, this results in information on the planetary orbit and radius. Finally, variations in the level of light throughout a planetary period will allow an estimation of the temperature of the dayside and the nightside of HAT-P-7’s planet.

In the ﬁnal chapter, all information is brought together. For HAT-P-7, I make an extensive comparison between the results obtained throughout the previous chapters and those that were obtained by the many other authors who have previously studied the system. The results obtained for HAT-P-7 and Sandra are put into context and the merits of performing a complete analysis and combining asteroseismology and transiting exoplanets are discussed.

Some additional information can be found in the appendices. As required, a Dutch summary is available under “Nederlandstalige samenvatting”. I have presented a number of the results of this work at a conference on asteroseismology, which was organized by the European Sci- ence Foundation in Obergurgl (Austria). Appendix B contains the science poster that was presented there. Finally, full lists of the transit times and the frequencies we have determined are presented in additional appendices.

Vincent Van Eylen Aarhus, Denmark Acknowledgements

I have spent the past year at the University of Aarhus in Denmark. The many results presented in this work would have never been possible without the excellent supervision that I have received throughout my stay. I am extremely grateful to my promotor, Prof. Hans Kjeldsen for the many hours spent in his oﬃce discussing the science. His enthusiasm about my progress has been inspiring throughout the year. He gave me the opportunity to work on restricted and extremely interesting Kepler data, for which I can only thank the entire Kepler team.

I would have never been able to run all the codes to model my stars without the guidance of Prof. Jørgen Christensen-Dalsgaard. Moreover, his many comments and advice have been extremely valuable. I am grateful that he pointed me in the direction of an ESF conference on “The Modern Era of Helio- and Asteroseismology” and encouraged me to participate by presenting my results there on a poster. I learned a lot from the many scientiﬁc talks and I had a number of interesting discussions about the results on my own poster. I have met many astronomers whose papers I have been reading in the past year, as well as many young scientists. I am grateful to all of them.

Despite the distance between Leuven and Aarhus, Prof. Conny Aerts has provided me with excellent guidance, both on an academic and on a more personal level. But first and foremost, I would like to thank her for making it possible to spend my final Master year abroad. Finishing my education in another country has proven to be extremely beneficial academically, and very fruitful on a personal level.

I cannot begin to name all the people I have met the past year, since this page might even be too short to list all their nationalities. I have learned an incredible amount of things from all of them and from all the diﬀerent cultures, and I owe them all a very big thank you. One person has been here with me from the start. Be it on a wonderful trip to Norway or on one of our weekly dinners, you have been my source of advice and inspiration from the ﬁrst day until the last. Whether in Spain or here in Denmark, I am sure you will become an excellent biologist. Thank you, Sandra.

To my parents and my family: thank you for all the support and encouragement, and the words of wisdom that make me who I am today. I am grateful for the wonderful roommates I have had this year, and for the (distant but not less valuable) support from my Belgian friends. My time in Leuven was shaped by student organisation Wina, and in Denmark I have gladly volunteered for Studenterhus Aarhus. The beneﬁts (free coﬀee and espresso) have certainly contributed to this thesis. I owe a thank you to everyone who has read parts of this report and provided me with immensely valuable feedback.

Finally, I would like to thank you, the reader of this dissertation. I hope you ﬁnd some of the joy and inspiration it has brought me, and learn a thing or two about our place among the stars and planets of the mighty universe. Contents

Preface iii

Acknowledgements v

Table of contents vii

1 Introduction 1 1.1 Stellar pulsations ...... 1 1.1.1 Overview ...... 1 1.1.2 Solar-like oscillations ...... 3 1.1.3 Observations ...... 6 1.2 Extrasolar planets ...... 8 1.2.1 History ...... 8 1.2.2 Exoplanet detection mechanisms ...... 9 1.2.3 Transiting exoplanets ...... 11 1.3 The Kepler satellite ...... 13 1.3.1 Mission design and time series ...... 13 1.3.2 Data normalisation ...... 14 1.4 The target systems: HAT-P-7 and KIC 8866102 (‘Sandra’) ...... 16 1.5 Objectives ...... 18

2 The pulsations of the host stars 20 2.1 Observing the frequencies ...... 20 2.1.1 Procedure ...... 21 2.1.2 Rotational splitting ...... 23 2.1.3 Results ...... 27 2.2 Frequency modeling ...... 29 2.2.1 Procedure ...... 29 2.2.2 Reliability ...... 30 2.2.3 Stellar properties ...... 31

3 The planetary transits 35 3.1 Planetary period ...... 35 3.1.1 Determining transit times ...... 35 3.1.2 Determining the orbital period ...... 37 3.2 Transit timing variation ...... 38 3.2.1 Hat-P-7 ...... 39 3.2.2 Sandra ...... 42

vi Contents vii

3.3 Transit depth ...... 44 3.3.1 Overview ...... 44 3.3.2 Results ...... 46 3.3.3 Planet mass ...... 48 3.4 Phase-dependent ﬂux variations (HAT-P-7) ...... 48 3.4.1 Albedo measurement and ellipsoidal light variations ...... 48 3.4.2 Dayside and nightside temperature ...... 50

4 Results and conclusions 53 4.1 Parameter overview ...... 53 4.2 Conclusions ...... 59

Afterword 61

A Nederlandstalige samenvatting 62

B ESF conference poster 65

C Transit times 67

D Observed frequencies 70

List of ﬁgures 75

List of tables 76

Bibliography 82

I wonder why. I wonder why. I wonder why I wonder. I wonder why I wonder why. I wonder why I wonder! Richard Feynman 1 Introduction

Before jumping into the exciting results that have been obtained throughout the thesis, this chapter will provide the ‘lay of the land’. The first two sections introduce the important aspects of the two fields of astronomy that are most relevant for understanding the remainder of this report: the first section introduces stellar pulsations (asteroseismology), the second focuses on extrasolar planets. An overview of the Kepler mission is given, as it has been extremely relevant for the development of both of these fields. It has also provided the high-quality observations of the stars that are discussed in this work. Subsequently, the stars HAT-P-7 and KIC 8866102 (‘Sandra’) are themselves introduced. We conclude the introduction with an overview of what is to be expected in the following chapters.

1.1 Stellar pulsations

The field of asteroseismology deals with stellar pulsations. In this section we provide a general overview and we focus on solar-like oscillations (which have the most relevance for HAT-P-7 and Sandra). We finally discuss the observational aspects of the field throughout the last few decades. The most complete work covering the various aspects of this field has been written by Aerts, Christensen-Dalsgaard, and Kurtz (2010). This section is partially based on this book, but leaves many aspects of the broad field largely untreated. It serves to introduce the concepts relevant for the rest of this work.

1.1.1 Overview Few astronomers will doubt the importance of understanding the structure of stars and their evolution. The way stars are patched together in structured galaxies traces the density structure in the early universe. Almost every chemical element originates from stars: the lighter elements are formed in the star’s interior due to nuclear fusion, the elements heavier than iron originate from Supernova explosions that occur in the ﬁnal stage of their life and from stars on the Asymptotic Giant Branch.

1 1.1. Stellar pulsations 2

In general terms, the stellar structure and evolution is understood. However, when it comes to the physics of stellar interiors, a significant level of uncertainty still remains. The classical techniques to study stellar evolution use spectroscopic and photometric observations, as well as astrometry and interferometry. Major steps forward in understanding the interior structure of the Sun came thanks to helioseismology: the study of solar oscillations. These oscillations are affected by the interior structure and thereby effectively offer a way to look at the inside of the Sun. In addition, the pulsations are sensitive to the solar rotation and magnetic field.

The next step was to move on from the Sun to other stars: changing helioseismology into asteroseismology, hoping to copy the achievements for the Sun to other stars. Pulsations have indeed been detected in many other stars, as can be seen from ﬁgure 1.1, showing a Hertzsprung-Russell (HR) diagram depicting for which stars pulsations have been found.

The stellar oscillations in stars are caused by sound waves (pressure waves) inside the stars. We see with sound. The comparison with ‘real’, audible sounds is instructive. The shape of a musical instrument, or its environment temperature, will influence the pressure waves and consequently the sound we hear. In the same way, the speed of sound inside a star depends on the stellar composition and the temperature of the gas, and this influences the observed oscillation frequencies. Stars are giant music instruments. As Aerts et al. (2010) put it: “Asteroseismology uses astronomical observations - photometric and spectroscopic ones - to extract the frequencies, amplitudes and phases of the sounds at a star’s surface. Then we use basic physics and mathematical models to infer the sound speed and density inside a star, throughout its interior, and thence the pressure. With reasonable assumptions about chemical composition and knowledge of appropriate equations of state, the temperature can then be derived. [...] We build up a picture in the 3-D theatre in our minds of what the inside of a star looks like. We see inside the star. The sounds tell us what the interior structure of the star has to be.” “Who has not been amazed to see a picture of the face of a foetus in the womb, imaged using ultrasound waves? Do you question the reality of that? No. That is a real picture of the baby before it is born. Identically, using infrasound from the stars, the pictures of their insides that we see using asteroseismology have this same reality.” The recent change from ground-based to space missions (see section 1.1.3) has led to huge improvements in the quality of data available for pulsations in stars other than the Sun. The past years, the field has matured and many hope that, during the next decade, some of the extraordinary successes that have been achieved for the Sun during the past half century, can be mimicked for other stars.

While many people work on improving our understanding of frequency spectra and stellar interiors, we focus on the use of the stellar pulsations to obtain speciﬁc parameters of stars. We apply the available methods of asteroseismology to model the stars and obtain their parameters. In other words: we do not focus on developing better analysis methods, but rather make use of existing ones to go beyond the level of interpretation presently available in the literature. 1.1. Stellar pulsations 3

Figure 1.1: A pulsation HR diagram showing many classes of pulsating stars for which asteroseismology is possible. Figure courtesy by Aerts et al. (2010).

1.1.2 Solar-like oscillations In the Sun, oscillations are stochastically excited by convection. Stars are said to oscillate solar-like whenever the excitation mechanism that creates the stellar oscillations is the same as in the solar case. Some of these stars may be very diﬀerent from the Sun. Indeed, solar-like oscillations have recently even been detected in red giant stars (De Ridder et al., 2009). Much in this section has been based on the overview paper by Bedding (2011).

Solar-like oscillations require an outer convection zone to get excited. In this situation, the star resonates in its outer layers: acoustic energy is transfered to energy of global oscillations in the outer convection zone, in a way that is very similar to how musical instruments resonate in a very noisy environment. The other well-known mechanism to excite stellar oscillations (the heat-engine mechanism or κ-mechanism) originates from opacity variations. In this case 1.1. Stellar pulsations 4

ionisation layers of hydrogen, helium or heavier elements block radiation, causing the gas to heat up. As a consequence, the pressure increases and the star swells past its equilibrium. When the gas gets ionized, the opacity is reduced and the star contracts again. From here on, we concentrate only on solar-like oscillations.

The acoustic (sound) waves are also called pressure modes (p-modes), because the restoring force (the force that brings the star back to equilibrium when it is perturbed due to the oscillations) comes from the pressure gradient. An alternative restoring force is buoyancy. These modes are termed gravity modes (g-modes). To further complicate matters, some stars have both p-modes and g-modes, and some even have mixed modes that have a p-mode and g-mode character at the same time. The pressure modes are very sensitive to the conditions in the outer layers of a star, while g-modes probe the conditions of stellar cores. As mixed modes are sensitive to both, they have been an area of intensive research leading to interesting results (e.g. Beck et al., 2012, who discovered rapid rotation of the core in red giant stars). The rest of this work will deal with p-modes only.

Each p-mode can be described by three integers: the radial order (n), the angular degree (l) and the azimuthal order (m). The radial order determines the number of nodal shells of the standing wave, while the angular degree speciﬁes the number of nodal lines visible on the surface of the star. Modes with l equal to zero are radial modes, while all others are called non-radial.

For every non-radial mode, there are always parts of the surface expanding while other parts of the stellar surface are contracting. Surfaces of stars other than the Sun are generally not resolved, which means only the integrated quantities across the surface are measured. Modes of high radial order are therefore very hard to observe, due to cancellation eﬀects. In general only the monopole, dipole and quadrupole (l = 0, 1, 2) modes are observable, as well as oc- tupole modes in certain cases.

Finally, the azimuthal order can take values from l to l. In case of spherical symmetry, the values are degenerate and the m-value can not be− observed. The presence of a magnetic ﬁeld or stellar rotation can break this symmetry and split a mode with given n and l values into l + 1 or 2l + 1 diﬀerent components. In case of rotational splitting, the frequency separation of the 2l + 1 components, and the relative amplitudes depend on the rotation frequency of the star and the inclination angle of the stellar rotation axis (compared to our line of sight).

Figure 1.2a shows the solar power spectrum. It shows a peak frequency νmax near 3100 µHz, surrounded by a broad envelope of regularly spaced frequencies. The bottom pannel shows a zoom of this spectrum, where all modes are labeled by their (n, l) values. We can see how l cycles repeatedly through the values 2, 0, 3 and 1. The large separation ∆ν shows the distance between consecutive radial overtones (modes that differ by n = 1 and have the same l value). The small separations δν are also indicated on the figure. They give the distance between pairs with l = 0 and l = 2 and between l = 1 and l = 3 pairs (note that these pairs have an n value that is different by one). The figure shows the small separations δν02 and δν13. Other small separations can be defined. As we will see, for the stars discussed in this work, only l = 0, 1 and 2 modes can be obtained and therefore only δν02 is observable for us. 1.1. Stellar pulsations 5

(a) (b)

Figure 1.2: Solar frequencies. Figure (a) shows the solar power spectrum, the lower panel shows a close-up labeled with (n, l) values for each mode. The dotted lines are the radial modes, the large and small separations are indicated. Figure (b) shows an ´echellediagram of the solar frequencies: the frequencies are stacked on top of each other by plotting them modulo the large separation. ∆ν = 135.0 µHz was used for the full symbols, the open symbols show the frequencies modulo a slightly smaller large separation (∆ν = 134.5 µHz). The frequencies line up vertically, with a ridge for each value of l = 0, 1, 2, 3 and 4. Figure courtesy by Bedding (2011).

The regular spacing of solar-like oscillations can be used to display oscillation frequencies in a very neat way. An ´echellediagram stacks frequencies of the same angular degree vertically, by dividing the spectrum into segments of length ∆ν. This is illustrated for the Sun in ﬁgure 1.2b.

As ﬁgure 1.2 shows, the large and small separations can be obtained relatively easily once enough frequencies have been observed. In fact, all frequencies can be expressed in terms of these separations:

1 ν ∆ν n + l + δν . (1.1) n,l ≈ 2 − 0l

In this equation, νn,l is the frequency with radial order n and angular degree l (neglecting rotational splitting so there is no dependence on m), and ∆ν and δν0l the large and small separations while is an oﬀset coming from second order and near-surface eﬀects. Equation (1.1) is known as the asymptotic relation and dates back to Tassoul (1980). The large separation is connected with the inverse of the sound speed (see e.g. Kjeldsen and Bedding, 1995): 1.1. Stellar pulsations 6

1 Z R dr− ∆ν 2 . ≈ 0 cs The square of the sound speed is proportional to the temperature, which is proportional to 2 mass over radius: cs T M/R. From this, it follows directly that the large separation is proportional to the mean∝ density∝ of a star, which makes it an extremely important quantity. Like many relations in asteroseismology, it is commonly expressed as a scaling relation (scaling stars to the Sun): s ∆ν M/M = . (1.2) ∆ν (R/R )3

The small separations depend largely on the sound speed gradient near the core, which makes them sensitive to the age of stars.

1.1.3 Observations There are two main types of observational techniques to detect stellar oscillations. Pulsa- tion frequencies can be obtained from photometric or from spectroscopic observations. With photometry one measures the ﬂux variations of the star. They are caused by temperature variations, which are caused by the pulsations. With spectroscopy one measures the velocity variations (Doppler shift) in the elements, as the pulsations cause the stellar surface to move.

In both types of observations, long time series of measurements are desired, as the actual information comes from the variations in ﬂux or velocity. The frequencies are always observed simultaneously. Time series analysis (e.g. Fourier analysis) is therefore required to obtain the individual frequencies. More details on how to treat this can be found in Aerts et al. (2010) which focuses largely on ground-based observations, while a more recent paper by Kjeldsen and Bedding (2011) surveys how to treat space-based time series. This work makes use of the software package PERIOD04 to transform time-series into frequency space (Lenz and Breger, 2005).

Until a few years ago, data were acquired solely from ground-based measurements. One of the main disadvantages of ground-based observations is the low duty cycle: there is always only a short fraction of time stars can actually be measured (due to weather conditions, day-night cycles, ...). To improve on this, complicated multi-site campaigns were set up. This required tremendous eﬀorts in getting the time stamps from each observatory correct, while signiﬁcant gaps in the data continued to exist, causing confusion in the interpretation of the frequency signal.

Earth-based observations have other disadvantages. Great care has to be taken to subtract all signal caused by the Earth’s orbit and rotation from the data. Photometric measurements based on Earth suffered from atmospheric effects, making it hard to obtain a high level of precision in photometric variability. It should therefore not be surprising that after the discovery of pulsations in the Sun, the transformation from helioseismology to asteroseismology took a slow start. The first efforts to detect solar-like oscillations in stars were made shortly 1.1. Stellar pulsations 7

after the ﬁrst detection of solar oscillations (which date back to Claverie et al., 1979). Several detection claims for stars have been made, but most were later judged unconvincing (Kjeldsen and Bedding, 1995).

Currently Procyon is considered to be the ﬁrst star other than the Sun where light variations due to oscillations have been found. They have been discovered by Brown et al. (1991) but it was only years later that the large separation could be determined (Mosser et al., 1998). Another important result was obtained by Kjeldsen et al. (1995), who detected oscillations in the subgiant η Boo. This was probably the ﬁrst star apart from the Sun with a clear detection of solar-like oscillations (Bedding, 2011). The star was also found to contain mixed modes.

A few other discoveries were made, among which the important observations of α Cen A by Bouchy and Carrier (2002). The real step forward came with the appearance of space missions. The first dedicated such mission was MOST (Microvariability and Oscillations of Stars), a 15-cm telescope in a low-earth orbit (Walker et al., 2003) and led to many discoveries in the field of heat-engine pulsators. The CoRoT (Convection Rotation et Transits planetaires) satellite was launched by the French Space Agency and reported first light in the start of 2007. It is a 27-cm telescope in a low-earth orbit. The mission has made the list of known stars with solar-like oscillations significantly longer and is still used extensively to study red giant stars, as well as subgiants and main-sequence stars (Bedding, 2011).

Only two years later an American mission was launched by NASA: the Kepler satellite. It began science operations in May 2009 and has revolutionised asteroseismology with its superior quality of photometric data. It operates from a heliocentric orbit trailing the Earth and can observe continuously without any interference from the Earth. It has a 95-cm telescope and can operate in a long-cadence mode (sampling 29.4 minutes) and a short-cadence mode sampling 58.8 seconds (Borucki et al., 2009a). The long-cadence mode is suitable for detecting solar-like oscillations in red giant stars, while the short-cadence mode can be used for main-sequence stars and subgiants. More details of the Kepler mission are discussed in section 1.3.

It is worth noting that the main goal of the Kepler mission is the detection of extra-solar planets. Indeed, throughout history, asteroseismology has beneﬁted enormously from the search for planets, which have similar observational requirements and goals as the study of pulsating stars. The CoRoT satellite aimed to detect extrasolar planets and many of the improvements in precision of ground-based spectroscopy have also been driven by the hunt for extrasolar planets.

While Kepler has shown the future for photometric observations most certainly lies in space, earth-based telescopes are expected to deliver multi-site spectroscopy for the foreseeable future. The most interesting in this regard is the SONG (Stellar Observations Network Group) project, which is planned to consist of eight telescope nodes across the Earth, each containing a high-resolution spectrograph (Grundahl et al., 2008). This mission has a shared goal between asteroseismology and exoplanets. The first node has been built in Tenerife, Spain (Grundahl et al., 2011). It is expected to produce its first scientific results around the publication time of this work. 1.2. Extrasolar planets 8

1.2 Extrasolar planets

The aim of this section is to give the reader an introduction to the field of extrasolar planets (exoplanets). We focus in particular on those planets that have been discovered by their planetary transits, as the systems studied in this work fall into this category. For an excellent general overview on the field of exoplanets, we refer the reader to the monograph by Seager and Dotson (2011). This section is largely based on the introductory chapter of this book (Seager and Lissauer, 2010) and on the chapter about planetary transits (Winn, 2010). It aims to introduce the concepts relevant for the results presented in this work and makes no attempt to be a complete introduction to the entire field.

1.2.1 History For thousands of years, the search for our place in the cosmos has fascinated humans. Along with it, questions have been raised about the existence of other planets, potentially even other planets that are very similar to the earth. Ultimately, one wonders if they could display signs of life. The idea that there might be worlds beyond the Earth is more than 2000 years old. Epicurus (ca. 300 BCE) already stated:

“There are inﬁnite worlds both like and unlike this world of ours... We must believe that in all worlds there are living creatures and plants and other things we see in this world.”

The subject was so interesting that by the time of Newton (17th century CE), entire books had been written on the topic. One of those comes from Huygens (1968), who wrote in beautiful English:

“A man that is of Copernicus’s Opinion, that this Earth of ours is a Planet, carry’d round and enlighten’d by the Sun, like the rest of the Planets, cannot but sometimes think, that it’s not improbable that the rest of the Planets have their Dress and Furniture, and perhaps their Inhabitants too as well as this Earth of ours.”

The ﬁrst published claim of a planet beyond our solar system was made for the binary star system 70 Ophiuchi over 150 years ago. It was published by Jacob in the Monthly Notices of the Royal Astronomical Society in 1855. It might surprise the reader that the scientiﬁc journal has been in continuous existence since as early as 1827.

However, the claim was later discredited (Moulton, 1899) and that was indeed the faith of many of the early ‘discoveries’. Infamous in this regard is Barnard’s star, which was announced to be surrounded by a 24-year-period planet about the size of Jupiter (van de Kamp, 1963). The signal was found to be caused by instrumental systematic errors.

Twenty years ago, the ﬁrst exoplanets were found to orbit the pulsar PSR 1257+12 (Wolszczan and Frail, 1992). Pulsar planets are often named “dead worlds”, because of the radiation they receive from their host stars. The major breakthrough ﬁnally occured a few years later, in 1995, with the discovery of a planet with a period of 4.2 days, orbiting the Sun-like star 1.2. Extrasolar planets 9

51 Peg (Mayor and Queloz, 1995). Thousands of years of wondering have resulted in an exciting answer, only less than twenty years ago: out there, around other stars, planets actually exist.

The discovery was exciting for another reason: the planet was very similar in size to Jupiter, but orbited its star about seven times closer than Mercury orbits the Sun. Over the years many more of this type of planets were found, and they were named “Hot Jupiters” because of their similar size to the solar planet, but much higher temperature due to their proximity to the host stars.

Hot Jupiters were unexpected as the planet formation theories suggested the presence of smaller, earth-like planets closer to stars, while large gas giants would occur only further away, where the environment is cool enough for ice to condense during the planet-formation phase. This matches what is found in our solar system, but clearly the theory is inadequate (or at least incomplete) to explain Hot Jupiters. Various migration theories have since been suggested: Jupiter-size planets would still be formed at large distance from the stars, but subsequently have migrated towards closer orbits. This is an excellent example of how the detection of exoplanets has contributed to our understanding of planetary formation mechanisms, and still does.

Since the first discovery, the number of exoplanets has been growing rapidly during the last two decennia. To date, we know several hundreds confirmed exoplanets that have been discovered using a number of different detection techniques. The next section discusses the different detection mechanisms.

1.2.2 Exoplanet detection mechanisms After the discovery of 51 Peg’s Hot Jupiter, many more planets have been detected over the past two decades. An overview of this is presented in figure 1.3a. The figure was taken from the NASA Exoplanet Archive (http://exoplanetarchive.ipac.caltech.edu), which lists 691 confirmed exoplanets (which were observed using a number of different techniques) as of May 2012.

The pulsar timing method makes use of the ultraprecise radiation that comes from millisecond pulsars. The variations in this radiation can be detected. They are caused by the pulsar rotating around the common center of mass of the star and the planet.

Direct imaging means spatially separating the star and the planet in the sky. Because planets orbit relatively close to the stars and the latter are far brighter, this is technically very diﬃcult to achieve. The technique has therefore not yet been able to result in a large number of discoveries.

Gravitational microlensing occurs when an observer looks at a distant star and a foreground star passes through the line of sight. The background star is magniﬁed through the foreground star (‘the lens’). A planet orbiting the lens star would perturb the image and is thereby detected. However, a lensing is a one-time event, which is a major weakness of this detection technique. However, microlensing is also sensitive to low-mass planets that orbit relatively far away from their star - something most other techniques are not very good at. 1.2. Extrasolar planets 10

(a)

(b)

Figure 1.3: An overview of the history and current status of the confirmed exoplanets, color- coded for the different detection techniques. The top figure sorts the number by their discovery year, the bottom figure gives an overview of period and mass of the planets. Figures taken from the NASA Exoplanet Archive (http://exoplanetarchive.ipac.caltech.edu). 1.2. Extrasolar planets 11

By far the most successful techniques to date have been radial velocity measurements and observations of planetary transits. Both techniques infer the planetary existence from changes in the host stars: the ﬁrst technique does so using spectroscopy, the second utilises photometry.

The radial velocity technique measures the motion of the star along the line-of-sight (through Doppler shifts), as the star moves around the center of mass of the star-planet system. The major disadvantage of this technique is that when used on its own, it only measures a minimum planetary mass as only the stellar motion along our line-of-sight can be observed.

Planetary transits occur when a planet moves in front of its host star, as observed from the Earth (or the telescope’s location). This causes the planet to block a part of the stellar light and this light variation can be measured if a star’s brightness is continuously being monitored. To observe a transit, the orbit of the planet must be aligned with our line-of-sight, and since the inclination of an orbit is expected to be a random factor, this requires observing a large number of stars to find those that accidentally have a suitable orbit. The chance of a transit occurring, increases if the planet orbits closer to its host star (lower semi-major axis a), or if the stellar radius R is larger: ∗ R P ∗ . tra ≈ a In addition, short-period planets will transit more frequently and planets with a larger radius will cause deeper transits (see next section), which makes them easier to observe. This causes the method to be biased towards detecting short-period planets close to their stars, often of relatively large mass. This can be seen from figure 1.3b, which gives the distribution of known planets over their periods and masses. The lack of observed planets that have a long period and low mass is expected to be a selection effect, as these are very hard to detect with the current methods (except for gravitational lensing, which is rare).

1.2.3 Transiting exoplanets We now turn our focus to transiting exoplanets. The general idea behind planetary transits is shown in ﬁgure 1.4. The ﬁgure shows the primary eclipse where the planet moves in front of the star, which is usually referred to as the transit. The secondary eclipse occurs when the planet disappears behind the star and is referred to as the occultation.

The percentage of ﬂux loss during transit is referred to as the transit depth. We can see from the ﬁgure that it will depend on the relative size of the planet compared to the star. A deeper transit can be caused by a smaller star, or by a larger planet. To obtain the absolute planetary radius, we need to be able to accurately estimate the stellar radius. This is one of the key results asteroseismology can deliver.

It is clear from the figure that long time series of stellar flux measurements will result in the observation of a series of transits. From this, we can obtain the period of the planet. If the orbit does not follow a fixed ellipse, there will be variations in the interval between transits. This can be caused by gravitational interaction and thereby reveal additional bodies in the system. The transit duration can give information about the exact inclination angle, which in any case is close to 90◦ (otherwise a transit can never be observed). 1.2. Extrasolar planets 12

Figure 1.4: Illustration of transits and occultation. During a transit, the ﬂux drops as a part of the starlight is blocked. During the occultation, the observed ﬂux drops slightly because the planetary thermal radiation is no longer observed. Figure courtesy by Seager and Deming (2010).

During the occultation, it is the planetary rather than (a part of) the stellar light that gets blocked. Since planets are intrinsically much fainter than stars, the occultation will be considerably smaller than the transit and might not always be detectable. The depth of the occultation can reveal information about the thermal radiation from the planet, and therefore about the planetary temperature on its dayside and nightside.

Planets are expected to have atmospheres. The opacity of these atmospheres will in general be wavelength-dependent, thus causing the transiting planet to appear slightly larger or smaller when measured in a diﬀerent wavelength. Measuring a planet in more than one wavelength, can therefore reveal information about the nature and the composition of the planetary atmosphere.

The planetary mass cannot be obtained from transit photometry alone. However, it can be measured accurately if the information can be combined with radial velocity data. Radial velocity data on their own provide only a lower limit on the mass, as only the star’s movement along the line of sight is measured. If the velocity measurements can be combined with transit data, the orbit of the planet is fully known, and a mass measurement can be obtained.

One other interesting use of radial velocity data for stars with transiting planets comes from the Rossiter-McLaughlin (RM) eﬀect. When spectral lines of a star are measured, they are broadened because of rotation. At any moment, a part of the star is moving away from the observer, while the other side of the star is approaching the observer, due to the star’s rotation. This causes a red- and a blueshift in the spectrum and causes spectroscopic lines to be rotationally broadened. At a certain moment during a transit, the planet blocks part of the blueshifted half of the stellar disk, causing the starlight to appear more red. As the transiting planet continues its orbit, it blocks the redshifted half of the stellar disk, causing the measured starlight to be blueshifted. By measuring this anomalous Doppler shift throughout the transit, it is possible to measure which part of the starlight is blocked, throughout the entire transit. This provides information on the angle between the planetary orbit and the stellar orbital axis. 1.3. The Kepler satellite 13

1.3 The Kepler satellite

The data that were used in this work to study HAT-P-7 and Sandra comes from the Kepler satellite. The long time series of data points is suitable for both asteroseismic analysis as well as a study of the transiting planet. This section describes the general outline of the Kepler mission and its speciﬁcations. More details on the dataset used in this work are given, as well as the procedure which we have followed to process and normalise it.

1.3.1 Mission design and time series The Kepler mission was launched on March 6, 2009. It was designed explicitly to be able to detect earth-size planets, including those that were present in the so-called habitable zone. These are planets where liquid water can exist on the surface (for our Sun, this requires a planetary distance of 0.95 to 1.37 astronomical units) and that can have an atmosphere to sustain life (a size of about 0.8 to 2.2 R ). For more information about habitability of planets, we refer to Kasting et al. (1993).

When the Kepler mission was being designed, more than 250 exoplanets had already been detected, but most were gas giants, often in short orbits and usually detected using the radial velocity technique. The earth-like planets the Kepler mission was developed for, are typically three hundred times less massive than Jupiter. The detection method for Kepler to achieve this goal, is measuring planetary transits to a high level of photometric precision (e.g. the 4 Earth would block about 10− of the light of the Sun). To improve the chances of detection, a large number of stars have to be monitored for a long period of time, and preferably continuously (Borucki et al., 2008).

Asteroseismology was the lucky bystander of the hunt for planets. The Kepler mission provided an immense opportunity for asteroseismic observations on a wide variety of stars. The precision required for the detection of earth-size planets is similar to that required for doing asteroseismology, while the large Kepler field ensures the observation of many different interesting targets. Asteroseismology was added to the Kepler mission as a second science goal. The Kepler Asteroseismic Investigation (KAI) is organised around the Kepler Asteroseismic Science Operations Centre (KASOC), which is led by the University of Aarhus. The asteroseismic observations are shared by all members of the Kepler Asteroseismic Science Consortium (KASC), which is organised in different working groups. More information on the asteroseismology part of the Kepler mission can be found in Christensen-Dalsgaard et al. (2007).

For the purpose of asteroseismic studies, a small sample of stars (around 500) in the Kepler ﬁeld are being observed in a short-cadence (SC) mode, sampling the star’s brightness every 58.8 seconds. A much larger number of stars are observed by Kepler in the long-cadence (LC) mode, which samples about 150 000 stars at a rate of 29.4 minutes. The Kepler mission is equipped with a 95-cm Schmidt telescope with a ﬁeld of view (FOV) having a diameter of 16◦. The photometer consists of 42 CCD cameras observing roughly in a wavelength band between 450 and 950 nanometer (Borucki et al., 2008). 1.3. The Kepler satellite 14

Kepler was launched in an earth-trailing heliocentric orbit (ETHO). This has several advan- tages over a low-Earth orbit (such as e.g. CoRoT), such as no passing in and out of the Earth’s shadow and heating from the Sun, or no varying earthshine on the telescope. The ﬁeld-of-view of the telescope had to be relatively close to the north or south pole, to avoid solar light blending in (a 55◦ Sun avoidance angle was chosen). The northern hemisphere was chosen and a region with a low amount of bright stars was selected. The ﬁnal position on the h m s sky is centered on the Cygnus-Lyra region (RA: 19 22 40 , DEC: 44◦ 300 ), which is above the Orion arm of the galaxy just above the galactic plane (Borucki et al., 2008). Choosing a single FOV has the advantage of assuring a high duty cycle, simplifying data processing and observing the same targets for an extended period of time (Koch et al., 2010).

Science operation began on May 13, 2009, after a commissioning period of 67 days. The data are divided into quarters. The ﬁrst two datasets are Q0 (9.7 days taken during commissioning) and Q1 (33.5 days of data). After this, quarter data contain 93 days of observations, after which the photometer is rolled 90◦ to keep the solar arrays pointed at the Sun. This causes a data gap of 42 hours. Once a month, data is downloaded to the Earth, causing an interruption of 26 hours (Haas et al., 2010).

For this work, data from Q0 until Q11 have been used for HAT-P-7 and Sandra. Both have been continuously observed by Kepler in its short-cadence mode. As of June 2012, data until Q6 has been made available to the public. The other data (Q7-Q11) used in this work will become public at latest in November 2012. The time series contains long-term trends that are caused by detector eﬀects and need to be removed before the data can be used for science. The next section discusses the data treatment procedure.

1.3.2 Data normalisation The raw Kepler data contain several significant long-term trends in the time series. These are caused by focus changes and thermal effects on the instrument. Step discontinuities can be caused by pointing offsets or thermal transients due to safe modes, or drops in pixel sensitivity (Jenkins et al., 2010). I have processed the data myself, using a moving median filter to remove the trends and normalise the data to arbitrary units.

For every data point, the median of the flux of the surrounding data points is calculated. Subsequently, every data point is divided by this median value. The result is a time series of data that is normalised to one. The result is shown in figure 1.5 for HAT-P-7, and in figure 1.6 for Sandra.

The size of the median filter must be chosen so that it is significantly larger than the gaps caused by the planetary transits, to assure no loss of information. For HAT-P-7, we also expect to see a variation in the flux on a period of the planet orbit. We pick the exact size of the median filter to be as close as possible to one planetary period. Therefore, the size of our median filter is taken to contain 6475 data points. 1.3. The Kepler satellite 15

(a) Raw data (b) Normalised data

Figure 1.5: Raw and normalised data for Hat-P-7. The bottom ﬁgures provide a subset of the data, to display more details.

This procedure is harder to apply for Sandra, for several reasons. This star has longer periods and significantly shallower transits, as can be seen by comparing figures 1.5d and 1.6d. To further complicate matters, there is a periodic data trend as can be seen in figure 1.6c, with a period of about 20 days. It is different from the planetary orbital period (see section 3.1.2).

Because of the shallow transits, we do not expect the planet to cause any other periodic variation in the flux level (e.g. a phase-dependent variation, or occultation). Therefore we are only interested in the transit information (and the stellar pulsations), and we chose to filter away the 20-day effect by filtering with a relatively short period. We apply a median filter which uses 800 data points (about 13 hour). This does not destroy any information about the transits (or the asteroseismology), but removes all longer trends present in the time series. 1.4. The target systems: HAT-P-7 and KIC 8866102 (‘Sandra’) 16

(a) Raw data (b) Normalised data

Figure 1.6: Raw and normalised data for Sandra. The bottom ﬁgures provide a subset of the data, to display more details.

1.4 The target systems: HAT-P-7 and KIC 8866102 (‘Sandra’)

In this work, two star-planet systems are studied in detail: HAT-P-7 (which is also known as KIC 10666592 or as Kepler-2; visual magnitude 10.46; spectral type F8) and KIC 8866102, which I have nicknamed ‘Sandra’ throughout this work (visual magnitude 9.50, spectral type G0). Both have been observed by the Kepler mission from the start in its short-cadence mode. Data from Q0 until Q11 was used in this work. Both stars host a transiting exoplanet and are suitable for asteroseismic studies.

HAT-P-7 is presumably one of the most extensively studied star-planet systems known to date. The planetary companion to the star (HAT-P-7b) was discovered by the HATNet Project by P´alet al. (2008), prior to the start of the Kepler mission. For this work, two diﬀerent studies of the system are extremely relevant: a study of the stellar pulsations by Christensen-Dalsgaard et al. (2010) and a study of the planet’s properties by Borucki et al. (2009b). Both have used early Kepler data. 1.4. The target systems: HAT-P-7 and KIC 8866102 (‘Sandra’) 17

The asteroseismic study (Christensen-Dalsgaard et al., 2010) performed a modeling of the star, by using the pulsation data from Kepler’s ﬁrst month of observations (Q0 and Q1). We will use the entire available Kepler dataset and improve on their initial stellar modeling. We will repeat the entire frequency determination based on the full dataset, and redo the frequency modeling following a procedure similar to the one used by Christensen-Dalsgaard et al.

Borucki et al. (2009b) discovered phase variations in the light curve of HAT-P-7 based on Kepler’s Q0 data only. They also discovered occultations and made an attempt to derive a planetary temperature. We will measure the phase variations and occultations with the currently available data, to obtain far more precise parameter estimates.

From the beginning, the system was very interesting because of the very close orbit of the planet. Welsh et al. (2010) discovered ellipsoidal variations in the light curve, caused by the non-sphericity of the star as a consequence of the closely orbiting planet. This effect was not incorporated in the phase variations and occultations as studied by Borucki et al. (2009b), as it was not known at that time, but we will include it in our work. As HAT-P-7 is the only star known to date where ellipsoidal effects are caused by a planet, we will also study the magnitude of the effect to improve on the results of Welsh et al. (2010). Finally, studying the phase variations will also lead us to new temperature estimates for HAT-P-7b, which we will compare with work done by Spiegel and Burrows (2010) and Christiansen et al. (2010), who have also studied the atmosphere of HAT-P-7’s planet.

Another consequence of the close orbit of HAT-P-7b is presumably the large planetary radius. This is modeled and explained as an eﬀect of coupling between the tidal and thermal evolution of a planetary system (Miller et al., 2009).

A very fascinating result obtained, deals with the inclination of HAT-P-7, which is found to have a retrograde or polar orbit (Winn et al., 2009). The authors came to this conclusion by analysing the Rossiter-McLaughlin effect in radial velocity data. A similar result seems to be obtained by an asteroseismic study of the full currently available photometry and simulations of amplitudes and splittings for different inclination angles, finding a nearly polar inclination for the star (Lundkvist and Lund, paper in preparation). We merely touch upon this subject in this work: our rotational splittings (or lack thereof) seem to support a polar orbit.

The exceptionally tilted orbit of the star (a 90◦ inclination would be natural in a star with a transiting exoplanet) is puzzling. Most potential explanations include some sort of encounter with a third body (another star or an additional planet). Winn et al. (2009) ﬁnd evidence in the radial velocity data that could support an additional, distant body. Ballard et al. (2011a) study the transit timings based on data from the EPOXI mission to place certain limits on the presence of additional planets, but ﬁnd no evidence of their existence. Direct imaging observations have discovered massive bodies that are companion candidates for HAT-P-7 (Narita et al., 2010; Narita, 2011). We will analyse the transit timings in the Kepler data to look for potential additional bodies.

That HAT-P-7 is an extremely interesting system that has been studied extensively, can be illustrated best by performing a search in the Astronomical Database (ADS) by NASA. This 1.5. Objectives 18 results in no less than 28 published papers dealing with this star or its planet, since 2008. This situation is entirely different for Sandra, where the same search returns zero results. This work represents the first results on this star and its planet. We study the host by doing a frequency determination and a full modeling of the star using these pulsations. The rotational splitting allows us to estimate the star’s rotation. In addition, we are the first to determine the planetary properties by studying the planetary transits. Studying the transit times and their potential variations, we look for the presence of third-bodies in the system. They are currently unknown.

With an increasing amount of Kepler data becoming available, it can be expected that a number of publications on this star will follow soon. The results that are reported in this work make use of non-public Kepler data. This report will therefore remain under embargo until the end of October 2012, after which the data become publicly accessible, as will this dissertation.

1.5 Objectives

We now know how asteroseismology can be used to deliver information on stars. Studying transits allows us to infer a wealth of information about the planet. Both of these ﬁelds use observations that can be acquired by the Kepler mission. HAT-P-7 and Sandra are two stars in the Kepler ﬁeld that have been observed for hundreds of days now, and oscillate solar-like while they also host a transiting planet.

It is clear from this introduction that both ﬁelds, asteroseismology and exoplanets, evolve so rapidly that it is hard to keep up with all progress. Some claim that asteroseismology has only become a mature science in the last few years, despite the fact that the theoretical founda- tions have been around for a few decades now. The ﬁeld of extrasolar planets has not even celebrated its twentieth birthday yet, so it is safe to say that the best is yet to come.

Kepler is responsible for bringing both domains to their golden age. Indeed, it has brought nothing less than a revolution. Some might argue that there are currently more Kepler data available than scientists to interpret them. This thesis work aims to show the power of the currently available datasets, by performing a complete analysis on two star-planet systems.

For HAT-P-7, a large number of publications have already been made. However, all of them have focused on a speciﬁc aspect of the system only. Christensen-Dalsgaard et al. (2010) analyzed the stellar pulsations, while Borucki et al. (2009b) looked at the planetary transits. This work shows that it is possible for one scientist to do both, within a reasonable amount of time. Not only did I redo the work of both groups and that of many others who have worked on this system, the quality of the currently available data has allowed me to improve the results of almost all of them.

Whereas HAT-P-7 oﬀers a lot of publications to improve on and compare with, quite the opposite is true for Sandra. All results on this star are completely new. Asteroseismology has allowed me to model the star in detail and determine its mass, radius, temperature, age, and a number of other parameters. Using the same Kepler data, I have studied the star’s planet and determined its period and its orbit. Both ﬁelds were combined to obtain the planetary radius, 1.5. Objectives 19 which shows the power of working on both aspects. At the end of this work, the reader will know both the star and the planet, whereas nothing was known before.

One other hot topic in planetary science is also treated in this work, since it was too exciting to leave out: the study of transit timing variations. Studying the times of transits and seeing how they might diﬀer from what is expected from the calculated orbit, I have looked for the presence of additional bodies in the system.

The following chapter treats the asteroseismology. I have carried out the observational aspect (obtaining the frequencies from the Kepler time series) as well as the modeling of this star. Improving on the currently available theory about stellar pulsations is outside the scope of this work. The main goal of this chapter is to use what is known in asteroseismology today, to obtain as many details about the star as possible.

Chapter three then uses this stellar information and combines it with everything that can be learned from the planetary transit. A planetary period, inclination and radius are the main take aways here. For HAT-P-7, the planet is large and orbits close to the star, so we can also study the temperature of its dayside and nightside. Finally, obtaining planetary masses requires additional radial velocity data that can not be obtained using Kepler. For HAT-P-7, such data are available and we have used them to reﬁne the mass estimate of the planet. When such observations will be performed for Sandra, the values quoted in this work can be used to determine the planetary mass.

The last chapter then puts everything into perspective and discusses the future of the field. The scope of this work is broad, but the following chapters show that with the quality of observations available today, it is entirely acceptable to be ambitious. The heavenly motions... are nothing but a continuous song for several voices, perceived not by the ear but by the intellect, a figured music which sets landmarks in the immeasurable flow of time. Johannes Kepler 2 The pulsations of the host stars

In this chapter, the asteroseismic aspects are discussed. This chapter contains two large sections. The ﬁrst section discusses how to obtain pulsation frequencies based on a time series of Kepler observations. Rotational splitting is also discussed, and the section concludes with a number of observed frequencies for both stars. The other section of this chapter deals with the modeling of these obtained frequencies. The procedure is discussed, with a focus on the reliability of the results. This brings us to the ultimate goal of this chapter, a number of accurately determined parameters for each star. The next chapter focuses on the planetary aspects of the systems, and thereby puts some of the obtained stellar parameters to direct use.

2.1 Observing the frequencies

To determine the frequencies based on the available Kepler data, we follow a procedure that is similar to the one followed by Christensen-Dalsgaard et al. (2010). In this paper the authors have performed a preliminary asteroseismic analysis of HAT-P-7 and two other stars, based on the ﬁrst month of Kepler data. With a much longer dataset available now, we will redo their analysis for HAT-P-7 to reach far greater precision.

For Sandra no asteroseismic analysis has yet been performed. Following the same procedure as for HAT-P-7, we will determine the frequencies and determine their radial order and angular degree. For Sandra, we also ﬁnd frequencies that are split due to the rotation of the star. We will use the splitting to estimate the stellar rotation.

Section 2.1.1 describes the procedure, the rotational splitting for Sandra is discussed in section 2.1.2 and the obtained frequencies are shown in section 2.1.3.

20 2.1. Observing the frequencies 21

2.1.1 Procedure We base our asteroseismology on the same dataset that we use for the planetary observations, but the transits have been removed before starting our analysis. The transits were removed by using median filtering. The filtering code was designed specifically for this process and will detect transit-like features, where it will use a short filter period (1 hour). In the remainder of the time series, a longer period is used for filtering to avoid destroying asteroseismic information. This filtering was designed by Hans Kjeldsen (paper in preparation).

The ﬁltered photometry is then used to calculate a power spectrum. We use the software package PERIOD04 for this purpose and calculate a spectrum with a sampling rate of 0.005 µHz. This means an oversampling of about three times, as HAT-P-7 and Sandra are both measured by Kepler in its short cadence mode of 58.85 seconds (a frequency of about 0.017 µHz). We show the power spectra for HAT-P-7 and Sandra in ﬁgure 2.1.

In a next step, we smooth this power spectrum by applying a median filter with a period of around 1 µHz. This removes the fine structure of the spectrum caused by the finite lifetime of the modes. We have experimented with different sizes for median filters to get the best smoothing result while still being able to identify all modes, and found this period to yield the best results.

The detectable p-modes in the two stars follow a regular pattern described by the well-known asymptotic relation (equation 1.1). Based on this relationship, the modes can now be identi- ﬁed from the smoothed spectrum. The angular degree can be determined without ambiguity, while the exact radial order will be ﬁxed only when comparing the observations to the models. An algorithm locates all maximum values in the smoothed spectrum, but picking the ones that belong to actual peaks was done manually. In certain cases, the average of two peak frequencies is used.

For Sandra, the l = 1 and l = 2 modes have been rotationally split and we detect respectively two and three frequencies for each radial order. In the case of l = 1, we use the average of the two frequencies. For l = 2, we obtain the most precise result by using the central frequency. For more details we refer to section 2.1.2.

It should be noted that determining frequencies to high precision is as much an art as a science. This is particularly true for HAT-P-7, where the short mode lifetime is causing signiﬁcant blending for the l = 0 and l = 2 modes. The exact mode lifetime can be calculated from the half width at half maximum (HWHM) of the frequency peak proﬁle:

1/2π 1 τ = = HWHM 2π HWHM ·

While we do not attempt to measure the exact mode lifetime, it is clear from the width of the l = 1 peaks of HAT-P-7 (see ﬁgure 2.2) that the lifetime is of the order of a few hours only. 2.1. Observing the frequencies 22

(a) HAT-P-7

(b) Sandra

Figure 2.1: Oversampled power spectrum (red) and smoothed curve (black), in relative intensities. HAT-P-7 shows a peak around 1100 µHz, Sandra’s peak is located at around 2000 µHz. 2.1. Observing the frequencies 23

Figure 2.2 illustrates the entire procedure that has been described here. It shows the oversampled power spectrum in red together with the smoothed spectrum in black. On the smoothed spectrum, all maxima are shown as small green dots. The relevant maxima are selected manually (illustrated by the colored lines). For the l = 2 mode, the average of two maxima is used for the frequency determination. The regular pattern which repeats l = 1, l = 2 and l = 0 frequencies, is determined by equation 1.1.

Figure 2.2: Mode identiﬁcation for three observed frequencies for HAT-P-7. The red oversampled power spectrum is smoothed (black line). An algorithm determines the maxima of the smoothed spectrum (green dots). The relevant maxima are selected manually using the large separation. Sometimes, an average of two maxima is used (e.g. l = 2 mode in this ﬁgure).

2.1.2 Rotational splitting The asymptotic equation (eq. 1.1) relates the frequencies for different modes, assuming there is no dependence on the azimuthal order m. Lacking rotation, there is a (2l + 1)-fold degeneracy for each mode. The rotation of stars lifts this degeneracy and splits the l = 1 and l = 2 modes into three and five frequencies, respectively. For a general description of the effects of rotation on stellar pulsations, we refer the reader to the relevant chapter of Aerts et al. (2010).

Assuming a rigid-body rotation and an approximation of ﬁrst order in the angular velocity Ω (assuming slow rotation), the frequency of a mode (n, l, m) is given by 2.1. Observing the frequencies 24

ω = ω + mΩ(1 C ). nlm nl − nl This means the frequency splitting depends on m and on the stellar angular velocity Ω. The factor Cnl is a dimensionless quantity that corrects for the Coriolis force, which we will ignore 2 in this discussion as it is expected that Cnl < 10− for p modes of solar-like stars (Gizon and Solanki, 2003). Similarly we will ignore second order eﬀects, such as a distortion of the equilibrium structure of the star due to centrifugal forces.

Gizon and Solanki (2003) argue that the frequencies we actually observe depend on the inclination of the rotating star. This fact is illustrated in figure 2.3 which is taken from their paper and shows power spectra for different angles of inclination. In our solar system, the difference between the inclination of the normal of the orbital plane (the planetary inclination) and of the solar rotational inclination is less than 10◦. It is the most natural to expect an inclination of about 90◦ for stars with transiting exoplanets, even though exceptions can exist. HAT-P-7 indeed seems to be such an exception.

Figure 2.3: Expectation value of the power spectrum for dipole and quadrupole multiplets as a function of the inclination angle i, for Ω = 6Ω . The bottom panels show the power for i = 30◦ and i = 80◦. Figure courtesy by Gizon and Solanki. (Gizon and Solanki, 2003) 2.1. Observing the frequencies 25

Frequency splitting: 2Ω (µHz) l = 1 4.38 0.18 l = 2 (left) 4.32 ± 0.35 l = 2 (right) 4.30 ± 0.18 Average 4.35 ± 0.13 ±

Table 2.1: Rotational splitting for Sandra. For l = 1 the value is based on 14 split frequency peaks, for l = 2 we have 7 peaks. For l = 2 we have calculated the diﬀerence between left and central frequency (left) and central and right frequency (right).

Assuming an inclination of 90◦, we expect to see two peaks for the l = 1 modes, split by 2Ω and three peaks for the l = 2 modes, also each split by 2Ω. This is indeed what we observe for Sandra. We show an example in ﬁgure 2.4. Gizon and Solanki (2003) argue that the amplitudes of the split frequencies can be used to determine precise orbital inclinations. This is beyond the scope of this work, but illustrates the power of using asteroseismology on exoplanet hosts.

For Sandra, we have determined the frequency diﬀerences between the two l = 1 peaks and between the left and central l = 2 peak and the central and right one. We have observed splitting for 14 l = 1 modes, and for 7 modes with l = 2. We present the results in table 2.1.

To determine the rotation rate of Sandra, we now assume the inclination is exactly 90◦. We then ﬁnd: 1 P = = 5.33 0.16 days. (2.1) Ω ± If we look back to the photometry of this star, we see that it displays a regular variability with a period of about 20 days. This is shown in ﬁgure 2.5. This could be interpreted as being caused by the star rotating in a period of 20 days. Since the frequency splitting features are very clear, there seems to be little doubt that the actual rotation rate is the one calculated in equation (2.1).

We therefore suspect that the features visible in the time series are caused by a background star (binary star) that blends with the photometry of Sandra. To conﬁrm this theory, we should look at the stars that are in the Kepler ﬁeld close to Sandra, but we make no attempt to do so in this work. An alternative, but much more exotic (and unlikely) explanation would be that the stellar inside of the star is rotating much more rapidly than the stellar surface.

In the case of HAT-P-7, we have not been able to detect any clearly split frequencies. This could mean that the small mode lifetime is causing all split frequencies to blend, so that the splitting is not observable. Another possible explanation is that we are looking at the star on its pole, which would make it very hard to detect any splitting. Simulations of the amplitudes of the modes and their splitting seem to conﬁrm this theory (Lund and Lundkvist, paper in preparation). A pole-on or retrograde orbit for the star has been suggested earlier by Winn et al. (2009). 2.1. Observing the frequencies 26

Figure 2.4: Power spectrum for Sandra, showing split frequencies for l = 1 and l = 2 modes.

Figure 2.5: Part of the time series of Sandra, showing a regular ﬂux variation with a period of 20 days. This could be caused by stellar rotation, but the rotation period derived from the stellar≈ pulsations rules out this interpretation. Instead, it is likely caused by variable ﬂux of a background object. 2.1. Observing the frequencies 27

2.1.3 Results The frequencies we obtain are summarised in an échellediagram. In this figure, we plot the frequencies versus the frequencies modulo the large separation. In the case of Sandra, the rotationally split average is used for the l = 1 and l = 2 modes. Assuming the asymptotic equation (eq. 1.1) is exact, we expect the frequencies to line up vertically in this diagram, showing three ridges for l = 0, 1, 2 (see also figure 1.2b, showing an échellediagram for the Sun).

In reality, the equation is not an exact one and we do not expect to find a straight line. Since the equation is correct up to first order, we expect to see a smooth shape (see e.g. the best model in section 2.2). The large separation is calculated as the average of the separations between the modes: we fit a straight line through all mode frequencies of the same angular degree and take the average of the values for the three different radial orders. The values are shown in table 2.2. The échellediagrams with the observations are shown in figure 2.6 and 2.7 for HAT-P-7 and Sandra.

We will use the échellediagram and the asymptotic relation to try to estimate the uncertainty on the determination of the frequencies. By assuming that the frequency separations are linear to first order, we can estimate the error of a frequency by seeing how much it differs from the average value of the frequencies with a lower and a higher n value. Mathematically, this becomes:

fn+2 + fn σ δfn+1 = fn+1 = + σ, 2 − √2 where we assumed that all frequencies have equal errors to ﬁrst order. We can use this to calculate error estimates for every frequency, except for the lowest and highest measured radial order n for each angular degree l. We can then average the error values of the diﬀerent frequencies and use this as an estimate for the true error. We have listed all observed frequencies, along with their errors, in tables C-1 and C-2 for HAT-P-7 and Sandra.

On average, we determine the frequencies of HAT-P-7 and Sandra up to 0.75 and 0.80 µHz. These are very accurate frequency values. Christensen-Dalsgaard et al. (2010) calculated the frequencies of HAT-P-7 to an error level of 1.40 µHz. In addition, in this work more frequencies are determined: while Christensen-Dalsgaard et al. determined the frequency values of 10 (l = 0), 12 (l = 1) and 11 (l = 2) modes, we do so for 14, 13 and 16 frequencies. Since their analysis was based on the ﬁrst month of Kepler data only, it is no surprise we can do far better with the currently available data.

∆ν0 (µHz) Hat-P-7 59.519 Sandra 94.55

Table 2.2: Average large frequency separation. 2.1. Observing the frequencies 28

Figure 2.6: Echelle´ diagram for HAT-P-7. Reduced frequency calculated modulo ∆ν = 59.519 µHz

Figure 2.7: Echelle´ diagram for Sandra. Reduced frequency calculated modulo ∆ν = 94.55 µHz 2.2. Frequency modeling 29

2.2 Frequency modeling

We attempt to ﬁnd a stellar model that produces frequencies that match the frequencies we observe. First, we calculate stellar evolution tracks with the Aarhus STellar Evolution Code (ASTEC). In a next step, we select relevant evolutionary phases and calculate frequencies for each model, using the Aarhus Adiabatic Oscillation Package (ADIPLS). We correct the models for near-surface eﬀects before comparing them with observations. This procedure is described in section 2.2.1.

In section 2.2.2, we discuss the accuracy of our models. We obtain stellar parameters and error estimates by doing a Monte Carlo analysis, introducing Gaussian noise on our observed frequencies. Finally we present the stellar parameters we obtain in section 2.2.3.

2.2.1 Procedure In a ﬁrst step, the Aarhus STellar Evolution Code (ASTEC) is used to calculate stellar evolution tracks for a range of initial mass and metallicity values. The code was originally designed to in- vestigate solar equilibrium models and is described in detail in Christensen-Dalsgaard (2008b). Its main use nowadays is to calculate models along evolution tracks, for which stellar frequencies can be calculated for time steps in the evolution. It includes a treatment of convective cores and core overshoot.

The initial abundances (by mass) X0 and Z0 of hydrogen and heavy elements are characterized by the value for [Fe/H]. We assume X0 = 0.7679 - 3Z0 to determine the Helium abundance Y0. For the convective core, diﬀerent possible overshoot values were included. In terms of pressure scale heights, we calculate evolution tracks for diﬀerent αov values (we have used 0, 0.1 and 0.2; see section 2.2.3).

The code uses the OPAL equation of state and the NACRE nuclear reaction parameters. Diﬀu- sion and settling of helium is not included. For more details we refer to Christensen-Dalsgaard (2008b).

Next we use the fact that the large separation is directly related to the mean density of the star: 1/2 ∆ν0 ρ (equation 1.2). However, this scaling relation is not an exact one. White et al. (2011)∝ hsuggest∗i the use of a slightly modiﬁed version, based on Kepler results for numerous solar-like pulsators:

2 ρ ∆ν 2 = (f(T ))− , ρ ∆ν eﬀ

with

T 2 T f(T ) = 4.29 eff + 4.84 eff 0.35, eff − 104 K 104 K − and we will use this modification to estimate the stellar density.

An estimate for ∆ν0 can easily be calculated from the measured frequencies (see equation 1.1). For each initial mass in our grid, we now select the time in the stellar evolution tracks 2.2. Frequency modeling 30

when the star has approximately the right mean density. We select the ten most relevant phases for each evolution track (each having its own mass and metallicity).

We subsequently use the Aarhus Adiabatic Oscillation Package (ADIPLS) on these relevant evolutionary phases to calculate oscillation frequencies. The full physics behind the numerical integration performed by this package, is described in detail in Christensen-Dalsgaard (2008a). This code, as any other pulsation code, does not model the near-surface layers very well. As a consequence, there is an oﬀset between observed and calculated frequencies for the Sun.

Based on this oﬀset, an empirical correction is calculated by Kjeldsen et al. (2008). The authors model the oﬀset between observed and best model frequencies with a power law:

b νobs(n) νobs(n) νbest(n) = a , − ν0

where ν0 is a reference frequency and a and b are parameters to be determined. In addition, they set νbest(n) = rνref (n), where r should be close to one as the reference model should be close to the initial best model. In practice, we set b = 4.90 (the solar value) and we ﬁnd the best ﬁt for r and a (for each model individually). We can then calculate the reference model, for comparison with the observed frequencies.

The best model is then picked by comparing the observed frequencies with the corrected model frequencies. Only those model frequencies for which an observational counterpart is detected, are used in this procedure. We use a χ2 minimisation to determine which model ﬁts best:

2 2 1 X (obs) (mod) χ = νnl νnl . (N 1)σν − − nl Here we have written σν in front of the summation, since we use the average observational error on the frequencies, as described in section 2.1.3.

2.2.2 Reliability Determining the reliability of parameters derived using ASTEC and ADIPLS is generally diﬃ- cult. The code introduces both errors as a consequence of ﬂaws in theoretical models as well as due to numerical calculations. In addition, errors on parameters are usually not independent. In most areas of astronomy error estimation is a poorly developed area, and asteroseismology is unfortunately no exception to this.

To overcome these difficulties, we run a small Monte Carlo analysis to estimate the stellar parameters and their error values. For this, we introduce Gaussian noise on the observed frequencies (centered on the observed frequency, with a width of the estimated observational error). In this way we create 1000 different datasets of observed frequencies. For each of these datasets, we will calculate the model that fits best, following the procedure in the previous section. Each of the best models has its own stellar parameters. We will now use the mean and the standard deviation of the 1000 sets of parameters as the true value and its error estimate.

This way of estimating the error assumes the models themself contain no errors. To illustrate this point, let us assume there exists one model that is far superior to other models, so that it 2.2. Frequency modeling 31 provides the best frequency ﬁt to all 1000 observational datasets. The standard deviation will now be zero, so in this case we would estimate the error as zero. If the parameters that are associated with this model are not entirely correct (due to imperfections in the asteroseismic theory), the error values are underestimated.

Another way to understand this point is by calculating errors only for one overshoot value (e.g. take overshoot to be zero). The models that ﬁt very well, will all be in a fairly tight range of stellar ages. However, since changing the overshoot value will aﬀect the age of the best models (and it is hard to discriminate which overshoot value is the most appropriate), the true age is less reliably determined (this is illustrated in section 2.2.3).

2.2.3 Stellar properties We have calculated different sets of parameters following the procedure described in the previous section. For three different values of overshoot, the results are shown in table 2.3. The error values come from having calculated 1000 different best models to different sets of observations (see section 2.2.2). The last column of table 2.3 gives an average parameter value, that has been calculated giving equal weight to all overshoot possibilities. The parameters and errors come from the average and standard deviation of all 3000 best models.

The stellar density ρ and the surface gravity log g are derived from the other parameters (as M/[(4πR3)/3] and GM/R2 respectively, with G the gravitational constant). They have been calculated for each best model individually, and averaged afterwards.

For HAT-P-7, the overshoot-free column has been calculated for a grid of models with a mass range of 1.33-1.70 M and a metallicity between 0.020 and 0.035. Between 1.33 and 1.45 M , additional metallicities from 0.015 to 0.020 were also calculated. For both overshoot values, a grid has been calculated using masses from 1.35 to 1.60 M , and metallicities from 0.015 to 0.033.

For Sandra, we used a grid of metallicities 0.010-0.030 and 0.70 to 1.30 M for αov = 0. In both cases that have used overshoot, the mass range was limited to 1.00-1.30 M and the metallicity range was 0.019 until 0.030. As we can see in the table, the overshoot cases ﬁnd metallicities up to the edge of our grid. It might be worth calculating higher metallicity values to obtain a more accurate model. The other parameters are of more interest to us for this work, and seem to be consistent with the no-overshoot case.

Arguments can be made for giving extra weight to a certain value of overshoot, or using one column entirely. These can be made either on physical grounds favoring a certain overshoot value, or based on the χ2 value that is better for a certain column. We provide all individual parameter sets in the table, to provide an unbiased result. For the planetary calculations in this work, we have chosen to use the averages in the rightmost column of table 2.3.

Comparing the various possibilities, we see that the diﬀerent values agree especially well for the radius. As we will see in the next chapter, this value is important for calculating the planetary radius. It is therefore encouraging to see that we can determine the stellar radius to excellent accuracy. The stellar masses are in fair agreement as well, so they are determined 2.2. Frequency modeling 32 to a good level of accuracy, as are the temperature values. The metallicities are harder to constrain. The main diﬀerence between the overshoot values is present in the values for the stellar age. It is therefore impossible to obtain a very accurate age without favoring a certain value of overshoot, although a relative precision of 10% for HAT-P-7 and of 25% for Sandra is still by far better than any other age estimation method.

Figures 2.8a and 2.8b provide an ´echellediagram comparing the best model to the observations, for overshoot zero (αov = 0). We see that the model for Sandra is excellent, which is also reﬂected in the χ2 value. For HAT-P-7, the overall agreement is very good, but the model fails to accurately mimic a number of higher order l = 1 frequencies.

It might be interesting to look into this and attempt to further improve the model for HAT- P-7. One possible explanation for the frequency deviation would be discontinuities in the star that are not modeled accurately, such as the base of the convection zone or the second helium ionisation zone. More detailed information and potential ways of modeling these ‘acoustic glitches’ can be found in Mazumdar (2005) and Mazumdar and Michel (2010), but this is outside the scope of this work.

This modeling ‘problem’ did not arise in the earlier work on this star by Christensen-Dalsgaard et al. (2010). The authors derived fewer frequencies and their observational errors were almost twice as large. It is in a way encouraging to see that we seem to have arrived at a level of precision that will challenge the quality of the models, a situation that resembles the decades of solar modeling in helioseismology. 2.2. Frequency modeling 33

HAT-P-7 αov = 0 αov = 0.1 αov = 0.2 Average (χ2 = 3.17) (χ2 = 3.62) (χ2 = 4.17) (χ2 = 3.65)

M? /M 1.369 0.027 1.362 0.018 1.3519 0.0091 1.361 0.021 Z 0.0165 ± 0.0021 0.0166 ± 0.0015 0.01656 ± 0.00051 0.0166 ± 0.0015 0 ± ± ± ± R? /R 1.907 0.014 1.904 0.010 1.9019 0.0036 1.904 0.010 ± ± ± ± Teﬀ (K) 6254 17 6263 34 6259 41 6259 32 Age (Gyr) 2.085 ±0.065 2.173 ±0.059 2.308 ±0.094 2.19 ±0.12 3 ± ± ± ± ρ (g/cm− ) 0.2785 0.0010 0.2784 0.0016 0.2774 0.0020 0.2781 0.0017 log g (cgs) 4.0140 ± 0.0024 4.0131 ± 0.0018 4.0110 ± 0.0028 4.0127 ± 0.0027 L/L 4.996 ± 0.057 5.010 ± 0.087 4.98 ± 0.13 4.996 ± 0.098 ± ± ± ±

Sandra αov = 0 αov = 0.1 αov = 0.2 Average (χ2 = 1.98) (χ2 = 2.82) (χ2 = 3.14) (χ2 = 2.65)

M? /M 1.255 0.041 1.2340 0.0099 1.2296 0.0031 1.240 0.027 Z 0.0244 ± 0.0055 0.0300 ± 0.0050 0.02966 ± 0.00047 0.0280 ± 0.0041 0 ± ± ± ± R? /R 1.373 0.020 1.3731 0.0029 1.3717 0.0011 1.373 0.012 ± ± ± ± Teﬀ (K) 6138 104 5884 27 5886 16 5970 135 Age (Gyr) 1.93 ±0.24 3.31 ±0.24 3.39 ±0.11 2.87 ±0.70 3 ± ± ± ± ρ (g/cm− ) 0.6839 0.0075 0.6727 0.0012 0.67260 0.00064 0.6764 0.0069 log g (cgs) 4.2614 ± 0.0026 4.2543 ± 0.0017 4.25370 ± 0.00054 4.2565 ± 0.0040 L/L 2.40 ± 0.11 2.031 ± 0.047 2.027 ± 0.022 2.15 ± 0.19 ± ± ± ±

Table 2.3: Parameter values for HAT-P-7 and Sandra. Three diﬀerent sets of parameters have been calculated, using diﬀerent values for the overshoot. The average has been calculated giving equal weight to all three overshoot calculations. 2.2. Frequency modeling 34

(a) HAT-P-7

(b) Sandra

Figure 2.8: Echelle´ diagrams showing the observed frequencies (red) and the best model (blue). We ever long for visions of beauty, We ever dream of unknown worlds. Maxim Gorky 3 The planetary transits

This chapter deals with HAT-P-7b and Sandra-b, the planets that are orbiting around the host stars analysed seismically in the previous chapter. In the ﬁrst section, we discuss how an accurate transit time can be obtained. This is then used to determine the planetary period. The next section studies transit timing variations to look for the presence of additional bodies in the system, or lack thereof. Subsequently the time series is folded and all transit data are combined to make an accurate transit model. From the depth and the duration, the planetary radius and inclination is derived. Finally, for HAT-P-7, the light curves show phase variations (ﬂux variations outside the transit) and occultations, which are used to study the temperature of the planet’s day- and nightside. Whenever stellar parameters are required to calculate planetary values, the parameters for the host star are taken from the asteroseismic results in the previous chapter.

3.1 Planetary period

As a ﬁrst step in analysing the time series containing planetary transits, we will measure the exact orbital period of the planet. Because the periods are of the order of days, with the data spanning years (largely without gaps), this measurement can be done to high precision. To determine the period, we need to accurately measure the individual transit times. In a next step, we will then assume a perfectly Keplerian orbit and linearly ﬁt the transit times to determine the period. Deviations from the Keplerian orbit could reveal the presence of additional bodies in the system, and the analysis of these transit timing variations is treated in section 3.2.

3.1.1 Determining transit times In a basic naive approach to determine the transit time, we just pick the deepest point of the transit to determine the time. Random noise renders this method relatively inaccurate, and

35 3.1. Planetary period 36

to improve on this method we try to reduce this noise, by using a running median ﬁlter with a short period on the data. Such a ﬁlter has been used in the data normalisation procedure as well (section 1.3.2), and this improves accuracy. It replaces a data point by the median of its surroundings. How many surrounding points are used depends on its period.

The exact shape of the bottom of the transit curve is governed by the limb darkening of the star. This can be modeled and we indeed did so to determine the transit depth (see section 3.3). On the other hand, the ingress and egress of the transit are always extremely well approximated by a straight line. We will therefore use these straight lines of the transit to estimate the transit times.

We fit a straight line to the left and right side of the (median filtered) transit data. In a first step, we determine the slope and offset for each left and right side of a transit. Since we expect all transits to have the same shape, we then average the slopes and use the average value of the slope for each transit, and we determine the offset. For HAT-P-7, we fit a straight line between a flux level of 0.9956 and 0.9992 of the normalised flux. We then determine the average of those lines at a depth of 0.9975. For Sandra, we fit between 0.99985 and 0.99995 and average at 0.99990. In figure 3.1 we illustrate this procedure with an example transit for each star.

(a) HAT-P-7 (b) Sandra

Figure 3.1: An individual transit for Hat-P-7. The left and right side of the transit are fitted by a straight line. The middle dot represents the average time between the left and right line calculated at a relative flux of 0.9975. It is obvious from this figure that this procedure gives more stable time information than using the minimum of the transit to determine its time.

We now set out to conﬁrm that this method indeed provides accurate time estimates. We calculate the errors on the linear ﬁt (y = mx + b) following Barlow (1989) and using s 1 P(y yˆ )2 σ = i − i m N 2 P(x x¯)2 − i − and 3.1. Planetary period 37

r 1 X σ = σ x2. b m N i

Here yi is the data point and yî is the corresponding value of the best fit, xi again corresponds to data points whereas x¯ is the average x-value of the best fit. We use the basic rule for error propagation: df V (f(x)) = ( )2V (x). dx From y = mx+b, we find x = y/m b/m and we calculate the variance V (x) = y2V (1/m)+ V (b/m). Here, y is the flux level of− the transit time derived from the method explained above. We find

y2 b2 1 V (x) = V (m) + V (m) + V (b). (3.1) m4 m4 m2 This determines the error on the time determination of the ingress and egress of the transit. We use this to ﬁnd the error on the average of those values, for each transit. The resulting transit times and their error estimates are found in the appendix in table C-1 and C-2 for 4 HAT-P-7 and for Sandra. The precisions on the transit times are of order 10− days and so our method is indeed very suitable. Because of the shallower transits, the precision is lower for Sandra than for HAT-P-7.

3.1.2 Determining the orbital period After we determined the times of each transit, we determine the planetary period. We number the transits and add a number for every gap in our data (where we miss a transit). Once this is done, we fit a straight line through the times as a function of the numbers. Doing this, we assume that the transits are perfectly linear, following a Keplerian orbit. We then find the period as the slope of the linear fit:

y = mx + b (transit time) = period (transit number) + offset. · The result for Hat-P-7 is shown in figure 3.2a, for Sandra we can see it in figure 3.2b. The difference in amount of transits is mainly caused by the longer planetary period for Sandra. Because of the high precision to which the time of individual transits can be calculated, we find the orbital period to high precision. Since we have more transits available for HAT-P-7, its period precision is higher. The results are shown in table 3.1.

Period (d) Error (d) Hat-P-7 2.20473506 0.00000011 Sandra 17.834067 0.000068

Table 3.1: Calculated orbital periods. 3.2. Transit timing variation 38

(a) HAT-P-7 (b) Sandra

Figure 3.2: The transits are numbered, taking gaps into account and a line is ﬁtted through the transit times to determine the period.

3.2 Transit timing variation

We can use measured transit times as a signal for detecting additional planets if they are present. In an ideal, single-planet system, we expect the planet to orbit on a perfectly Keple- rian orbit around its host star. If another planet would be present in the system, it might not necessarily transit but the gravitational interaction with the transiting planet can be measurable.

In calculating the period of the transiting planet, we assumed a Keplerian orbit. Predicting the transit times for a single-planet system, we calculate an O-C diagram, plotting the observed minus the calculated transit time for each transit. A secondary planet is expected to cause a periodic signal on this diagram.

Ten years after the first exoplanet detection (and five years after the first transiting one), the possibility of detecting additional planets through transit timing variations (TTVs) was described. One of the earliest projected uses was the detection of planets as low as earth-size mass (Holman and Murray, 2005). In the same year an attempt to detect additional bodies based on eleven transits in the TrES-1 system yielded inconclusive results (Steffen and Agol, 2005).

It is important to stress that transit times can vary due to other reasons. The decay of the orbital semi-major axis due to tidal interaction is one of those (Holman and Murray, 2005). Starspots being transited can cause apparent variations or the perturbations could be caused by another body, such as a brown dwarf. Dynamic interactions, known for binary stars as the Applegate eﬀect, can be applied to exoplanets and cause TTVs that can be confused with a signature that is the result of planetary interactions (Watson and Marsh, 2010). An intriguing alternative to planetary interaction is gravitational interaction due to exomoons orbiting the planet, as these objects are potentially more habitable than the planets themselves, and are extremely hard to detect in other ways. The eﬀects on the transit timing are described by 3.2. Transit timing variation 39

Kipping (2009) and the author points out that the exomoons should also cause a measurable variation in the duration of a transit.

Presently, the more interesting results coming from the use of this TTV technique have been those ruling out possible planets, placing upper limits on the mass of additional bodies. An interesting example here is discussed in Ballard et al. (2011a), placing upper limits on additional planets for five systems, including HAT-P-7. Upper limits on additional bodies for CoRoT-1 are discussed in Bean (2009). That caution must be in place, is illustrated by the case of OGLE-TR-111, where, based on only a few transits, an additional body was first suspected, but could not be confirmed later using additional transit data (Adams et al., 2010).

For WASP-3b (Maciejewski et al., 2010) and Kepler-19b (Ballard et al., 2011b), transit timing variations are detected and seem to be a consequence of perturbations by an additional planet. Both of these examples illustrate the difficulty of constraining the parameters of the additional body, if it is not found to be transiting as well. Many different scenarios are known to be causing similar TTVs. Usually, three-body simulations are performed to confirm the plausibil- ity of possible orbits. Such detailed analysis is often focused on inner or outer mean-motion resonance, for example placing the additional body close to a 2:1 orbit with the transiting planet.

Perhaps the most interesting case known today is Kepler-11 (Lissauer et al., 2011). This system is known to contain six transiting exoplanets, which in itself makes it highly interesting. The planets each aﬀect one another, and the combination of the transit timing variations of the diﬀerent transiting planets makes it possible to estimate planetary masses without using radial velocity data.

The Kepler mission allows to analyse systems containing transiting exoplanets for many months, with several systems that today have been studied for over two years, while observations are still continuing. As these datasets grow longer and are studied systematically, the TTV technique can be expected to yield an increasing amount of planet detections as variations can be more uniquely interpreted. Where no variations are detected, increasingly strict upper mass limits can be placed on additional bodies.

In this work, we discuss both scenarios. For HAT-P-7, we detect no periodic variation and we place strict upper limits on the mass of potential additional bodies in the system. Analysing the transit times for Sandra, we ﬁnd a variation which we suspect is caused by an additional and previously unknown planet. In both cases, our conclusions are heavily based on similar systems studied in the literature, as full dynamical simulations are outside the scope of this work. These would, however, be extremely interesting to perform, and indeed should be done to conﬁrm our preliminary conclusions.

3.2.1 Hat-P-7 The variations of the transit times for HAT-P-7 are shown in an O-C plot in ﬁgure 3.3. With a few exceptions (corresponding to a known increased error level in the Kepler observations), variations are lower than about 0.0002 days (17 seconds). Calculating a Fourier transform 3.2. Transit timing variation 40 of this data, no periodic signal was detected. The measurement errors of a single observed transit are typically around 0.00015 days (see table C-1 for a full list), so we suspect that the variations we see are caused by random errors.

Figure 3.3: A so-called O-C plot, giving the observed minus the calculated transit times for HAT-P-7. No sinusoidal pattern is detected, to the (very low) level of the noise (observational error level: 0.00015 days). ≈ We would like to place an upper limit on the mass of additional planets. Planets with a higher mass would be expected to cause a detectable TTV signal, and can therefore not be present in the system. Planets that are of lower mass could still be causing a signal that is ‘hiding’ in the noise. Of course this mass limit depends on the period of the potential additional planet: closer planets, as well as planets in resonant orbits, would cause higher perturbations, as was found by Holman and Murray (2005) and later conﬁrmed by many other TTV studies.

The normal way to proceed would be to run three-body simulations for third bodies on different orbits, and find out how large the TTV signal they produce is, for different masses, so an upper limit can be determined for different orbits. This procedure has been followed by Bean (2009) for CoRoT-1b. The parameters of this system are very similar to those for HAT-P-7 (see table 3.2), so in this work we will limit ourselves to rescaling their results to estimate the upper limits in our system.

Bean (2009) determines that no stable planet can exist in orbits interior to the orbit of the transiting planet. For the period of outer orbits they run simulations to determine an upper mass limit and they ﬁnd the most strict limit in a 1:2 orbit. We will scale the limits they ﬁnd to our period, so that their limits for 1:2, 1:3 and 1:4 orbits match these same integers for our system. 3.2. Transit timing variation 41

HAT-P-7 CoRoT-1 O-C (s) 3.82 25.2 | |max Mp (MJ ) 1.743 0.029 1.03 0.12 ± 7 ± ap (R ) 8.04173271 (2.6 10− ) 5.46 0.26 P (d) 2.20473506 ±0.00000011· 1.5089557 ± 0.0000064 ± ±

Table 3.2: A comparison between HAT-P-7 and CoRoT-1 of the parameters relevant for TTV. The systems are very similar. CoRoT-1’s O-C value comes from Bean (2009), the other parameters from Barge et al. (2008). HAT-P-7’s value comes from this work.

In addition, we will rescale their upper limits. The rescaling depends on the mass of the planets, since a higher planetary mass will be subject to a lower variability and therefore allow for a less strict limit. We will also scale with the square of the semi-major axis, since higher distances in the system will mean less gravitational inﬂuence and therefore less strict limits. Additionally, we will scale with the inverse of the period, since a longer period will give more time to perturb the planet.

Finally, we calculate the Fourier spectrum and ﬁnd the highest amplitude of O-C variations that can hide in our dataset. Here, we see from the table that our dataset gives much more strict limits, mostly because we use a longer time series which makes it harder for a large signal to hide in the noise.

The ﬁnal rescaling then becomes:

2 O C HAT P 7 MHAT P 7 aHAT P 7 PCoRoT 1b MHAT P 7,upper = MCoRoT 1b,upper | − | − − − − − − − . − − − O C CoRoT 1b MCoRoT 1b aCoRoT 1b PHAT P 7 | − | − − − − − Using the values from table 3.2, we find a rescaling factor of 0.38. This means that we obtain more strict limits than Bean (2009) for CoRoT-1b. The rescaled limits for a range of periods are shown in figure 3.4. At a period of 10 days, we determine that no planet with a mass higher than about 4 Jupiter masses can exist. At the 2:1 orbit, we can be as strict as to place the upper limit as 1.6 M . ⊕ We emphasise that this method only provides a first attempt to place mass limits. The strict limits that we find show that it should be very interesting to perform a detailed numerical analysis. Doing so should confirm whether the limits we place are indeed correct, and that we are right in assuming no inner exoplanets can exist in this system.

It is interesting to compare these results with Winn et al. (2009). The authors ﬁnd evidence for a third body in this system, based on radial velocity data. While their limited data do not allow them to determine an exact period, they ﬁnd their data to be consistent with periods longer than a few months. This shows that radial velocity variations will continue to be useful for detecting distant exoplanets, while the TTVs can be used mainly to detect additional bodies in close orbit to the host stars. 3.2. Transit timing variation 42

Figure 3.4: An upper limit for the mass of additional planets in the system of HAT-P-7. The most strict limit is in the 2:1 resonance orbit and is about 2M ⊕

Another method to detect additional bodies is of course by observing their photometric transits. We have manually analysed our time series and found no such evidence. A detailed analysis based on EPOXI data was also performed by Ballard et al. (2011a), who found no evidence of additional transiting bodies either.

3.2.2 Sandra In the case of Sandra, we find a significant sinusoidal deviation of the transit times as compared to the predicted linear ephemeris. We show the O-C diagram in figure 3.5. We have used PERIOD04 to determine a best fitted sinus and find a frequency of 0.001621 0.000094 cycles per day (a period of 616 36 days) and an amplitude of 0.00541 0.00077± days (7 1 minutes). This fit is shown in± figure 3.5a and the residuals to the fit are± shown in figure 3.5b.±

The fit lowers the residuals from 0.0049 days to 0.0032 days. With the individual transit times determined to an uncertainty of 0.0015 days on average, this still seems to be a factor two too high to be caused by purely random noise. We attempted to fit additional frequencies to the residuals, but found none that significantly lowered the residuals.

The important ﬁrst question to ask is whether this deviation from linearity is caused by a planet or has another nature. It is useful here to compare with a similar system that has already been investigated. Kepler-19 (Ballard et al., 2011b) presents such a case. The transiting planet has a mass of 2.2 R and a period of 9.3 days, and they ﬁnd a transit timing variation with a period of 316 days and⊕ an amplitude of 5 minutes. 3.2. Transit timing variation 43

(a) O-C plot

(b) O-C plot (residuals)

Figure 3.5: OC plot for Sandra, giving the observed minus the calculated transit times. The level of divergence from zero is an order of magnitude higher than for HAT-P-7. The best fit to transit timing variations is shown in the top figure, the bottom one shows the residuals to the fit. 3.3. Transit depth 44

Ballard et al. spend considerable time in excluding other scenarios that could cause timing variations. They exclude stellar activity (the Applegate eﬀect, Watson and Marsh (2010)) as it is only likely for planets that orbit their host star very closely. They also exclude rotation of the planetary orbit’s apsidal line about the star. In both cases this exclusion seems to apply for our system as well.

They also consider two scenarios in which the variations are caused by a star. One possibility is light time delay as caused by the reﬂex motion of the system as it is orbited by a third body, another one is a dynamical signal owing to perturbations in the system by another body (a second star or brown dwarf). They exclude both scenarios using radial velocity data (and not detecting a large signal in it). Lacking such observations of Sandra, both of these possibilities should be considered. Future radial velocity measurements of the star should conﬁrm or disprove this possibility fairly easily.

Ballard et al. (2011b) then consider five categories of planetary scenarios: orbits with the period of the TTV signal, resonant perturbers, orbits near first-order mean-motion resonances, orbits near higher-order resonances, and satellite scenarios. It is difficult to distinguish between these options, and radial velocity data are required for this to be efficient. If the additional residuals can be modeled, this might provide additional distinguishing features that are not present in the current single-sine fit.

Despite being unable to say much about the exact nature of the transit deviations that we detect, they are unmistakably there. We emphasize that it was never the goal of this work to search for additional planets in our system, which is why we leave it at this basic discussion. On the other hand, this result is very interesting and deﬁnitely requires follow-up research - in the ﬁrst place by spectroscopic observations of the star.

3.3 Transit depth

3.3.1 Overview After the planetary period has been calculated, the time series can now be folded on this period. This will eﬀectively place all transit data on top of each other, providing us with a very well-described transit shape. All other ﬂux variations that are dependent on the phase (including the occultation), are discussed in the next section (section 3.4).

Assuming a perfectly homogeneous star, the loss of light during a planetary transit can be described by two main physical parameters: the impact parameter b and the transit depth δ. This is illustrated in ﬁgure 3.6. Throughout this section we assume an eccentricity of zero.

The total transit duration depends on the semi-major axis and the planetary period (which are known from the transit times) and the stellar radius (known from asteroseismology). The main parameter which can then be used to ﬁt the observed transit width, is the impact parameter. It is itself directly related to the inclination (i), once the semi-major axis and the stellar radius are known (b = a cos i/R ). The expressions describing the transit width are (Winn, 2010): ∗ 3.3. Transit depth 45

Figure 3.6: Illustration of a transit, assuming a star with a homogeneous brightness. Four contact points are shown, the location of which depends on the impact parameter b. The transit depth depends on the relative size of the star and the planet. Figure courtesy by Winn (2010).

p 2 2 ! P 1 R (1 + k) b T = t t = sin− ∗ − , (3.2) tot IV − I π a sin i

p 2 2 ! P 1 R (1 k) b T = t t = sin− ∗ − − . (3.3) full III − II π a sin i

Here the symbol k was used as a short version of Rp/R . The transit depth is directly related to the relative planetary radius: ∗

R 2 I (t ) R 2 δ = p 1 p tra p . R − I ≈ R ∗ ∗ ∗ The approximation is valid when the light from the planetary nightside is negligible. We will use this assumption for Sandra, but use the full equation for HAT-P-7. The nightside ﬂux for HAT-P-7 is estimated in the next section.

When it comes to measuring the depth of the transit, figure 3.6 assumes the star has a flux which is homogeneous over its entire surface. This is not quite true for real stars. A real stellar disk is brighter at the center, and fainter at the edge (limb). This is an optical effect, as in the center of the disk we look deeper into the (hotter) interior of the star. Near the center of the star, the planet will therefore cause a deeper transit than when the planet is near the limb. A quadratic intensity profile can be defined:

I(X,Y ) 1 u (1 µ) u (1 µ)2, ∝ − 1 − − 2 − 3.3. Transit depth 46

with µ = √1 X2 Y 2 (see figure 3.6). The coefficients u are constants defining the precise shape of the limb-darkening− − law. Theoretical models exist, but often they are used as fitting parameters to a transit light curve. In principle, a planet should be modeled by a limb darkening law too. However, the data precision does not require this detail (Winn, 2010).

3.3.2 Results We ﬁnally ﬁt the bottom of the transit using

I(X, b) R2 I (X, b) = 1 p , t − R I(X, Y)dXdY R2 ∗ where the (X,Y ) coordinates and the impact parameter b are shown in figure 3.6, and we take the flux outside the transit to be constant. This assumes an infinitely small planet (vertical transit sides, or tI = tII and tIII = tIV in figure 3.6. The X-coordinate follows through the transit and depends on the impact parameter. It is converted to a time using the expected transit duration from equations 3.2 and 3.3.

From these formulas, τ (see ﬁgure 3.6) is also calculated and this length is used to smooth the sides of the transit. Every data point is replaced by the mean of its τ surroundings. This results in the tilted slope of the sides of the transit. It is encouraging to see that the proper choice for impact parameter does not only get the width of the transit correct, but also the slope of the sides of the transit.

The results for HAT-P-7 and Sandra are shown in figures 3.7 and 3.8. It is immediately visible from the labels in the figures that the transits for Sandra are much shallower than for HAT- P-7. This is the reason why no phase variations in the flux can be seen outside the transit for Sandra, nor are occultations visible (for HAT-P-7, they are treated in section 3.4).

Table 3.3 lists the values used in the ﬁtting formula. The values for u1 and u2 (the limb darkening parameters) were used as ﬁtting parameters, and we did not attempt to derive them from any theoretical models. The important parameters are the inclination angle (which follows from the impact parameter, and therefore from the transit width) and the planetary radius (which follows from the transit depth, but requires the stellar radius to be known!).

HAT-P-7 Sandra

u1 0.53 0.79 u2 0.10 0.30 b 0.24 0.01 0.48 0.01 δ 0.00550 ± 0.00002 0.00020 ± 0.00002 ± ± i (C◦) 86.68 0.14 88.778 0.028 R (R ) 1.431 ±0.011 0.2026 ±0.0049 p J ± ±

Table 3.3: Parameters describing the transit of HAT-P-7 and Sandra. The limb darkening constants (u1 and u2), the impact parameter (b) and the transit depth (δ) go into the transit ﬁt. The inclination (i) and the planetary radius (Rp) are the important parameters that can be derived from this. 3.3. Transit depth 47

Figure 3.7: Transit for HAT-P-7. The blue dots represent the combination of the data points for all observed transits, the black line is the transit model.

Figure 3.8: Transit for Sandra. The blue dots represent the combination of the data points for all observed transits, the black line is the transit model. 3.4. Phase-dependent ﬂux variations (HAT-P-7) 48

3.3.3 Planet mass The planetary mass can be calculated if the host star has been observed spectroscopically. Assuming zero eccentricity and Mp << M we find (Seager and Dotson, 2011): ∗ 1/3 K P 2/3 Mp = ∗ M . sin i 2πG ∗ Here G is the gravitational constant and i the inclination, K is the velocity semiamplitude. P and M are the planetary orbital period and the stellar mass.∗ Radial velocity data do not exist for Sandra,∗ but Pálet al. (2008) have observed HAT-P-7 spectroscopically and found K = 213.5 1.9 m/s. Using this value together with the period and inclination determined ∗ ± in this chapter and the stellar mass from the asteroseismology, we find Mp = 1.741 0.028. A mass estimate for Sandra will have to await a spectroscopic observation of the system.±

3.4 Phase-dependent ﬂux variations (HAT-P-7)

3.4.1 Albedo measurement and ellipsoidal light variations When observing a system containing a star and a transiting planet, the observed flux is always coming from a combination of star and planet. During the transit, a part of the star light is blocked. Just outside the planetary transit, we observe flux coming from the star and the planetary nightside. Planets orbiting their host star in a very close orbit are tidally locked - the star always ‘sees’ the same side of the planet (the same is true for the Earth and the moon). When the planet continues its orbit, the observed flux increases as the dayside comes into view and reaches a maximum just before occultation, when the planet hides behind the star. This is illustrated in figure 3.9a.

For Sandra, the transiting planet is too small for this effect to be measurable. For HAT-P-7, both the phase-dependent flux variations and the occultations can be observed. In this specific case, the story is complicated slightly due to ellipsoidal shape of the star (Welsh et al., 2010). The authors argue that the planet orbits so close to the star (only four stellar radii) that it induces a tidal distortion: rather than being oblate due to rotation the star will have its longest axis towards the planet and its shortest axis perpendicular to the orbital plane. This causes ellipsoidal variations peaking at phases near 0.25 and 0.75, and should be added to the phase variations we described above. A modeling for HAT-P-7 was done by Welsh et al. (2010) based on the first month of Kepler data only (Quarter 0 and 1). The result is shown in figure 3.9b.

We model the light curve of HAT-P-7 in a similar way. We confirm that it is only possible to obtain a good fit by including the ellipsoidal variations. The result is shown in figure 3.10. The fitting formula in this figure is given by:

R 2 sin(α) + (π α) cos(α) cos(2α) f(Φ, i) = A p − b , g a π − π with the orbital phase Φ [0,1] taken 0 at maximum radial velocity of the star, so that the phase angle α is deﬁned as:∈ 3.4. Phase-dependent ﬂux variations (HAT-P-7) 49

(a) Figure courtesy by Winn (b) Figure courtesy by Welsh et al.

Figure 3.9: (a) Illustration of transits and occultation. During a transit, the flux drops. Afterwards, the flux rises as the dayside comes into view. The flux drops again when the planet is occulted by the star. Figure taken from Winn (2010). (b) Phase folded Kepler light curve (Q0 and Q1) for HAT-P-7. The red dashed line is the planet-only model (as shown in figure 3.9a). Welsh et al. (2010) include a ellipsoidal model for the star (dotted blue) to come to a best fit of the curve variation (orange solid line). Figure taken from Welsh et al. (2010).

cos α = sin i sin 2πΦ. − The first part of f(Φ, i) is the Lambert law for a sphere (Charbonneau et al., 1999). A non- zero value for b includes the ellipsoidal variations peaking at orbital phases 0.25 and 0.75 as shown in figure 3.9b. We find best fit values giving a geometric albedo Ag = 0.193 0.002 and an ellipsoidal variation b of 59 1 parts per million (the errors are rough estimations).± ± Welsh et al. (2010) find an albedo of 0.18 and an ellipsoidal variation of 37 ppm (no errors quoted) applying the same methods that were used in this work, but with a much shorter time series. As for the value of the albedo, the authors describe that it is comparable to what is found for CoRoT-1b (0.20) but significantly higher than for some other planets (CoRoT-2b, 0.06 0.06). On the other hand, it is still much lower than what we observe for giants in our solar± system (e.g. 0.52 for Jupiter). The extrasolar Hot Jupiters are too hot to allow for condensation of a reflective layer of clouds (Christiansen et al., 2010), which explains the large difference in albedo.

It is no surprise that the ellipsoidal variation is more hard to measure, as the albedo follows essentially from the ﬂux level just outside occultation and just outside transit. Despite the quantitative discrepancy, we conﬁrm the presence of ellipsoidal variations in HAT-P-7. Welsh et al. (2010) based their quantitative measurements on only initial Kepler data (Q0 and Q1), while we now have a far superior quality of observations available.

Ellipsoidal variations for exoplanets have been described in detail by Pfahl et al. (2008). HAT- P-7 is the only system known to date where they have been observed (caused by a planet, not by a binary star system). The variations arise as a consequence of gravity on a luminous ﬂuid body. The amplitude depends on the mass ratio, inclination, orbital period and stellar 3.4. Phase-dependent ﬂux variations (HAT-P-7) 50

Figure 3.10: A fit to the phase variation, including both the stellar and the planetary effect (see figure 3.9b). The occultation is clearly visible.

radius and can therefore be used to provide independent constraints on these values. Welsh et al. (2010) found that they could not distinguish between diﬀerent approximations, based on one month of Kepler data. We might be at a level of data quality where this is possible now. While certainly very interesting, a detailed analysis of this eﬀect is outside the scope of this work.

3.4.2 Dayside and nightside temperature We can also attempt to derive a planetary temperature based on the depth of the eclipse. As can be seen in figure 3.9a, the relative depth δflux(λ) of the occultation reveals the difference between seeing the planetary dayside and seeing the star only. Comparing the minimal flux of the occultation with the flux levels just outside transits gives information about the planetary nightside. Following Winn (2010), we find

2 Rp Bλ(Tp) δﬂux(λ) = , R Bλ(T?) ∗ where Bλ(T ) is the Planck function

2hc2 1 Bλ(T ) = . λ5 ehc/(λkbT ) 1 − The Kepler satellite observes in a bandpass between 450 and 900 nanometer. We can inte- grate the Planck function between these values and obtain a planetary nightside temperature, based on the values we obtained earlier for the stellar temperature, stellar radius and planetary radius, and the different in flux levels between outside the transit and inside the occultation. We obtain this value simply by selecting all points inside the occultation and just outside the transit (in the folded time series) and averaging them. We find a δflux of 4.90 0.25 ppm which leads to a temperature of 1772 10 K. The calculation of this temperature± value (and its error) assumes the planet is a perfect± blackbody and is therefore described by the above 3.4. Phase-dependent flux variations (HAT-P-7) 51

Planck function. Under this assumption, the temperature is determined very precisely. How- ever, the planet cannot be described entirely as a blackbody. The temperature level should therefore be treated a a ﬁrst preliminary estimate, and the formal error value is expected to be a serious underestimation.

The level of flux just outside occultation has already been used in the previous section to estimate the albedo or planetary reflection from the star light. There we assumed that the planetary equilibrium temperature is given by its nightside temperature. We can do the opposite and assume an albedo of zero. In that case the entire occultation would be caused by the disappearance of the planetary radiation. We can then obtain a maximum dayside temperature. The planetary occultation is shown in figure 3.11.

Figure 3.11: Folded ﬂux zoomed in on the occultation. The average ﬂux level inside (green) and just outside (yellow) the occultation is calculated to measure the occultation depth.

We ﬁnd an occultation depth of 71.85 0.23 parts per million. This is obtained by calculating the average of data points inside the occultation± and just outside (see ﬁgure 3.11). The high number of data points makes it possible to do this to high precision. The maximum dayside temperature that can be calculated from this is 2470 10 K. The word of caution about the assumption of a blackbody-spectrum and the underestimated± error for the nightside temperature, applies equally here.

Figure 3.11 shows what is presumably one of the best-observed occultations available to date. It is interesting to compare it with the discovery of HAT-P-7’s occultation, as was published by Borucki et al. (2009b) based only on Kepler’s Q0 (ﬁrst 10 days). The authors estimated the occultation depth to be 130 11 ppm (leading to a dayside temperature estimate of 2650 100 K). Our current value± is signiﬁcantly lower. Based on those data, it was not possible± to detect the ellipsoidal light variations and presumably that led to a wrong value for the occultation depth.

Welsh et al. (2010) later discovered the ellipsoidal variations and quote an occultation depth of 85.8 ppm based on Q0 and Q1 of Kepler data, and a nightside planet contribution of 22.1 ppm. While closer to our values, these still seem to be an overestimation. The authors also 3.4. Phase-dependent flux variations (HAT-P-7) 52 model the lightcurve using the Eclipsing Light Curve (ELC) code that was developed by Orosz and Hauschildt (2000) for eclipsing binary stars. In this way they obtain a temperature value of 2885 100 K, hotter than their (and our) maximum equilibrium temperature. ± Assuming the ELC code actually gives reliable temperature estimates, they speculate on three possible reasons for this. The first two include the presence of an additional body: tidal heating of the planet due to an encounter with another object or light from a third body contami- nating the flux. They suggested that transit timing variations might confirm this hypothesis, but our analysis of the transit times have not found any evidence for additional bodies in a close orbit around the star. The third option would be that the planet is not a blackbody, and that at other wavelengths, other temperatures would be measured. This can be tested by observing occultations using observations in different wavelengths and this has indeed been done. Christiansen et al. (2010) and Spiegel and Burrows (2010) have both attempted to model the atmosphere of HAT-P-7 by combining Spitzer (infrared) with Kepler and EPOXI (optical) observations. They find best-fits to data that are not a blackbody, and even contain thermal inversion. In this case, the upper atmosphere is hotter than the lower, due to the very close orbit of the planet around its star and the large irradiation it receives as a consequence.

Both Christiansen et al. (2010) and Spiegel and Burrows (2010) use realistic atmosphere models and attempt to model which elements are present in the planet’s atmosphere, each causing their absorption lines. Both ﬁnd an extremely hot upper atmosphere with thermal inversion and little day-night redistribution. However, both groups base their Kepler occultation depth on Borucki et al. (2009b). Christiansen et al. (2010) ﬁnd a brightness temperature of 3175 40 K based on the Kepler occultation measurements by Borucki et al. (2009b). Since this depth± seems to have been overestimated, it should be interesting to see how their atmospherical models compare with our revised occultation depth. Theories crumble, but good observations never fade. Harlow Shapley 4 Results and conclusions

This final chapter provides an overview of all the results that have been obtained in the previous two chapters. All obtained parameters are brought together in a table and discussed in a general context. For HAT-P-7, they are compared to the extensive list of available literature. The final section of this work draws a number of conclusions and reflects on the future for the research on the two stars and in the two fields that were the subject of this work.

4.1 Parameter overview

In this section we present all parameters that have been obtained on HAT-P-7 and Sandra throughout this work. To place the stars we have studied into context, it is instructive to look at the HR-diagram that is shown in figure 4.1. It gives an overview of the star classifications on the main sequence, based on parameters from Carroll and Ostlie (1996). We have placed the calculated evolution tracks for HAT-P-7 and Sandra on the figure as well, which start from their zero-age main sequence point (ZAMS). The diagram shows how the stars are currently making their way on their evolution track on the main sequence. Both of them are slightly more massive than the Sun (also shown in the diagram).

For Sandra, we have summarized all values obtained throughout this work in table 4.1. It shows a star not too diﬀerent from a younger version of our Sun. Rotational splitting of the frequencies leads to a determination of the star’s rotation rate, which is about ﬁve times as fast as the rotation rate of the Sun ( 25 days). The transiting planet is relatively small, but still orbiting closer to the star than≈ any solar planet. Lacking radial velocity observations, we cannot determine its exact mass. Transit timing variations suggest the presence of an additional body in the system, possibly an additional planet.

53 4.1. Parameter overview 54

Figure 4.1: An HR-diagram showing the location of HAT-P-7 and Sandra. Black (green) line represents the best evolution track for HAT-P-7 (Sandra), with the best model on the track shown in yellow. We acknowledge Michael Zingale for this plot, stellar properties taken from Carroll and Ostlie (1996). The track for HAT-P-7 is shown in more detail in ﬁgure 4.2. 4.1. Parameter overview 55

Sandra Value Mass (M ) 1.240 0.027 ± Z0 0.0280 0.0041 Age (Gyr) 2.87 ± 0.70 Radius (R ) 1.373 ± 0.012 Temperature (K) 5970± 135 3 ± Density (g/cm− ) 0.6764 0.0069 Luminosity (L ) 2.15 ± 0.19 log (g) (cgs) 4.2565 ± 0.0040 Rotation (days) 5.33 ± 0.16 ± Sandra-b Value Orbital period (d) 17.834067 0.000068 ± Orbital inclination i (◦) 88.778 0.028 ± Mp (MJ ) / R (M ) 0.2026 0.0049 p J ± Table 4.1: An overview of all values derived for Sandra throughout this work.

HAT-P-7 is a star that is (in general terms) fairly similar to Sandra. However, the planetary system is entirely diﬀerent and an extreme example of a Hot Jupiter: the planet orbits very closely around the star and is very hot, and it is considerably larger and more massive than Jupiter. An overview of all obtained parameters can be found in table 4.2, which also lists values that were previously obtained by a number of other authors.

Figure 4.2 shows an HR-diagram which compares luminosities and temperatures of different authors. The results presented in this work are shown in blue. Three evolution tracks are shown, representing the best fitting evolution tracks for each overshoot value, while the dots indicate the best model on the track. The error square shows the actual parameter determination (from Monte Carlo analysis). We compare with the values of Christensen-Dalsgaard et al. (2010) (no error bars were provided), Pálet al. (2008) (one-sigma error was plotted) and Ammler-von Eiff et al. (2009) (only temperature values available; one-sigma levels were plotted). The figure shows the temperature and luminosity we have obtained is lower than what Christensen-Dalsgaard et al. (2010) have obtained using the same methods but lower quality data (they performed asteroseismology on Kepler Q0 and Q1 data). We are in fair agreement with the spectroscopic determination by Pálet al. (2008), but not with Ammler-von Eiff et al. (2009) (also using spectroscopy).

Looking at the other stellar parameters in table 4.2, we notice a mild inconsistence in almost all parameters as compared to Christensen-Dalsgaard et al. (2010). The stellar age is in excellent agreement, but we find a lower mass and metallicity, and a slightly lower radius, together with the earlier quoted difference in temperature values. This leads to some disagreement in the derived parameters (density, luminosity, log g). It should be interesting to make a detailed comparison of their asteroseismic analysis with the procedure that has been followed in this work, to see if the discrepancy is caused by different observations (different pulsation frequencies), or if other reasons come into play. 4.1. Parameter overview 56

Figure 4.2: An HR-diagram showing our values for HAT-P-7 compared with the literature (see table 4.2 for values). For our own work, we show the evolution track that best fits the original data for different overshoot values (blue tracks), along with the model selected on the track (blue dots). The error square shows the Monte Carlo parameter averages presented in this work (blue square). For (Christensen-Dalsgaard et al., 2010, (a)), no error bars were quoted in their work, so only their parameters are indicated (black cross). (Pálet al., 2008, (b)) is plotted using one-sigma error levels (red square). (Ammler-von Eiff et al., 2009, (c)) did not present luminosity values, the bars indicate the one-sigma error on their temperature value (green bars).

It seems clear that the error estimates in one of the studies (or both) are too low. This points towards a need for a more developed error estimation in asteroseismology. Our mass and radius values are consistent with a number of other determinations, but we obtain a considerably more accurate determination than those quoted in the literature.

Winn et al. (2009) ﬁnd evidence for a retrograde or polar orbit for HAT-P-7, which is unexpected given that we observe a transiting planet. This is hard to conﬁrm or disprove based on the asteroseismic measurements we have available. However, the lack of observed rotational splittings is consistent with a pole-on inclination of the star.

Metallicity values that are obtained through spectroscopy are usually quoted as [Fe/H] values, which means they are relative to the solar metallicity. Unfortunately, the solar metallicity is not that well established. For comparison, we use the value 0.0142 for solar metallicity as quoted by Asplund et al. (2009). This results in the conversion formula: Z = 0.0142 10[Fe/H]. We have converted our absolute value into a logarithmic value relative to the Sun,∗ using this 4.1. Parameter overview 57

formula and find 0.13 (because of the uncertainty in the solar value, we do not quote an error estimate). Using the same conversion for the asteroseismology by Christensen-Dalsgaard et al. (2010), their [Fe/H] value would become 0.28. This latter value is within the range determined spectroscopically (Pálet al., 2008; Ammler-von Eiff et al., 2009), whereas our value is not consistent within a 1σ error level.

As a final point for HAT-P-7, we have confirmed the presence of ellipsoidal variations, which imply a non-spherical shape of the star caused by a gravitational interaction with the closely orbiting planet. We find a different magnitude for the variations as compared to Welsh et al. (2010), whose analysis was based on a much smaller data set.

Turning to HAT-P-7’s planet, we see that the period determination is done at least an order of magnitude better than any other determination (some of the other values are mildly inconsistent). This is a consequence of the higher data quality that was used, and is relevant for the hunt for additional planets in the system. We have placed strict limits on the possible masses of such additional bodies, and in a 2:1 resonance orbit no planet larger than 2 M can exist. For more distant orbits, transit timing variations are not suitable for excluding low-mass⊕ planets.

The inclination of HAT-P-7b is inconsistent with values derived earlier (Winn et al., 2009; Southworth, 2011; Welsh et al., 2010), which are mutually fairly consistent. However, our inclination is consistent with the (less accurate) determination by Christiansen et al. (2010).

We find a planetary radius which is within error regions with most other radius determinations (Pálet al., 2008; Southworth, 2011; Welsh et al., 2010), but gives a slightly larger radius than Christiansen et al. (2010). Our mass derivation makes use of the derived inclination, period and stellar mass from this work, combined with the radial velocity data by Pálet al. (2008) and it is therefore no surprise that our values are consistent. We also find good agreement with Welsh et al. (2010). We derive a slightly higher mass than Winn et al. (2009) (the authors quote only a relative mass, we calculated the absolute planetary mass using our stellar mass estimate and therefore quote no error bars).

Finally, there remains a large inconsistency in the planetary day- and nightside temperatures. We find a lower dayside temperature value than other authors (Pálet al., 2008; Borucki et al., 2009b; Welsh et al., 2010; Christiansen et al., 2010), who are among them also mutually inconsistent. We also find a lower nightside temperature compared to Welsh et al. (2010). Most temperature determinations have been based on the occultation depth quoted by Borucki et al. (2009b), who find a significantly larger occultation than what we observe. We believe their value to be overestimated. Our temperature values have been based on a simple black-body model of the planet, which is likely to be an oversimplification and causes our formal errors to be underestimated. 4.1. Parameter overview 58

HAT-P-7 This work Others Mass (M ) 1.361 0.021 1.520 0.036a, 1.47 0.08b ± ± a ± Z0 0.01904 0.0015 0.0270 ± a b c [Fe/H] 0.13 0.28 ∗∗, 0.26 0.08 , 0.31 0.07 Age (Gyr) 2.19 0.12 2.14± 0.26a ± Radius (R ) 1.904 ± 0.010 1.991 0.018±a, 1.84 0.23b, ± ±f ± h 1.92 ∗, 1.824 0.089 Temperature (K) 6259 32 6379a, 6350 80b±, 6525 61c 3 ± ± a ± Density (g/cm− ) 0.2781 0.0017 0.2712 0.0032 Luminosity (L ) 4.996 ± 0.098 5.87a, 4.9± 1.1b log (g) (cgs) 4.98 ± 0.13 5.87a, 4.07 0.06b±, 4.09 0.08c Spin-orbit angle (deg) ±/ 182.5± 9.4d ± Ellips. var. (ppm) 59 1 37.3±g ± HAT-P-7b This work Others Orbital period (d) 2.20473506 (11) 2.2047304 (24)d, 2.204802 (63)e, 2.2047304 (24)f , 2.204733 (10)g, 2.2047308 (25)h d f Orbital inclination i (◦) 86.68 0.14 80.8 2.8 , 83.40 0.12 , ± 83.1± 0.5g, 85.7 ± 3.5h ±b d ± g Mp (MJ ) 1.741 0.028 1.78 0.08 , 1.61 ∗, 1.82 0.03 ± ± b f ± Rp (RJ ) 1.431 0.011 1.36 0.20 , 1.47 ∗, ± 1.50 0.02± g, 1.342 0.068h ± b ± e Tday (K) 2470 10∗∗∗ 2730 150 , 2650 100 , ± 2885± 100g, 3175± 40h ± g ± Tnight (K) 1772 10∗∗∗ 2570 95 A 0.193 ± 0.002 0.57 0.05±g, 0.13h g ± ± ≤ Table 4.2: An overview of all values that were obtained for HAT-P-7 in this work, and a comparison with the available literature. Some error values have been symmetrized, the bracketed numbers for the period represent errors on the last two digits. a: Christensen-Dalsgaard et al. (2010) b: P´alet al. (2008) c: Ammler-von Eiﬀ et al. (2009) d: Winn et al. (2009) e: Borucki et al. (2009b) f: Southworth (2011) uses fractional stellar and planetary radius... g: Welsh et al. (2010) h: Christiansen et al. (2010) *: only a relative radius or mass was provided in this source, we calculated an absolute radius based on the values in their work. **: only an absolute metallicity was quoted in this work, we calculated this value using a solar metallicity of 0.0142. ***: formal error is an underestimation 4.2. Conclusions 59

4.2 Conclusions

HAT-P-7 presented itself as a very interesting study object. While some discrepancies remain between parameters quoted by diﬀerent authors, both the star and the planet have now been described to an excellent level of detail. Three major uncertainties remain: the inclination of the host star, the presence of additional bodies and the temperature of HAT-P-7b.

The ﬁrst and second points are likely to improve only when additional and more accurate radial velocity data become available. The excellent quality of Kepler data has allowed us to conclude that no large objects exist in close orbits, but cannot impose strong limits on potential planets in orbits far away from the host star. The asteroseismology is consistent with a polar inclination for the star, but is inconclusive on this matter. A more detailed study of the Rossiter-McLaughlin eﬀect might provide more decisive conclusions.

Different authors disagree on the planetary temperature. The improved quality of photometric data and the inclusion of ellipsoidal variations in the modeling of the stellar flux, leads us to a smaller occultation, and therefore a lower planet temperature. A blackbody approximation for the planet is likely to be insufficient and the planet should remain interesting for future atmosphere models because of its close orbit. The planetary nightside temperature might help in determining how heat is transfered from the day- to the nightside of planets. The situation for HAT-P-7 is complicated by the ellipsoidal variations the planet has caused on the star. The level of accuracy with which they have now been described makes them interesting to be studied in their own regard.

Finally, the asteroseismic modeling of this star allows us to give very accurate determinations of the stellar parameters, but we have not found any models that reproduce all frequencies within the level of observational errors. This should be an interesting topic for future research, in particular a detailed comparison between the methods used in this work and those used to ﬁnd the asteroseismic parameters in Christensen-Dalsgaard et al. (2010). In general, it seems that the Kepler data quality has caught up with the accuracy of theoretical models. This should lead to a number of improvements in the near future, which, among other things, could make asteroseismology an even more powerful tool for studying planet host stars. In particular, an improvement of the treatment and understanding of near-surface eﬀects and of mixing processes on the frequencies is desirable.

Our parameter determination for Sandra (KIC 8866102) cannot be compared with any independent literature, but that situation is likely to change in the near future. Both the star and the planet are now well-described, but the planetary mass can only be determined after the object has been observed spectroscopically. The periodic variability in the photometry should be looked at closer, and the ﬁeld around the star should be studied to see if this signal can indeed be attributed to an eclipsing binary system in the background.

The most interesting and intriguing result for this star is the discovery of transit timing variations. More research is required to exclude other possibilities, but there is a reasonable likeli- hood that the variations are caused by an additional body. An even longer photometric time series will likely be useful, but radial velocity data will be indispensable to draw real conclusions. 4.2. Conclusions 60

It is clear from this work that much can be gained from asteroseismic studies of planet host stars. Every planetary parameter derived in this work (except the planet’s period) is dependent on one or more stellar parameters. Asteroseismology is an excellent tool in this regard. The Kepler mission has undoubtedly revolutionized both asteroseismology and exoplanet research. High-quality spectroscopy is the observational missing piece. Extremely interesting in this regard is the SONG-network (Grundahl et al., 2008), which will start spectroscopic observations from the Canary Islands in the next weeks. A second telescope is being built, and the network should contain telescopes spread over the diﬀerent continents once it is fully deployed. Together with the recent extension of the Kepler mission until 2016, there seems to be no doubt that this will lead to a number of interesting discoveries and results in the next few years. Sometimes I think we’re alone in this world. Sometimes I think we’re not. In either case, the prospect is staggering! Arthur C. Clarke Afterword

This paper has dealt with two ﬁelds in astronomy, whose names derive from Ancient Greek: exoplanets (ξω, outer, external; πλανητης, planet, wanderer) and asteroseismology (αστηρ, star; σισµoς, earthquake). I have studied the earthquakes of stars and their outer wanderers. At the start of this dissertation, I exclaimed that what these ﬁelds are capable of is truly amazing, and I feel the past few chapters have lived up to that promise.

I hope the previous chapters show some of the incredible progress that has been made in the past few years by scientists much greater than myself. Asteroseismology, while very worth- while in its own regard, also provides a very useful tool to study and understand stars and their interiors. To me it is clear that no deep understanding of the planets will ever be achieved without also understanding the stars.

Many years ago, it was my childhood’s dream to discover extraterrestrial life. It has taken me many years and required me to wander from Belgium into the land of the Vikings - but here in Denmark I feel I have come closer to that dream than I had ever imagined. Certainly the past chapters do not show the signatures of any aliens waving at us, neither have they discovered E.T.’s place to phone home to. However, through a study of both stars and planets, I feel I have contributed a few tiny steps towards formulating an answer to some fascinating questions, about the uniqueness of the life on our Earth.

My contribution to this has been only very small, but it was with great pleasure that I have experienced “the thrill of discovery, the incredible, visceral feeling of doing something no one has ever done before, seen things no one has seen before, know something no one else has ever known” ( Phil Plait). I hope the reader has learned something new, but above all has ∼ picked up some of the inspiration this thesis work has brought me in the past year.

It was less than two decades ago that humanity could not even formulate an answer to the question if exoplanets even existed. With a bit of luck astronomers will, within years, start studying the planetary atmospheres and look for spectroscopic signatures of life. A real dream remains: the direct imaging of distant planets and stars, to see them in the same way we could see Venus transit the Sun. Who knows what more to expect from the future?

When asked about which planet or discovery is my favorite one, I like to reply that it is the next one. - Sara Seager

61 A Nederlandstalige samenvatting

De combinatie van twee jonge domeinen in de sterrenkunde kan ons heel wat leren over sterren en planeten. Asteroseismologie bestudeert stertrillingen en door het modelleren van dit ‘stergeluid’ kunnen allerlei parameters van de ster worden afgeleid (bv. temperatuur, leeftijd, massa, straal, ...). Aan de andere kant kan men exoplaneten (planeten rond andere sterren dan de zon) bestuderen aan de hand van de transit of overgang van de moederster (‘host star’) waarrond ze cirkelen. Als deze informatie gecombineerd wordt met wat asteroseismologie ons leert over de moederster, komt men heel wat te weten over de planeten (bv. periode, straal, inclinatiehoek, massa, temperatuur, ...).

Beide domeinen maken gebruiken van variaties in de lichtkracht van sterren. In asteroseismologie leiden de variaties in temperatuur die worden veroorzaakt door de stertrillingen tot variaties in de helderheid van de ster; een overgang van een exoplaneet blokkeert dan weer tijdelijk een fractie van het sterlicht. In maart 2009 is de satelliet Kepler gelanceerd, die de helderheid van sterren aan een fractie van de hemel (in de zomerdriehoek bij de Zwaan en de h m s Lier; RA: 19 22 40 , DEC: 44◦ 300 ) bestudeert voor lange tijd. De superieure kwaliteit van deze metingen, vergeleken met de korte grondobservaties uit het verleden, heeft geleid tot een revolutie in zowel asteroseismologie als exoplaneetonderzoek tijdens de voorbije drie jaar.

Twee stersystemen die allebei al elf kwartalen door deze satelliet werden waargenomen, werden in dit werk bestudeerd. Van beide was bekend dat asteroseismologische studie mogelijk is en dat er planeetovergangen plaatsvinden. HAT-P-7 (een F8-ster met visuele magnitude 10.46) is een systeem dat al heel wat werd bestudeerd en waarover dan ook al heel wat bekend is. De kwaliteit van de recente observaties maakt het mogelijk om heel wat parameters een stuk nauwkeuriger te bepalen. De andere bestudeerde ster heet KIC 8866102 (visuele magnitude 9.50, spectraaltype G0). Omwille van de leesbaarheid wordt ze in dit werk ‘Sandra’ genoemd. Over deze ster is nog geen wetenschappelijk werk gepubliceerd en zijn alle resultaten volledig nieuw. Beide systemen worden dus in detail bestudeerd. De opzet van dit werk is een analyse van A tot Z: zowel de sterren als hun planeten worden geanalyseerd en deze resultaten worden gecombineerd.

62 63

In hoofdstuk 2 worden de frequenties van de stertrillingen bepaald en vervolgens gemodelleerd. Aangezien in een ster heel wat trillingen tegelijk optreden, wordt gebruik gemaakt van Fourier- analyse om de indivuele frequenties te bepalen aan de hand van de totale lichtkrachtvariaties. Het bepalen van de uiteindelijke frequenties wordt sterk geholpen door de bestaande theorie: de zogeheten asymptotische relatie (zie vergelijking 1.1) beschrijft de reguliere afstand tussen frequenties onderling. Hierdoor kunnen de gevonden frequenties handig worden weergegeven in een ladderdiagram (‘échellediagram’). In zo’n figuur worden frequenties boven elkaar weergegeven in plaats van naast elkaar. De gevonden frequenties worden getoond in figuren 2.6 en 2.7 voor HAT-P-7 en Sandra. Tabellen C-1 en C-2 geven een volledige lijst.

De volgende stap in dit hoofdstuk is het modelleren van deze stertrillingen aan de hand van the- oretische modellen. De Aarhus codes (Christensen-Dalsgaard, 2008a,b) worden gebruikt om de sterparameters te bepalen. In een eerste stap wordt een rooster van evolutiesporen berekend voor sterren met verschillende massa en metalliciteit, vervolgens wordt aan de hand van de dichtheid van de ster gekozen voor welke fases in de evolutie frequenties worden uitgerekend. Deze frequenties worden ten slotte vergeleken met de stertrillingen die we voor HAT-P-7 en Sandra hebben geobserveerd, en aan de hand hiervan wordt bepaald welke modellen het best bij de ster passen. Dit leidt uiteindelijk tot sterparameters die staan opgelijst in tabel 2.3. In het geval van Sandra kan uit de speciﬁeke manier waarop de frequenties zijn opgesplitst (‘rotational splitting’) ook de rotatiesnelheid van de ster worden bepaald. Voor HAT-P-7 is dit niet mogelijk; een mogelijke verklaring is dat we deze ster op zijn noord- of zuidpool observeren.

De planeetovergangen worden bestudeerd in hoofdstuk 3. In een eerste stap wordt de precieze timing van elke planeetovergang bepaald en aan de hand hiervan wordt de periode van de planeten berekend. De verwachte transittijden worden vergeleken met de observaties, en aan de hand hiervan zoeken we of er zich andere lichamen in het systeem bevinden. Bij HAT-P-7 vinden we geen transit timing variations (TTV’s). We plaatsen daarom een bovenlimiet op de massa van een mogelijke extra planeet (zie ﬁguur 3.4). Bij Sandra worden wel TTV’s gevonden, die mogelijk wijzen op een extra planeet (of ster) in dit systeem. Deze interessante ontdekking dient wel verder te worden onderzocht om andere oorzaken voor de variaties uit te sluiten.

Eens de periode van de planeet bepaald is, kunnen de observaties ‘opgeplooid’ worden: alle observaties worden over elkaar gelegd zodat alle metingen tijdens de planeetovergangen bovenop elkaar komen te liggen. Op deze manier is de overgang zeer nauwkeurig observationeel beschreven en kan deze ook worden gemodelleerd. Het model houdt rekening met de rand- verduistering (‘limb darkening’) van de ster, waardoor de overgang in het midden dieper is. Uit de breedte van de overgang wordt de precieze inclinatiehoek van de planeet bepaald. De diepte geeft dan weer informatie over het oppervlak van de planeet, relatief ten opzichte van de moederster. Gecombineerd met de informatie over de straal van de moederster (uit het hoofdstuk over asteroseismologie), wordt dan de straal van de planeet bepaald. Voor HAT-P-7 gebruiken we eerdere spectroscopische observaties van de ster om ook de massa van de planeet te bepalen, voor Sandra zijn zulke waarnemingen nog niet verricht.

Omdat HAT-P-7b een bijzonder kleine periode heeft en dus zeer dicht bij de moederster be- weegt, is deze planeet uitzonderlijk warm. Hierdoor wordt ook planetaire straling waargenomen. 64

Deze straling wordt echter geblokkeerd door de ster wanneer de planeet achter de moederster verdwijnt: een occultatie. Uit de occultatie wordt de temperatuur voor de dagkant en de nachtzijde van de planeet berekend. Daarenboven is de ster niet bolvormig door de gravitationele aantrekking van de dichtbijgelegen planeet, en ook dit periodisch effect wordt geobserveerd. Bij dubbelsterren wordt dit wel vaker waargenomen, maar HAT-P-7 is de enige gekende ster waar dit effect het gevolg is van een planeet. Deze gehele analyse wordt niet uitgevoerd bij Sandra, omdat deze effecten niet meetbaar zijn, vanwege de grotere afstand van de planeet en de aanzienlijk kleinere omvang, vergeleken met de planeet bij HAT-P-7.

Hoofdstuk 4 brengt alle gevonden parameters samen. Zowel de waarden voor de ster als voor de planeet zijn te vinden in tabellen 4.1 en 4.2, voor Sandra en HAT-P-7. Voor Sandra zijn alle gevonden waarden volledig nieuw. De parameters van HAT-P-7 worden vergeleken met de verscheidenheid aan eerdere publicaties voor dit systeem. De parameters die in dit werk werden bepaald, hebben een aanzienlijk grotere nauwkeurigheid dan eerdere resultaten. Er zijn enkele discrepanties tussen de sterparameters die gevonden werden door Christensen-Dalsgaard et al. (2010), op basis van een asteroseismische analyse die gebruik maakte van de eerste maand observaties door Kepler. Verder vinden we een aanzienlijk lagere temperatuur voor de planeet, als gevolg van een minder diepe occultatie dan de originele occultatiebepaling door Borucki et al. (2009b) op basis van de eerste 10 dagen aan Kepler-metingen.

Daar waar eerder onderzoek zich gewoonlijk toespitst op een speciﬁek aspect van de ster of de planeet, werd in deze masterthesis het ganse systeem geanalyseerd. Het is duidelijk dat asteroseismologie sinds de lancering van de Kepler-missie in een gouden tijdperk is terecht gekomen. Dit zal in de komende jaren tot heel wat resultaten leiden, maar is tevens bijzonder nuttig om sterren die planeten herbergen te analyseren. Het planeetonderzoek is wat betreft niet-solaire planeten nog geen twintig jaar oud, maar heeft al voor heel wat verrassingen gezorgd. Het zal dat ongetwijfeld in de komende jaren ook blijven doen.

De kwaliteit van spectroscopische observaties zal de komende jaren sterk toenemen met de uitbouw van het SONG-netwerk (Grundahl et al., 2008), een netwerk van grondtelescopen verspreid over de verschillende werelddelen. Samen met de verlenging van de Kepler-missie tot 2016, zal dit in de komende jaren tot heel wat interessante nieuwe observaties leiden. Exoplanetair onderzoek en asteroseismologie zijn op elkaar aangewezen wat betreft de obser- vationele vereisten. Dit werk toont aan dat dit huwelijk voor beide partners ook erg boeiende resultaten kan opleveren. B ESF conference poster

A number of results of this thesis were presented at a conference on asteroseismology. The conference was titled “The Modern Era of Helio- and Asteroseismology” and took place in Obergurgl, Otz¨ Valley, near Innsbruck, Austria, 20-25 May 2012. It was organised by the European Science Foundation (ESF), in partnership with the Leopold-Franzens-Universit¨at Innsbruck (LFUI). Dr. Markus Roth was the conference chair and Dr. Katrien Uytterhoeven co-chaired the event.

This appendix contains the science poster that was selected for the conference. It contains a large part of the thesis results that used Kepler data that were public at that time. Conference proceedings will be published in Astronomical Notes/Astronomische Nachrichten, in December 2012 or March 2013.

65 Properties of extrasolar planets and their host stars Vincent Van Eylen1,2, Hans Kjeldsen1, Jørgen Christensen-Dalsgaard1, Conny Aerts2 1: Dep. of Physics and Astronomy, Aarhus University, Denmark; 2: Instituut voor Sterrenkunde, KU Leuven, Belgium [email protected]

Introduction Results

2 2 We use data from the Kepler satellite (Q0-Q6) to study HAT-P-7 and KIC 8866102. We perform an HAT-P-7 (σν = 0.75 µHz, χ = 2.62) KIC 8866102 (σν = 0.80 µHz, χ = 1.23) asteroseismic study of these stars and use the obtained parameters to deduce properties of the transiting exoplanets. The properties of the host stars are derived using the Aarhus Evolution and Pulsations Code and these parameters are combined with transit observations to study the planets.

Planetary transits Stellar pulsations 1. Planetary period We observe the frequencies: We measure the transit times and fit them linearly Consider datasets without transits • (assuming a Keplerian orbit) to obtain the period. Calculate oversampled power spectrum using • PERIOD04 (shown in red) Median filter the power spectrum to smooth • the finite mode lifetime (shown in dark blue) Automatically detect all maxima in the • smoothed power spectrum, manually select Fig. 5: Echelle diagrams showing the frequencies of the best model (blue) and observations (red). the real peaks based on the asymptotic equation: 1 Stellar parameters Planetary parameters νn,l = ∆ν(n + l + ) δν0l. Fig. 1: Measuring the transit times (fig. HAT-P-7) 2 − The above echelle diagrams show the frequen- The mass is calculated using radial velocity In case of frequency splitting (due to rotation), • 2 cies that best fit according to a χ minimisa- • data by Pálet al. (2008) for HAT-P-7. 2. Planetary radius • use the average (e.g. l = 1 in figure) We fold the dataset on the period to model tion. We find a good general agreement for The radius is determined to high accuracy. • the transit. We use a quadratic limb-darkening both stars. No occultations or out-of-transit variations are The parameter values are obtained by calcu- • model to accurately determine the transit depth. • observed for KIC 8866102 because of the lating 1000 perturbed datasets by introducing small size of the planet. Gaussian noise on the observations. For each HAT-P-7: we determine a nightside effec- of them a best model is calculated and the • tive temperature by comparing occultation parameter average and standard deviation are flux with the flux just outside transit, and a presented in the table. maximum dayside temperature by assum- Overshoot is kept at zero in this analysis. This • ing no stellar reflection: 1750 and 2450 K. might lead to an underestimation of the errors. We find a maximum albedo≈ of 0.180 0.004. For KIC 8866102 we observe split l = 1 (14) ± • and l = 2 (7) frequencies. We determine a fre- Planet HAT-P-7b KIC 8866102 Fig. 2: Modeling the transit (fig. HAT-P-7) Period 2.2047351 0.00000011 17.8341 0.00022 quency splitting of 4.35 0.13 µHz (2Ω) and ± ± ± Mp /MJ 1.743 0.029 / 3. Temperature and albedo for HAT-P-7 Fig. 4: Mode identification for three observed assume the rotation axis lies in the plane of the R /R 1.482 ± 0.017 0.2026 0.0037 frequencies for KIC 8866102. p J We can measure the out-of-transit variation and sky. For HAT-P-7 we detect no splitting. ± ± the occultation, providing information about the We model the frequencies similar to the proce- Star HAT-P-7 KIC 8866102 The results for HAT-P-7 are in general agree- day- and nightside of the planet. For KIC 8866102, dure in Christensen-Dalsgaard et al. (2008ab, 2010): M? /M 1.434 0.029 1.255 0.041 ment with previously known values from vari- ± ± Z0 0.0209 0.0024 0.0244 0.0055 the planet is too small to show these effects. Stellar evolution tracks were calculated using ± ± ous sources but provide higher accuracy. • R? /R 1.940 0.014 1.373 0.020 the Aarhus Stellar Evolution Code (ASTEC) ± ± Teff (K) 6234 24 6183 104 For KIC 8866102, these are the first obtained for a range of masses Age (Gyr) 1.954 ±0.068 1.93 ±0.24 Relevant evolutionary phases were picked by Rot. (d) ±/ 5.33 ± 0.16 values for both the star and the planet. ± • selecting on mean density using the large separation References Acknowledgements We calculate model frequencies using the • [1] Christensen-Dalsgaard. Ap&SS, 316:13–24, August 2008a. Mia Lundkvist and Jan Van Haaren have provided valuable input Aarhus Adiabatic Pulsation Code (ADIPLS) [2] Christensen-Dalsgaard. Ap&SS, 316:113–120, August 2008b. with regards to the design of this poster. [3] Christensen-Dalsgaard et al. ApJ, 713:L164–L168, April The research leading to these results has received funding from We combine observation and model using a correc- 2010. the European Research Council under the European Commu- tion for near-surface layers, according to the proce- [4] Kjeldsen et al. ApJ, 683:L175–L178, August 2008. nity’s Seventh Framework Programme (FP7/2007–2013)/ERC Fig. 3: Folded time series (zoomed in): grant agreement n◦227224 (PROSPERITY) and n◦267864 (AS- flux variation and occultation dure by Kjeldsen et al. (2008). [5] Pál et al. ApJ, 680:1450–1456, June 2008. TERISK). C Transit times

Number Transit time (HJD) Error 64 55095.461178 0.00018 0 54954.358009 0.00013 65 55097.665619 0.00021 1 54956.562941 0.00012 66 55099.870468 0.00013 2 54958.767634 8.3e-05 67 55102.075403 0.00011 3 54960.972403 0.00011 68 55104.28022 0.00011 4 54963.177204 9.7e-05 69 55106.484749 0.0001 5 54965.381825 9.9e-05 70 55108.689656 9.9e-05 6 54967.586539 0.00011 71 55110.894526 0.00015 7 54969.791394 0.00012 72 55113.099084 0.00015 8 54971.996145 0.00016 73 55115.30376 0.00011 9 54974.20095 0.0001 74 55117.508537 0.00015 10 54976.405426 9.6e-05 75 55119.713513 0.00016 11 54978.610471 0.00011 76 55121.917778 0.00011 12 54980.815023 0.0001 78 55126.328139 0.00017 13 54983.019687 0.00012 79 55128.531686 0.0002 14 54985.224511 0.0001 80 55130.736997 0.00014 15 54987.429292 0.00011 81 55132.94122 0.00017 16 54989.634051 9.7e-05 82 55135.147159 0.00028 17 54991.838667 0.0001 84 55139.556017 0.00013 18 54994.043357 9.8e-05 85 55141.760242 0.00018 19 54996.248016 0.00013 86 55143.965466 0.00014 22 55002.862405 0.00011 87 55146.170007 0.00014 23 55005.067056 9.8e-05 88 55148.374189 0.00016 24 55007.271845 0.00012 89 55150.580003 0.00018 25 55009.476671 0.00014 90 55152.783737 0.00023 26 55011.681191 0.00012 92 55157.194161 0.00015 27 55013.886218 0.00012 93 55159.398342 0.00021 29 55018.295478 0.00015 94 55161.603197 0.00018 30 55020.499971 7.7e-05 95 55163.808249 0.00015 31 55022.704894 9.9e-05 96 55166.012815 0.00016 32 55024.909825 9.4e-05 97 55168.217753 0.00013 33 55027.114427 0.00011 98 55170.422298 0.00014 34 55029.319134 7.2e-05 99 55172.626846 0.00016 35 55031.523922 0.00015 100 55174.831396 0.00016 36 55033.728588 0.00015 101 55177.036316 0.00018 37 55035.933782 0.00015 102 55179.241371 0.00013 38 55038.138191 9.3e-05 103 55181.446015 0.00015 39 55040.342728 0.00011 105 55185.855517 0.00013 40 55042.547541 0.00012 106 55188.05989 8.1e-05 41 55044.752248 8.2e-05 107 55190.264752 0.00011 42 55046.956865 8.9e-05 108 55192.469727 0.00011 43 55049.161716 0.00011 109 55194.674184 8.6e-05 44 55051.366704 0.00011 110 55196.879019 0.0001 45 55053.571213 7.5e-05 111 55199.083545 8.1e-05 46 55055.775978 0.00012 112 55201.288389 0.00016 47 55057.980504 0.0001 113 55203.493222 8e-05 48 55060.185492 9.6e-05 114 55205.698095 9.8e-05 49 55062.390224 0.00011 115 55207.902655 0.00011 50 55064.594952 0.00013 116 55210.107229 0.00012 51 55066.799531 8.5e-05 117 55212.312237 9.6e-05 52 55069.00463 0.00011 118 55214.516844 0.00012 53 55071.209274 9.7e-05 120 55218.926337 0.00011 54 55073.413836 9.8e-05 121 55221.131008 9.8e-05 55 55075.618635 8.3e-05 122 55223.335553 9.9e-05 56 55077.823397 9.2e-05 123 55225.540792 0.00011 57 55080.027935 0.0001 124 55227.74504 9.5e-05 58 55082.232618 0.0001 127 55234.359584 9.7e-05 59 55084.437472 9.5e-05 128 55236.564184 0.00011 60 55086.642426 9.5e-05 129 55238.769067 0.00011 62 55091.051615 0.00012 130 55240.973702 8.9e-05

67 68

131 55243.178596 9.9e-05 219 55437.194973 0.00011 132 55245.383126 0.00013 220 55439.399803 9.1e-05 133 55247.588019 0.00012 221 55441.604453 0.00014 134 55249.792571 0.00011 222 55443.809352 0.00011 135 55251.997405 0.0001 223 55446.014192 0.00011 136 55254.202273 0.00012 224 55448.218715 0.00013 137 55256.407064 0.00011 225 55450.423576 9.1e-05 138 55258.611343 9e-05 226 55452.628221 8.2e-05 139 55260.816366 0.00012 227 55454.832933 9.3e-05 140 55263.020823 9.9e-05 228 55457.037665 9.6e-05 141 55265.225621 0.00012 229 55459.242512 0.0001 142 55267.430531 0.00013 230 55461.447402 0.00011 143 55269.635082 9.7e-05 231 55463.652236 0.00012 144 55271.840032 0.00012 232 55465.856812 9.4e-05 145 55274.044702 0.00013 233 55468.061422 0.00013 147 55278.454173 0.00011 234 55470.266152 0.00011 148 55280.658996 0.00013 235 55472.470953 0.00011 149 55282.863587 9.9e-05 236 55474.675552 0.00015 150 55285.068289 0.00013 237 55476.880408 0.00013 151 55287.273241 0.00013 238 55479.08529 9.9e-05 152 55289.477851 0.00011 239 55481.290068 9.5e-05 153 55291.682699 0.00012 240 55483.494629 0.00011 154 55293.887397 0.0001 241 55485.699412 9.6e-05 155 55296.091926 0.00011 242 55487.904257 9.3e-05 156 55298.296943 0.00013 243 55490.108985 9.3e-05 157 55300.501486 8.3e-05 244 55492.313464 0.00013 158 55302.706231 9.7e-05 245 55494.518402 9.6e-05 159 55304.911507 9.5e-05 246 55496.722992 0.00011 160 55307.115678 8.7e-05 247 55498.92774 9e-05 162 55311.525005 0.00011 248 55501.132274 9.6e-05 163 55313.729924 0.00012 249 55503.337017 9.7e-05 164 55315.93471 9.3e-05 250 55505.541967 0.00011 165 55318.139488 7.6e-05 251 55507.746571 0.00013 166 55320.344367 9.3e-05 252 55509.951539 0.00012 167 55322.548963 0.00011 253 55512.156159 0.00012 168 55324.753648 0.0001 254 55514.361029 0.00012 169 55326.958077 0.00014 255 55516.565602 0.00011 170 55329.162969 0.00011 256 55518.770497 0.00013 171 55331.367947 0.00011 257 55520.975039 8.9e-05 172 55333.572587 9.5e-05 259 55525.384453 0.00012 173 55335.777482 0.00015 260 55527.589356 0.00011 174 55337.982143 8.3e-05 261 55529.793955 9e-05 175 55340.186863 9.1e-05 262 55531.998692 0.00012 176 55342.391497 0.00011 263 55534.203438 0.0001 177 55344.596175 0.00011 264 55536.407994 8.7e-05 178 55346.801195 9.3e-05 265 55538.613101 9.3e-05 179 55349.005802 0.00014 266 55540.817647 0.0001 180 55351.210436 0.00013 267 55543.022514 0.00011 181 55353.415113 0.00012 268 55545.227164 9.1e-05 182 55355.620335 0.0001 269 55547.431765 9.4e-05 183 55357.824796 0.00011 270 55549.63659 0.0001 184 55360.029524 0.00011 271 55551.841269 0.00013 185 55362.234018 0.00012 279 55569.479149 0.00011 186 55364.438835 0.00011 280 55571.684144 8.4e-05 187 55366.643655 0.00011 281 55573.88854 0.00014 188 55368.848409 0.00011 282 55576.093591 0.00011 189 55371.052957 0.0001 283 55578.29802 0.00012 191 55375.462441 9.5e-05 284 55580.502811 0.00011 192 55377.667261 0.00011 285 55582.707684 0.0001 193 55379.872098 0.00015 286 55584.912295 8.7e-05 194 55382.07682 0.00013 287 55587.117153 0.0001 195 55384.281626 9e-05 288 55589.321553 9.7e-05 196 55386.486156 0.00011 289 55591.526764 0.00011 197 55388.69109 0.00012 290 55593.73147 0.00011 198 55390.89573 0.00011 292 55598.140776 0.00013 199 55393.100308 9.8e-05 293 55600.345417 9.8e-05 200 55395.305109 0.00011 294 55602.550236 9.7e-05 201 55397.510018 0.00014 295 55604.75504 0.00011 203 55401.919387 0.00013 296 55606.959717 0.00011 204 55404.124164 0.0001 297 55609.164374 0.0001 205 55406.328889 8.2e-05 298 55611.369271 9.4e-05 206 55408.533636 0.00012 299 55613.573898 8.9e-05 207 55410.738214 9.4e-05 300 55615.778694 0.00013 208 55412.942968 9.8e-05 301 55617.983558 0.00011 209 55415.147903 0.00012 302 55620.18839 8e-05 210 55417.352574 0.0001 303 55622.392704 9e-05 211 55419.55721 0.00011 304 55624.597492 0.0001 212 55421.762168 0.0001 305 55626.802404 0.0001 213 55423.966522 9.3e-05 306 55629.007216 9.6e-05 214 55426.171527 0.00012 307 55631.211765 8.9e-05 215 55428.376124 8.9e-05 308 55633.416642 0.00011 216 55430.581152 8.9e-05 217 55432.785581 7.2e-05 218 55434.990502 0.00017 Table C-1: Measured transit times for HAT- P-7 69

Number Transit time (HJD) Error 0 54978.562346 0.00091 7 55103.39276 0.0014 9 55139.061257 0.0012 10 55156.89799 0.0024 11 55174.734113 0.00055 13 55210.404813 0.00083 14 55228.23097 0.00064 15 55246.062157 0.00061 16 55263.905081 0.00067 17 55281.735094 0.00042 18 55299.568241 0.016 19 55317.409891 0.00056 20 55335.236498 0.00045 21 55353.070012 0.00034 22 55370.908602 0.00053 23 55388.748882 0.00044 24 55406.57692 0.0017 25 55424.409859 0.0013 26 55442.24754 0.0010 27 55460.083025 0.00057 28 55477.919417 0.00065 29 55495.749773 0.00085 30 55513.591947 0.0018 31 55531.419378 0.00043 34 55584.927468 0.00066 36 55620.593617 0.0069 38 55656.257869 0.0010 39 55674.088296 0.0039 40 55691.920165 0.0017 41 55709.759213 0.0007 42 55727.587154 0.00037 43 55745.417724 0.00029 44 55763.254999 0.00081 45 55781.084281 0.0012 46 55798.918068 0.002 47 55816.755889 0.00047 Table C-2: Measured transit times for Sandra D Observed frequencies

Frequencies for l = 0 (µHz). Estimated error: 0.73 µHz 768.1 826.5 886.2 945.8 1005.7 1065.8 1122.8 1181.7 1241.2 1302.1 1362.0 1425.1 1485.2 1545.1 Frequencies for l = 1 (µHz). Estimated error: 0.51 µHz 682.6 740.4 797.2 854.0 912.1 971.1 1031.7 1090.2 1149.8 1208.0

70 71

1266.9 1326.8 1387.2 1448.1 1509.9 1569.7 Frequencies for l = 2 (µHz). Estimated error: 1.0 µHz 763.9 822.4 940.5 1000.9 1061.3 1118.7 1178.5 1236.7 1298.0 1357.1 1419.8 1479.4 1539.6

Table D-1: Observed frequencies for HAT-P-7 (average error: 0.75 µHz)

Frequencies for l = 0 (µHz). Estimated error: 0.82 µHz 1435.4 1528.7 1623.0 1717.2 1808.6 1902.3 1996.3 2091.2 2187.0 2282.3 2377.1 2470.9 2568.5 2670.1 Frequencies for l = 1 (µHz). Estimated error: 0.80 µHz 1385.1 1477.4 1572.6 1666.3 1759.7 72

1851.9 1945.8 2040.4 2135.2 2229.9 2326.6 2420.2 2515.8 2610.0 2708.4 2806.1 Frequencies for l = 2 (µHz). Estimated error: 0.77 µHz 1615.4 1709.6 1802.9 1894.8 1987.8 2084.6 2179.4 2275.0 2368.8 2463.2 2560.0

Table D-2: Observed frequencies for Sandra (average error 0.80 µHz) List of Figures

1.1 A pulsation HR diagram showing many classes of pulsating stars for which asteroseismology is possible. Figure courtesy by Aerts et al. (2010)...... 3 1.2 Solar frequencies. Figure (a) shows the solar power spectrum, the lower panel shows a close-up labeled with (n, l) values for each mode. The dotted lines are the radial modes, the large and small separations are indicated. Figure (b) shows an échellediagram of the solar frequencies: the frequencies are stacked on top of each other by plotting them modulo the large separation. ∆ν = 135.0 µHz was used for the full symbols, the open symbols show the frequencies modulo a slightly smaller large separation (∆ν = 134.5 µHz). The frequencies line up vertically, with a ridge for each value of l = 0, 1, 2, 3 and 4. Figure courtesy by Bedding (2011)...... 5 1.3 An overview of the history and current status of the confirmed exoplanets, color-coded for the different detection techniques. The top figure sorts the number by their discovery year, the bottom figure gives an overview of period and mass of the planets. Figures taken from the NASA Exoplanet Archive (http://exoplanetarchive.ipac.caltech.edu)...... 10 1.4 Illustration of transits and occultation. During a transit, the flux drops as a part of the starlight is blocked. During the occultation, the observed flux drops slightly because the planetary thermal radiation is no longer observed. Figure courtesy by Seager and Deming (2010)...... 12 1.5 Raw and normalised data for Hat-P-7. The bottom figures provide a subset of the data, to display more details...... 15 1.6 Raw and normalised data for Sandra. The bottom figures provide a subset of the data, to display more details...... 16

2.1 Oversampled power spectrum (red) and smoothed curve (black), in relative intensities. HAT-P-7 shows a peak around 1100 µHz, Sandra’s peak is located at around 2000 µHz...... 22 2.2 Mode identiﬁcation for three observed frequencies for HAT-P-7. The red oversampled power spectrum is smoothed (black line). An algorithm determines the maxima of the smoothed spectrum (green dots). The relevant maxima are selected manually using the large separation. Sometimes, an average of two maxima is used (e.g. l = 2 mode in this ﬁgure)...... 23 2.3 Expectation value of the power spectrum for dipole and quadrupole multiplets as a function of the inclination angle i, for Ω = 6Ω . The bottom panels show

the power for i = 30◦ and i = 80◦. Figure courtesy by Gizon and Solanki. (Gizon and Solanki, 2003) ...... 24 2.4 Power spectrum for Sandra, showing split frequencies for l = 1 and l = 2 modes. 26

73 List of Figures 74

2.5 Part of the time series of Sandra, showing a regular ﬂux variation with a period of 20 days. This could be caused by stellar rotation, but the rotation period derived≈ from the stellar pulsations rules out this interpretation. Instead, it is likely caused by variable ﬂux of a background object...... 26 2.6 Echelle´ diagram for HAT-P-7. Reduced frequency calculated modulo ∆ν = 59.519 µHz...... 28 2.7 Echelle´ diagram for Sandra. Reduced frequency calculated modulo ∆ν = 94.55 µHz ...... 28 2.8 Echelle´ diagrams showing the observed frequencies (red) and the best model (blue)...... 34

3.1 An individual transit for Hat-P-7. The left and right side of the transit are fitted by a straight line. The middle dot represents the average time between the left and right line calculated at a relative flux of 0.9975. It is obvious from this figure that this procedure gives more stable time information than using the minimum of the transit to determine its time...... 36 3.2 The transits are numbered, taking gaps into account and a line is fitted through the transit times to determine the period...... 38 3.3 A so-called O-C plot, giving the observed minus the calculated transit times for HAT-P-7. No sinusoidal pattern is detected, to the (very low) level of the noise (observational error level: 0.00015 days)...... 40 3.4 An upper limit for the mass of additional≈ planets in the system of HAT-P-7. The most strict limit is in the 2:1 resonance orbit and is about 2M ...... 42 3.5 OC plot for Sandra, giving the observed minus the calculated transit⊕ times. The level of divergence from zero is an order of magnitude higher than for HAT-P-7. The best fit to transit timing variations is shown in the top figure, the bottom one shows the residuals to the fit...... 43 3.6 Illustration of a transit, assuming a star with a homogeneous brightness. Four contact points are shown, the location of which depends on the impact parameter b. The transit depth depends on the relative size of the star and the planet. Figure courtesy by Winn (2010)...... 45 3.7 Transit for HAT-P-7. The blue dots represent the combination of the data points for all observed transits, the black line is the transit model...... 47 3.8 Transit for Sandra. The blue dots represent the combination of the data points for all observed transits, the black line is the transit model...... 47 3.9 (a) Illustration of transits and occultation. During a transit, the flux drops. Afterwards, the flux rises as the dayside comes into view. The flux drops again when the planet is occulted by the star. Figure taken from Winn (2010). (b) Phase folded Kepler light curve (Q0 and Q1) for HAT-P-7. The red dashed line is the planet-only model (as shown in figure 3.9a). Welsh et al. (2010) include a ellipsoidal model for the star (dotted blue) to come to a best fit of the curve variation (orange solid line). Figure taken from Welsh et al. (2010). . 49 3.10 A fit to the phase variation, including both the stellar and the planetary effect (see figure 3.9b). The occultation is clearly visible...... 50 3.11 Folded flux zoomed in on the occultation. The average flux level inside (green) and just outside (yellow) the occultation is calculated to measure the occultation depth...... 51 List of Figures 75

4.1 An HR-diagram showing the location of HAT-P-7 and Sandra. Black (green) line represents the best evolution track for HAT-P-7 (Sandra), with the best model on the track shown in yellow. We acknowledge Michael Zingale for this plot, stellar properties taken from Carroll and Ostlie (1996). The track for HAT-P-7 is shown in more detail in figure 4.2...... 54 4.2 An HR-diagram showing our values for HAT-P-7 compared with the literature (see table 4.2 for values). For our own work, we show the evolution track that best fits the original data for different overshoot values (blue tracks), along with the model selected on the track (blue dots). The error square shows the Monte Carlo parameter averages presented in this work (blue square). For (Christensen-Dalsgaard et al., 2010, (a)), no error bars were quoted in their work, so only their parameters are indicated (black cross). (Pálet al., 2008, (b)) is plotted using one-sigma error levels (red square). (Ammler-von Eiff et al., 2009, (c)) did not present luminosity values, the bars indicate the one- sigma error on their temperature value (green bars)...... 56 List of Tables

2.1 Rotational splitting for Sandra. For l = 1 the value is based on 14 split frequency peaks, for l = 2 we have 7 peaks. For l = 2 we have calculated the difference between left and central frequency (left) and central and right frequency (right)...... 25 2.2 Average large frequency separation...... 27 2.3 Parameter values for HAT-P-7 and Sandra. Three different sets of parameters have been calculated, using different values for the overshoot. The average has been calculated giving equal weight to all three overshoot calculations. . . 33

3.1 Calculated orbital periods...... 37 3.2 A comparison between HAT-P-7 and CoRoT-1 of the parameters relevant for TTV. The systems are very similar. CoRoT-1’s O-C value comes from Bean (2009), the other parameters from Barge et al. (2008). HAT-P-7’s value comes from this work...... 41 3.3 Parameters describing the transit of HAT-P-7 and Sandra. The limb darkening constants (u1 and u2), the impact parameter (b) and the transit depth (δ) go into the transit ﬁt. The inclination (i) and the planetary radius (Rp) are the important parameters that can be derived from this...... 46

4.1 An overview of all values derived for Sandra throughout this work...... 55 4.2 An overview of all values that were obtained for HAT-P-7 in this work, and a comparison with the available literature. Some error values have been symmetrized, the bracketed numbers for the period represent errors on the last two digits. a: Christensen-Dalsgaard et al. (2010) b: P´alet al. (2008) c: Ammler-von Eiﬀ et al. (2009) d: Winn et al. (2009) e: Borucki et al. (2009b) f: Southworth (2011) uses fractional stellar and planetary radius... g: Welsh et al. (2010) h: Christiansen et al. (2010) *: only a relative radius or mass was provided in this source, we calculated an absolute radius based on the values in their work. **: only an absolute metallicity was quoted in this work, we calculated this value using a solar metallicity of 0.0142. ***: formal error is an underestimation ...... 58

C-1 Measured transit times for HAT-P-7 ...... 68 C-2 Measured transit times for Sandra ...... 69

D-1 Observed frequencies for HAT-P-7 (average error: 0.75 µHz) ...... 71 D-2 Observed frequencies for Sandra (average error 0.80 µHz) ...... 72

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