S t u d y a n d C h a r a c t e r i z a t i o n o f E x o p l a n e t s ( S t u d i e r a f e x o p l a n e t e r s e g e n s k a b e r )

Transcription

1 S t u d y a n d C h a r a c t e r i z a t i o n o f E x o p l a n e t s ( S t u d i e r a f e x o p l a n e t e r s e g e n s k a b e r ) S p e c i a l e a f h a n d l i n g i A s t r o n o m i M i c h a e l L i n d h o l m N i e l s e n V e j l e d e r : H a n s K j e l d s e n 2 4. a u g u s t I n s t i t u t f o r F y s i k o g A s t r o n o m i A a r h u s U n i v e r s i t e t

2

3 Abstract The study and characterization of extrasolar planets, or exoplanets, is an exciting field that brings together different disciplines from astronomy, physics and chemistry. Several open questions remain in respect to the stability of determining the exoplanet orbital period and temperature. The latter study is further complicated by possible variations in the exoplanet surface or atmospheric reflection and emission. The stability of the orbital period determination has been investigated using a model by Mandel and Agol (2002) together with data from the Kepler satellite. The study involves three filtering techniques: locally weighted scatterplot smoothing, a cubic smoothing spline and a spectral filter designed to isolate the exoplanet signature in the frequency domain. The analysis show that the stability is dependent on the utilized method. The locally weighted scatterplot smoothing is generally the most accurate of the three methods. It is often also more precise than the cubic smoothing spline. Both methods are, however, still sensitive to outliers and stellar variability, including frequencies from stellar oscillations. The spectral filter typically yields less accurate but more precise results. It is not influenced by outliers and is less sensitive to stellar oscillations. The stability of the planetary temperature determination has been investigated using a high-precision time series photometry program together with data of HAT-P-7 from the Spitzer space telescope. The program includes two stellar position algorithms, three types of aperture masks and an outlier filter. In addition a series of statistical tests has been performed on the sky background. The analysis show that the stability is dependent on the utilized method. The most stable method is based on a modified Lorentzian profile which is folded by a sub-pixel aperture mask. The stellar position algorithms were found to yield similar results. The sky background was also found to be stable on timescales relevant for the observation. Some noticeable features could not be detected by the outlier filter. The features are most likely due to instrumental effects. The stability of the surface or atmospheric reflection and emission has been studied by measuring variations in the exoplanet occultation depth. The study is based on data from the Kepler satellite. The analysis show that the exoplanet surface or atmospheric reflection and emission can change over time. An upward trend from a lower occultation depth in mid 200 to a larger occultation depth in early 20 was found for HAT-P-7b. This thesis has resulted in a potentially powerful spectral filter, a high-precision time series photometry program and some interesting observations of variations in the occultation depth of HAT-P-7b. All of which merits further studies. i

4

5 Contents Abstract Contents i iii Introduction. Orbital Stability Atmospheric Conditions Goals of This Project Theoretical Background 5 2. Transit Method Reflection and Emission Kepler Data Analysis 7 3. Kepler Data Planetary Timing Reflected and Emitted Light Spitzer Data Reduction and Analysis Spitzer Data Data Reduction Emitted Light Discussion The Orbital Period The Planetary Temperature Reflection and Emission Future Prospects Acknowledgements 65 A Appendix 67 A. Planetary Timing A.2 Reflected and Emitted Light iii

6 iv CONTENTS Bibliography 7

7 Chapter Introduction The study and characterization of extrasolar planets, or exoplanets, is an exciting field that brings together different disciplines from astronomy, physics and chemistry. The first exoplanet to be discovered around a main sequence star was 5 Pegasi b. This peculiar object was found to orbit the star in just 4.2 days and with a minimum mass of 0.5 M Jupiter it defined a new class of objects now known as hot Jupiters (Mayor and Queloz, 995). In broader terms, a hot Jupiter has a mass roughly similar to that of Jupiter and it orbits the host star closer than 0. AU (Perryman, 20, Ch 6). As of August 202 over 700 exoplanets have been detected of which a smaller population are hot Jupiters (Schneider, 202). A study of these objects can lead to a better understanding of their orbital stability and surface or atmospheric conditions. Hot Jupiters are generally thought to belong to single planet systems (Szabó et al., 202). However, recent studies points to possible orbital variations in some of these systems (Szabó et al., 202). The variations might be caused by other planetary bodies, exomoons or they might be caused by other effects. These effects includes virtual variations due to the observational sampling, stellar rotation or activity of the host star (Szabó et al., 202). One of the open questions is how stable the orbital period determination is. The host star also has a great impact on the exoplanet surface or atmospheric conditions. One of the highly irradiated hot Jupiters is HAT-P-7b which receives 4.5 ˆ 0 9 erg s cm 2 from the host star (Christiansen et al., 200). It has been suggested that the high incident flux can lead to a gas-phase of TiO or VO (Fortney et al., 2008). In the gas phase, these molecules are strong absorbers of the incident light (Seager, 200, Ch 4.2). The strong absorption can lead to a temperature inversion in the upper atmosphere as has been reported for HAT-P-7b (Christiansen et al., 200). One of the open questions is how stable the planetary temperature determination is. The stellar heating also drives mass motion in the planet atmosphere (Seager, 200, Ch 2.7). In the case of strong atmospheric circulation the absorbed energy can be redistributed from the planetary dayside to the nightside through large-scale movements of gas (Seager, 200, Ch 0). In the presence of clouds such large-scale movements could change the reflectivity of the exoplanet (Seager, 200, Ch 0.5). One of the open questions is how stable the surface or atmospheric reflection and emission is for hot Jupiters.

8 2 CHAPTER. INTRODUCTION. Orbital Stability The discovery of close-in hot Jupiters came as a surprise to planetary scientists. In comparison, Mercury, has an orbit which is about ten times further away relative to the orbit of HAT-P-7b (Seager, 200). It is, however, believed that hot Jupiters are formed further away from the star and later move to a closer orbit due to inward migration. At least three mechanisms are theorized to lead to a significant orbital evolution (Perryman, 20, Ch 0.8).. Gas disk migration, involving tidal interactions between a Jupiter mass planet and the residual protoplanetary disk. 2. Planetesimal disk migration, where the planet interacts with remnant planetesimals. 3. Planet-planet scattering due to interactions between other orbiting planets. At some point a halting mechanism must set in, otherwise the hot Jupiter would terminally fall into the star. Several halting mechanisms have been suggested but the debate is still out on which scenario is most likely (Perryman, 20, Ch 0.8.2). Models suggests that a significant fraction of existing terrestrial sized planets can survive the inward migration of a giant planet (Mandell and Sigurdsson, 2003). In addition if the giant planet forms and migrates quickly, the planetesimals can survive and smaller planets can still have time to form in the disk (Perryman, 20, Ch 0.8.2). If two such orbiting planets interact in a regular and periodic interval, orbit resonance effects can occur (Perryman, 20, Ch 2.6.2). Such effects can lead to regular changes in the planetary timing and this can be observed using repeated observations of an exoplanet that passes in front of the host star. This observational method is called the transit method and is further discussed in the next chapter..2 Atmospheric Conditions There is a great deal of diversity between hot Jupiters. From the darkest world, TrES- 2b, which reportedly reflects less than % of the incident stellar light (Kipping and Spiegel, 20) to the brighter Kepler-7b with a relative reflection of 32% (Demory et al., 20). Models suggests that the atmospheric conditions depend, to a great extent, on the incident flux from the host star (Fortney et al., 2008). In relation to this, two classes of hot Jupiters have been suggested. pm: a very hot Jupiter where absorption by TiO and VO gases leads to a temperature inversion in the upper atmosphere. The absorbed energy is instantaneously reemitted and the day- to nightside planetary contrast is predicted to be high. pl: a hot Jupiter where the absorption is dominated by Na and K. In this case the absorbed energy is more readily redistributed through atmospheric dynamics and the dayside is predicted to be cooler (Fortney et al., 2008). The simple picture outlined above is further complicated by the possibility for clouds. Clouds play an important role in the atmospheric energy balance. Some types of clouds are highly reflective while other types, such as the hazes on Jupiter and Saturn, absorb radiation at shorter wavelengths (Seager, 200, Ch 4.4.). Figure. shows a simple illustration of elements that can contribute to the planetary energy balance. The atmospheric reflectance and emission spectrum can be studied through the geometric albedo and the brightness temperature, respectively. These parameters are further discussed in the next chapter.

9 .3. GOALS OF THIS PROJECT 3 Incident Flux Absorbed Absorbed Reflected Reflected Outgoing IR flux Haze Emitted by atmosphere Clouds Surface Figure.: Simple illustration of elements that can contribute to the planetary energy balance. After: Seager (200, Fig 6.)..3 Goals of This Project As described above there are several unanswered questions relating to the stability of the planetary orbit, reflection and emission. It is therefore of central interest to investigate these parameters further. To be specific this project contours the following questions:. How stable is the orbital period determination? 2. How stable is the planetary temperature determination? 3. How stable is the surface or atmospheric reflection and emission? To answer these questions I will make an independent analysis of data containing measurements of three hot Jupiters: TrES-2b, HAT-P-7b and Kepler-7b, supplemented by measurements of the hot super Earth, Kepler-9b where orbital variations have recently been detected (Ballard et al., 20). The data primarily consists of photometric measurements from the Kepler satellite. In addition data from the Spitzer space telescope is used to study the planetary temperature for HAT-P-7b. Chapter two introduces the theoretical background behind the transit method. In addition concepts relevant for the description of reflected and emitted light is introduced. Chapter three is on the stability of the orbital period determination and on the reflection and emission from hot Jupiters. The orbital stability is studied using a model by Mandel and Agol (2002) together with three methods of filtering transit signatures. These methods includes locally weighted scatterplot smoothing, a cubic smoothing spline and a spectral filter designed to isolate the transit signatures from the frequency domain. The three methods are also applied to data from the four exoplanets. The stability of the surface or atmospheric reflection and emission is studied using nearly two and a half years of continued observations from the Kepler satellite. Chapter four is on the stability of the planetary temperature determination. The study incorporates different methods of reducing the photometric data. These methods includes two stellar position algorithms, three approaches on determining the stellar flux and a semi-automatic outlier filter. In addition a series of statistical tests are performed on the global and local sky background determination. The thesis is rounded off with a discussion on the main results.

10

11 Chapter 2 Theoretical Background This chapter describes the relevant theoretical background. The chapter is divided into two parts. The first part introduces concepts relevant for the transit method. The second part concentrates on the reflected and emitted light from exoplanets. 2. Transit Method This part introduces concepts relevant for the description of the transit method. The first subsection introduces the basic principle behind a planetary transit and occultation. The second subsection is on the planetary timing. Transit and Occultation It is of interest to study the flux from a stellar system that has an orbiting exoplanet which passes in front or behind the host star. The primary passage can give information about the orbital period while the occultation can yield information about the conditions of an exoplanet atmosphere or surface. This subsection introduces the principle behind a planetary transit and occultation. The Primary Transit The planetary transit occurs when an orbiting planet passes in front of its host star. In this case, the planet blocks part of the stellar light and this leads to a drop in the flux from the stellar system. There are four observables which characterize the primary transit. The period P, the transit depth δ, the total transit time t tot and the full transit time t full (Perryman, 20, Ch 6.4.). The primary transit and three of the observables are illustrated in Figure 2.. The time between t and t 2 is known as the ingress while the time between t 3 and t 4 is known as the egress. In observations performed at shorter wavelengths the primary transit is typically rounder in the bottom (Perryman, 20, Ch 6.4.2). This effect is caused by stellar limb darkening and results from variations in the stellar temperature and opacity as a function of radius (Winn, 200, Ch 2.5). In a circular orbit and with no limb darkening, the four observables are related through a set 5

12 6 CHAPTER 2. THEORETICAL BACKGROUND Y b X Flux δ Time t full t t 2 t 3 t 4 t tot Figure 2.: Simple illustration of the primary transit. The conjunction occurs at X 0. After: Winn (200, Fig 2). of geometric equations (Winn, 200, Ch 2) 2 ˆRplanet δ k 2 R «t tot P a sin R p ` kq2 b 2 π a sin i «t full P sin R π a sin i ff (2.) (2.2) a ff p kq2 b 2. (2.3) Where R is the stellar radius, R planet and a is the planet radius and semi-major axis, respectively. The inclination angle i is defined as the angle between the plane of the sky and the observer. The impact parameter b describes the sky projected distance at conjunction. This occurs at X 0 shown in Figure 2.. For a circular orbit the impact parameter is given by (Winn, 200, Ch 2.) b a R cos i. (2.4) In addition the orbital period P can be obtained from repeated observations of the primary transit. This is further discussed in the subsection on planetary timing.

13 2.. TRANSIT METHOD 7 Flux occultation star + planet dayside star alone Time star - planet shadow star + planet nightside transit Figure 2.2: Illustration of the transit and occultation. From: Winn (200, Fig ). The Secondary Transit The planetary occultation occurs when the planet passes behind the host star. In this case, the illuminated planet is hidden from the observers point of view. This results in a second drop in the flux from the stellar system (Perryman, 20, Ch 6.4). There are four observables which characterize the secondary transit. The period P occ, the transit depth δ occ, the total transit time t tot,occ and the full transit time t full,occ (Winn, 200, Ch 2). The added subscript refers to the occultation or secondary transit. The primary and secondary transit is illustrated in Figure 2.2. After the primary transit, the planetary dayside becomes progressively visible as the planet orbits the star. The brighter dayside leads to an increase in the total flux from the stellar system. This culminates in the secondary transit where the planetary dayside is hidden by the star. The nightside contribution can be calculated using the difference between the wings just outside the primary transit and the occultation depth. A study of the phase curve and the secondary transit in particular, can give valuable information about the conditions of the planetary atmosphere or surface. This is further discussed in the section on reflection and emission.

14 8 CHAPTER 2. THEORETICAL BACKGROUND Planetary Timing It is of interest to study the planetary timing as this can give information about the orbital period and indicate the presence of other planets in the respective stellar system. This subsection focuses on the orbital period and on transit timing variations. Orbital Period The orbital period is used to compare and combine a sequence of transits. The period can be determined by timing individual transits using (Winn, 200, Ch 3.2) t c pnq np ` t c p0q. (2.5) Where t c pnq is the time of conjunction of the nth event and P is the orbital period. The error in P varies inversely as the total number of observed transits. Thus an extraordinary precision on the orbital period can be achieved by repeatedly observing the transiting event (Winn, 200, Ch 3.2). Transit Timing Variations Various effects can lead to transit timing variations. These effects include perturbations due to other gravitating bodies, tidal forces and relativistic precession (Perryman, 20, Ch 6.4.3). The timing variations can be studied by comparing the observed and calculated times of conjunction. 2.2 Reflection and Emission This part introduces concepts relevant for the description of light reflected off, and emitted by an exoplanet. The first subsection introduces the planetary phase angle and various albedos as a measure of surface or atmospheric reflectivity. The second subsection introduces two planetary temperatures. The final subsection is on the theoretical fluxratio between an exoplanet and its host star. Planetary Albedos It is of interest to study the reflectance spectrum of the planetary dayside as this can give information about the properties of an exoplanet atmosphere. Several closely related planetary albedos are typically used as a measure of surface or atmospheric reflectivity. This subsection introduces the phase angle and two types of albedos. This includes the geometric albedo and the bond albedo. The nature of the scattering particles is discussed in the final part of this subsection. The Phase Angle The phase angle is used to quantify the planetary phase. It is defined as the angle α between the planet and the host star, relative to an observer (Seager, 200, Ch 3.4.2). The phase angle is illustrated in Figure 2.3. At full phase α 0 0 and the planet is fully illuminated by the star. In contrast, the planet is at conjunction when α 80 0.

15 2.2. REFLECTION AND EMISSION 9 Star Planet α Observer Figure 2.3: Illustration of the phase angle. After: Seager (200, Fig 3.5). Geometric Albedo The most directly measurable parameter is the geometric albedo. The geometric albedo is a global planetary parameter. It is defined as the ratio of the planet s flux at zero phase angle relative to the flux from an ideal diffusely reflecting surface, also known as a Lambert surface, which is located at the same distance and with the same cross-sectional area as the planet (Seager, 200, Ch 3.4.3). The planet to star flux ratio is related to the geometric albedo by (Perryman, 20, Ch 6.4.7) εpα, λq F planetpα, λq F pλq A g pλq ˆRplanet a 2 Φpα, λq. (2.6) Where A g is the geometric albedo at the relevant wavelength λ. R planet is the planetary radius, a the semi major axis, α the phase angle and Φpα, λq is the relevant phase function. At shorter wavelengths, starlight scattered off an exoplanet is expected to dominate over thermal emission. In this case, the geometric albedo can be estimated from the exoplanet occultation using (Winn, 200, Ch 3.4) δ occ pλq» A g pλq ˆRplanet a 2. (2.7) Where δ occ is the wavelength dependent occultation depth. In the ideal case of a Lambert sphere the geometric albedo is (Seager, 200, Ch 3.4.3) A g,lambert pλq 2 3. In this case, one third of the incident radiation is scattered out of the line of sight. Bond Albedo The planetary energy balance can be studied through the bond albedo. The bond albedo is a global planetary parameter. It describes the fraction of incident stellar radiation that is scattered back into space. The bond albedo is related to the geometric albedo by (Seager, 200, Ch 3.4.4) A B ż 8 0 A g pλqqpλqdλ. (2.8)

16 0 CHAPTER 2. THEORETICAL BACKGROUND Where A g pλq is the geometric albedo and qpλq is the phase integral qpλq 2 ż π 0 Φpα, λq sin αdα. (2.9) Where Φpα, λq is the relevant phase function and α is the phase angle defined previously. By definition, the bond albedo has a value between zero and one. A bond albedo of zero or one corresponds to the case where all the incident radiation is absorbed or scattered, respectively. The latter case can be exemplified by the Lambert phase function Seager (200, Ch 3.6) Φ Lambert pαq psin α ` pπ αq cos αq. (2.0) π Inserting this into Equation 2.9 and solving the integral gives ż π qpλq 2 psin α ` pπ αq cos αq sin αdα π 0 qpλq 2 α π 2 3 sin 2α pπ αq cos 2α 8 4 qpλq 2 π π 2 p j 4 πq qpλq 3 2. Inserting this into Equation 2.8 and noting that the geometric albedo for a Lambert sphere is A g,lambert pλq 2{3 leads to a bond albedo of one. In summary, the bond albedo depends on the geometric albedo and the phase integral. The phase integral is specified through the phase function which in turn depends on the nature of the scattering particles. Nature of Scattering Particles The global planetary albedo depends on the scattering properties of individual gas, cloud and surface particles. The fraction of incident light that is scattered by a given particle is determined by the single scattering albedo ω. The single scattering albedo is a wavelength dependent parameter and has a value between zero and one (Seager, 200, j π 0 I inc I sc Θ Figure 2.4: Schematic illustration of single particle scatter. Θ is the scattering angle, I inc and I sc is the incident and scattered light, respectively. After Seager (200, Fig 5.).

17 2.2. REFLECTION AND EMISSION Figure 2.5: The single scattering albedo and scattering asymmetry parameter at 5500 Å for enstatite, iron and corundum as a function of particle size. From: Seager et al. (2000, Fig 5). Ch 3.4.). A value of zero or one corresponds to the case where all the incident radiation is absorbed or scattered, respectively. The global planetary albedo also depends on the directional scattering properties of the individual particles. These properties can be described by the scattering asymmetry parameter (Seager, 200, Ch 8.5.) g cos Θ. (2.) Where Θ is the scattering angle illustrated in Figure 2.4. The scattering asymmetry parameter has a value between ď g ď. The extreme cases corresponds to g : g 0 : g : Strong forward scattering. Isotropic scattering. Strong backward scattering. It is of interest to study the single scattering albedo and scattering asymmetry parameter for cloud particles that are potentially present in the atmospheres of hot Jupiters. The relevant parameters have been studied for enstatite (MgSiO 3 ), iron (Fe) and corundum (Al 2 O 3 ) by Seager et al. (2000). The study utilizes Mie s scattering theory which assumes that the particles are spherical in nature (Seager, 200, Ch 8.5.3). Figure 2.5 shows the single scattering albedo and scattering asymmetry parameter at 5500 Å for the three condensates as a function of particle size. Rayleigh scattering occurs for small particles compared to the wavelength of light. In this case, the forward and backward scattering average out and the resulting scattering is isotropic. This is observed for particles with r» 0.0µm. Moreover for r» 0.0µm, corundum and iron have ω» 0 which means that absorption dominates over scattering. Forward scattering is generally observed for large particles compared to the wavelength of light. This is seen for particles with r» 0µm. Furthermore, variations in g and ω are observed near r» 0.05µm for corundum and enstatite. The variations are caused by interference effects and by light that refract twice through the spherical particle (Seager et al., 2000).

18 2 CHAPTER 2. THEORETICAL BACKGROUND In general, light scattering depends on two things. First, the particle size compared to the wavelength of light. Second, on the different nature of the particles which is controlled by opacities (Seager, 200, 8.). Ultimately the global planetary albedo is specified through complex multiple scattering between various particles. The problem involves solving the radiative transfer equation which is beyond the scope of this thesis. Planetary Temperatures It is of interest to study the emission spectrum of the planetary dayside as this can give information about the properties of an exoplanet atmosphere. This subsection introduces the brightness temperature as a measure of the planetary emissivity at various wavelengths. In addition the equilibrium temperature is introduced as a measure of the planetary energy balance. Brightness Temperature The most directly measurable parameter is the brightness temperature. It is defined as the temperature of a black body at the same distance, of the same shape and with the same flux within a given spectral range as the planet. The planetary surface flux is related to the brightness temperature by Stefan-Boltzmann s law (Seager, 200, Ch 3.3.3) F s, planet pλq σ R T b pλq 4. (2.2) Where σ R is the Stefan-Boltzmann constant and T b is the brightness temperature in the relevant wavelength range λ. In contrast to a black body radiator, which has a constant brightness temperature, the brightness temperature of a planet can vary greatly with respect to wavelength (Seager, 200, Ch 3.3.3). This is exemplified for Earth in Figure 2.6. The brightness temperature is further discussed in the final part of this chapter. Equilibrium Temperature The planetary energy balance can be studied through the equilibrium temperature. The equilibrium temperature is defined as the temperature of an isothermal planet after it has reached equilibrium with the incident radiation from the host star (Seager, 200, Ch 3.3.2). The equilibrium temperature is related to the bond albedo by (Perryman, 20, Ch 6.5.) {2 ˆR T eq, planet T rfp A B qs {4. (2.3) 2a Where T is the effective stellar temperature defined as the temperature of a black body at the same distance, of the same shape and with the same total flux as the star. R is the stellar radius, a the semi-major axis and A B is the bond albedo. The geometric factor f describes the effectiveness of atmospheric circulation and the degree to which the absorbed energy is redistributed from the planet s day to nightside (Perryman, 20, Ch 6.5.). Readers interested in the subject should refer to chapters 5, 6 and 8 in Seager (200).

19 2.2. REFLECTION AND EMISSION 3 Figure 2.6: Illustration of the brightness temperature for Earth s atmosphere at different wavelengths. The solid curve shows the surface flux from Earth as observed by Mars Global Surveyor. The dashed curves are the black body curves for brightness temperatures at T b p2µmq 270 K and T b p4.2µmq 25 K. From: Seager (200, Figure 3.4). The extreme cases corresponds to f : f 2 : Uniform redistribution. Instantaneous reradiation. Uniform redistribution can occur if the planet is a fast rotator or if it is tidally locked and dominated by strong atmospheric circulation. Instantaneous reradiation can occur if the planet is a slow rotator or if it is tidally locked and does not have a strong atmospheric circulation (Seager, 200; Perryman, 20, Ch 3.3.2; Ch 6.5.). Beyond the two extreme cases, f can be ascertained from detailed observations or from atmospheric circulation models (Seager, 200, Ch 3.3.2). Finally it should be noted that the equilibrium temperature does not include additional heat sources such as tidal deformation, radiogenic decay or effects from greenhouse gases.

20 4 CHAPTER 2. THEORETICAL BACKGROUND Flux Ratios It is of interest to compare the theoretical flux ratios between relevant exoplanets and their host star. This subsection introduces a theoretical spectrum as a measure of the planet to star flux ratio. The measured occultation depth is also discussed. Finally the Rayleigh-Jeans region is discussed in respect to the brightness temperature. Theoretical Spectrum The exoplanet to host star emission flux ratio can be studied using the black body spectrum. A black body in thermal equilibrium absorbs all the incident light and reemits it in a spectrum corresponding to the equilibrium temperature of the black body. The black body flux is related to the Planck function by (Seager, 200, Ch 2.8) F B pt, λq 2πhc2 λ 5 hc exp λk B T. (2.4) Where h is the Planck constant, c the speed of light in vacuum and k B is the Boltzmann constant. The flux is calculated at the respective equilibrium temperature T and wavelength λ. The flux from light scattered off an exoplanet atmosphere or surface should also be included in the total flux. Thus the total observed planet to star flux ratio is given by (Seager, 200, Ch 3.5)» fi F obs, planet pλq exp hc λk B T 2 2 ˆRplanet ˆRplanet fl ` A g pλq. (2.5) F obs, pλq hc exp λk B T eq, planet R looooooooomooooooooon a looooooooooooooooooooooomooooooooooooooooooooooon emission scattering Where R is the stellar radius and R planet and a is the planet radius and semi-major axis, respectively. The two terms describes the black body thermal emission and the incident light which is scattered off an exoplanet atmosphere or surface. Figure 2.7 shows the simplified spectrum of TrES-2b, HAT-P-7b and Kepler-7b compared to a Sun-like star. The values from Table 2.2 were used to calculate the spectrum. A significant part of the emitted light contributes to the total flux in the visual range. The is particularly noticeable for HAT-P-7b. This means that flux observed at visual wavelengths is a combination of reflected and emitted light. Also notice that the flux ratio is on the order of 0 5 in the visual while in the mid-infrared it is 0 3. This underlines the difficulty in observing these objects. Table 2.: Host star parameters. The values for the stellar temperature and radius are from Schneider (202). Star T [K] R s TrES HAT-P Kepler

21 2.2. REFLECTION AND EMISSION Sun TrES 2b HAT P 7b Kepler 7b 0 2 Normalized Flux Wavelength [nm] Figure 2.7: Theoretical spectrum of TrES-2b, HAT-P-7b, Kepler-7b compared to a Sun-like star. The solid lines shows the combined flux from the respective body. The dashed and dotted lines shows the contribution from thermal emission and scattering, respectively. The values from Table 2.2 were used to calculate the spectrum with the geometric albedo set as constant. Table 2.2: Exoplanet parameters. TrES-2b: Equilibrium temperature calculated using Equation 2.3 with f, A B 0, T 5850 K and R The geometric albedo is from Kipping and Spiegel (20). Kepler-7b: Equilibrium temperature and geometric albedo from Latham et al. (200) and Demory et al. (20), respectively. HAT-P-7b: Equilibrium temperature and geometric albedo from Nymeyer et al. (2009) and Christiansen et al. (200), respectively. The values for the exoplanet semi-major axis and radius are from Schneider (202). Exoplanet A g T eq, planet [K] R planet rr Jupiter s a [AU] TrES-2b ă HAT-P-7b À ` Kepler-7b

22 6 CHAPTER 2. THEORETICAL BACKGROUND Occultation Depth The theoretical spectrum can also be used to predict the occultation depth for an exoplanet in thermal equilibrium. The detected flux depends on the instrumental throughput by (Bradt, 2004, Ch 8.3) F obs, detector ż 8 0 F obs pλqɛpλqdλ. (2.6) Where ɛpλq is the relevant spectral response function. Assuming that the stellar flux is constant over timescales relevant for the occultation, the planet to star flux ratio is directly related to the occultation depth by (Winn, 200, Ch 2.4) F planet F» δ occ. (2.7) In the case of TrES-2b the geometric albedo is predicted to be below % (Kipping and Spiegel, 20). This information is used to estimate the lower limit on the occultation depth in the following calculation. The relevant stellar and planetary parameters, from Table 2. and 2.2, are inserted into Equation 2.5 together with A g pλq 0. Next Equation 2.6 is applied in the numerator and denominator together with the Kepler spectral response function shown in Figure 3.. This gives an estimated lower limit on the occultation depth of δ occ.2ppm. Rayleigh-Jeans Region The brightness temperature can be approximated from the observed planet to star flux ratio in the Rayleigh-Jeans region. The Rayleigh-Jeans region applies to high temperature black bodies in the mid-infrared where the exponent in Equation 2.4 is hc{λk B T! and this leads to the approximation (Seager, 200, Ch 3.5.) F obs, planet pλq F obs, pλq Teq, planet pλq» T pλq j ˆRplanet R 2. (2.8) The black body equilibrium temperature should be replaced by the wavelength dependent brightness temperature. This follows from the prior discussion relevant to Figure 2.6. In this case, the brightness temperature can be estimated from infrared observations of the exoplanet occultation using Tb, planet pλq δ occ pλq» T b, pλq j ˆRplanet R 2. (2.9)

23 Chapter 3 Kepler Data Analysis This chapter describes the steps involved in the Kepler data analysis. The chapter is divided into three main parts. The first part introduces the Kepler data and known characteristics of the datasets. The second part concentrates on the planetary timing and orbital period determination. The final part is on the analysis of the reflected and emitted light from hot Jupiters. 3. Kepler Data The Kepler satellite was launched on March and is in continued operation from an Earth-trailing orbit around the Sun. The primary scientific goal of the Kepler mission is to search for Earth sized planets in the habitable zone of solar-like stars. For this purpose the satellite has been equipped with 42 CCD detectors that continuously observe over 00,000 stars simultaneously. The observations are made in the visual range where solar-like stars emit most of their radiation. Figure 3. shows the relevant spectral response curve. The combined optical design covers most of the visual and part of the near infrared spectrum. The blue and red cutoffs were selected to avoid stellar variability and internal optical fringing, respectively (Koch et al., 200). Raw pixel data is downlinked on a monthly basis from the satellite and is subsequently processed in a science pipeline. First the calibration module produces calibrated target and background pixels. Next these pixels are processed in the photometric analysis module where background sky is removed and simple aperture photometry is performed. The raw light curves are subsequently processed by the presearch data conditioning module. This module corrects for systematic errors including pointing errors, focus changes and thermal effects. The module also identifies outliers and fills gaps within a quarterly segment to make it contiguous. The corrected data is next analyzed by the transiting planet search and data validation modules to identify possible exoplanet candidates. The resulting raw and corrected light curves are available for download when made public by NASA (Jenkins et al., 200). Two types of data are available from the Kepler satellite. The long cadence data consists of 29.4 minutes sampled data. This data is primarily used for planet detection. In addition a series of short cadence 58.8 seconds sampled data is available for a limited As of this writing 40 CCD detectors are still in operation. 7

24 8 CHAPTER 3. KEPLER DATA ANALYSIS 0.8 Spectral Response Wavelength [nm] Figure 3.: Kepler CCD spectral response curve. number of targets. This data is primarily used for asteroseismology and for improving the timing of planetary transits (Koch et al., 200). Both the long cadence (LC) and short cadence (SC) time series have gaps from when the satellite is in save mode, performing a quarterly roll maneuver or transferring data back to Earth. In addition momentum dumping can lead to a few bad data points (Haas et al., 200). An example of a long and short cadence time series is shown for HAT-P-7 in Figure 3.2. The figure illustrates various gaps in the time domain and describes their relevant causes. The long-term trend is primarily caused by instrumental drift. The spikes results from the transiting close-in exoplanet HAT-P-7b. x Q2 Flux [Counts/sec] Q0 Q Safe Mode Event Quarter Roll Safe Mode Event Downlink May Jun Jul Aug Sep Time [Days] Figure 3.2: HAT-P-7 time series. Only Q0-Q2 is shown for clarity. Blue: short cadence data. Green: long cadence data.

25 3.2. PLANETARY TIMING Planetary Timing This part deals with the planetary timing analysis. Precise determination of the planetary timing is important for specifying the orbital period, studying transit timing variations and for phase folding the data. The first subsection introduces a transit model which is used to test three methods of filtering the transit signatures. The various methods are introduced in the second subsection. The methods are applied to TrES-2b, HAT-P-7b, Kepler-7b and Kepler-9b in the final subsection. Transit Model It is of interest to create a model transit light curve to test three methods of filtering the transit signatures. The transit light curve should account for the geometry of the primary transit as well as the effects of stellar limb darkening, described previously. For a quadratic limb-darkening law the light curve is (Mandel and Agol, 2002) F pp, zq 4Ω tp c 2qλ e ` c 2 λ d ` 2 Θpp zq 3 j c 4 η d u. (3.) Where c n are the limb darkening coefficients, p is the ratio between the exoplanet and stellar radius, z is the normalized separation between the exoplanet and stellar center, Θ is the Heaviside step function $ & 0, pp zq ă 0, Θ {2, pp zq 0, %, pp zq ą 0, Ω 4ÿ n 0 c n n ` 4, $ λ & e : 0, ` p ă z, b λ e pp, zq λ e 2 : p 2 4z κ π 0 ` κ 2 p`z2 p2 q 2 4 % λ e 3 : p 2, z ď p, ˆp2 κ 0 cos ` z 2 2pz ˆ κ cos p2 ` z 2. 2z j, p ă z ď ` p, (3.2) λ d and η d are case sensitive functions described in Table of Mandel and Agol (2002). In a circular orbit z can be determined as a function of time by (Mandel and Agol, 2002) zptq a c psin 2π R P tq2 ` pcos i cos 2π P tq2. (3.3) Where R is the stellar radius, R planet, a, P and i is the planet radius, semi-major axis, orbital period and inclination angle, respectively. The quadratic-limb darkening law is used to create a model light curve using the set of values in Table 3.. The temporal grid is based on the long and short cadence data of HAT-P-7 from Q0 - Q0. The geometric setup and a single transit is shown in Figure 3.3.

26 20 CHAPTER 3. KEPLER DATA ANALYSIS Table 3.: Model parameters. The relevant limb darkening coefficients are calculated using c 2 γ ` 2γ 2, c 4 γ 2, c c 3 0 and c 0 c c 2 c 3 c 4. The values for γ and γ 2 represents HD (Mandel and Agol, 2002). Model Sampling R planet a P i γ γ 2 LC 29.4 min 0.R 0R 3 days SC 58.8 s 0.R 0R 3 days R * =pr R p * d=z R * e λ e λ 2 e λ 3 e λ 2 e λ.002 Flux [Arb. Units] % 50% 25% Time [Days] Figure 3.3: Top: geometric setup. Bottom blue: raw model based on Mandel and Agol (2002). The model parameters are given in Table 3.. Green: raw model with Gaussian noise and a 5 min simple harmonic oscillator added. Red: horizontal lines marking 25%, 50% and 75% above the minimum light curve level. Black: solid and dashed vertical lines marks the change in λ e as described in Equation 3.2. Black: dotted vertical line marks the central transit time.

27 3.2. PLANETARY TIMING 2 Methods This subsection introduces a method of determining the planetary timing and three methods of filtering the transit signatures. This includes locally weighted scatterplot smoothing, a cubic smoothing spline and a spectral filter. The three methods are tested in the final part. Determining The Planetary Timing A precise determination of the orbital period is dependent on a precise identification of the individual central transit times. The central transit time can be ascertained from the timing of ingress and egress using tpcttq j ř N i tpingressq i ` tpegressq i. (3.4) N Where N is the number of points used near ingress and egress of transit j to determine the central transit time tpcttq j. Real data is affected by different noise sources such as measurement noise, photon noise and stellar variability (Koch et al., 200). To make a simple simulation of these noise sources, randomly distributed data and a 5 min simple harmonic oscillator is added to the model light curve. The result is shown in bottom Figure 3.3. The noise should be reduced during the analysis to accurately determine the timing of ingress and egress. In the following subsections three different filtering methods will be described and applied to the simulated time series to test the efficiency in reconstructing the transit signature. Locally Weighted Scatterplot Smoothing One way to filter the transit signatures involves the use of local regression. The LOcally WEighted Scatterplot Smoothing (LOWESS) is of particular interest because of its simplicity. A weighted linear least square fit is applied over a given local span. This results in a noise reduced time series that potentially retains the local transit signatures. The regression weights for each data point is given by (MathWorks, 200, smooth) w i t t i ˇ dptq ˇ 3 3. (3.5) Where t is the temporal predictor value, t i are the nearest neighbors of t and dptq is the distance from t to the most distant predictor value within the span. The free parameter is the span which defines the number of data points used in the local regression. An extreme case is when the span is equal to one. Here the lines pass through all the data points and is in essence a linear interpolation. At the other extreme, when the span is comparable to the number of elements in the time series, the fit will not retain the local transit signatures. An example of locally weighted scatterplot smoothing is shown in Figure 3.4. The black lines illustrate the extreme cases. The magenta line shows a case where the noise has been reduced and the transit signature is retained.

28 22 CHAPTER 3. KEPLER DATA ANALYSIS Flux [Arb. Units] Time [Days] Time [Days] Figure 3.4: Blue: raw model based on Mandel and Agol (2002). The model parameters are given in Table 3.. Green: model with Gaussian noise and a 5 min simple harmonic oscillator added. Black: full and partially drawn lines are locally weighted scatterplot smoothing fits with span and span 44050, respectively. Magneta: locally weighted scatterplot smoothing fit with span equal to ten. Flux [Arb. Units] Time [Days] Time [Days] Figure 3.5: Blue: raw model based on Mandel and Agol (2002). The model parameters are given in Table 3.. Green: model with Gaussian noise and a 5 min simple harmonic oscillator added. Black: full and partially drawn lines are cubic smoothing spline fits with the smooth factor p equal to one and zero, respectively. Magneta: cubic smoothing spline fit with p Smoothing Spline The transit signatures can also be filtered using a cubic smoothing spline. The smoothing spline minimizes (MathWorks, 200, csaps) p Nÿ i ż w i py i fpt i qq 2 ` p pq pf 2 ptqq 2 dt. (3.6)

29 3.2. PLANETARY TIMING 23 Where p is the smooth factor, w i weights, y i data points and f is the cubic smoothing spline. The weights are estimated using w i. (3.7) σi 2 Where σ i is the standard deviation of each data point. The free parameter is the smooth factor which specifies the relative weight between having a smooth spline versus having a spline close to the data. An extreme case is when the smooth factor is zero. In this case, f is a linear least square fit to the data. At the other extreme when p the resulting fit is a natural cubic spline that pass through all the data points. An example of cubic spline smoothing is shown in Figure 3.5. The black lines illustrate the extreme cases. The magenta line shows a case where the noise has been reduced and the transit signature is retained. Spectral Filter The transit signatures can also be filtered using information from the least squares spectrum. The least squares spectrum minimizes Rpν j q Nÿ py i rα sin pν j t i q ` β cos pν j t i qsq 2. (3.8) i Where ν j is the angular frequency. The minimum is found when BR Bα BR Bβ The solution is (Karoff, 2008; Frandsen et al., 995, Ch 2.4.2) 0. (3.9) s cc c sc αpν j q ss cc sc 2 (3.0) c ss s sc βpν j q ss cc sc 2 (3.) Nÿ s y i sin pν j t i q (3.2) c ss cc sc i Nÿ y i cos pν j t i q (3.3) i Nÿ sin pν j t i q 2 (3.4) i Nÿ cos pν j t i q 2 (3.5) i Nÿ sin pν j t i q cos pν j t i q. (3.6) i Finally the least squares amplitude spectrum is calculated by b Apν j q αpν j q 2 ` βpν j q 2. (3.7)

30 24 CHAPTER 3. KEPLER DATA ANALYSIS Fundamental LC: f Nyquist Peak SHM 20 Amplitude [ppm] Frequency [Days ] Frequency [Days ] Figure 3.6: Blue: least squares spectrum of SC model time series with Gaussian noise and a 5 min Simple Harmonic Oscillator (SHM). The model parameters are given in Table 3.. Black: least squares spectrum of same model time series which has been sinusoidally modulated by 20 sec. A single peak marked in the spectrum to the right is shown to the left. Red: fundamental and overtones from the transiting planet. The first 333 tones are shown for clarity. Green: Nyquist limit for the LC model. The least squares amplitude spectrum for the SC model time series is illustrated in Figure 3.6. The Nyquist frequency for the LC model time series is also shown. The Nyquist frequency marks the upper limit for which one can measure oscillations for an equidistant sampled time series and it is given by (Kurtz, 983) f Nyquist 2 T. (3.8) Where T is the time between two neighboring data points. For this thesis the Nyquist frequency is used as an upper limit although the data points are not strictly equidistant in time due to gaps, as described previously. The fundamental and overtones from the periodic transiting event and the 5 min simple harmonic oscillator is also present in the spectrum. At higher frequencies the spectrum is dominated by white noise. The frequencies associated with the transiting events can be isolated using the iterative CLEAN algorithm (Karoff, 2008; Frandsen et al., 995, Ch 2.4.5). The CLEAN algorithm follows three steps. First the frequency with the highest peak is ascertained from the spectrum. Next the wave associated with this peak is removed from the time series using y i y i0 αpν 0 q sin pν 0 t i q βpν 0 q cos pν 0 t i q. (3.9) Finally a new spectrum is calculated based on the time series that no longer contains the wave associated with the peak. The process is continued iteratively until all peaks above 4 S/N has been detected and the associated waves has been removed. The CLEAN algorithm is used in local intervals around the frequencies associated with the transiting events. This leads to a time series which no longer contains the

31 3.2. PLANETARY TIMING 25 transit signatures. Moreover the α and β coefficients relevant for the CLEAN process are used to construct a time series which only contains the planetary signal. The time series is constructed by y i nÿ αpν j q sin pν j t i q ` βpν j q cos pν j t i q. (3.20) j Where n is the number of frequencies from the local CLEAN procedure. The relevant free parameters are the number of overtones used to reconstruct the transit signatures, the frequency sampling and the local interval width. The interval width may be defined in respect to the width of a peak which can be estimated from (Christensen-Dalsgaard, 2003, Ch. 2.2) δν» 2π T. (3.2) Where δν is the estimated peak width in units of angular frequency and T is the observational time. The frequency sampling is defined in units of the angular frequency resolution ν 2π T. (3.22) Furthermore the isolated overtones should be below the upper limit marked by the Nyquist frequency. The frequency sampling, interval width and number of overtones used to reconstruct the transit signatures is explored in Figures Flux [Arb. Units] Time [Days] Raw Model Model + Noise δν 2δν 4δν 6δν Time [Days] Figure 3.7: Blue: raw model based on Mandel and Agol (2002). The model parameters are given in Table 3.. Green: model with Gaussian noise and a 5 min simple harmonic oscillator added. Reconstructed transit signature based on 75 overtones, a frequency sampling of 00 ν in local intervals of δν, 2δν, 4δν and 6δν, respectively. Notice the overlapping lines.

32 26 CHAPTER 3. KEPLER DATA ANALYSIS Flux [Arb. Units] Raw Model Model + Noise 75 Tones 50 Tones 300 Tones 600 Tones Time [Days] Time [Days] Figure 3.8: Blue: raw model based on Mandel and Agol (2002). The model parameters are given in Table 3.. Green: model with Gaussian noise and a 5 min simple harmonic oscillator added. Reconstructed transit signature based on a frequency sampling of 00 ν in local intervals of δν using 75, 50, 300 and 600 overtones, respectively Flux [Arb. Units] Raw Model Model + Noise ν /50 ν /00 ν /200 ν Time [Days] Time [Days] Figure 3.9: Blue: raw model based on Mandel and Agol (2002). The model parameters are given in Table 3.. Green: model with Gaussian noise and a 5 min simple harmonic oscillator added. Reconstructed transit signature based on 600 overtones in local intervals of δν and a frequency sampling of ν, 50 ν, 00 ν and 200 ν, respectively.

33 3.2. PLANETARY TIMING 27 Testing the different methods The locally weighted scatterplot smoothing, smoothing spline and spectral filter is applied to the model light curve to reconstruct the transit signatures, calculate the central transit times and test the stability in determining the orbital period. First the central transit times are determined using Equation 3.4. The number of points used to ascertain the central transit time has been varied in two ways. First using two points at 50% above the minimum light curve level near the reconstructed ingress and egress, respectively. Second using multiple points between 25% and 75% above the minimum light curve level near the reconstructed ingress and egress, respectively. The levels are comparable to the illustration in Figure 3.3. Next the transit times are fitted against the transit number using Equation 2.5. This gives the orbital period for the particular case. The procedure is repeated for the method of locally weighted scatterplot smoothing and smoothing spline using an incrementing span or smooth parameter, respectively. The procedure is stopped when the transit signature is no longer detectable due to a large span or when the smooth parameter reaches a value of one. The free parameters for the spectral filter is set to the maximum number of detectable overtones above 4 S/N and a frequency sampling of ν in local intervals of 5δν. The spectral filtered data is 00 subsequently smoothed using a simple moving average with a given span. The span acts as a free parameter and the procedure described previously is again applied repeatedly using an incrementing span until the transit signature is no longer detectable. In each case the central transit times, period and the associated standard error is saved. The best fit periods are outlined in Table 3.2. Table 3.2: Orbital period determined for model transit light curve with added Gaussian noise and a 5 min simple harmonic oscillator. The model parameters are outlined in Table 3.. In/Eg illustrates the level at which points from the ingress and egress were used to determine the central transit times, respectively. The levels are comparable to the illustration in Figure 3.3. Long Cadence (LC) results are based on the LC model. Short Cadence (SC) results are based on the SC model. The orbital periods are in units of days. In/Eg LC LOWESS LC Smoothing Spline LC Spectral Filter 50% Ñ 75% In/Eg SC LOWESS SC Smoothing Spline SC Spectral Filter 50% Ñ 75% It is also of interest to test the orbital period determination in the case of transit timing variations. Thus a 20 s modulated sinusoidal signal has been applied to the model time series to create a periodic difference in the timing of the planetary transits. The resulting amplitude spectrum is shown in right Figure 3.6. The left figure illustrates a single peak which is compared to the original case with no modulation. The difference between the observed and calculated central transit times is shown in Figure 3.0. In this case, the spectral filter is tested in two cases. First using 350 overtones. Second

34 28 CHAPTER 3. KEPLER DATA ANALYSIS using 75 overtones. In both cases a frequency sampling of ν in local intervals of 5δν 00 is utilized. The respective central transit times are based on the best fit periods for each case. The best fit periods are outlined in Table Observed Calculated [min] Transit Number Figure 3.0: Difference between observed and calculated central transit times for SC model transit light curve with added Gaussian noise and a 5 min simple harmonic oscillator. The model parameters are outlined in Table 3.. Blue: based on locally weighted scatterplot smoothing. Green: based on smoothing spline. Red: based on spectral filter with 350 overtones. Magenta: based on spectral filter with 75 overtones. Black solid line: reference based on raw model light curve without noise. Black horizontal dashed line: zero point reference. Table 3.3: Orbital period determined for SC model transit light curve with added Gaussian noise and a 5 min simple harmonic oscillator. The model parameters are outlined in Table 3.. A 20 s modulated sinusoidal signal has been applied to the model time series to create a periodic difference in the timing of the planetary transits. Multiple points between 25% and 75% above the minimum light curve level near the reconstructed ingress and egress were used to determine the central transit time. The levels are comparable to the illustration in Figure 3.3. Method Period [days] LOWESS Smoothing Spline Spectral Filter (350 Tones) Spectral Filter (75 Tones)

35 3.2. PLANETARY TIMING 29 TrES-2b, HAT-P-7b, Kepler-7b and Kepler-9b This subsection describes the determination of the planetary timing and orbital periods for TrES-2b, HAT-P-7b, Kepler-7b and Kepler-9b. This includes the application of the locally weighted scatterplot smoothing, smoothing spline and spectral filter on the raw long and short cadence Kepler data. Long Cadence Orbital Period First the transit signature is detected using a boxcar filter on the raw long cadence data. The boxcar filter is calculated by Dpt i q Cpt i q Wpt i q ` W 2 pt i q. (3.23) 2 Where Cpt i q, W pt i q and W 2 pt i q are mean values in the boxes shown in Figure 3.. The box width for Cpt i q is B while for W pt i q and W 2 pt i q the box width is B 2 2 B. The free parameter B is estimated by calculating the total transit time for a close-in planet, orbiting a Sun-like star. The total transit time for a planet orbiting in a circular orbit with an inclination angle of ninety degrees is (Perryman, 20, Ch 6.4.) ˆ {2 M a t tot» 3 AU {2 ˆ R hours. (3.24) Where M is the stellar mass relative to the solar mass R is the stellar radius relative to the solar radius and a is the semi-major axis in astronomical units. In this case, M and R Moreover for a hot Jupiter a ď 0. AU while for a very hot Jupiter a ď AU (Perryman, 20, Ch. 6). Thus an intermediate value of a 0.05 AU is used to estimate the timescale of the transit duration. This leads to a boxcar width of B t tot» 3 hours. The boxcar filtered data is placed on a grid with equidistant temporal points using nearest neighbor interpolation. The.002 W (t ) i C(t ) i W 2 (t ) i Flux t i :00 2:00 8:00 00:00 06:00 2:00 Time [Hours] Figure 3.: Illustration of boxcar filter. In this case, B 4 hours.

36 30 CHAPTER 3. KEPLER DATA ANALYSIS First Peak Correlation Coefficient Harmonics Relative Lag x 0 4 Figure 3.2: Autocorrelation for truncated boxcar filtered data of HAT-P-7b. Blue: correlation coefficients as a function of relative lag. Red: peaks from the periodic transiting events. temporal sampling is set to 29.4 minutes. Missing data and data with values above zero are replaced by zero values. This results in a time series with a comb-like structure. The time series is subsequently processed by the autocorrelation function to search for repeated signatures. The autocorrelation function is calculated by (Scargle, 98, Ch 3) ř N k i rpkq px i x ÑN k q px i`k x `kñn q (3.25) N k Where x j is the j th component in the dataset, x sñt is the mean value in the respective interval, N the total number of data points and k is the time lag. Normalization of the correlation coefficients is performed using rpkq{rpk 0q which implies that r : r 0 : r : x i and x i`k are strongly correlated. x i and x i`k are not correlated. x i and x i`k are strongly anti-correlated. The relative lag and magnitude of the first relevant strongly correlated peak is ascertained and used to detect the upper harmonics. An example of this is shown for HAT-P-7b in Figure 3.2. The respective peaks are used to estimate the orbital period using ř P AC T ř i k iw i. (3.26) Where T is the temporal sampling, k i the relative lag and w i are weights equal to the ith correlation coefficients. The long cadence data for HAT-P-7b, TrES-2b and Kepler-7b is subsequently normalized using a median filter with a span equivalent to P AC. For Kepler-9b the data is normalized using a median filter with a span equivalent to 0.5 days. This effectively removes the long-term trend exemplified in Figure 3.2. Next the locally weighted scatterplot smoothing, smoothing spline and spectral filtered data is phase folded with the i iw i

37 3.2. PLANETARY TIMING 3 Normalized Flux % 50% 25% 7:00 8:00 9:00 20:00 2:00 22:00 23:00 00:00 7:00 8:00 9:00 20:00 2:00 22:00 23:00 00:00 Time [Hours] d/dt(normalized Flux) Figure 3.3: Single transit of HAT-P-7b. Blue: normalized long cadence data. Solid red line: locally weighted scatterplot smoothing with a span equal to one. Horizontal dotted red lines: level above global minimum light curve level. Vertical dotted red lines: ingress and egress cutoffs. Solid black lines: data points near the ingress and egress, used to ascertain the central transit time. Horizontal dotted black line: global minimum light curve level. Vertical dotted black line: central transit time. Top green: cubic spline of reference data. Bottom green: absolute differentiated cubic spline of reference data. Magenta: approximate location of the ingress and egress, ascertained from the maximum slopes on the cubic spline fit. period P AC using Equation Individual phase observations are isolated and the approximate location of the local ingress and egress is ascertained from the maximum slopes of a cubic spline on a set of reference data. In all cases the spectral filtered data is used as reference data. Multiple points between 25% and 75% above the minimum light curve level near the reconstructed ingress and egress are used to determine the central transit time using Equation 3.4. The procedure is exemplified for HAT-P-7b in Figure 3.3. Next the transit times are fitted against the transit number using Equation 2.5. The procedure is repeated for the method of locally weighted scatterplot smoothing and smoothing spline using an incrementing span or smooth parameter, respectively. The procedure is stopped when the transit signature is no longer detectable due to a large span or when the smooth parameter reaches a value of one. The free parameters for the spectral filter is set to the maximum number of detectable overtones above 4 S/N and a frequency sampling of ν in local intervals of 5δν. The spectral filtered data is 00 subsequently smoothed using a simple moving average with a given span. The span acts as a free parameter and the procedure described previously is again applied repeatedly using an incrementing span until the transit signature is no longer detectable. In each case the central transit times, period and the associated standard error is saved. The best fit periods are outlined in Table 3.4.

38 32 CHAPTER 3. KEPLER DATA ANALYSIS Short Cadence Orbital Period In the case of HAT-P-7b, TrES-2b and Kepler-7b the raw short cadence data is first normalized using a median filter with a span equivalent to the best determined orbital period from the long cadence data. For Kepler-9b the data is normalized using a median filter with a span equivalent to 0.5 days. This effectively removes the longterm trend exemplified in Figure 3.2. Next the locally weighted scatterplot smoothing, smoothing spline and spectral filter is applied to the normalized data. The filtered data is again used to determine the planetary timing and orbital period of the respective exoplanets. This procedure is similar to the description from the long cadence orbital period determination. The best fit periods are outlined in Table 3.4. Table 3.4: Long and short cadence orbital period determined using locally weighted scatterplot smoothing, smoothing spline and spectral filter. Points between 25 75% on the ingress and egress were used to determine the individual central transit times. This level is comparable to the illustration in Figure 3.3. Long Cadence (LC) and Short Cadence (SC) results are based on Q0 - Q0 for HAT-P-7b and TrES-2b. For Kepler-7b, LC is from Q0 - Q8 and SC is from Q3 - Q8. For Kepler-9b, LC is from Q0 - Q and SC is from Q3 - Q. The orbital period is in units of days. Exoplanet LC LOWESS LC Smoothing Spline LC Spectral Filter HAT-P-7b TrES-2b Kepler-7b Kepler-9b Exoplanet SC LOWESS SC Smoothing Spline SC Spectral Filter HAT-P-7b TrES-2b Kepler-7b Kepler-9b Transit Timing Variations The linear ephemerides is also used to search for variations in the planetary timing. The difference between the observed and calculated central transit times for long cadence data is exemplified for HAT-P-7b in Figure 3.4. In this case, the transit times are based on locally weighted scatterplot smoothing with three different spans of sizes one, three and four. A single sine function has been fitted to the results based on the lowest span equal to one. Transit timing variations has also been studied for Kepler-9b. The difference between the observed and calculated central transit times for long and short cadence data of Kepler-9b is shown in Figure 3.5. In this case, the transit times are associated with the best fit periods from Table 3.4. A single sine function has been fitted to the results based on the locally weighted scatterplot smoothing. The sine fit parameters are included in the appendix together with planetary timing figures for TrES-2b, HAT-P-7b and Kepler-7b.

39 3.2. PLANETARY TIMING 33 Observed Calculated [min] Transit Number Figure 3.4: Difference between observed and calculated central transit times for long cadence data of HAT-P-7b. The calculated transit times are based on locally weighted scatterplot smoothing. Blue, green and red: illustrates cases where the span is equal to one, three and four, respectively. Magenta: fit based on a sine function. Only the first fifty instances are shown for clarity. 5 0 Observed Calculated [min] Transit Number Figure 3.5: Difference between observed and calculated transit times for long and short cadence data of Kepler-9b. Black and blue: based on long cadence data utilizing the method of locally weighted scatterplot smoothing and smoothing spline, respectively. Red: based on short cadence data utilizing locally weighted scatterplot smoothing. Green and cyan: based on long and short cadence data utilizing the spectral filter. Magenta: fit based on a sine function.

40 34 CHAPTER 3. KEPLER DATA ANALYSIS 3.3 Reflected and Emitted Light This part deals with the analysis of the reflected and emitted light from hot Jupiters. The first subsection introduces the principle behind phase folding the time series data. The second subsection is on the planet to star flux ratio for TrES-2b, HAT-P-7b and Kepler-7b. Phase Folded Data It is of interest to study the phase folded data as this can give information about the reflected and emitted light from an exoplanet atmosphere or surface. The data is phase folded using t p,i t i modp ` t p,0. (3.27) P Where t i is the ith time index, t p,0 is a constant that shifts the primary transit to zero phase and P is the orbital period. The long and short cadence data for TrES-2b, HAT- P-7b and Kepler-7b is phase folded using the respective best fit periods from Table 3.4. Next 4σ outliers are removed using the Hampel filter, which is described in the next chapter. The phase folded data is exemplified for HAT-P-7b in Figure 3.6. The data is utilized in the next subsection to determine the occultation depth for the respective exoplanets. Flux Ratio It is of interest to study the planetary occultation as this can give information about the planet to star flux ratio. First the occultation is isolated from the phase folded data. A weighted linear least squares fit is applied to data just outside the occultation and HAT P 7b Normalized Flux Phase Normalized Flux Figure 3.6: Black: binned data in 5 minute intervals. Red: running median with a span equivalent to 30 minutes. The primary and secondary transit occurs near zero and half phase, respectively.

41 3.3. REFLECTED AND EMITTED LIGHT 35 the resulting fit is used to renormalize the data near the occultation. Next the weighted mean and median is calculated within the renormalized full occultation. The results are used to determine the occultation depth by (Jackson et al., 202) δ occ F occ. (3.28) In this case, F occ is the renormalized flux near the full occultation. The procedure is exemplified for HAT-P-7b in Figure 3.7. The final results are outlined in Table 3.5 where upper limits for the geometric albedos are calculated using the respective weighted mean together with Equation 2.7 and the planet radius and semi-major axis from Table 2.2. The nightside contribution is calculated using the difference between the wings just outside the primary transit and the occultation depth. The nightside contribution has been subtracted in the calculation of the geometric albedo for HAT-P-7b. The lower limit on the thermal occultation depth of δ occ.2ppm has been subtracted in the calculation of the geometric albedo for TrES-2b. This value follows from Chapter 2 in the subsection on flux ratios. Table 3.5: Median and weighted mean occultation (occ) depth and nightside (ns) contribution for TrES-2b, HAT-P-7b and Kepler-7b. TrES-2b and HAT-P-7b: calculated using short cadence data from Q0 - Q0. Kepler-7b: calculated using long cadence data from Q0 - Q2 combined with short cadence data from Q3 - Q8. The upper limits for the geometric albedos are calculated using the respective weighted means together with Equation 2.7 and the planet radius and semi-major axis from Table 2.2. The nightside contribution has been subtracted in the calculation for HAT-P-7b. The lower limit on the thermal occultation depth of δ occ.2ppm has been subtracted in the calculation of the geometric albedo for TrES-2b. This value follows from Chapter 2 in the subsection on flux ratios. Exoplanet median(δ occ ) δ occ median(δ ns ) δ ns A g TrES-2b.7 ppm ppm À 0.04 HAT-P-7b 7.29 ppm ppm 2.5 ppm ppm À 0.8 Kepler-7b 43.9 ppm ppm À 0.33 The occultation can also be used to study for variations in the planet to star flux ratio. In this case, the full time series is first divided into segments of N days and the segments are phase folded using Equation Next renormalization is performed on each segment following the procedure described previously. The weighted mean and median is again calculated within the respective renormalized full occultation. The results are again used to determine the occultation depth using Equation The procedure has been performed for long and short cadence data of HAT-P-7b with N «90 days and the result is shown in Figure 3.8. The procedure has also been performed for long and short cadence data of TrES-2b with N «293 days. The result is shown in Figure A.4.

42 36 CHAPTER 3. KEPLER DATA ANALYSIS.000 HAT P 7b Normalized Flux Renormalized Flux Phase Figure 3.7: HAT-P-7b occultation. Short cadence data from Q0 - Q0. Black: binned data in 5 minute intervals. Red: running median with a span equivalent to 30 minutes. Blue vertical lines: cutoff to data outside occultation. Green vertical lines: cutoff to data inside full occultation. Magenta: linear least squares fit to data outside occultation. 85 Occultation Depth HAT P 7b 80 Occultation Depth [ppm] Time [Years] Figure 3.8: Occultation depth for HAT-P-7b. Red and blue: based on weighted mean near the renormalized full occultation of long and short cadence data, respectively. Magenta and green: based on median near the renormalized full occultation of long and short cadence data, respectively. The individual data points are calculated using approximately ninety days of phase folded data. The solid horizontal lines are based on the weighted mean and median near the renormalized full occultation using data from Q0 - Q0. Here the dotted lines are one sigma weighted standard deviations of the weighted means.

43 Chapter 4 Spitzer Data Reduction and Analysis This chapter describes the steps involved in the Spitzer data reduction and analysis. The chapter is divided into three parts. The first part introduces the Spitzer data and known problems with the datasets. The second part gives an overview of the steps involved in the data reduction. The final part deals with the emitted light from hot Jupiters. 4. Spitzer Data The Spitzer Space Telescope was launched on the 25th of August It is an infrared observatory with three onboard science instruments. This includes an InfraRed Spectrograph (IRS), a Multiband Imaging Photometer for Spitzer (MIPS) and the InfraRed Array Camera (IRAC). The IRAC instrument is an imaging camera capable of making simultaneous 5.2 ˆ5.2 images in 3.6, 4.5, 5.8 and 8.0 µm (Kwok et al., 2004). Figure 4. shows the respective spectral response curves. For this project data from this instrument is used to determine the photometric variability during a planetary occultation. 0.7 Spectral Response µm 4.5µm 5.8µm 8.0µm Wavelenght [µm] Figure 4.: Spitzer CCD spectral response curves for the respective channels. As of this writing only the IRAC instrument remains in partial operation. 37

44 38 CHAPTER 4. SPITZER DATA REDUCTION AND ANALYSIS In general the Spitzer IRAC data is subject to an under-sampled Point Spread Function (PSF) (JPL, 20; Howell, 2000, Ch 2.2.2; Ch. 5.6). For the 5.8 and 8.0 µm channels there is a time-dependent flux sensitivity which is caused by an increase in the effective pixel gain over time. The effect can be attenuated using the pre-flash technique, in which a bright diffuse region is observed prior to the target observations. This leads to an increase in the effective gain of the pixels whereby the time-dependent flux sensitivity is attenuated. The remaining variation can be corrected by fitting to (Knutson et al., 2009; Christiansen et al., 200) F ptq F 0 pc ` c 2 lnpdt ` 0.02qq. (4.) Where the free parameters are c c 2, F 0 is the original stellar flux, F ptq is the measured flux and dt is the elapsed time in days since the start of the observations. The Spitzer data is available from the Spitzer Heritage Archive (JPL, 20). The data contains several different Flexible Image Transport System (FITS) files. The Basic Calibrated Data (BCD) contains the flux calibrated individual images which have been corrected for known instrumental signatures including bias and flat-field (JPL, 20, Ch. 5.). The Basic UNCertainty (BUNC) data contains conservative error estimates on the individual pixels in each image and the Basic Image MaSK (BIMSK) contains information on erroneous pixels (JPL, 20, Ch. 6). 4.2 Data Reduction This part deals with the steps involved in the data reduction. The first subsection gives an overview of a program designed to perform high-precision time series photometry using the available Spitzer data. This is followed by a description of the relevant photometry algorithms. Finally the algorithms are tested against simulated data and data from HAT-P-7. Photometry Program The photometry program is developed in MathWorks MATLAB R200a. The program is designed to perform high-precision time series photometry using the available information from the individual Spitzer frames. It is also flexible enough to allow for photometry on other scientific images. In addition it utilizes a Graphical User Interface (GUI) which allows the user to inspect the intermediate and final results of the data reduction. The principle parts of the program are shown in Figure 4.2. First the relevant Flexible Image Transport System (FITS) files must be loaded into memory (a). Next the user can inspect the BCD files for unwanted artifacts which may be present in the individual images (b). If the individual images are of low quality, a segment of images can be combined to create a median, mean and if the individual BUNC files are available, a weighted mean image (c). After the optional combination of images the user must select a reference image (d). At this point the program automatically detects stellar candidates and the user have the option to perform simultaneous time-series photometry on all stellar candidates or select the individual star(s) using the computer mouse (e). After this intermediate step the user should click "Perform Photometry" whereby the program automatically performs the necessary data reduction (f). Following the data

45 4.2. DATA REDUCTION 39 a Load FITS Files b Investigate Images c Combine Images d Choose Reference e Select Star(s) f Perform Photometry g Detect Outliers h Investigate Results (a) Block Diagram. (b) Basic Astrophotometry Program. Figure 4.2: Principle parts of the photometry program. In (a) the open and closed arrows indicate optional and required steps, respectively. The red labels corresponds to labels in (b) which shows the relevant buttons, panels and axes in the GUI. reduction, the resulting time-series is shown. A built-in filter can be activated to detect possible outliers (g). Finally the user can investigate the resulting time series and save the result as an ASCII text file on the hard drive (h). Further details on the image combination-, stellar position-, photometry- and outlier algorithms are introduced in the following subsection. Program Algorithms This subsection introduces the algorithms used in the time series photometry program. This includes an algorithm for combining FITS images, determining stellar positions, calculating the local sky background, stellar flux, signal to noise calculation and finally an algorithm for detecting outliers in a time series. Combining FITS Images It is of interest to determine the relative shift between the images. The relative shift is determined using the method of phase correlation. First the normalized cross-power spectrum is calculated using (Druckmüller, 2009) R F F 2 F F 2. (4.2) Where F and F 2 are the two-dimensional fast Fourier transforms of the two images to be compared and F 2 is the complex conjugate of the respective Fourier transform. Next the cross-correlation is found using the inverse Fourier transform rpx, yq F prq. (4.3)