Transit timing analysis of the hot Jupiters WASP-43b and WASP-46b and the super Earth GJ1214b

Transcription

1 Transit timing analysis of the hot Jupiters WASP-43b and WASP-46b and the super Earth GJ1214b Mathias Polfliet Promotors: Michaël Gillon, Maarten Baes 1

2 Abstract Transit timing analysis is proving to be a promising method to detect new planetary partners in systems which already have known transiting planets, particularly in the orbital resonances of the system. In these resonances we might be able to detect Earth-mass objects well below the current detection and even theoretical (due to stellar variability) thresholds of the radial velocity method. We present four new transits for WASP-46b, four new transits for WASP-43b and eight new transits for GJ1214b observed with the robotic telescope TRAPPIST located at ESO La Silla Observatory, Chile. Modelling the data was done using several Markov Chain Monte Carlo (MCMC) simulations of the new transits with old data and a collection of transit timings for GJ1214b from published papers. For the hot Jupiters this lead to a general increase in accuracy for the physical parameters of the system (for the mass and period we found: 2.034±0.052 M Jup and ± days and 2.03±0.13 M Jup and ± days for WASP-43b and WASP-46b respectively). For GJ1214b this was not the case given the limited photometric precision of TRAPPIST. The additional timings however allowed us to constrain the period to ± days and the RMS of the TTVs to 16 seconds. We investigated given systems for Transit Timing Variations (TTVs) and variations in the other transit parameters and found no significant (3σ ) deviations. Based on the RMS of the TTVs we designed a tool using the MERCURY package. In doing so we were able to exclude super-earth massed planets in the resonances for the hot Jupiters, WASP-43b and WASP-46b and down to a tenth of an Earth mass for GJ1214b. 2

3 If I have seen further it is by standing on the shoulders of giants 3

4 Acknowledgments I would like to thank the university of Ghent and Maarten Baes to make this thesis possible and for the provided education over the past few years. My gratitude goes out to the university of Liège and the people working there for their hospitality. I wish to thank Sandrine Sohy for her help regarding all sorts of computational issues and Brice-Olivier Demory for the helpful discussions about TTVs and the development of my program. Most of all I would like to thank Michaël Gillon for his continuous support and help during the course of this thesis and his readiness to aid me when I required it. 4

5 Contents 1 Other Worlds Radial velocity Transiting exoplanets Exoplanets: The global picture Transit Timing Variations Introduction Inner planet Non-resonant outer planet Resonant planet Analytic approach Exomoon Other TTV signals Observations and data reduction Data description Data analysis TTV simulator Results and discussion WASP-43b WASP-46b GJ1214b Conclusion 54 5

6 Introduction In this work, we will use our extensive data sets to determine precisely the parameters of the transiting systems WASP-43, WASP-46 and GJ1214. We will compare our results to the ones presented in these system s discovery papers Hellier et al. [2011], Anderson et al. [2012] and Charbonneau et al. [2009]. Additionally, we will investigate the given exoplanets further to reveal possible planetary partners using the method of Transit Timing Variations (TTVs). TTVs are a promising method to detect planets down to several Earth masses for systems which already have known transiting planets, particularly in the orbital resonances of the system. We look for deviations in the transit timings that would be caused by an additional body orbiting the star. To achieve this we have created a program using the MERCURY package. The program is designed to exclude possible partners based on the RMS of the signal we find using the Markov Chain Monte Carlo simulations. Observations are made primarily by the TRAPPIST telescope. We present four new transits for WASP-46b, four new transits for WASP-43b and eight new transits for GJ1214b. in chapter 1, I will give an overview of the exoplanet findings and of the complementarity of the so-called transit and RV methods for studying exoplanets in details. In chapter 2 I describe the TTV method. The method we use to reduce, analyze and interpret the data are presented in chapter 3. Eventually we will present our results in chapter 4 and our conclusion in chapter 5. 1 Other Worlds To date more than 750 exoplanets in little over 600 planetary systems are know and a fraction of them have been characterized using numerous techniques [Schneider et al., 2011]. This new branch of astronomy has literally boomed since the first discoveries in the nineties, as can be seen in Figure 1. Figure 2 shows that in recent years, we have begun to reach the precision to detect earth-like planets. The most successful detection techniques are the radial velocity and transit methods, but other techniques have also demonstrated their efficiency: microlensing, pulsar timing, direct imaging... Going through all of these techniques is out of the scope of this thesis, and we will only discuss below the two most relevant techniques for our work, i.e. radial velocity and transits. Afterwards we will discuss some of the most important results in the field of exoplanetology. 1.1 Radial velocity Introduction It was already thought in the 1950s that the reflex stellar velocity for an edge-on orbit could be around 2 km s 1 for a planet ten times the mass of Jupiter [Struve, 1952]. The field of exoplanetary science had to wait, however, until 1989 for the first claimed discovery of an exoplanet [Latham et al., 1989]. The planet with minimal mass of 11 M Jup and a period of 84 days had a velocity amplitude of 600 m s 1. With a precision of 400 m s 1 hundreds of measurements were required to achieve a decent Signal to Noise Ratio (SNR). The object was named HD b after the star it orbits and has been considered to be a brown dwarf since the uncertainty in the inclination of the system (see below). Four years later the spectrograph ELODIE was used with a precision of 13 m s 1 to detect a Jupiter-like planet around 51 Pegasi with a reflex motion of 59± 3 m s 1 and a period of 4.23 days (see figure 3 taken from Mayor and Queloz [1995]). Among the scientific community there remained some skepticism regarding the nature of the source of this signal for two reasons. First of all there wasn t a single 6

7 Figure 1: Histogram displaying the number of peer-reviewed exoplanetary discoveries per year Figure 2: Plot displaying the (line of sight) mass of exoplanetary discoveries per year 7

8 planetary formation theory that predicted the existence of such objects so close to their host star. Gas giants can t form so close to the star since there is not enough mass in the inner part of the protoplanetary disk, let alone enough hydrogen. And thus planetary physics underwent a revolution by introducing the concept of inward migration caused by gravitational interaction between the protoplanet and the surrounding gaseous disk (Goldreich and Tremaine [1979], Ward [1997], Tanaka et al. [2002], Tanaka and Ward [2004]). It is worth mentioning that the notion of inward migration had already been proposed in 1979 by Goldreich and Tremaine but remained unnoticed by the planetary community until the discovery of 51 Peg b. Secondly since we only measure the radial velocity, one cannot determine the true mass as will be shown later. All of this lead to the suggestion that the signal could be caused by a non-resolved binary or a object in the gray zone between star and planet, the so-called brown dwarfs. These objects are not massive enough to start Hydrogen- 1 fusion (the object would need a mass of Jupiter masses), but would only support deuterium fusion (13 Jupiter masses) in their cores. The eventual confirmation of the planet-like nature of the objects came when the eleventh gas giant that was discovered using radial velocity measurements, HD b, also yielded a transit feature detected by ground-based photometry at the exact time predicted by the radial velocity data (see figure 4 taken from Charbonneau et al. [2000]). The discovery removed all skepticism and confirmed that the wobble observed by the radial velocity was indeed caused by a planet. Figure 3: Original phased radial velocity diagram for 51 peg b Observing a transit feature meant that one could measure the inclination of the system and determine the true mass of the planetary companion instead of the minimal mass. Since then many more planets have been discovered and radial velocity remains the technique with the most discoveries after its name. The state-of-the-art instrument in this field is the HARPS spectrograph [Mayor et al., 2003] that can achieve radial velocity precisions of a few dozens of cm/s 1 on bright and quiet stars. Among its achievements is the detection of Gl581e, which is one of the lightest exoplanets known at the moment with a mass of 1.7 ± 0.2 Earth masses[mayor et al., 2009]. The Doppler effect The effect that is used to measure the velocity changes is the well-known Doppler effect. The effect describes a change in frequency and wavelength of a wave for an observer moving relative to the source of the wave. The Doppler effect has many application not only within radial velocity and astronomy but also in the fields of radar, medical imaging, satellite communications, etc. Most of the time however in astronomy we will use the general-relativistic Doppler effect which considers the influence of the Doppler effect on electromagnetic waves in the framework of general relativity. The wavelength that will be detected by an observer,λ, moving from a source with a relative velocity, v, and k the unit vector pointing to the source in a gravitational potential Φ (neglecting terms of the order of c 4 ): 8

9 Figure 4: Original transit light curve for HD209458b λ = λ c k.v 1 Φ c² v² 2c² (1) After spectrographic observations have been made, we can use this formula to obtain the measured radial velocity of the star and it is more precise for slow rotating and cold stars (since the absorption lines are sharper). The semi-amplitude The technique of radial velocity focuses on the dynamical properties from stars. A planet orbiting around a star describes a Keplerian orbit around the center of mass (c.o.m) of the system where a is the semi-major axis for the relative motion and a pl for the motion around the c.o.m: a pl = m m pl + m a (2) But the same notion applies to the star as well, and thus the star orbits around the common center of mass. In polar coordinates the keplerian orbit described by the star becomes: r = a (1 e 2 ) 1 + e cos f = m pl m pl + m a(1 e 2 ) 1 + e cos f (3) With r the distance from the c.o.m for the star, e the eccentricity and f the true anomaly. After differentiating (2), we find: 9

10 r = m pl a(1 e 2 ) m pl + m f sin f (1 + e cos f ) 2 = er2 f sin f a (1 e 2 ) (4) If the star is light enough or the planet close or massive enough, this wobble of the star might be detected using the Doppler effect and a spectrometer. The semi-amplitude of this wobble, K, is derived in the following paragraph. For the position and velocity vector we have in a Cartesian coordinate system with the x-axis pointing towards the periastron and the origin at the center of mass: ṙ = r = ( ) r cos f r sin f ( ) ṙ cos f r f sin f ṙ sin f + r f cos f (5) (6) When we insert equation 4 into 6 we find that: ṙ = = r 2 f ( sin f ) a (1 e²) cos f + e h ( sin f ) m a (1 e²) cos f + e (7) (8) We also have for the relative orbital momentum of the star h : If we substitute 10 in 8: h = m m + m pl h (9) = Gm2 m 4 pla(1 e²) (m + m pl ) 3 (10) Gm ṙ = 2 pl (m + m pl )a(1 e²) ( ) sin f cos f + e (11) Our next task is to project these vectors onto the line of sight of the observer. This time with the z-axis perpendicular to the orbital plane. Here is the angle between the orbital plane and the plane of the sky (perpendicular to the orbital plane) is the inclination angle. We thus find for the unit vector of the line of sight, k: k = sin ω sin i cos ω sin i cos i (12) 10

11 Table 1: Radial velocity signals for several (hypothetical) planets and the planets studied in this work. Planet Mass a (AU) K 1 (m s 1 ) WASP-43b M Jup WASP-46b M Jup GJ1214b M Earth ±1.6 Jupiter 1M Jup Jupiter 1M Jup Jupiter 1M Jup Earth 1 M Earth Earth 1 M Earth Once we project them, we find the radial velocity equation: v r, = r.k = G (m + m pl )a(1 e²) m pl sin i(cos(ω + f ) + e cos ω) (13) From here it is trivial to see the semi-amplitude, K, as (v r,max -v r,min )/2 G K 1 = (m + m pl )a(1 e²) m pl sin i (14) As one can see from the equation 14, it is impossible to determine the true mass of the planet using radial velocity on its own since the inclination angle of the system is unknown. This derivation has been taken from [Lovis and Fischer, 2010] and typical RV amplitudes for exoplanets can be found in table 1. Stellar limitations Besides the instrumental challenges that come with the increasing requirements of radial velocity precision, there are also limitations to the precision caused by short and/or long term variations of the stars. These phenomena usually arise in the atmosphere of the star and are called stellar noise. P-mode oscillations named after their restoring force, pressure, are acoustic waves that find their origin in the turbulent nature of the outer convective zone of the stars. These oscillations have periods in the order of minutes and amplitudes of decimeters per second per mode in sun-like stars (Bouchy and Carrier [2001], Kjeldsen et al. [2005]). Given that several modes are observed at the same time, the observed RV signal is of the magnitude of several meters per second. The frequency of the oscillations scales with the square root of the mean stellar density and the RV amplitude with the luminosity over mass ratio. As a consequence low-mass non evolved stars are intrinsically better targets to observe since they have a lower amount of noise due to these p-mode oscillations [Kjeldsen et al., 2005]. Another solution is, since the signal is of such short nature, to average it out using a exposure time of 15 minutes or more. Granulation and supergranulation are effects with a similar amplitude that are caused by the large-scale convective movements of the outer convective layers of the star. In figure 5, one can see the granulae in the sun. In the middle of the granulae the rising and hotter plasma can be seen, whilst on the edges the cooler and descending plasma is present. This confirms our views that the effect is caused by convective 11

12 Figure 5: Granulae on the sun motion. On the sun the amplitude of these granulae is 1-2 km s 1 and since there are a large number of them they average out but a jitter of 1m/s 1 remains and proves to be the biggest challenge for ultrahigh-precision RV measurements (Palle et al. [1995],Dravins [1990]). The typical timescale of occurrence for these granulae is 10 minutes for the sun, but over longer timescales the so-called supergranulae play a role. Another problem that arises when trying to make radial velocity measurements of stars, is their magnetic fields (Saar and Donahue [1997],Santos et al. [2000],Wright [2005]). These magnetic fields are responsible for several phenomena that occur on the surface of the star. The most important one is stellar spots (after flares, coronal mass ejections, etc.). These are spots that cause the star to appear brighter or darker then it really is. They evolve in time and are carried across the star by stellar rotation causing spectral lines to change shape. To diagnose the problems with dark spots a quantity is introduced as the line bisector. This quantifies the level of asymmetry of the mean line of the spectrum. Another indicator for the activity of the star is the Ca II H&K chromospheric index. All of these can be used to make sure we differentiate between stellar noise and dynamical radial velocity signals. One can see that it is advised to select stars with a slow rotation velocity and thus an older age. In young stars the dark star spots can cause variations of the amplitude of m s 1, making it even hard to detect hot Jupiters. A possible solution to this problem is to observe in the infrared where the spots are a lot less prominent. Dedicated spectrographs in the IR have and are being designed but have not reached the precision achieved by spectrographs in the optical like HARPS. This chapter on radial velocity was largely taken after the review of Lovis and Fischer on Radial Velocities Techniques for Exoplanets [Lovis and Fischer, 2010]. 1.2 Transiting exoplanets Introduction If a planet, laying in the line of sight defined by the observer and the observed star, passes in front of its parent star, a dip in the apparent brightness can be observed. This event is what we call a primary eclipse or transit. This event may be repeated half an orbit later, though not in same strength, when the planet is covered by the star. This is called a secondary eclipse or an occultation. Where radial velocity focused primarily around spectroscopic measurements, the search for transiting exoplanets focuses on (relative) flux measurements. 12

13 As it was mentioned in the previous chapter, the first observed transit was predicted by the radial velocity technique and in doing so confirmed the planetary nature of the objects causing the radial velocity signals. When we search for transits we have two options available to us. Either we search the stars which already have a radial velocity signal present or we search thousand stars at random in a survey. Both of these techniques can be used in space and ground based telescopes. We will go over some of the most successful projects when it comes to detecting transits. Recently the Kepler telescope, which was funded by NASA, has had a lot of success detecting transiting planets [Koch et al., 2010]. Kepler is a space-based survey telescope with a single photometric instrument which observes the brightness of more than main sequence stars in a field of view of 115 deg². At the moment the number of confirmed planets detected by Kepler is only 61, but the database contains 2326 more planetary candidates and has allowed astronomers for the first time to really use statistics as a tool to test planetary formation theories. Given the very good precision of Kepler Howard et al. [2011] were able to construct an almost complete sample for planets around solar-like stars within 0.25 AU. They found occurrence rates (after correcting for biases) for all planets with orbital periods less than 50 days of ± 0.008, ± 0.003, and ± planets per star for planets with radii 2 4, 4 8, and 8 32 R earth suggesting that exoplanets are not as rare as was first thought. Using the extensive database of planetary candidates Fabrycky et al. [2012] where able to conclude that planets in multiplanet systems are generally well aligned to within a few degrees. They were also able to conclude that these planets are usually non-resonant but show a peak just after the important resonances ( 2:1 and 3:2 ). This and many more statistically relevant conclusions can be drawn from the database. COROT (Convection Rotation et Transits planétaires) was the first space-based telescope dedicated to the search for exoplanets and asteroseismology and was quite successful in doing so [Baglin et al., 2006]. It detected 22 exoplanets and a brown dwarf. Recently however the data processing unit for the first two CCDs (A1 and E1) broke down which reduced the field of view by 50%, luckily without reducing any of the actual quality of the data. COROT is a collaboration between the French space agency (CNES) and ESA and was launched into a polar orbit around the Earth compared to Kepler which is in an earthtrailing orbit Another successful ( for the moment even the most successful ) project regarding transiting exoplanets discoveries is definitely the WASP (Wide-Angle Search for Planets) program which is a collaboration of several British universities and other international institutes [Pollacco et al., 2006]. The project has Figure 6: Phased WASP-south photometry for WASP-43b two separate observatories: one in the Roque de los Muchachos Observatory in La Palma (WASP-North) and one in South African Astronomical Observatory in South Africa (WASP- South). The WASP survey has produced the initial photometry leading to the detection of 65 exoplanets, where WASP-43b (see figure 6) and WASP-46b are two of. The success of this mission is most likely explained by the huge sky coverage of 500 square degrees (compared to the moons approximate 0.5 square degrees). Given the success of WASP, part of its consortium (together with other international institutes) is planning to set up a new project in Paranal. This project, the Next Generation Transit Survey (NGTS), will continue on the same path as WASP and use a wide FOV and several telescopes. NGTS will aim to 13

14 detect objects the size of Neptune around relatively bright stars (magnitude V < 13). The project has already been tested in La Palma, using only one telescope. The prototype produced some nice results for the hot Neptune GJ436b as can be seen in figure 7. The HATNet (Hungarian-made Automated Telescope Network) project consists of six small telescopes which aims to detect and characterize transiting planets [Bakos et al., 2002]. In 2009 three more telescopes were added to the network in the southern hemisphere. The discovery count for HAT stands at 29 including the three co-discoveries with WASP. Located on Maui, Hawaii the XO telescope consists of a pair of 20cm lenses with the hardware only costing US dollars to construct and not surprisingly being surpassed by the cost of the software [McCullough et al., 2005]. The XO telescope lead to the discovery of four hot Jupiters and a brown dwarf. Similar to XO is the TrES (Trans-Atlantic Exoplanet Survey) project located at the Lowell Observatory in Arizona, the Palomar Observatory in California, in Texas and in the Canary Islands with one 10cm telescope at each location [Alonso et al., 2004]. TrES has discovered five hot Jupiters. Eventually we come to the MEarth project which has one very important planetary discovery, the super- Earth GJ1214b. MEarth consists of eight 40cm telescopes (Nutzman and Charbonneau [2008],Irwin et al. [2009]) and was designed specifically to detect super-earths around the brightest M-dwarfs to allow for atmospheric characterization. Obviously other surveys exist but to summarize them all would lead us to far and thus we only discussed the most important. The general set of equations In principle it is possible to geometrically reconstruct a transit light curve using three parameters: the transit depth, df, the total transit duration from first to fourth contact, t T, and the transit shape, which is the ratio of the duration of the flat part of the transit over the total transit duration. The flat part of the transit is from second till third contact (see figure 8). If we want to make a closed set of equations we are going to have to make a number of assumptions which will simplify the equations: The orbit of the planet is circular. The planet does not emit light itself. The mass of the planet is small compared to the mass of the star. The stellar mass-radius relation is known. The light comes from a single star. Figure 7: NGTS prototype photometry for GJ436b, taken from 14

15 The transit is non-grazing, meaning the planet s disk is completely superimposed on the stellar disk. And thus the three geometrical parameters can we written as [Seager and Mallén-Ornelas, 2003]: df = F NoTransit F Transit F NoTransit = ( R pl R ) 2 (15) t T = P π arcsin( R a [ (1 + R pl R )² ( R a cos i)² ] 1 2 ) (16) 1 cos ²i R t F arcsin( = t T a [ (1 R pl arcsin( R a [ (1+ R )² ( R a cos i)² 1 cos ²i ] 1 2 ) R pl R )² ( R a cos i)² 1 cos ²i ] 1 2 ) (17) The last parameter that can be obtained is the period (need of two consecutive transits or RV measurements) and can be related to the other physical parameters using the well known third law of Kepler: P² = 4π²a3 GM (18) The physical parameters Eventually we want to obtain the five physical parameters M, R, a, i and R pl from the four equations from above. We do this by rewriting and simplifying the above equations so we can solve them using only the observable parameters F, t T, t F and P [Winn, 2009]. We see that we are able to find four combinations of the physical parameters. The planet-star radius ratio: R pl R = df (19) The impact parameter, b, which can be defined as the projected distance between the centrum of the star and planet at mid-transit: The scaled stellar radius R a : b a R cos i 1 df t T t F (20) R a π tt t F df 1 4 P (21) and the stellar density ρ : ρ ρ 3P df π 2 G ( ) 3 2 (22) t T t F 15

16 Figure 8: Transit geometry with the contact points labeled 1 through 4 taken from Seager and Mallén-Ornelas [2003] 16

17 It is clear now that we have five unknown parameters: the inclination, i, the stellar mass and radius, M and R, the semi major axis, a, and the planetary radius, R pl, but only four equations. We can solve this degeneracy by obtaining the stellar mass in number of possible ways. One of which is modeling the stellar mass using as input the stellar density and the effective temperature and metalicity obtained from spectroscopy as was done for WASP-43b by Gillon et al. [2012]. It is also possible to assume a empirical mass-radius relation for the star as was done for WASP-46b[Anderson et al., 2012]. Here we used the Enoch relationship which takes the same input and has been shown to be in good agreement with other models for the WASP subset[enoch et al., 2010]. Once the mass and radius of the star have been determined we can continue and find for the other physical parameters: And the definition of the impact parameter: a = [ P²GM ] 1 3 (23) 4π² Most importantly, we want to know the radius of the planet: i = cos 1 (b R a ) (24) R pl R = R R df (25) Limb Darkening Limb darkening is the effect that a stellar disk seems to be brighter in the middle and dimmer on the edges. This is causes by two effects. First of all we have to consider that the density and temperature decreases as the distance from the core increases. And secondly that the line of sight is more and more oblique compared to the normal of the stellar surface. This causes higher altitude and thus cooler shells to be probed near the limb of the star. Limb darkening has several effects on the light curve. As one can see in figure 9 it gives a parabolic nature instead of the typical boxlike form. One can see as well that the effect is far less pronounced in the red part of the spectrum. It is possible to include limb darkening in the mathematical description, leading to lengthy algebra. In this work we have used a quadratic approach to model the limb darkening in the MCMC code, using the following formula: I I 0 = 1 c 1 (1 µ) c 2 (1 µ) 2 (26) Figure 9: Limb darkening in HD209458b where the color of the light curves indicates the observed wavelengths take from Knutson et al. [2007] 17

18 Where c 1 and c 2 are coefficients that can be calculated from a sufficiently precise light curve and µ = cos γ with γ the angle between the line of sight of the observer and a line normal to the stellar surface. Followup observations Transiting exoplanets are an interesting subset of the known exoplanets since they allow for a large number of followup measurements. Among others there is the possibility of transmission transit spectroscopy, occultation emission spectroscopy and the Rossiter-McLaughlin (RM) effect. Transmission spectroscopy allows us to probe the atmospheric contents of the transiting planet. In the previous sections we have silently assumed that the planet has a sharp edge. In reality however and especially for gas giants this is not the case. The atmosphere of the planets have a different opacity for a given wavelength seeing that absorption of the stellar light takes place. Meaning that when we would observe a transit at a certain wavelength for which we know absorption takes place, we would absorb a larger flux deficit and thus a larger transit depth. Converting this observational signal into a theoretical atmospheric model is difficult task for which one must follow the radiative transfer of the stellar flux through the atmosphere of the exoplanet. Occultation spectroscopy provides complementary information about the planet s atmosphere. Planetary radiation knows two sources: first of all there is the thermal emission and secondly there is the radiation reflected from the star. Since we only measure the stars radiation during an occultation we can reconstruct with a negative measurement the thermal emission and reflected spectrum. From the thermal emission we can reconstruct how much of the absorbed stellar flux is redistributed to the night side of the planet, whether or not an inversion layer is present, etc. One of the largest uncertainties when constructing atmospheric models is the occurrence of clouds, which might increase the albedo of the planet by a significant amount. When measuring the reflectance spectrum we can effectively determine the albedo at a given wavelength. Figure 10: Radial velocity measurements and best fit RM model for WASP-15b as was obtained by Triaud et al. [2010] The RM effect can be used to determine how the planet is aligned compared to the stellar spin axis. If we were to observe a star during transit using the radial velocity method, we would measure a redshift is the planet is blocking the approaching part of the star and vice versa. A good sampling would then allow us to determine the angle between the stellar spin axis and the normal of the orbital plane as can be seen in figure Exoplanets: The global picture As it was mentioned before one of the first puzzling discoveries in the extrasolar field are the so called hot Jupiters. These are Jupiter-massed planets in a very short (peaked at days) orbits around their 18

19 parent stars. The objects invoked a revolution in the field of planetary evolution and formation theories when they were discovered and required astronomers to renew their theories. It is now thought that they came to such short orbits through the notion of inward migration. More often than not the orbits have been circularized and are tidally locked always facing the side to their parent star. Giant exoplanets with a period of more than ten days, however, often show an orbit which is not consistent with a circular orbit and in the extremest case have orbits with an eccentricity of up to 0.93 [Naef et al., 2001]). There are two main theories for inward migration. One is centered around interactions in the planetary disk to cause planets to fall inward and the other is centered around gravitational scattering and tidal circularization afterwards. The theory of disk migration would mean that any inner planetisimals will most likely be scattered by the much heavier inward falling gas giant. Although recent simulations have shown that this might not be as destructive as it was first thought and would even lead to planets up to two earth masses in the habitable zone containing plenty of water [Fogg and Nelson, 2007]. Observations show however that these planetary partners in hot Jupiter systems are rare. Planetary scattering theories could explain the misalignment of the orbits of hot Jupiters and the large eccentricities found for planets on long orbits. More than half of the sample of hot Jupiters that has been investigated using the RM effect have been shown to be misaligned with their stellar spin axis confirming migration theories with planetary scattering causing hot Jupiters to get misaligned by close interactions with other bodies in the system [Triaud et al., 2010]. Hot Jupiters are known to have large dispersion regarding their radii and some being even too large for a full Hydrogen model. Astronomers have come to pose a number of reasons for these discrepancies. One of the proposed theories is that tidal heat dissipation, caused by the circularization process of the planet, lays at the origin of inflation. More generally though hot Jupiter s radii tend to show a strong correlation with incident flux from the star[enoch et al., 2012]. The way this energy is dissipated is still open to debate with two main options: Ohmic heating which is caused by the magnetic drag of ions in the atmosphere [Batygin and Stevenson, 2010]and kinetic heating where a part of the incident flux is transferred to kinetic and afterwards to thermal energy[showman and Guillot, 2002]. Other planets show a very large density requiring a very dense or large core. Most likely is that the outer layers of the planet have been blown away by the stellar winds and leave behind a planet which has a larger core than we would usually suspect or simply, given the strong correlation between the stellar metalicity and hot Jupiters densities, that they formed with a larger core. Actual Jupiter analogs appear to be quite rare from RV surveys carried out of a time span of more than ten years. Wittenmyer et al. [2011] come to an occurrence rate of 3.3 ± 1.4% fully consists with the findings of Cumming et al. [2008] who found a value of 2.7 ± 0.8%. Hot Jupiters are the easiest systems to detect because of their large RV amplitudes, short periods and large radii and for transits even possible with amateur equipment. Recently detection thresholds have gone to the limit of Earth sized/mass objects. Objects not quite there are called super-earths and are informally defined of having a mass between 1 [Valencia et al., 2007], 1.9 [Fortney et al., 2007] or 5 [Charbonneau et al., 2009] to 10 Earth masses. Notice that the definition only refers to the mass and does not reflect any other properties they might have in common with Earth. Since the Kepler mission is not able to obtain the mass of their planetary candidates without radial velocity or a mass radius relationship, they inferred another definition for a super-earth as an object with a radius between 1.25 and 2 Earth radii[borucki et al., 2011]. While in our solar system planets fall in two distinct categories (the terrestrial planets and the gas giants), super-earths fall in between them and pose a great mystery regarding their composition. Since these planets are often observed at the brick of the detection thresholds it is very difficult to obtain a strong constrain on their mass and radius 19

20 if you are lucky enough to have both available. Even if we would have a perfect determination of the mass and radius, we would not be able to put a strong constraint on the interior structure since these solutions are often degenerate regarding configuration and composition of the different layers. A way to lift this partially lift this degeneracy is to observe the atmospheric composition Super-Earths have always been of particular interest since they could allow for the harboring of alien lifeforms primarily depending on whether they are located in the habitable zone. The habitable zone is defined by the region around the star for which liquid water on the surface on the planet is possible. The first detection of a super-earth around a main-sequence star was Gliese 876d with a mass of 7.5 Earth masses in 2005 [Rivera et al., 2005]. Not much later, in 2007, the first detection of a super-earth in the habitable zone was announced. Gliese 581c and 581d are located just on the edges of the habitable zone of their parent star [Udry et al., 2007]. With the statistical data available from the Kepler observations Borucki et al. [2011] found that each stars hosts on average planetary candidates. Of which are Earth sized and are super-earth sized. From the 1202 planetary candidates they found 54 to be in the habitable zone and six of those are less than twice the size of the Earth. To suggest an occurrence rate for this subset of planets asks for complex bias removal and care should be taken not to extrapolate these results. 20

21 2 Transit Timing Variations 2.1 Introduction Transit Timing Variations (TTVs) are proving to be a valid method for detecting or characterizing exoplanets (Holman and Murray [2005], Agol et al. [2005]). The transits of a planet in a Keplerian orbit around a star are strictly periodic. This is no more the case if a third body is present, as the orbits are no more Keplerian, and the time between consecutive transits varies. These TTVs thus represent an opportunity for detecting a second planet around a star. Recently, the TTV technique has been employed to derive the mass of known transiting planets (Holman et al. [2010], Lissauer et al. [2011], Cochran et al. [2011]) and constrain possible orbital configurations for non-transiting planets[ballard et al., 2011]. TTVs are an effective method for these Kepler systems since the semi-amplitude of the RV signal is often quiet small, possibly noisy because of stellar variability or the star might be too faint to perform a decent spectroscopic analysis on it. Studies of a sample of 822 Kepler candidates observed in the first seventeen months of Kepler observations show that 35 ( 4.1 % of the data set) planets have a strong TTV signal and 145 ( 18% of the data set ) show a weaker TTV signal. In 60-76% of the multiplanet systems observations of TTV signals have been made and would thus allow for a very strong mass determination ( see figure 11 taken from Ford et al. [2012]). In the following chapters we will go over the most common causes of TTVs in which we will always assume edge-on coplanar orbits which is a reasonable assumption as was shown by Fabrycky et al. [2012] in their analysis of multi-planetary Kepler systems. Figure 11: TTVs for long term trends observed by Kepler. Panel c and d have already been confirmed as Kepler-9b and Kepler-9c. 21

22 Figure 12: Source of a TTV signal in a system where the outer far out planet is transiting. Taken from Agol et al. [2005]. 2.2 Inner planet The simplest case of TTVs is probably where you have a far out transiting planet and a close in noninteracting perturbing planet. The unique nature of the TTVs in this case is that they are not (primarily) caused by planet-planet interactions. They are caused by the star s motion around the inner binary s c.o.m and making the timings appear early or late. Figure 12 shows an example of this situation. The TTVs can also easily be described mathematically if we neglect planet-planet interactions as was stated before. We can do this when the periapse of the outer planet is much larger then the apoapse of the inner planet ( i.e. (1 e trans )a trans (1 + e pert )a pert ). If we investigate the equations for circular orbits we find that the inner planet displaces the star from the barycenter by: x = a pert µ pert sin(2π(t t 0 )/P pert ) (27) And for the mth transit we find a timing deviation of: x δt P transa pert µ pert sin(2π(mp trans t 0 )/P pert ) (28) v trans v 2πa trans where we have neglected v since it is typically much smaller then v trans and µ i = deviation of the TTV signal we find: σ = 12 (δt) 2 = m i m i. For the standard P transa pert µ pert πa trans (29) 22

23 The case of an inner perturbing planet is on its own a less interesting case since most detection methods are biased towards closer in planets and have a much easier time detecting them. If there is already an outer transiting planet in the system it is very likely that the inner planet is also transiting given that most multiplanet systems appear to be coplanar. 2.3 Non-resonant outer planet The amplitude of TTVs in planets on nearly circular orbits can be calculated using perturbation theory. Since planets interact most strongly at conjunction the amplitude of the TTV signal is primarily determined by the resonant forcing terms. The transiting planet gets, similar to the slingshot technique used by spacecraft, a radial kick at conjunction inducing a change in eccentricity. Since the planets are not in resonance the longitude of the conjunction shifts after each orbit and starts to cancel out after the longitude of the conjunction has increased by π. The changes in eccentricity lead to changes in the semi-major axis and thus are the cause for the observed TTVs. For two planets near a j : j + 1 resonance and ɛ = 1 (1 + j 1 ) P trans P pert < 1 being the fractional distance to the resonance one can calculate the TTVs using the change in orbital frequency to the first order in the eccentricity: θ = n(1 + e cos f )² (1 e²) 3 2 n 0 + δn + 2en 0 cos(λ ϖ) (30) Where n is the mean motion, n 0 is the unperturbed mean motion, f the true anomaly, λ the mean longitude and ϖ is the longitude of the periapse. In this equation there are two perturbations on the mean motion. The first can be found after applying the Tisserand relation: da de = 2a 5 2 e a (31) and leads to: δt ( m pert m trans )µ 2 transɛ 3 P trans (32) and the other term which is eccentricity dominated gives a timing variation ( where we have assumed that the heavier planet is the transiting one): δt µ pert ɛ 1 P trans (33) Planets with a larger period ratio have the timing deviations become proportional to: δt µ pert P 3 trans P 2 pert (34) One can note that for all of these equations (in the case of M M pert ) the TTV is directly proportional to the perturbing mass. 23

24 Figure 13: TTVs for Kepler-18c (left) and Kepler-18d (right) in a 2:1 mean motion resonance[cochran et al., 2011]. 2.4 Resonant planet The TTVs for planets in mean-motion resonances are especially interesting since the signal is largest here. This is caused by the conjunctions that take place at the same longitude each orbit. These interactions will cause a change in eccentricity and semi-major axis and thus lead to a shift in the longitude of conjunctions. These perturbations start to cancel out once the longitude of conjunction has shifted π and the period of this phenomena is called the libration period. If one approaches the TTVs in a j : j + 1 resonance on a qualitative manner it can be found that the amplitude and libration period are proportional to (once again for near-circular orbits): δt P m pert (35) 4.5j m pert + m tran P lib 0.5j 4 3 µ 2 3 P (36) An example of this situation is presented in figure 13. The equations form the previous two sections were taken from Agol et al. [2005]. 2.5 Analytic approach When trying to reproduce a given TTV signal the simplest way to do so would be to run a N-body simulation testing the parameter space. If we were to assume in such a simulation that the signal is caused by a single coplanar perturber, we would have to explore six different parameters of the perturber: The mass The period 24

25 Figure 14: TTV (top) and TDV (bottom) signal for KOI [Nesvorny et al., 2012]. The eccentricity The longitude of pericenter The longitude at a certain epoch The precession of the eccentricity Exploring this parameter space is a heavy task and requires a high amount of CPU time. Nesvorný and Morbidelli [2008] have developed a method and accompanying code which avoids the need for orbital integrations and is based on perturbation theory. To go into the theory of this method would lead us too far and we will only discuss the advantages and the two major shortcomings. Working with a perturbation theory asks of the expansion terms that they are convergent and when this is not the case the method diverges and fails to reproduce the results from numerical N-body simulations. This occurs when the pericenter distance of the perturbing planet approaches the semi-major axis of the transiting planet. The shortcoming proves to be of a non-critical nature since such systems may not have a long term stability. Another problem is that the method fails for the mean motion resonances, the places where the TTV detection method is most sensitive. As it was the case in the previous shortcoming divergent expansion terms have been discarded. Major advantage is that it reduces the computational time by 10 4 and the strength of the method has been shown in KOI-872 where it has lead to the first characterization of an exoplanet with 0.37 Jupiter masses using the method of TTVs [Nesvorny et al., 2012]. The TTVs and TDVs (see next section) are presented in figure

26 Figure 15: Coordinate system used for the derivation of TTVs and TDVs caused by exomoons from Kipping [2009] 2.6 Exomoon Not only planetary perturbers cause variations in the transit parameters. Transit Durations Variations (TDVs) and TTVs are a promising method to detect the first exomoon [Kipping, 2009]. The first exomoon detection would not only bring great prestige, it is also crucial to the understanding of the formation of our and other planetary systems. When we derive the amplitude of the signal we assume that the system is edge-on and coplanar. Notation conventions are taken from figure 15. Because of the moon the planet is displaced by an amount x 2 and the timing would thus be off by x 2 divided by the planet-moon orbital velocity projected on x 2, v B. δt( f ) = x 2 ( f ) v B (37) where f is the true anomaly for the planet in the moon-planet system and for the RMS, δ TTV, we find: δ TTV = a w 2vB (38) where the subscript W denotes the orbit of the planet in the planet-moon system. After a very lengthy amount of algebra one can find that the RMS can be written as: δ TTV = 1 a 1 2 P a S M S MPRV 1 ζ T (e P, ϖ P ) 2 G(M + M PRV ) Υ(e P, ϖ P ) (39) 26

27 where: ζ T = (1 e2 S ) 1 4 e S e 2 S + cos(2ϖ S)(2(1 e 2 S ) e 2 S ) (40) Υ = cos(arctan( e P cos ϖ P 1 + e P sin ϖ P )) 2(1 + e P sin ϖ P ) (1 e 2 P ) 1 (41) where the subscript P denotes the exoplanet s orbit around the central star and the subscript S denotes the exomoon s orbit around the exoplanet. M PRV is the combined mass of the exomoon and exoplanet. The same goes for the transit duration, τ: where: τ 1 v P (42) v P = v B + v W (43) Here it is clear that v B remains relatively constant whilst, depending on the moon s position, v W can change and will determine the period and amplitude of the TDVs. Similar to the case of TTVs we find for the RMS for TDVs: δ TDV = ap a S MS 2 τ ζ D (e S, ϖ S ) M PRV (M PRV + M ) 2 Υ(e P, ϖ P ) (44) where: ζ D = 1 + e 2 S e2 S cos(2ϖ S) 1 e 2 S (45) If we were to consider the TTV effect alone we would find that the signal is degenerate and only solves for M S a s. But if we were to combine the TTVs with the TDVs, we would break the degeneracy and be able to solve for the orbital parameters of the exomoon. Typically, however, errors on the transit duration are two or three times larger than the errors on the mid transit time and it would be difficult to obtain a clear signal for this. To illustrate how big the these timing deviations practically are, we will provide some values calculated by Kipping [2009] for some prime candidates. For Gl436b, a Neptune sized planet on an eccentric orbit, the RMS of the TTV signal of an exomoon could be seconds. Compared to HD209458b for which only a RMS of 2.97 seconds is predicted if is it orbited by an exomoon. One can see that excellent photometric precision is required to be able to constrain these orbits. 2.7 Other TTV signals In the previous sections we have talked about relatively short term TTV signals. When we look at a longer timescale we see that precession of the longitude of the pericenter occurs. This requires of 27

28 course a non-zero eccentricity for the transiting planet. The maximum timing deviation caused by the precession of the pericenter due to another planet is (for e 1) [Miralda-Escudé, 2002]: δt = ep trans π (46) Precession of the orbit s pericenter can be caused by an number of other phenomena as well. Main causes for this are general relativity, oblateness of the star and tidal distortion of the planet. But generally these effects take place at much longer timescales or require extreme systems to be observed. Agol et al. [2005] calculated that for Gl876c the precession of the pericenter would have go through with a speed of -41 per year and an RMS of 1.87 days more than 5 percent of the total period of the planet due to Gl876b orbiting the star as well. Compared to when one would observe transits of Jupiter which would have an RMS of 24.1 seconds caused by the precession due to interactions with Earth. When investigating a system for TTVs it is possible that some sort of signal is observed, but it is not caused by a planetary perturber. Most often then not TTVs are caused by stellar variability and appear as a scatter on the O-C plot rather than a clear (non-) sinusoidal signal. Knutson et al. [2011] have shown that timings obtained from light curves observed in the infrared, where spots are less pronounced, are in fact a better fit to a linear ephemeris and show less scatter as can be seen from figure 16. It should be noted that most of the equations given in these sections should not be used to analyze actual transit timings but are rather to estimate and compute putative TTV signals. Figure 16: TTVs for GJ436b observed by Knutson et al. [2011] at different wavelengths (blue for visible, red for IR) This chapter on TTVs was largely taken after Agol et al. [2005]. 28

29 3 Observations and data reduction 3.1 Data description TRAPPIST photometry Nearly all measurements used in this work were made by TRAPPIST (TRAnsiting Planets and PlanetesImals Small Telescope). TRAPPIST is a small 60 cm telescope based at ESO La Silla Observatory, Chile, and dedicated to the study and detection of exoplanets and comets. The telescope is fully automated and is protected by a 5 meter diameter dome with weather station. Observations for exoplanetary transits are made in an Astrodon blue-blocking and an I + z filter and a Sloan z for occultations on a thermoelectrically-cooled 2k 2k CCD camera with a field of view of (pixel scale = 0.65 ) (Jehin et al. [2011], Gillon et al. [2012]). The software guiding system on TRAPPIST is able to keep the observed star at the same position on the detector to within a few pixels. We observed four new transits for WASP-43b, four new transits for WASP-46b and eight new transits for GJ1214b. EulerCAM and CORALIE Three transits for WASP-43b and one transit for WASP-46b were observed with the EulerCAM CCD camera at the 1.2-m Euler Swiss telescope. These observations were made in a Gunn-r filter on a nitrogen-cooled 4k 4k CCD camera with a field of view (pixel scale=0.23 ). All the radial velocity measurements that are used in this work, were produced by CORALIE, the spectrographic instrument on the Euler telescope. Transit Timings GJ1214b GJ1214b is a well studied planet due to its low mass and has been observed many times before by other research groups. We went through the literature data and collected the timings for all the observed transits (see tables 2 and 3). We used this data to set a really strong constraint on the period and the TTVs. 3.2 Data analysis Introduction to Markov Chain Monte Carlo For modeling the transits we used the latest version of the Markov chain Monte Carlo code (Ford [2005], Ford [2006]) developed by M. Gillon (Gillon et al. [2010] and references therein) to determine the parameters of a transiting system using as input photometric and radial velocity data. MCMC simulations are based on Bayesian inference meaning it is a method of drawing conclusions from data subject to random variations such as observational data using Bayes rule. Bayes rule is known in statistics to relate the odds for different events before and after conditioning another event. Let us start with a joint probability distribution p(x,y) and after integrating over y we find a marginalized probability distribution p(x). Using Bayes theorem we find then that: p(x, y) = p(x)p(y x) = p(y)p(x y) (47) 29

30 Table 2: Transit Timings from literature Epoch Timing BJD TBD Reference ± Kundurthy et al. [2011] ± Kundurthy et al. [2011] ± Kundurthy et al. [2011] ± Kundurthy et al. [2011] ± Kundurthy et al. [2011] ± Kundurthy et al. [2011] ± Kundurthy et al. [2011] ± Kundurthy et al. [2011] ± Kundurthy et al. [2011] ± Berta et al. [2011] ± Berta et al. [2011] ± Berta et al. [2011] ± Berta et al. [2011] ± Berta et al. [2011] ± Berta et al. [2011] ± Berta et al. [2011] ± Sada et al. [2010] ± Sada et al. [2010] ± Sada et al. [2010] ± Sada et al. [2010] ± Bean et al. [2010] ± Carter et al. [2011] ± Carter et al. [2011] ± Carter et al. [2011] ± Carter et al. [2011] ± Carter et al. [2011] ± Carter et al. [2011] ± Carter et al. [2011] ± Carter et al. [2011] ± Carter et al. [2011] ± Carter et al. [2011] ± Carter et al. [2011] ± Carter et al. [2011] ± Carter et al. [2011] ± Carter et al. [2011] ± Carter et al. [2011] ± Carter et al. [2011] ± Désert et al. [2011] ± Désert et al. [2011] ± Croll et al. [2011] ± Croll et al. [2011] ± Croll et al. [2011] ± Croll et al. [2011] ± Croll et al. [2011] 30

31 Figure 17: The TRAPPIST telescope at La Silla. See ˆ p(x) = p(x, y)dy (48) p(y x) = p(x, y) p(x) = p(y)p(x y) (49) p(y)p(x y)dy If we now identify x with observational data ( d ) and y with a set of model parameters ( θ ), we see that we can find p( θ d ), called the posterior probability distribution, which gives the probability that a given set of observations leads to a certain set of model parameters. The goal of MCMC simulations is to construct a chain of states each with a given set of model parameters. After a chain has been completed we can calculate the medians of the model parameters and their uncertainties from all the states (except the first 20% since they are considered to be in the burn-in phase and thus discarded). We thus only take into account the sampled model parameters for the posterior probability function after convergence has been reached, which is after the burn-in phase. The Monte Carlo aspect of the method implies that each set of model parameters is randomly generated (within their respective probability distributions of course). Whereas the Markov chain aspect requires that each new state with its unique set of model parameters depends solely of the state before it and no further correlation is allowed. As an example as to how a (Metropolis-Hasting) MCMC simulation with one chain and n tot steps works we will give a flow chart to illustrate it better: 1. Initialize a Markov chain with a set of model parameter x 0 obtained from a prior probability distribution and set n = 0 2. Generate a new set of parameters x obtained from a candidate transition probability function q(x x n ) which is a Gaussian centered around x n. 3. Acquire the goodness of the fit by calculating χ 2 (x ) and χ 2 (x n ) 31