Transit Timing Analysis of the Hot Jupiters WASP-43B and WASP-46B and the Super Earth Gj1214b

Home , Despina (moon), Galatea (moon), HD 80606 b

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 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 MJup and 0.81347460±0.00000048 days and 2.03±0.13 MJup and 1.4303723±0.0000011 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 1.580404695±0.000000084 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 (3σv) 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 If I have seen further it is by standing on the shoulders of giants

3 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 Contents

1 Other Worlds 6 1.1 Radial velocity...... 6 1.2 Transiting exoplanets...... 12 1.3 Exoplanets: The global picture...... 18

2 Transit Timing Variations 21 2.1 Introduction...... 21 2.2 Inner planet...... 22 2.3 Non-resonant outer planet...... 23 2.4 Resonant planet...... 24 2.5 Analytic approach...... 24 2.6 Exomoon...... 26 2.7 Other TTV signals...... 27

3 Observations and data reduction 29 3.1 Data description...... 29 3.2 Data analysis...... 29 3.3 TTV simulator...... 34

4 Results and discussion 36 4.1 WASP-43b...... 36 4.2 WASP-46b...... 42 4.3 GJ1214b...... 47

5 Conclusion 54

5 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 com- plementarity 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 −1 planet with minimal mass of 11 MJup and a period of 84 days had a velocity amplitude of 600 m s . 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 114762 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 skepti- cism regarding the nature of the source of this signal for two reasons. First of all there wasn’t a single

6 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 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 75- 80 Jupiter masses), but would only support deu- terium 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 209458b, 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 skepti- Figure 3: Original phased radial velocity diagram for 51 cism and confirmed that the wobble observed by peg b the radial velocity was indeed caused by a planet. 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 Figure 4: Original transit light curve for HD209458b

1 + 1 k.v = c λ λ0 Φ v² (1) 1 − c² − 2c²

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 apl for the motion around the c.o.m:

m∗ apl = a (2) mpl + m∗

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:

2 2 a∗(1 − e ) mpl a(1 − e ) r∗ = = (3) 1 + e cos f mpl + m∗ 1 + e cos f

With r∗ the distance from the c.o.m for the star, e the eccentricity and f the true anomaly. After differen- tiating (2), we ﬁnd:

9 2 2 mpl a(1 − e ) f˙ sin f er f˙ sin f = = ∗ r˙∗ 2 2 (4) mpl + m∗ (1 + e cos f ) a∗(1 − e )

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:

r∗ cos f r∗ = (5) r∗ sin f

! r˙∗ cos f − r∗ f˙ sin f ˙r∗ = (6) r˙∗ sin f + r∗ f˙ cos f

When we insert equation 4 into 6 we ﬁnd that:

2 r∗ f˙ − sin f ˙r∗ = (7) a∗(1 − e²) cos f + e

h∗ − sin f = (8) m∗a∗(1 − e²) cos f + e

We also have for the relative orbital momentum of the star h∗:

m∗ h∗ = h (9) m∗ + mpl

v u 2 4 u Gm∗m a(1 − e²) = t pl 3 (10) (m∗ + mpl)

If we substitute 10 in 8:

v u 2 u Gmpl − sin f ˙r∗ = t (11) (m∗ + mpl)a(1 − e²) cos f + e

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:

  sin ω sin i   k =  cos ω sin i  (12)   cos i

10 Table 1: Radial velocity signals for several (hypothetical) planets and the planets studied in this work.

−1 Planet Mass a (AU) K1(m s ) +0.052 +0.00018 +5.6 WASP-43b 2.034−0.052MJup 0.01526−0.00018 547.9−5.5 +0.13 +0.00071 +11 WASP-46b 2.03−0.13MJup 0.02409−0.00071 386−11 +0.52 +0.00056 GJ1214b 6.47−0.54MEarth 0.01433−0.00062 12.2±1.6 Jupiter 1MJup 0.1 89.8 Jupiter 1MJup 1.0 28.4 Jupiter 1MJup 5.0 12.7 Earth 1 MEarth 0.1 0.28 Earth 1 MEarth 1.0 0.09

Once we project them, we ﬁnd the radial velocity equation:

s G vr,∗ = ˙r∗.k = mpl sin i(cos(ω + f ) + e cos ω) (13) (m∗ + mpl)a(1 − e²)

From here it is trivial to see the semi-amplitude, K, as (vr,max-vr,min)/2

s G K1 = mpl sin i (14) (m∗ + mpl)a(1 − e²)

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 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 ultra- high-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 re- sponsible 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 10-100 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 Veloc- ities 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 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 145 000 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 0.130 ± 0.008, 0.023 ± 0.003, and 0.013 ± 0.002 planets per star for planets with radii 2–4,

4–8, and 8–32 Rearth 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 earth- trailing 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 univer- sities 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 plan- ning 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 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 60 000 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, tT, 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 www.ngtransits.org

14 • 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]:

F − F Rpl dF = NoTransit Transit = ( )2 (15) FNoTransit R∗

R ( + pl ) − ( a i) P R∗ 1 R∗ ² R∗ cos ² 1 t = arcsin( [ ] 2 ) (16) T π a 1 − cos ²i

R pl a (1− R )²−( R cos i)² 1 R∗ ∗ ∗ 2 tF arcsin( a [ 1−cos ²i ] ) = R (17) t pl a T (1+ R )²−( R cos i)² 1 R∗ ∗ ∗ 2 arcsin( a [ 1−cos ²i ] )

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 Ke- pler:

4π²a3 P² = (18) GM∗

The physical parameters

Eventually we want to obtain the ﬁve physical parameters M∗,R∗, a, i and Rpl from the four equations from above. We do this by rewriting and simplifying the above equations so we can solve them us-

ing only the observable parameters F, tT, tF and P [Winn, 2009]. We see that we are able to ﬁnd four combinations of the physical parameters. The planet-star radius ratio:

Rpl √ = dF (19) R∗

The impact parameter, b, which can be deﬁned as the projected distance between the centrum of the star and planet at mid-transit:

a √ t b ≡ cos i ≈ 1 − dF T (20) R∗ tF

R∗ The scaled stellar radius a : √ R∗ π tTtF ≈ 1 (21) a dF 4 P and the stellar density ρ∗: √ ρ∗ 3P dF 3 ≈ ( ) 2 2 (22) ρ π G tTtF

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

16 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, Rpl, 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:

P²GM∗ 1 a = [ ] 3 (23) 4π²

And the deﬁnition of the impact parameter:

R∗ i = cos−1(b ) (24) a

Most importantly, we want to know the radius of the planet:

Rpl R∗ √ = dF (25) R~ R~

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 math- ematical 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: Figure 9: Limb darkening in HD209458b where the color of the light curves indicates the observed wavelengths I 2 take from Knutson et al.[2007] = 1 − c1(1 − µ) − c2(1 − µ) (26) I0

17 Where c1 and c2 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 complemen- tary 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 con- structing atmospheric models is the occurrence of clouds, which might increase the albedo of the planet by a significant amount. When measuring Figure 10: Radial velocity measurements and best fit RM the reflectance spectrum we can effectively deter- model for WASP-15b as was obtained by Triaud et al. mine the albedo at a given wavelength. [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 10.

1.3 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 3-3.5 days) orbits around their

18 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]. Observa- tions 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 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 0.341 planetary candidates. Of which 0.054 are Earth sized and 0.068 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 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 op- portunity 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 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 non- interacting 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 − etrans)atrans (1 + epert)apert). If we investigate the equations for circular orbits we ﬁnd that the inner planet displaces the star from the barycenter by:

x∗ = −apertµpert sin(2π(t − t0)/Ppert) (27)

And for the mth transit we ﬁnd a timing deviation of:

x∗ Ptransapertµpert sin(2π(mPtrans − t0)/Ppert) δt ≈ − ≈ − (28) vtrans − v∗ 2πatrans

mi where we have neglected v∗ since it is typically much smaller then vtrans and µ = . For the standard i ∑ mi deviation of the TTV signal we ﬁnd:

1 D 2E 2 Ptransapertµpert σ = (δt) = 3 (29) 2 2 πatrans

22 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 e = 1 − (1 + j−1) Ptrans < 1 being the fractional distance Ppert to the resonance one can calculate the TTVs using the change in orbital frequency to the ﬁrst order in the eccentricity:

( + ) ˙ n 1 e cos f ² θ = 3 ≈ n0 + δn + 2en0 cos(λ − v) (30) (1 − e²) 2

Where n is the mean motion, n0 is the unperturbed mean motion, f the true anomaly, λ the mean longitude and v 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:

5 da 2a 2 e = 3 (31) de a 2 − 1 and leads to:

mpert 2 −3 δt ' ( )µtranse Ptrans (32) mtrans and the other term which is eccentricity dominated gives a timing variation ( where we have assumed that the heavier planet is the transiting one):

−1 δt ' µperte Ptrans (33)

Planets with a larger period ratio have the timing deviations become proportional to:

P3 ∼ trans δt µpert 2 (34) Ppert

One can note that for all of these equations (in the case of M∗ Mpert) the TTV is directly proportional to the perturbing mass.

23 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):

P mpert δt ∼ (35) 4.5j mpert + mtran

−4 −2 Plib ∼ 0.5j 3 µ 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 Figure 14: TTV (top) and TDV (bottom) signal for KOI-872.02 [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 104 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 14.

25 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. No- tation conventions are taken from ﬁgure 15. Because of the moon the planet is displaced by an amount 0 0 0 x2 and the timing would thus be off by x2 divided by the planet-moon orbital velocity projected on x2, vB⊥.

0 x ( f ) δt( f ) = 2 (37) vB⊥

where f is the true anomaly for the planet in the moon-planet system and for the RMS, δTTV, we ﬁnd:

aw δTTV = √ (38) 2vB⊥ 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:

1 2 −1 1 aP aS MS MPRV ζT(eP, vP) δTTV = √ p (39) 2 G(M∗ + MPRV ) Υ(eP, vP)

26 where:

2 1 (1 − e ) 4 q 3 S 2 2 2 2 ζT = eS + cos(2vS)(2(1 − eS) − 2 + 3eS) (40) eS s −e cos v 2(1 + e sin v ) Υ = cos(arctan( P P )) P P − 1 (41) + 2 1 eP sin vP (1 − eP)

where the subscript P denotes the exoplanet’s orbit around the central star and the subscript S de-

notes the exomoon’s orbit around the exoplanet. MPRVis the combined mass of the exomoon and exoplanet.

The same goes for the transit duration, τ:

1 τ ∝ (42) vP⊥

where:

vP⊥ = vB⊥ + vW⊥ (43)

Here it is clear that vB⊥remains relatively constant whilst, depending on the moon’s position, vW⊥ 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:

s r 2 aP MS τ¯ ζD(eS, vS) δTDV = √ (44) aS MPRV (MPRV + M∗) 2 Υ(eP, vP) where:

s 1 + e2 − e2 cos(2v ) = S S S ζD 2 (45) 1 − eS

If we were to consider the TTV effect alone we would ﬁnd that the signal is degenerate and only solves

for MSas. 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 14.12 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 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]:

eP δt = trans (46) π

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 an- Figure 16: TTVs for GJ436b observed by Knutson et al. [2011] at different wavelengths (blue for visible, red for alyze actual transit timings but are rather to esti- IR) mate and compute putative TTV signals.

This chapter on TTVs was largely taken after Agol et al.[2005].

28 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 Plan- etesImals 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 22’ × 22’ (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 15’ × 15’ 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 Table 2: Transit Timings from literature

Epoch Timing BJDTBD Reference 0 4964.944779 ± 0.000799 Kundurthy et al.[2011] 10 4980.748721 ± 0.000263 Kundurthy et al.[2011] 10 4980.748740 ± 0.000390 Kundurthy et al.[2011] 12 4983.909490 ± 0.000228 Kundurthy et al.[2011] 12 4983.909552 ± 0.000381 Kundurthy et al.[2011] 22 4999.713442 ± 0.000253 Kundurthy et al.[2011] 217 5307.892454 ± 0.000271 Kundurthy et al.[2011] 246 5353.724539 ± 0.000307 Kundurthy et al.[2011] 265 5383.752143 ± 0.000260 Kundurthy et al.[2011] 1 4966.525207 ± 0.000351 Berta et al.[2011] 10 4980.748682 ± 0.000104 Berta et al.[2011] 12 4983.909507 ± 0.000090 Berta et al.[2011] 22 4999.713448 ± 0.000155 Berta et al.[2011] 222 5315.794564 ± 0.000066 Berta et al.[2011] 229 5326.857404 ± 0.000110 Berta et al.[2011] 234 5334.759334 ± 0.000066 Berta et al.[2011] 222 5315.794196 ± 0.00042 Sada et al.[2010] 241 5345.822026 ± 0.00014 Sada et al.[2010] 246 5353.724086 ± 0.00036 Sada et al.[2010] 253 5364.787456 ± 0.00029 Sada et al.[2010] 222 5315.794502 ± 0.000047 Bean et al.[2010] 10 4980.748570 ± 0.00015 Carter et al.[2011] 12 4983.909820 ± 0.00016 Carter et al.[2011] 15 4988.650808 ± 0.000049 Carter et al.[2011] 24 5002.874670 ± 0.00019 Carter et al.[2011] 193 5269.962990 ± 0.00016 Carter et al.[2011] 205 5288.928200 ± 0.0011 Carter et al.[2011] 210 5296.830130 ± 0.00023 Carter et al.[2011] 222 5315.794850 ± 0.00023 Carter et al.[2011] 222 5315.794693 ± 0.000080 Carter et al.[2011] 224 5318.955230 ± 0.00017 Carter et al.[2011] 246 5353.723870 ± 0.00018 Carter et al.[2011] 248 5356.884950 ± 0.00015 Carter et al.[2011] 253 5364.787000 ± 0.00015 Carter et al.[2011] 260 5375.849970 ± 0.00013 Carter et al.[2011] 265 5383.752050 ± 0.00013 Carter et al.[2011] 270 5391.654105 ± 0.000059 Carter et al.[2011] 220 5312.633877 ± 0.000085 Désert et al.[2011] 221 5314.214231 ± 0.000098 Désert et al.[2011] 260 5375.850100 ± 0.0001 Croll et al.[2011] 286 5416.940400 ± 0.0001 Croll et al.[2011] 286 5416.940200 ± 0.0004 Croll et al.[2011] 291 5424.842300 ± 0.0001 Croll et al.[2011] 315 5462.772200 ± 0.0001 Croll et al.[2011] 30 Figure 17: The TRAPPIST telescope at La Silla. See http://www.ati.ulg.ac.be/TRAPPIST

p(x) = p(x, y)dy (48) ˆ

p(x, y) p(y)p(x|y) p(y|x) = = (49) p(x) 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 ntot steps works we will give a ﬂow chart to illustrate it better:

1. Initialize a Markov chain with a set of model parameter x0 obtained from a prior probability distribution and set n = 0

2. Generate a new set of parameters x0 obtained from a candidate transition probability function 0 q(x |xn) which is a Gaussian centered around xn.

2 0 2 3. Acquire the goodness of the ﬁt by calculating χ (x ) and χ (xn)

31 Table 3: Continued from table 2

220 5312.633890 ± 0.000090 Gillon et al., in prep. 222 5314.214249 ± 0.000085 Gillon et al., in prep. 453 5680.86801 ± 0.00011 Gillon et al., in prep. 454 5682.448411 ± 0.000091 Gillon et al., in prep. 455 5684.02890 ± 0.00010 Gillon et al., in prep. 456 5685.60932 ± 0.00012 Gillon et al., in prep. 457 5687.189798 ± 0.000083 Gillon et al., in prep. 458 5688.770014 ± 0.000091 Gillon et al., in prep. 459 5690.350548 ± 0.000082 Gillon et al., in prep. 460 5691.930995 ± 0.000093 Gillon et al., in prep. 461 5693.511322 ± 0.000070 Gillon et al., in prep. 463 5696.67215 ± 0.00011 Gillon et al., in prep. 464 5698.25240 ± 0.00012 Gillon et al., in prep. 465 5699.832751 ± 0.000088 Gillon et al., in prep. 466 5701.413274 ± 0.000092 Gillon et al., in prep. 574 5872.09687 ± 0.00010 Gillon et al., in prep. 575 5873.67734 ± 0.00011 Gillon et al., in prep.

0 4. Determine the acceptance probability α(x |xn):

1 α(x0|x ) = min exp(− (χ2(x0) − χ2(x ))), 1 (50) n 2 n

5. Draw a random number from a uniform distribution between 0 and 1 0 0 0 6. If u 6 α(x |xn) then xn+1 = x , if u > α(x |xn) then xn+1 = xn 7. Set n = n + 1

8. Go to step 2 until n > ntot If the Markov chain is reversible, i.e. if:

f (x)p(x|x0) = f (x0)p(x0|x) (51) where f (x) is the equilibrium distribution and p(x|x0) = q(x|x0)α(x|x0), and aperiodic, i.e. if there is a non-zero probability for the current state to be the same as the previous, and irreducible, i.e. if it is possible to get from the current state to every other state with a non-zero probability, then it can be proven that the chain will converge to a stable equilibrium distribution for the model parameters. Once a chain is completed we would like to run another one to check whether or not the MCMC simulation has converged and is well-mixed. We can achieve this by using a Gelman-Rubin test [Gelman and Rubin, 1992]. Eventually after the convergence has been checked and found to be good, we end up with −→ −→ a posterior probability function p( θ | d ) which can then be marginalized over all but two parameters to check for correlations and over all but one to obtain the probability distribution of a certain parameter. And here lays the one of the two disadvantages of the MCMC method. The evaluation of the integral in equation 49 is proving to be a huge computational effort. The other and more criticized disadvantage is that the MCMC approach requires a prior probability distribution. Most of the time, however, the

32 observational data provides a strong enough constrain on itself so that the choice of the prior does not inﬂuence the posterior. When one suspects the data does not constrain the model in a strong manner it is required to check whether the posterior depends on the choice of the prior.

To implement a MCMC algorithm we need a set of model parameters to ﬁt the data to. The model parameters, often called jump parameters in the framework of MCMC algorithms, used in our analysis are:

• The transit depth, dF cos i b0 = a p • The modiﬁed impact parameter, R∗ • The transit width, W

• The mid-transit time T0 • The orbital period, P

1 • The modiﬁed RV semi amplitude, K2 = KP 3

• The limb darkening coefficients for each filter c1and c2 The code uses as jump parameters specific combinations of parameters to reduce the correlations and speed up the convergence of the algorithm. For each light curve separately we assumed a trend caused by some sort of systematic effects. Examples for this are the drift on the detector pixels, meridian flips, long-term variability of the star, ... To model these effects we often assume a quadratic trend in the affected parameters ( dx, dy, dt, ... ) and the coefficients for these parameters are also adjusted at each step. Most of the time though not all sources of the noise can be modeled and we are left with a fair amount of correlated, often called red, noise. Taking into account this noise is required to achieve reliable uncertainties on the model parameters. If we would measure uncorrelated, often called white, noise we would find for the standard deviation for the residuals of the observed flux compared to the calculated flux a value σ1. After binning the residuals in M bins containing N points the expected standard deviation would be:

r σ1 M σN = √ (52) N M − 1

Usually σN is larger since we do have some contribution from correlated noise. To account for this we introduce a factor βr by which the observed standard deviation is larger than the expected. To get an accurate estimate of the factor we run each MCMC simulation twice. Once to determine the photometric properties and to estimate the βr and afterwards to obtain the model parameters and their correct uncertainties.

Individual analysis

We start our data analysis by investigating the variability of the derived parameters for each individual light curve. For each light curve the jump parameters were dF, b, W and T0. Since the period cannot be derived from a single light curve we had to ﬁx the period of the exoplanet in our simulations. In all of our simulations we kept the eccentricity to zero because of knowledge obtained in previous papers on the planet under investigation. For each individual light curve a baseline is chosen and we will see that for most transits a quadratic trend in time is enough to achieve a minimal BIC (Bayesian Information Criterium)[Schwarz, 1978]. The BIC describes the likelihood of a given model similar to chi-square.

33 It differs from the fact that the BIC incorporates Occam’s razor where the simplest model often is the best:

BIC = χ2 + k log N (53)

Where k is the number of free parameters and N the number of data points.

Global analysis

Secondly we want to put the strongest constraint on our system and start a global MCMC analysis using all data available. Using all the observed transits, (if available) radial velocity measurements and timings we will try to determine the system parameters with the best precision. As it is the case with all the MCMC analysis we ran the simulations twice. First to determine the scaling factors and secondly to retrieve the system parameters.

Transit analysis

Eventually we run another MCMC simulation using only the transit light curves and timings obtained by other research groups.We ﬁx the period and T0 to the values as they were obtained form the global analysis. We do this to obtain the TTV signal and to remove any possible biases from other techniques.

3.3 TTV simulator

During the course of this thesis we developed a program that is able to exclude planetary perturbers at a four sigma level based on the RMS of the observed signal. The goal was to put constraints on the system and to determine the sensitivity of the TTV technique in the given systems.

We started out by learning the details of the N-body simulator, MERCURY, developed by Chambers and Migliorini[1997]. For our simulations we worked in a central reference frame with the star as central body orbited by two additional bodies: the transiting planet in mid-transit and thus in the line of sight between the star and observer and a planetary perturber in anti-conjunction. We choose to simulate an outer perturbing planet since other detection techniques are biased to closer-in exoplanets. Since planets appear to be well aligned is it very likely all planetary partners closer to the star can be detected if they are above the observation threshold. The period for the perturbing exoplanet ranged from 1.5 to 3.1 times the period of the transiting planet with typical steps of 0.005 period ratios. We choose this period range based on figure 18. We can see that periods below the 3:2 resonance have a fairly low occurrence rate but mostly because the systems produce long term instability. Above the 3:1 resonance there is another cutoff for which we no longer saw the need to simulate. In addition is the TTV signal produced by perturbers on such large orbits very small and difficult to detect for secondary planets lighter than Jupiter. The mass of the perturber was chosen large enough so that the simulated TTV signal compared to the RMS caused by computational errors. But we needed to choose it small enough not to cause any significant changes in the orbit of the transiting planet or any chaotic behavior. Another reason from choosing the perturber mass small enough is to allow for scaling with the RMS of the TTV signal in the resonances (explained in the next paragraph). After a single 3-body simulation was ran and we obtained the output from MERCURY at set time steps we search for the correct minimum in the coordinate (since

34 Figure 18: Period statistics from all planet pairs binned in a histogram from Fabrycky et al.[2012]

we have a central coordinate system = 0, is the spatial coordinate for the mid transit time). From this local minimum we calculated the mid transit timings using Newton’s method of minimization where we tried to minimize the projected distance from centrum of the star. Which amounts to solving:

g(x, x˙, y, y˙) = xx˙ + yy˙ = 0 (54)

Where x and y are the coordinates in the plane of the sky. With Newton’s method we calculate the time step for which we have to integrate our system forward or backward in time:

∂g δt = −g( )−1 (55) ∂t

Once the x and y coordinates converges we have found our mid transit time. With these transit timings we calculated the period and the RMS on the period with a χ2 minimization method assuming a linear ﬁt.

To summarize: our program, written in FORTRAN 90, initializes all the required input files for MER- CURY and then calls MERCURY to simulate a given system. Afterwards Newton’s method is used to find the mid transit timings and with a χ2 minimization we find the period of the transiting planet and the RMS of the TTV signal. Eventually we rewrite the input files for a new simulation and the process start all over again until the final period is reached after which the output of the software is written as the RMS of the simulated TTV signal is function of the period of the perturber for a pre-set perturber mass.

To produce the plot we made a short script in Python where we assumed the RMS of the TTV signal to be linearly proportional to the mass of the perturbing planet and scaled to be four times the observed RMS, thus leading to the exclusion at a four sigma level of the planetary perturbers above the curve produced in the plots. If we take a look at equations 32, 33 and 34 we notice that for non-resonant planets the assumption of the TTV signal to be directly proportional to the mass of the perturbing planet is valid. In the case of resonant planets, however, we found that this is only the case if the mass of the perturber is negligible compared to the mass of the perturbing planet (equation 36). That is why we assume small masses for the perturbing planets in our simulations.

35 4 Results and discussion

4.1 WASP-43b

The first planet we will investigate is WASP-43b which is primarily known for its ultra-short period of 0.8134746 days. The planet orbits a K7V star with an apparent magnitude of 12.4 in the constellation Sextans. It is an interesting planet because whilst studying it we might learn more about planetary migration theory given its rather unique period. The discovery of the planet goes to WASP (as the name might suggest) with follow-up photometry by TRAPPIST and EulerCAM and RV measurements by the spectrograph CORALIE [Hellier et al., 2011]. The planet has been extensively studied by Gillon et al. [2012] in which is was shown that the original proposed solution for the stellar mass and radius was not correct leading to relatively large errors on the other physical parameters. Among other things they investigated the atmospheric properties of WASP-43b and suggested a model in which there is a poor heat redistribution to the night side of the planet. With a new season of observations we hope to confirm and possibly improve the findings of both papers regarding the system parameters.

Photometry

We present a total 27 transits: three already published from Euler, twenty already published from TRAP- PIST and four new transits by TRAPPIST (Hellier et al.[2011], Gillon et al.[2012]). We use 23 already published RV measurements from CORALIE (Hellier et al.[2011], Gillon et al.[2012]).

WASP-43 is a special case compared to the other stars under study regarding the photometry. Normally when observing a star we want to keep that star on the same few pixels to improve photometric precision. This is achieved by the software guiding system installed on TRAPPIST which send pointing corrections directly to the telescope. The software, however, could not be used on WASP-43 since it was in an area of the sky that was not covered in the used catalogue (GSC1.1). This resulted in a drift of the star on the detector camera. As it was mentioned in the section 3.2, the MCMC code allows us to model this by introducing a spatial trend in the transit model. In table 4 we can see that for most transits it is enough to introduce a quadratic trend in time to achieve the lowest BIC. We see however that the observations of the new season require a more complex baseline in all cases and have a high correlated noise. For the first new transit we observed at a rather high air mass ranging from 2.5 to 1.1 and observed for a very long time the out-of-transit flux requiring the focus to be reset and causing slight deviations. The next three transits are all observed with a meridian flip. The second and third transit show some severe focus issues with the width of the stellar image (PSF) jumping to 20 pixels, most likely caused by clouds. For the second transit this lead to the closing of the dome only to reopen a couple minutes later and in effect removing the egress from the observation.

System elements

After a standard data reduction (bias, dark, flatfield) was applied on the original observational data we investigated each transit separately using the MCMC code to look for any sort of variability. The results from this analysis, where we kept the period fixed to 0.8134775 ± 0.0000007 [Gillon et al., 2012], can be found in table 5 and in figure 19. For our global analysis we assume that the star has a mass of 0.717 ± 0.025 M as was derived by the Geneva stellar evolution tracks. As can be seen from table 6 (and figure 20) the uncertainties for all the model parameters have gone down thanks to the additional

36 Table 4: 27 transits for WASP-43b where for the baseline function tN denotes a N-order polynomial function in N N time, xy in x and y positions, f in the FWHM and o denotes an offset caused by a meridian ﬂip. NP denotes the number of data points in the light curve and BB is a blue-blocking ﬁlter.

Epoch Instrument Filter Np Baseline -232 TRAPPIST I+z 408 t² -221 TRAPPIST I+z 466 t² -216 TRAPPIST I+z 484 t² -205 TRAPPIST I+z 572 t² -205 EulerCAM Gunn-r’ 111 t²+xy² -200 TRAPPIST I+z 420 t² -194 TRAPPIST I+z 284 t² -184 TRAPPIST I+z 246 t² -178 TRAPPIST I+z 289 t² -173 TRAPPIST I+z 290 t² -167 EulerCAM Gunn-r’ 114 t²+xy -156 EulerCAM Gunn-r’ 107 t² -146 TRAPPIST I+z 312 t² -119 TRAPPIST I+z 290 t² -103 TRAPPIST I+z 323 t² -102 TRAPPIST I+z 238 t² -91 TRAPPIST I+z 246 t² -75 TRAPPIST I+z 292 t²+xy² -70 TRAPPIST I+z 292 t² -54 TRAPPIST I+z 329 t² -43 TRAPPIST I+z 282 t² -32 TRAPPIST I+z 250 t² 0 TRAPPIST I+z 608 t² 224 TRAPPIST I+z 654 t²+f+xy² 245 TRAPPIST BB 626 t²+o 256 TRAPPIST I+z 573 t²+xy+o 299 TRAPPIST I+z 359 t²+o

37 Table 5: Model parameters from the individual MCMC analysis. Epochs were calculated relative to 5726.54336 BJDTBD

Epoch dF b W (days) T0(BJDTBD) +0.0018 +0.036 +0.0018 +0.00046 -232 0.0290−0.0018 0.670−0.047 0.0526−0.0018 5537.81630−0.00054 +0.0011 +0.032 +0.0012 +0.00022 -221 0.0254−0.0010 0.660−0.033 0.0514−0.0012 5546.76488−0.00022 +0.0011 +0.025 +0.0010 +0.00018 -216 0.02689−0.00099 0.715−0.028 0.05195−0.00096 5550.83218−0.00018 +0.00090 +0.030 +0.0010 +0.00020 -205 0.02549−0.00092 0.691−0.035 0.05133−0.00092 5559.78059−0.00020 +0.00074 +0.030 +0.00068 +0.00016 -205 0.02433−0.00072 0.642−0.033 0.04924−0.00070 5559.78089−0.00016 +0.00090 +0.052 +0.00086 +0.00016 -200 0.02168−0.00089 0.550−0.069 0.04794−0.00080 5563.84781−0.00015 +0.0012 +0.052 +0.0012 +0.00020 -194 0.0248−0.0010 0.626−0.053 0.0513−0.0011 5568.72831−0.00021 +0.00075 +0.025 +0.00071 +0.00017 -184 0.02531−0.00078 0.643−0.030 0.05027−0.00075 5576.86378−0.00017 +0.00082 +0.029 +0.00071 +0.00014 -178 0.02388−0.00079 0.624−0.037 0.05035−0.00068 5581.74408−0.00015 +0.00099 +0.035 +0.0013 +0.00022 -173 0.02512−0.00097 0.675−0.046 0.0495−0.0013 5585.81263−0.00023 +0.00091 +0.056 +0.00085 +0.00017 -167 0.02358−0.00095 0.535−0.077 0.04814−0.00090 5590.69247−0.00017 +0.00089 +0.028 +0.00086 +0.00018 -156 0.02659−0.00083 0.658−0.034 0.05241−0.00078 5599.64040−0.00018 0.00062 0.028 0.00063 0.00014 -146 0.02475−0.00071 0.642−0.035 0.04902−0.00060 5607.77520−0.00014 0.00075 0.034 0.00076 0.00016 -119 0.02439−0.00078 0.612−0.041 0.05059−0.00075 5629.74012−0.00016 0.00079 0.026 0.00088 0.00021 -103 0.02566−0.00084 0.699−0.031 0.05090−0.00088 5642.75429−0.00020 0.0011 0.027 0.0010 0.00023 -102 0.0283−0.0011 0.661−0.036 0.05212−0.00099 5643.56861−0.00024 0.0014 0.040 0.0014 0.00031 -91 0.0240−0.0013 0.706−0.050 0.0513−0.0014 5652.51564−0.00032 0.0018 0.034 0.00097 0.00024 -75 0.0277−0.0017 0.606−0.048 0.04992−0.00099 5665.53282−0.00026 0.00095 0.029 0.0013 0.00025 -70 0.02727−0.00089 0.676−0.034 0.0522−0.0012 5669.59974−0.00026 0.0010 0.047 0.00094 0.00020 -54 0.02402−0.00097 0.604−0.065 0.05051−0.00095 5682.61591−0.00020 0.00062 0.018 0.00057 0.00012 -43 0.02738−0.00060 0.669−0.020 0.05032−0.00063 5691.56379−0.00012 0.00079 0.024 0.00071 0.00014 -32 0.02300−0.00073 0.663−0.029 0.05050−0.00064 5700.51245−0.00015 0.00090 0.032 0.0011 0.00028 0 0.02503−0.00098 0.669−0.042 0.0508−0.0012 5726.54388−0.00026 0.0019 0.080 0.0020 0.00037 224 0.0185−0.0021 0.611−0.21 0.0457−0.0022 5908.76164−0.00038 0.0013 0.046 0.0011 0.00032 245 0.0228−0.0012 0.624−0.057 0.04912−0.00098 5925.84398−0.00030 0.0020 0.027 0.0013 0.00023 256 0.0414−0.0018 0.642−0.030 0.0562−0.0012 5934.79155−0.00024 0.0013 0.044 0.0011 0.00027 299 0.0276−0.0012 0.638−0.056 0.0502−0.0010 5969.77108−0.00028

observations compared to the results for the discovery paper. Since the observations in the new season were generally of poor quality no improvement in model parameters, except for the period thanks to the large baseline, can be seen compared to the paper by Gillon et al.[2012]. Since the largest fraction of the errors on physical parameters is caused by the uncertainty of the stellar model, their uncertainties have been reduced drastically with the use of a ﬁxed and corrected stellar mass compared to those of the discovery paper. After comparing our results with those obtained by Gillon et al.[2012] we can conclude that the agreement is excellent and the four new transits were able to only improve the transit ephemeris. As it was done in the discovery paper we will calculate the tidal inspiral time using:

0 ∼ 1 Q∗ a 5 M∗ τa = ( ) (56) 48 n R∗ MP from Levrard et al.[2009] to confirm their findings. Using three different values for the quality factor ( 106, 107and 108) we find respectively 8.1 Myr, 81 Myr and 810 Myr confirming the findings of the discovery paper. These findings will be discussed in the next section.

38 Figure 19: Variations in the impact parameter b (top) and transit duration W (bottom) for WASP-43b compared to the values and their one sigma intervals (solid and dashed lines) for TRAPPIST transits (blue) and EulerCAM (red).

Table 6: MCMC system parameters for WASP-43(b) as they were obtained from the global analysis

Deduced parameters Hellier et al. (2011) Gillon et al. (2012) Orbital period P (d) 0.81347460±0.00000048 0.813475±0.000001 0.81347753± 0.00000071 +0.000079 T0 - 2450000 (BJDTDB) 5726.542919−0.000083 5528.86774±0.00014 5726.54336±0.00012 +0.00023 Transit width W (d) 0.05034±0.00026 0.0483±0.0011 0.05037−0.00021 +0.011 +0.04 Impact parameter b (R∗) 0.662−0.012 0.66−0.07 0.656 ± 0.010 +0.00027 +0.00024 Transit depth dF 0.02541−0.00028 0.0255±0.0012 0.02542−0.00025 −1 +5.6 +5.5 RV semi-amplitude K (m s ) 547.9−5.5 550.3±6.7 547.9−5.4 +0.077 +0.61 +0.079 Stellar density ρ∗(ρ ) 2.378−0.075 2.70−0.36 2.410−0.075 +0.059 Surface gravity log g∗(cgs) 4.641±0.011 4.646−0.044 4.645±0.011 Stellar mass M∗(M ) 0.717±0.025(fixed) 0.58±0.05 0.717±0.025(fixed) +0.034 +0.010 Stellar radius R∗(R ) 0.671±0.011 0.598−0.042 0.667−0.011 Roche limit ar(AU) 0.00772±0.00017 0.00768± 0.00016 Orbital semi-major axis a (AU) 0.01526±0.00018 0.0142±0.0004 0.01526± 0.00018 +0.032 +0.030 a/ar 1.977−0.031 1.986−0.029 +0.053 a/R∗ 4.896±0.052 4.918−0.051 +0.22 +1.3 Orbital inclination i (°) 82.23−0.21 82.6−0.9 82.33± 0.20 +0.0060 Eccentricity 0 (fixed) 0 (fixed) 0.0035−0.0025 +0.089 +0.73 +0.084 Planetary density ρpl(ρJup) 1.802−0.083 2.21−0.41 1.826−0.078 +0.052 +0.052 Planetary mass Mpl(MJup) 2.034−0.052 1.78±0.10 2.034−0.051 +0.07 Planetary radius Rpl(RJup) 1.041±0.020 0.93−0.09 1.036± 0.019 +40 +40 Teq(K) 1443−39 1370±70 1440−39

39 Figure 20: Top: MCMC transit model superimposed on the photometry binned per interval of two minutes and folded on the best-ﬁt transit ephemeris. Bottom: residuals. Their standard deviation is 307 ppm when considering the data between -0.05 and 0.05 dt, demonstrating the excellent photometric precision of this combined light curve.

TTVs

For the TTVs we did a transit analysis of the light curves with the period ﬁxed to 0.81347460±0.00000048

days and the T0 to 5726.542919 ± 0.000082 BJDTDB as was determined from the global analysis and found the RMS of the TTV signal to be 44.7 seconds, fully consistent with the linear regression we performed on the timings obtained from the individual analysis (0.81347455± 0.00000030 days and

5726.542926 ± 0.000048 BJDTDB) to construct the O-C diagram in ﬁgure 21. The transit analysis provided us with similar values and are displayed in the same plot. We notice that there are not any large differences between the two plots. For a couple of timings the accuracy increased but for most of them the errors are bigger. This is because forcing every transit into the same model leads to a general increase in red noise compared to the individual analysis. In the individual analysis of the transits the red noise is partly ’swallowed’ by the transit model and possible baseline.

One can see that several timings show a slight deviation from the linear ephemeris. We can suggest that the reason for this is stellar variability. It has already been shown by Knutson et al. (see figure 16) that transit timings obtained from light curves observed in the infrared (where stellar spots are less pronounced) are a better fit to the linear ephemeris. To exclude the possibility of a hidden periodic signal caused by a perturber we investigated the variability of the impact parameter, transit duration, transit timing residuals and radial velocity residuals using a Lomb-Scargle periodogram [Scargle, 1982]. Not a single significant period (except for some artifacts) could be determined from this analysis. Using our software and based on the previous results we constructed a detectability domain for WASP-43b in figure 22 where we have simulated the systems for a duration of 366 days spanning the total length of observations and a perturber of 30 Earth masses. One can see that we can exclude super-Earth massed perturbers in the resonances.

40 Figure 21: TTVs obtained from individual (top) and transit (bottom) analysis for WASP-43b for TRAPPIST transits (blue) and EulerCAM (red).

Figure 22: Detectability domain for WASP-43b for a perturber with e = 0 (blue) and e = 0.05 (red)

41 Table 7: Seven transits for WASP-46b where for the baseline function tN denotes a N-order polynomial function in N N time, f in the FWHM and a denotes in the air mass. NP denotes the number of data points in the light curve and BB is a blue-blocking ﬁlter.

Epoch Instrument Filter Np Baseline 0 TRAPPIST I+z 129 t² 37 EulerCAM Gunn-r’ 76 t² 205 TRAPPIST BB 105 t² 295 TRAPPIST BB 549 t²+f² 302 TRAPPIST BB 408 t² 323 TRAPPIST BB 480 t² 330 TRAPPIST BB 394 t²+a²

4.2 WASP-46b

The next planet we investigate is WASP-46b orbiting a mV = 12.9, G6V star located in the constellation Indus. The discovery of this planet goes to WASP with follow-up measurements from TRAPPIST, EulerCAM and CORALIE [Anderson et al., 2012]. WASP-46b is a heavy planet with a rather short period (2.101 MJup and 1.4303700 days) for which a strong rotation modulation was found, together with the projected trotational velocity this lead to the determination of the inclination of the stellar spin axis (40±41°). Another thing to note is the star showed weak emission in the Ca H+K regions of its spectrum, indicative of an active stellar chromosphere.

Photometry

As stated before a total of seven transits were used: two already published transits from TRAPPIST, one already published from EulerCAM, four new transits from TRAPPIST and sixteen already published radial velocity measurements by CORALIE [Anderson et al., 2012]. The observations are in general rather good except the last one which showed a slight deviation at the egress. Observations were made at a rather high air mass causing the deviation. Modelling this was easy using the MCMC where we simply had to introduce this trend in the baseline.

System elements

For the individual analysis of the transits we kept the period fixed to 1.4303723 ± 0.0000011 days as was obtained from a preliminary global analysis and the results are given in table 8 and figure 23. After a final global analysis was performed (of which the results are displayed in table 9 and figure 24) we compared the values for the system parameters to the ones from the individual analysis and no significant deviations could be found. The global analysis provided a stronger constraint on the model parameters than the values obtained in Anderson et al.[2012]. But if we compare the derived physical values such as planetary mass, stellar radius, semi-major axis, etc we see that the errors are much larger in our analysis. This can be explained by the empirical stellar model of Enoch we used which might have a different error propagation than the one in Anderson et al.[2012]. If we compare the values we find for the planetary mass for both hot Jupiters under study, we see that they are (coincidentally) almost identical, the values for the radius however differ heavily. This can be caused by a denser core present in WASP-43b which is a reasonable assumption since the metalicity of the its parent star is much higher than WASP-46. Another contributing factor for the larger radius is the higher incoming stellar

42 Table 8: Model parameters for the individual MCMC analysis where the epoch was calculated relative to 5396.60722

Epoch dF b (R∗) W (d) T0-2450000 (BJDTBD) +0.00097 +0.035 +0.0016 +0.00035 0 0.02018−0.0011 0.668−0.046 0.0675−0.0017 5396.60748−0.00036 +0.00097 +0.022 +0.0015 +0.00031 37 0.02119−0.0011 0.737−0.031 0.0704−0.0014 5449.53086−0.00031 +0.0028 +0.066 +0.0046 +0.0017 205 0.0186−0.0024 0.704−0.095 0.0709−0.0040 5689.8355−0.0016 +0.00062 +0.020 +0.0013 +0.00027 295 0.02055−0.00060 0.727−0.023 0.0694−0.0013 5818.56663−0.00026 +0.00051 +0.018 +0.0011 +0.00024 302 0.02000−0.00051 0.739−0.019 0.0713−0.0011 5828.57948−0.00024 +0.00072 +0.020 +0.0018 +0.00037 323 0.02025−0.00072 0.767−0.023 0.0686−0.0017 5858.61791−0.00037 +0.0014 +0.030 +0.0024 +0.00065 330 0.0211−0.0013 0.770−0.033 0.0722−0.0023 5868.63066−0.00063

ﬂux which causes the planet to bloat. A good indicator for the stellar ﬂux is the equilibrium temperature of the planet. Using the relation found by Enoch et al.[2012]:

0.9 R ∝ Teq (57) we see that indeed at least a part of the larger radius can be explained by this mechanism given that the equilibrium temperature of WASP-46b is indeed higher. In our simulations the equilibrium temperature of a planet is given by:

r R∗ T = T (58) eq e f f 2a

Where we have assumed a bond albedo of 0 and Te f f is the effective temperature of the star. Putting the bond albedo to zero makes sure all the incident radiation is absorbed and none is scattered back into outer space. For the orbital decay timescale we find 37 Myr, 370 Myr and 3.7 Gyr for the same values for the quality factor. The very short timescales given by the calculations for the orbital decay timescale suggest that a quality factor of 106 or 107 might not be justified for WASP-43b and/or WASP- 46b conform with the findings of Penev and Sasselov[2011] which obtained values for main-sequence stars of 108-109 using numerical simulations. But contradicts with the value, 106, that has been obtained from our solar system by Zhang and Hamilton[2008] for the sun.

TTV’s

Eventually we investigated WASP-46b for TTVs using the timings obtained from the individual analysis after applying linear regression to them and from the transit analysis as one can see in ﬁgure 25. For

the transit analysis we kept the period and T0 fixed to 1.4303723 ± 0.0000011 days and 5396.60722 ± 0.00029 BJDTDB. When applying linear regression on the timings from the individual analysis we find the period and T0 to be 1.43037190± 0.00000097 and 5396.60726± 0.00024 BJDTDB which is once again fully consistent with the transit analysis result. One can see that the timings errors are worse for most timings but better for the rather inaccurate third transit in the case of global analysis for the same reason as it was for WASP-43b. Different in this case is that not a single timing deviates more than one sigma from the expected linear ephemeris. This rather nice fit causes the RMS of the signal, 36.5 seconds, to be lower than that of WASP-43b. Once again several periodograms were plotted but none found a period with a significant p-value. As it was the case for WASP-43b we use our software to plot the detectability domain in figure 26. We simulated the systems over a timescale of 473 days spanning the total time over

43 Figure 23: Variations in the impact parameter b (top) and transit duration W (bottom) for WASP-46b compared to the values and their one sigma intervals (solid and dashed lines) for TRAPPIST transits (blue) and EulerCAM (red).

Table 9: MCMC system parameters for WASP-46(b) as they were obtained from the global analysis

Deduced parameters Anderson et al. 2012 Orbital period P (d) 1.4303723±0.0000011 1.4303700 ± 0.0000023 +0.00027 T0 - 2450000 (BJDTDB) 5396.60722−0.00028 5 392.31553 ± 0.00020 Transit width W (d) 0.07010±0.00067 0.06973 ± 0.00090 +0.011 Impact parameter b (R∗) 0.740−0.013 0.737 ± 0.019 +0.00035 Transit depth dF 0.02034−0.00034 0.02155 ± 0.00049 RV semi-amplitude K (m s−1) 386±11 387 ± 10 +0.067 Stellar density ρ∗(ρ ) 1.186−0.061 1.24 ± 0.10 +0.020 Surface gravity log g∗(cgs) 4.474−0.020 4.493 ± 0.023 +0.082 Stellar mass M∗(M ) 0.911−0.078 0.956 ± 0.034 Stellar radius R∗(R ) 0.915±0.032 0.917 ± 0.028 +0.00050 Roche limit ar(AU) 0.01022−0.00048 Orbital semi-major axis a (AU) 0.02409±0.00071 0.02448 ± 0.00028 +0.067 a/ar 2.356−0.064 +0.11 a/R∗ 5.657−0.098 5.74 ± 0.15 +0.26 Orbital inclination i (°) 82.49−0.24 82.63 ± 0.38 Eccentricity 0 (ﬁxed) 0 (ﬁxed) +0.086 Planetary density ρpl(ρJup) 0.986−0.078 0.94 ± 0.11 Planetary mass Mpl(MJup) 2.03±0.13 2.101 ± 0.073 +0.050 Planetary radius Rpl(RJup) 1.272−0.049 1.310 ± 0.051 Teq,pl(K) 1671±50 1654 ± 50

44 Figure 24: Top: MCMC transit model superimposed on the photometry binned per interval of two minutes and folded on the best-ﬁt transit ephemeris. Bottom: residuals. Their standard deviation is 405 ppm when considering the data between -0.05 and 0.05 dt, demonstrating the excellent photometric precision of this combined light curve. which observations were conducted and a perturber of 30 Earth masses. This time we see in ﬁgure 26 that we can exclude planetary partner down to a couple of Earth masses in the early resonances.

45 Figure 25: TTVs obtained from individual (top) and transit (bottom) analysis for WASP-46b for TRAPPIST transits (blue) and EulerCAM (red).

Figure 26: Detectability domain for WASP-46b assuming a planetary perturber with e = 0 (blue) and e = 0.05 (red)

46 Figure 27: Transmission spectrum for GJ1214b as it was obtained by the HST. Taken from Berta et al.[2012]

Figure 28: Transmission spectrum for GJ1214b at shorter wavelengths obtained by Bean et al.[2010]

4.3 GJ1214b

The final planet we will investigate is GJ1214b and has already been extensively studied by other groups +0.54 primarily because of its nature as super-Earth (6.55−0.52 MEarth) on one of the shortest orbits known at +0.00056 the moment (0.001433−0.00062 AU). GJ1214b was one of the few super-Earths detected before the Kepler- era. The exoplanet orbits a rather dim M4.5 red dwarf with a V-band magnitude of 14.67. GJ1214b has been discovered by the MEarth project with follow-up photometry from a nearby 1.2 meter telescope and radial velocity measurements by HARPS[Charbonneau et al., 2009]. GJ1214b has been observed several times using transmission transit spectroscopy leading to the results as seen in figure 27 and 28. Figure 27 shows the transmission spectrum observed with the Hubble space telescope which concludes that the atmosphere consists for more than 50% of H2O or have a mean molecular weight of µ > 4 [Berta et al., 2012]. These observations were consistent with the observations by several other groups (Bean et al.[2010], Bean et al.[2011], Désert et al.[2011], Crossfield et al.[2011]) but difficult to reconcile with Croll et al.[2011] which suggested a lower mean molecular weight. It is also possible that a flat spectrum is obtained because clouds hide the underlying atmosphere and could cause the spectrum to be erased up to a certain pressure.

47 Table 10: Eight transits for GJ1214b where for the baseline function tN denotes a N-order polynomial function in time. NP denotes the number of data points in the light curve.

Epoch Instrument Filter Np Baseline 422 TRAPPIST I+z 201 t² 434 TRAPPIST I+z 246 t² 446 TRAPPIST I+z 235 t² 451 TRAPPIST I+z 159 t² 453 TRAPPIST I+z 284 t² 463 TRAPPIST I+z 229 t² 465 TRAPPIST I+z 305 t² 670 TRAPPIST I+z 366 t²

Photometry

The photometry obtained for GJ1214b behaved nicely and required no further trending then the usual quadratic temporal trend to account for systematics and low-amplitude stellar variability as can be seen from table 10.

System elements

Once again we performed eight individual transits analysis whilst keeping the period fixed to 1.5804049± 0.0000001 taken from Gillon et al, in prep. The results for this individual analysis are given in table 11 and figure 29. We notice that the individual transits are fully consistent with the global analysis, but we note this is primarily due to the large errors. For the global analysis the results are given in table 12 and figure 30. We note that almost all errors are larger than those obtained by Charbonneau et al.[2009] except for the period. We thank this to the huge baseline over which observations were made and the additional timings that were used from other works.

Using our obtained values we will once again calculate the tidal inspiral timescale. Given the unique nature of GJ1214b, however, we will ﬁrst calculate if it a priori will decay. To do this we will compare the critical and total angular momentum as is described by Levrard et al.[2009] in equations 1 and 2:

2 3 3 G M∗ Mpl 1 LC = 4( (Cpl + C∗)) 4 (59) 27 M∗ + Mpl

M∗ Mpl q Ltot = C∗ω∗ + q Ga(1 − e²) (60) M∗ + Mpl

Where C∗ and Cpl denote the polar moments of inertia. If we neglect the moment of inertia of the planet, which is a reasonable assumption since the mass and radius are negligible compared to those of the star, 2 and assume that C∗ = kM∗R∗ with k set to 0.06 for centrally condensed stars we ﬁnd[Levrard et al., 2009]:

kg.m2 L = 1.77x1040 (61) C s

48 Table 11: Model parameters for the individual MCMC analysis where the epoch was calculated relative to 4964.944747 BJDTBD

Epoch dF b (R∗) W (d) T0-2450000 (BJDTBD) +0.00052 +0.17 +0.00051 +0.00018 422 0.01158−0.00050 0.25−0.16 0.03613−0.00062 5631.87600−0.00018 +0.00083 +0.13 +0.0011 +0.00021 434 0.01443−0.00071 0.46−0.22 0.03778−0.00092 5650.84041−0.00021 +0.0050 +0.16 +0.0061 +0.00015 446 0.01353−0.00043 0.26−0.16 0.03636−0.0046 5669.80545−0.00014 +0.0010 +0.20 +0.00095 +0.00035 451 0.01168−0.00095 0.29−0.19 0.03436−0.00093 5677.70760−0.00033 +0.00071 +0.019 +0.00096 +0.00025 453 0.01411−0.00060 0.31−0.019 0.03806−0.00083 5680.96766−0.00024 +0.00091 +0.19 +0.0013 +0.00024 463 0.01342−0.00072 0.42−0.25 0.03685−0.00081 5696.67195−0.00021 +0.00066 +0.13 +0.0011 +0.00017 465 0.01381−0.00061 0.43−0.21 0.03718−0.00083 5699.83282−0.00017 +0.00067 +0.16 +0.00078 +0.00016 670 0.01338−0.00053 0.40−0.22 0.03639−0.00062 6023.81608−0.00015

kg.m2 L = 8.74x1039 (62) tot s

Where we chose the planetary rotational period to be 53 days as was suggested by Berta et al.[2011].

And indeed the condition for tidal decay is fulfilled ( LC> Ltot) and the tidal inspiral timescale is calculated to be 77.7 Gyr, 777 Gyr or 7.77 Tyr for a quality factor of respectively 106, 107and 108. The tidal inspiral timescale is of a very long nature and vastly exceeds the age of the universe. This suggests that the use of large quality factors is not justified in this case or that the orbit is very stable. Even though it is tempting to suggest that the planet was captured in this configuration and circularized afterwards, care should be taken not to extrapolate these results backwards in time since both the stellar as the planetary parameters could have evolved over the course of their existence.

TTV’s

As a ﬁnal step we did a transit analysis for which we set the period and T0 to 1.580404695±0.000000084 days and 4964.944727 ± 0.0000274 BJDTDB as was determined from the global analysis. From the linear regression we applied to the individual transit timings we found 1.5804048 ± 0.00000078 days and

4964.94477 ± 0.00038 BJDTDB which is again fully consistent with the results from the global analysis and the results from these two are displayed in figure 31. The periodogram for all the timings residuals and other model parameters showed no significant period. After the transit analysis had been performed we found the RMS for all TTVs to be as low as 16 seconds over a period of 1024 days. This very long observation time and low RMS allowed us to put very strong constraints on possible planetary perturbers as can be seen from figure 32. We are even able to eliminate planetary perturbers well down below a tenth of an Earth mass in the 2:1 resonance, a very satisfying result.

Note should be taken when interpreting this figure though. As one can see in figure 21, there is a small gap or jump at the 2.15 period ratio. This is caused by the simulations afterwards being carried out by a perturbing planet of ten Earth masses compared to the single Earth mass the simulation started with. Simulating the systems over a long timescale causes the computational errors to accumulate and overtake the actual signal in significance. To remedy this we enlarged the mass of the perturber to guaranty a larger TTV signal and a negligible signal from the computational errors. This trick however brought a new issue to the horizon. As it was explained in the previous chapter we choose this planetary perturbing mass low to allow the TTV signal to be scaled with the mass in the resonances. In what

49 Figure 29: Variations in the impact parameter b (top) and transit duration W (bottom) for GJ1214b compared to the values and their one sigma intervals (solid and dashed lines).

Table 12: MCMC system parameters for GJ1214(b) as they were obtained from the global analysis

Deduced parameters Charbonneau et al. (2009) Orbital period P (d) 1.580404695±0.000000084 1.5803925± 0.000017

T0 - 2450000 (BJDTDB) 4964.944727±0.000028 5964.944208± 0.000403 +0.00041 Transit width W (d) 0.03664−0.00038 +0.084 +0.061 Impact parameter b (R∗) 0.370−0.14 0.354−0.082 +0.00034 Transit depth dF 0.01346−0.00034 RV semi-amplitude K (m s−1) 12.2±1.6(fixed) 12.2± 1.6 +2.3 Stellar density ρ∗(ρ ) 16.3−2.1 +0.041 Surface gravity log g∗(cgs) 4.978−0.044 4.991 ± 0.029 Stellar mass M∗(M ) 0.157±0.019(fixed) 0.157 ± 0.019 +0.014 Stellar radius R∗(R ) 0.213−0.013 0.2110 ± 0.0097 +0.00050 Roche limit ar(AU) 0.01022−0.00048 +0.00056 Orbital semi-major axis a (AU) 0.01433−0.00062 a/ar 2.88±0.17 a/R∗ 14.47±0.65 14.66± 0.41 +0.59 +0.35 Orbital inclination i (°) 88.55−0.47 88.62−0.28 Eccentricity 0 (fixed) <0.27 (95% confidence) +0.068 Planetary density ρpl(ρEarth) 0.332−0.045 +0.52 Planetary mass Mpl(MEarth) 6.47−0.54 6.55 ± 0.98 +0.19 Planetary radius Rpl(REarth) 2.69−0.18 2.678 ± 0.13 +28 Teq,pl(K) 563−27 555

50 Figure 30: MCMC transit model superimposed on the photometry binned per interval over 7.2 minutes and folded over the best-ﬁt transit ephemeris (top) and its residuals (bottom). The RMS of the residuals is 0.487 relative mmags.

follows we will show how this affects the depth of the exclusion region in the resonances. Choose the

mass for the transiting planet Mtrans, perturbing planet Mpert and mass to be scaled to Mscale and the RMS of the signal before, RMS, and after scaling, RMS’. We have:

P Mpert RMS ∼ (63) 4.5j Mpert + Mtran

Thus:

0 Mscale RMS M +M = scale trans (64) RMS Mpert Mpert+Mtrans

If we then choose:

RMS0 = αRMS (65)

Mscale = βMpert (66)

We ﬁnd after inserting these relationships in the previous equation:

M + M = pert trans α β 1 (67) β Mpert + Mtrans

This function is plotted in ﬁgure 33 and shows us that the depth of the resonances is overestimated in our plots when the mass of the perturbing planet is too high compared to the transiting planet. For

51 Figure 31: TTVs obtained from individual (top) and transit (bottom) analysis for GJ1214b for TRAPPIST transits (blue) and transits listen in table 2 and 3 (red). example in the 3:1 resonance the RMS is approximately scaled by a factor 200 and our python script scales the mass by as much whilst the mass should only be scaled by a factor of 80 according to 67. We can conclude that when we try to scale the mass of the perturbing planet which is non-negligible compared to the transiting planet in the resonances, errors are introduced and should be avoided if possible.

52 Figure 32: Detectability domain for GJ1214b assuming a planetary perturber with e = 0 (blue) and e = 0.05 (red).

Figure 33: The RMS scaling factor α in function of the mass scaling factor β in the GJ1214 system with a perturbing planet of ten Earth masses.

53 5 Conclusion

We have analyzed extensive transit data sets for WASP-43b, WASP-46b and GJ1214b to assess the periodicity of their transits. For these three planets, no TTV signal demonstrating the presence of a third body is ﬁrmly detected, and the transit timings are fully consistent with a perfect periodicity. During this analysis we were able to vastly improve the transit parameters for WASP-46b and WASP-43b compared to their discovery papers (Anderson et al.[2012], Hellier et al.[2011]) and put very strong constraints on the period of GJ1214b (1.580404695±0.000000084 days). We failed to improve the model parameters for WASP-43b compared to those obtained by Gillon et al.[2012] even though we had an additional four transits to our disposal. The recent transits for WASP-43b were in general of poor quality providing no additional information except for an improvement for the period. Because of the limited photometric precision of TRAPPIST we were not able to improve the system parameters of GJ1214b but our results are fully consistent with the ﬁndings of Charbonneau et al.[2009]. We have developed a program able to determine the detectability domain of the TTV method for a given transiting system based on the observed RMS. Its application shows that our data sets could have detected a super-Earth in the resonances for the hot Jupiters under study and perturbers lighter than Mars for GJ1214b.

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