Searching for Exoplanets in K2 Data

Transcription

1 Searching for Exoplanets in K2 Data Iskra Georgieva Space Engineering, master's level (120 credits) 2018 Luleå University of Technology Department of Computer Science, Electrical and Space Engineering

2 Abstract The field of extrasolar planets is undoubtedly one of the most exciting and fast-moving in astronomy. Thanks to the Kepler Space Telescope, which has given us the Kepler and K2 missions, we now have thousands of planets to study and thousands more candidates waiting to be confirmed. For this thesis work, I used K2 data in the form of stellar light curves for Campaign 15 the 15 th observation field of this mission to search for transiting exoplanets. I present one way to produce a viable list of planetary candidates, which is the first step to exoplanet discovery. I do this by first applying a package of subroutines called EXOTRANS to the light curves. EXOTRANS uses two wavelet-based filter routines: VARLET and PHALET. VARLET is used to remove stellar variability and abrupt discontinuities in the light curve. Since a transit appears box-like, EXOTRANS utilises a box-fitting least-squares algorithm to extract the transit event by fitting a square box. PHALET removes disturbances of known frequencies (and their harmonics) and is used to search the light curve for additional planets. Once EXOTRANS finishes its run, I examine the resulting plots and flag the ones, which contain a transit feature that does not appear to be a false positive. I then perform calculations on the shortlisted candidates to further refine their quality. This resulted in a list of 30 exoplanet candidates. Finally, for eight of them, I used a light curve detrending routine (Exotrending) and another software package, Pyaneti, for transit data fitting. Pyaneti uses MCMC sampling with a Bayesian approach to derive the most accurate orbital and candidate parameters. Based on these estimates and combined with stellar parameters from the Ecliptic Plane Input Catalogue, I comment on the eight candidates and their host stars. However, these comments are only preliminary and speculative until follow-up investigation has been conducted. The most widely used method to do this is the radial velocity method, through which more detailed information is obtained about the host star and in turn, about the candidate. This information, specifically the planetary mass, allows for the bulk density to be estimated, which can give indication about a planet s composition. Although the Kepler Space Telescope is at the end of its life, new missions with at least a partial focus on exoplanets, are either ongoing (Transiting Exoplanets Survey Satellite TESS) or upcoming (Characterising Exoplanets Satellite CHEOPS, James Webb Space Telescope JWST, Planetary Transits and Oscillations PLATO). They will add thousands of new planets, providing unprecedented accuracy on the transit parameters and will make significant advances in the field of exoplanet characterisation. The methods used in this work are as applicable to these missions as they have been for the now retired Convection, Rotation et Transits planétaires (CoRoT) the first space mission dedicated to exoplanet research, and Kepler. I

3 Acknowledgements Firstly, I would like to thank my supervisors, Dr Carina Persson and Prof. Malcolm Fridlund, for giving me the opportunity to have a glimpse of extrasolar planets as a research field. I thank them for their support, day and night, weekday and weekend; for the many inspiring conversations and for being the best supervisors anyone could have. You made me feel like a real scientist. I thank Oscar for being available to answer all my questions and helping me understand and improve my results. I thank Andreas for the countless conversations when I needed a second opinion and ideas, for supporting and encouraging me and for being a good boy. I met some excellent human beings in Kiruna, both lecturers and friends, I thank them for the friendship and the support. Finally, I thank my family for good advice and putting up with me through all my mistakes. Thank you all. I ve learned a lot. II

4 Contents 1. Introduction Classification of Stellar Spectra History of Exoplanet Discovery Kepler and K Planet Detection and Related Challenges The Transit Method Theoretical Background Governing Equations Errors Geometric Probability Stellar Characteristics Limb Darkening Transits, Occultations and False Positives Data Reduction and Light Curve Generation Kepler Data Processing Pipeline Vanderburg Light Curves EVEREST Light Curves Transit Detection and Transit Modelling EXOTRANS Wavelets and the Stationary Wavelet Transform VARLET PHALET False Positives and Signal Detection Efficiency Pyaneti Other Methods Results Discussion Validation and Follow-up Observation Error Sources and Uncertainties Summary and Conclusion References Appendix III

5 List of Figures FIGURE 1: THE HERTZSPRUNG-RUSSELL DIAGRAM. MOST STARS ARE FOUND IN A DIAGONAL BAND, CALLED THE MAIN SEQUENCE. TOP-LEFT END OF THIS BAND IS INHABITED BY THE HOT AND BRIGHT O-STARS, AND THE COOL AND DIM M-DWARFS ARE IN THE BOTTOM RIGHT END. ~90% OF ALL STARS ARE ON THE MAIN SEQUENCE. THE DIFFERENT GROUPS OF GIANTS ARE IN THE UPPER RIGHT CORNER AND THE WHITE DWARFS ARE LOCATED IN THE LOWER LEFT. SOME WELL-KNOWN STARS ARE ALSO SHOWN. THE TEMPERATURE ON THE BOTTOM HORIZONTAL AXIS INCREASES TO THE RIGHT, SPECTRAL TYPE IS ON THE TOP HORIZONTAL AXIS AND VERTICAL AXIS SHOWS THE LUMINOSITY NORMALISED RELATIVE TO OUR SUN FIGURE 2: THE EXOPLANET POPULATION DISTRIBUTION BY PLANET RADIUS, EXPRESSED IN TERMS OF JUPITER RADII (RJ). THE HIGHEST PEAK CORRESPONDS TO ~ RJ, OR ~1.38 EARTH RADII (RE). THIS MEANS THAT THE MOST COMMON PLANETS DISCOVERED SO FAR ARE IN THE SUPER-EARTH RANGE FIGURE 3: THE ELEMENTS AND PHASES OF A TRANSIT. T14 CORRESPONDS TO THE FULL TRANSIT TIME AND T23 TO THE TRANSIT AFTER INGRESS AND BEFORE EGRESS, OR WHEN THE PLANETARY DISC IS FULLY WITHIN THE STELLAR DISC. TWO DIFFERENT TRANSIT GEOMETRIES ARE SHOWN, CORRESPONDING TO TWO DIFFERENT IMPACT PARAMETERS AND INCLINATIONS. R, RP AND ΔF/F ARE ALSO NOTED. IMAGE ADAPTED FROM SEAGER & MALLEN-ORNELAS (2003)... 6 FIGURE 4: THE FAVOURABLE GEOMETRY FOR A TRANSIT TO BE VISIBLE. LEFT: TO WITNESS A TRANSIT, THE OBSERVER MUST BE LOCATED WITHIN THE SHADED REGION. RIGHT: A CLOSE-UP OF THIS GEOMETRY. THE GRAZING REGION, ENCLOSED BY THE THICK LINES IS THE PENUMBRA AND THE FULL REGION, SUBTENDED BY THE THIN LINES, CORRESPONDS TO THE ANTUMBRA, WHERE A FULL TRANSIT WOULD BE VISIBLE. IMAGE FROM WINN (2010) FIGURE 5: A SCHEMATIC OF THE EFFECT OF LIMB DARKENING. FOR THE SAME OPTICAL DEPTH (HERE ΤΛ=2/3), AN OBSERVER IS ABLE TO SEE INTO DEEPER, HOTTER LAYERS AT THE CENTRE, THAN AT THE EDGE OF THE DISK, AS THE ANGLE, Θ, INCREASES. R IS THE RADIAL DISTANCE FROM THE STAR S CENTRE. IMAGE FROM (CARROLL & OSTLIE, 2007) FIGURE 6: A SCHEMATIC OF A TRANSIT AND A SECONDARY ECLIPSE (OCCULTATION). THE MOST INTENSITY IS MISSING DURING THE TRANSIT. THE GREATEST INTENSITY IS RECORDED JUST BEFORE THE ECLIPSING OBJECT MOVES BEHIND THE TARGET. THIS IS BECAUSE THE DAY SIDE OF THE ECLIPSING OBJECT IS ILLUMINATED BY THE STAR, WHICH ADDS TO THE TOTAL DETECTED BRIGHTNESS. IMAGE FROM WINN (2010) FIGURE 7: THERE ARE THE FOUR TYPES OF ECLIPSING SYSTEMS. ALL, APART FROM THE BOTTOM RIGHT CASE, ARE CONSIDERED FALSE POSITIVE DETECTIONS. IMAGE FROM CAMERON (2016) FIGURE 8: RAW (BLUE) AND CORRECTED (YELLOW) LIGHT CURVE AFTER APPLYING THE ABOVE-DESCRIBED PROCEDURE. THE IMPROVEMENT IS CLEAR. THE VERTICAL LINES IN THE RAW LIGHT CURVE ARE DUE TO THE SPACECRAFT DRIFT AND LOSS OF POINTING. A TRANSIT WITH DURATION OF 2.37 DAYS IS VISIBLE IN THE CORRECTED LIGHT CURVE (SHORT VERTICAL LINES). THE TARGET STAR WAS OBSERVED DURING K2 CAMPAIGN 3. IMAGE FROM VANDERBURG, ET AL. (2016) FIGURE 9: FOUR TYPICAL PLOTS FOR VISUAL INSPECTION DERIVED BY EXOTRANS. THE LIGHT CURVES ARE PHASEFOLDED AND BINNED. THE VERTICAL AXES CORRESPOND TO RELATIVE FLUX AND HORIZONTAL AXES REPRESENT THE PHASE (0 TO 1). EACH RED BAR REPRESENTS BINNED VALUES, CENTRED ON THEIR AVERAGE. THE HEIGHT OF THE BARS DEPENDS ON HOW FAR APART THE VALUES IN EACH BIN ARE. THE LIGHT CURVES BELONG TO: EPIC (TOP LEFT), EPIC (TOP RIGHT), EPIC (BOTTOM LEFT), EPIC (BOTTOM RIGHT). TOP AND BOTTOM LEFT DO NOT CONTAIN A DISCERNIBLE TRANSIT FEATURE. THE TWO DIPS IN THE TOP RIGHT (ONE SHALLOWER THAN THE OTHER) MAY SIGNIFY AN EB, AS DESCRIBED IN SECTION 2.3. THE LIGHT CURVE IN THE BOTTOM RIGHT CORNER CONTAINS A CLEAR TRANSIT FEATURE, CONSISTENT WITH A PLANETARY TRANSIT. THIS TYPE OF CANDIDATE ENTERS THE NEXT STAGE OF ANALYSIS FIGURE 10: TOP PANEL: PRE-PROCESSED LIGHT CURVE OF EPIC WITH FIVE PROMINENT TRANSITS. THE FIFTH TRANSIT EVENT WAS REMOVED BECAUSE OF A DISTURBANCE. BOTTOM PANEL: CANDIDATE IS CONSISTENT WITH A WARM JUPITER. P = DAYS, T14 = 3.5 HOURS, RP = 8.5 RE, A = 0.15 AU FIGURE 11: TOP PANEL: PRE-PROCESSED LIGHT CURVE OF EPIC SMALL DIPS CORRESPONDING TO THE TRANSITS ARE VISIBLE. BOTTOM PANEL: PYANETI FIT OF THE EIGHT TRANSITS OF EPIC B: P = DAYS, T14 = 3.5 HOURS, RP = 6.8 RE, A = 0.22 AU FIGURE 12: TOP PANEL: PRE-PROCESSED LIGHT CURVE OF EPIC THE NINE SMALL TRANSITS ARE BARELY VISIBLE. THE MOST NOTICEABLE FEATURE ARE THE TWO TRANSIT EVENTS OF WHAT COULD BE A JUPITER PLANET WITH A PERIOD ~5 TIMES LARGER THAN THE CANDIDATE UNDER INVESTIGATION. BOTTOM PANEL: TRANSIT FIT CORRESPONDING TO THE MULTIPLE SHALLOW TRANSITS IN THE PRECEDING LIGHT CURVE. THE CANDIDATE IS CONSISTENT WITH A SUPER-EARTH IN A TIGHT ORBIT. P = 9.21 DAYS, T14 = 4.86 HOURS, RP = 2.7 RE, A = 0.08 AU IV

6 FIGURE 13: TOP PANEL: PRE-PROCESSED LIGHT CURVE OF EPIC DIPS OF TWO DIFFERENT CANDIDATES CAN BE IDENTIFIED. THE TWO POTENTIAL PLANETS ARE IN A 2:1 ORBITAL RESONANCE. MIDDLE PANEL: P = 9.95 DAYS, T14 = 3.88 HOURS, RP = 5.7 RE, A = 0.19 AU. BOTTOM PANEL: P = DAYS, T14 = 4.88 HOURS, RP = 7.8 RE, A = 0.31 AU FIGURE 14: TOP PANEL: PRE-PROCESSED LIGHT CURVE OF EPIC , A VERY ACTIVE STAR, WITH INTENSITY CHANGES AS HIGH AS ~1%. NEVERTHELESS, THE SEVEN DIPS ARE VISIBLE. BOTTOM PANEL: TRANSIT FIT FOR A CANDIDATE THE SIZE OF A SMALL NEPTUNE. P = DAYS, T14 = 3.88 HOURS, RP = 3.45 RE, A = 0.11 AU FIGURE 15: TOP PANEL: PRE-PROCESSED LIGHT CURVE OF EPIC BOTTOM PANEL: TRANSIT FIT FOR A LARGE JUPITER. P = DAYS, T14 = 3.56 HOURS, RP = 13 RE, A = 0.21 AU FIGURE 16: TOP PANEL: PRE-PROCESSED LIGHT CURVE OF EPIC BOTTOM PANEL: TRANSIT FIT FOR ANOTHER SUPER- EARTH IN A CLOSE ORBIT. ELEVEN TRANSITS FOR A CANDIDATE WITH P = 7.82 DAYS, T14 = 2.82 HOURS, RP = 2.25 RE AT A = 0.07 AU FIGURE 17: PERIOD DISTRIBUTION OF THE 30 EXOTRANS DETECTIONS AND THE MODELLED TRANSITS FIGURE 18: METALLICITY DISTRIBUTION OF THE 30 EXOTRANS CANDIDATES FIGURE 19: LEFT: EFFECTIVE TEMPERATURE DISTRIBUTION OF TARGETS FROM CAMPAIGN 15, FOR WHICH THERE WERE VALUES IN EPIC. THREE CLEAR PEAKS ARE VISIBLE AT ~4000K, ~5000K AND ~6200K. RIGHT: DISTRIBUTION OF THE 27 HOSTS IN TABLE 1. UNSURPRISINGLY, THIS DISTRIBUTION DOES NOT MIMIC THE ONE ON THE LEFT AS THE SAMPLE IS MUCH SMALLER FIGURE 20: LEFT: THE SPECTRUM OF A B9 STAR (TOP). ONLY A SINGLE WIDE AND SHALLOW SPECTRAL LINE CAN BE SEEN. THE SPECTRUM OF A K5 STAR CAN BE SEEN AT THE BOTTOM. IN THIS CASE, THERE ARE MANY LINES, ALL DEEP AND NARROW, WHICH MAKES IT EASIER TO DETECT A DOPPLER SHIFT. THE HORIZONTAL AXIS GIVES THE WAVELENGTH IN UNITS OF ÅNGSTRÖM (1Å=0.1 NM) VS THE RELATIVE FLUX ON THE VERTICAL. RIGHT: THE RV ERROR AS A FUNCTION OF STELLAR SPECTRAL TYPE. THE DASHED HORIZONTAL LINE GIVES THE NOMINAL 10 M/S NEEDED FOR THE DETECTION OF A JUPITER-SIZED PLANET AT 5AU FROM A SOLAR-TYPE STAR. THE RV ERROR IS WELL BELOW THIS NOMINAL VALUE FOR STARS AROUND F6 AND LATER AND INCREASES FAST FOR EARLIER TYPES. IMAGE ADAPTED FROM V

7 List of Tables TABLE 1: PERIOD, FLUX CHANGE, TRANSIT DURATION AND TIME OF TRANSIT FOR 30 PLANETARY CANDIDATES (27 TARGETS) AS DETECTED BY EXOTRANS AND TEFF WITH UNCERTAINTIES FROM EPIC TABLE 2: SUMMARY OF THE PARAMETERS AND UNCERTAINTIES OF THE EIGHT PLANETARY CANDIDATES AS CALCULATED BY PYANETI. 30 TABLE 3A: THE M-K SYSTEM OF SPECTRAL CLASSIFICATION TABLE 4A: THE SEVEN MAIN SPECTRAL CLASSES, THEIR ASSOCIATED TEMPERATURES AND COLOURS VI

8 List of Abbreviations BJD Barycentric Julian Data BLS Box-fitting Least Squares CCD Charge Coupled Device CHEOPS Characterising Exoplanets Satellite CoRoT Convection, Rotation et Transits planétaires DT Doppler Tomography DWT Discreet Wavelet Transform EB Eclipsing Binary EPIC Ecliptic Plane Input Catalog EVEREST EPIC Variability Extraction and Removal for Exoplanet Science Targets FOV Field of View FT Fourier Transform HR Hertzsprung Russel (diagram) HZ Habitable Zone JWST James Webb Space Telescope KELT - Kilodegree Extremely Little Telescope KST Kepler Space Telescope MAST Mikulski Archive for Space Telescopes MCMC Markov Chain Monte Carlo PLATO PLAnetary Transits and Oscillations PLD Pixel Level Decorrelation PRF Pixel Response Function PSF Point Spread Function RMS Root Mean Square RV Radial Velocity SDE Signal Detection Efficiency SNR Signal-to-Noise Ratio SSF Single Star Fraction SWT Stationary Wavelet Transform TCE Threshold Crossing Event TESS Transiting Exoplanets Survey Satellite TTV Transit Timing Variation WASP Wide Angle Search for Planets WT Wavelet Transform VII

9 1. Introduction The field of extrasolar planets and planetary systems is relatively new, exciting and fast-moving. It is so interesting because it provides a glimpse of worlds beyond our own and allows us to speculate about what the discoveries yielded by the exoplanet search efforts could mean for our understanding of our galaxy, our own Solar system and our place in it, as well as about implications for the habitability of distant planets. For this thesis project, I worked on the first stages of exoplanet detection in data from K2 Kepler Space Telescope s (KST) Second Light, which utilises high-precision photometry and the transit method to realise the detection. This included visually examining the data in the form of plots processed by a computer algorithm using wavelets and then trying to identify false positives. Once the best candidates were identified, I employed a transit fitting algorithm to obtain the best values of the key parameters. In this work, I describe in detail the process I followed in order to do this, along with the theoretical background needed to understand it. I begin by introducing some information about stellar classification, exoplanet discovery, including specifics about the nominal KST mission, Kepler (as far as exoplanets are concerned), and the K2 mission, which succeeded it, followed by typical challenges associated with the transit method. In Section 2, I present the theory background of the transit method. Next, I describe how three types of light curves are created by pre-processing the raw data from the spacecraft. In Section 4, I talk about the transit detection and modelling algorithms I used, followed by results I obtained in Section 5 and a discussion in Section 6. Sections 7 and 8 cover the follow-up stage once a list of the most viable candidates is decided, and a summary of the uncertainties in the observation, detection and modelling of transits, respectively. Section 9 contains the conclusion Classification of Stellar Spectra The Harvard classification scheme is a spectral taxonomy developed by scientists at Harvard during the late 19 th and early 20 th century by labelling spectra by the strength and width of their hydrogen absorption lines. The labelling was done using capital letters ("O B A F G K M"), followed by decimal subdivisions (0-9). This resulted in a temperature sequence with O being the largest and hottest blue (early-type) stars, and M the coolest red dwarf (latetype) stars. Thus, a B2 star is an early B star, while a F9 star is a late F star. More recently, additional spectral types were added very cool stars and brown dwarfs, carbon stars and more. Brown dwarfs are not stars in the conventional sense because they are too cool to undergo any significant nuclear reactions in their cores. Such low-temperature bodies were designated by the letters (L, T, C ) thus extending the scheme, but for this work the original seven types will suffice. However, it was not until understanding of the quantum atom was gained in the early 20 th century, that the physical foundation of the Harvard classification scheme began to be uncovered. It became clear that the difference in the spectral lines and temperatures of stars was dependent on the atomic orbitals electrons occupy in the stellar atmospheres. This is because absorption lines are formed when an atom absorbs a photon of specific energy. When this happens, an electron passes from a lower to a higher orbital. The reverse process takes place in emission: an electron passes from a higher to a lower orbital, thus emitting a photon carrying the lost energy. The photon s wavelength is dependent on the energies associated with the orbitals involved in the process. Spectral line formation is generally quite complicated, 1

10 because an electron can occupy any orbital and an atom can be in different stages of ionisation, affecting the number of orbitals present. As data was gathered, the wide range of stellar absolute magnitudes (the apparent magnitude of a star if it were located at 10 pc) and luminosities (emitted energy per unit time) became clear. In 1905, Ejnar Hertzsprung, discovered that G and later type stars can have different magnitudes. The obvious conclusion was that since their spectral type was the same (and so their temperature), the brighter stars must be larger 1. Henry Norris Russell, meanwhile, came to the same conclusion, but unlike Hertzsprung s tabulated results, made a diagram. Thus, the widely used Hertzsprung-Russell (or HP) diagram, shown on Figure 1, was created. Figure 1: The Hertzsprung-Russell diagram. Most stars are found in a diagonal band, called the main sequence. Top-left end of this band is inhabited by the hot and bright O-stars, and the cool and dim M-dwarfs are in the bottom right end. ~90% of all stars are on the main sequence. The different groups of giants are in the upper right corner and the white dwarfs are located in the lower left. Some well-known stars are also shown. The temperature on the bottom horizontal axis increases to the right, spectral type is on the top horizontal axis and the vertical axis shows the luminosity normalised relative to our Sun. Image from: The most prominent feature of this image is the band running from the bottom right to the top left, where most stars, including our Sun (a G2 star), are located. This band is called the main sequence and contains ~90% of all stars in the galaxy. This stellar population is in the main part of its life, fusing hydrogen into helium for fuel, with radius range, from top to bottom, 1 From Stefan-Boltzmann s law: R = 1/T eff L/4πσ, where R is the radius of the star, T eff the effective temperature, L is the luminosity and σ is the Stefan-Boltzmann constant, equal to W/m 2 K 4. 2

11 of 20Rʘ to 0.1Rʘ. On the other hand, the giant stars at the top right have ranges of Rʘ (e.g. Aldebaran in the constellation Taurus with 45Rʘ), and the supergiant stars are hundreds or even thousands of times larger than the Sun (e.g. Betelgeuse Rʘ, in the constellation Orion). From the HR diagram it can be seen that there is a clear relation between the temperature and the luminosity (and thus, the size) 2 of stars on the main sequence. The unique factor which determines their placement is the stellar mass. Hertzsprung, with the help of other astronomers, set the foundation for the work of astronomers William Morgan and Phillip Keenan, who described the influence of temperature and luminosity on stellar spectra. This is how the M-K (from Morgan-Keenan) system for spectral classification was established. It is expressed as a Roman numeral added at the end of the Harvard classification scheme identifier, representing the luminosity class of the star. To illustrate, the Sun is a G2V-type star, where the V signifies that the star is on the main sequence (Carroll & Ostlie, 2007). Tables of the luminosity classes and the basic form of the Harvard classification scheme can be found in Appendix A History of exoplanet discovery The existence of extrasolar planets had been suspected and speculated about since the nineteenth century, however the first confirmed detection of planets orbiting another star was made by Wolszczan & Frail (1992), where the presence of at least two planets around a pulsar was inferred via precise timing measurements of the pulses using the Arecibo radio telescope. This was followed by Mayor & Queloz (1995) with the first discovery of a planet around a main sequence star: a Jupiter-mass planet orbiting the star 51 Pegasi, a solar-type star. This was done using radial velocity (RV) measurements: measuring the motion of a star induced by the gravitational influence of one or more planets. Understandably, such effects are greatest for the largest planets, and so initially the first exoplanet discoveries were of giant planets. However, advancements in technology, specifically in high-resolution spectroscopy (RV method) and later in high-precision photometry (the transit method), which allowed for planets of various sizes and masses orbiting stars of various types, to be discovered. In fact, these two methods combined provide independent estimates of an extrasolar planet s mass and radius, respectively. Exoplanet discoveries are realised via ground-based and space-based missions among the most notable of which are Kepler and K2 (Borucki, et al, 2010; Howell, et al, 2014), CoRoT (French: Convection, Rotation et Transits planétaires; English: Convection, Rotation and planetary Transits) (Bordé, et al, 2003), the Wide Angle Search for Planets (WASP) (Pollacco, et al, 2006) and the Kilodegree Extremely Little Telescope (KELT) (Pepper, et al, 2004, 2007). Thanks to these telescopes, and the instruments and techniques, used by the other methods for exoplanet detection, the population of confirmed exoplanets to date has the distribution shown in Figure The relations between spectral type, luminosity, temperature and other parameters, can be found at: Information from these tables will be used later in this work. 3 The data in Figure 2 was drawn using the NASA Exoplanet Archive plotting tool, available at: 3

12 Figure 2: The exoplanet population distribution by planet radius, expressed in terms of Jupiter radii (R J). The highest peak corresponds to ~ R J, or ~1.38 Earth radii (R E). This means that the most common planets discovered so far are in the Super-Earth range. The larger, left part of the distribution encompasses radius ranges of slightly larger than 1 RE to ~1.74 RE. The highest peak in the above distribution shows that, in terms of size, the most commonly found exoplanet is a Super-Earth (1.38 RE). The second peak on the right corresponds to planets slightly larger than Jupiter Kepler and K2 So far, the most significant contribution to extrasolar planet discovery has been made by KST. It was launched in 2009 in an Earth-trailing heliocentric orbit with the purpose of collecting wide-field photometry with a field of view (FOV) pointed at the constellations of Cygnus and Lyra. As of July 30 th, 2018, it is responsible for ~70% of all confirmed exoplanets 4. This was accomplished during the KST nominal mission, Kepler. The discoveries made in the four years of observations made it possible to acquire information about the distribution and frequency of exoplanets and some key orbital and planetary, as well as stellar, parameters. The data has shown that planets around other stars are indeed common, with stars often hosting multiple planets, many of which Earth- or Super-Earth-sized ( R ) 5 (Ricker, et al, 2014; Borucki, 2017). Unfortunately, by 2013 KST had lost two reaction wheels. The loss of the second one nearly put an end to the mission as at least three reaction wheels are needed to maintain the precision in the spacecraft s pointing. A proposal adapting the crippled spacecraft to the new situation was accepted in This gave the beginning of the K2 mission, which utilised the same proven and well-established large FOV with high precision, long-baseline and high cadence photometry as Kepler. This was made possible by using the spacecraft s thrusters as a makeshift third reaction wheel. To minimise the disturbance and balance the effect of the solar pressure-generated torque around the roll axis it was necessary to change the above-mentioned original pointing of the spacecraft to observe independent fields in the ecliptic plane. Timed thruster firings are used to dump the accumulated momentum. The different fields, called Campaigns, are observed in periods of about 80 days, after which the 4 KST is responsible for the detection of 2,650 (Kepler + K2) out of a total of 3,774 confirmed exoplanets. 2,244 (Kepler) (K2) await confirmation. Numbers as per the NASA Exoplanet Archive: 5 The Super-Earth range is as per Ricker, et al. (2014). 4

13 attitude of the spacecraft is adjusted to avoid sunlight entering the telescope s aperture (Howell, et al, 2014). Most of the targets for the Kepler mission were selected before the launch. With the primary objective being to determine how common are Earth-like planets orbiting Sun-like stars, it was decided that observing at least 170,000 stars for four years would allow the detection of at least four transits of 50 planets (Borucki, 2017). The K2 targets were chosen differently: targets were exclusively selected via proposals submitted by the astronomical community through a Guest Observer program (Mayo, et al, 2018). Huber, et al. (2016) found that K2 targets K-M dwarfs ( 41%), F-G dwarfs ( 36%) and K giants ( 21%) to search for exoplanet transits and investigate galactic archaeology Planet detection and related challenges There are various methods that can be used to look for planets orbiting other stars: direct imaging, microlensing, RV, astrometry, transit photometry, among others. The latter has yielded thousands of confirmed planets and is currently the most effective method for detecting extrasolar planets. When a planet transits the face of its host star, it blocks a small part of the light reaching the observer. With the help of high-precision photometry, we can detect this change in flux. There are many reasons why the transit method has become the method of choice for so many exoplanet surveys. The biggest advantages are: The radius of the planet can be directly inferred from the depth of the transit. When combined with RV data, the mass of the planet can be derived with good accuracy, which then makes it possible for an estimate of the planetary density to be derived. Also, if the transit is observed at different wavelengths, the difference in depth can provide information about the absorption spectrum, indicating an atmosphere (Haswell, 2010). Detecting a transit from the recorded stellar intensity, however, is far from simple. Variations in stellar activity are a major setback in obtaining clean enough data to do transit analysis. Stellar variability can be of periodic as well as non-periodic character: stellar rotation sunspots move along the surface of the star as the star rotates, thus mimicking a transit; stellar pulsations can give a periodic change in flux; solar flares can produce a non-periodic increase in brightness. While stellar variations usually cause a less steep change in flux than transit events, they can often exceed the depth of a transit greatly. Different sources of noise and discontinuities to the stellar light curve are another unavoidable problem: instrument noise, photon-counting noise, Charge Coupled Device (CCD) saturation due to energetic particles colliding with the detector. The transit depth is also small (for a solar type star, between ~1% for Jupiter-sized and ~0.01% for Earth-sized planets) which makes it easy for any of the above to mask an existing planet or mimic the presence of one (Grziwa & Pätzold, 2016). 2. The Transit Method 2.1. Theoretical Background 5

14 Here follows the theory foundation of the transit method. The description is based on Carole Haswell s book Transiting Exoplanets (2010), Seager & Mallen-Ornelas (2003) and Winn (2010). A more in-depth discussion, derivations and alternative forms of these equations can be found in Seager & Mallen-Ornelas (2003). There are five parameters (stellar mass M, stellar radius R, radius of candidate Rp, semi-major axis a, and orbital inclination i) which constrain a star system with an observed transit. Adapted from Seager & Mallen-Ornelas (2003), Figure 3 illustrates these parameters and the elements and phases of a transit. The impact parameter, b, is the shortest distance from the centre of the stellar disc to the locus of the planet, with 0 b 1. It is equal to zero when the planet passes through the middle of the stellar disc, which corresponds to a maximum transit duration. Thus, it is clear that the duration gets shorter as b grows larger. The first and last (1 and 4) phases, called contacts, correspond to the ingress and egress of a transit. A transit commences (first contact) when the limb of the planet coincides with the limb of the star; second contact is the first instant when the planetary disc is fully within the stellar disc. Similarly, contacts 3 and 4 correspond to the last moment the full planetary disc is within the star and the instant when the trailing limb of the planet coincides with the stellar limb, respectively. Figure 3: The elements and phases of a transit. t 14 corresponds to the full transit time and t 23 to the transit after ingress and before egress, or when the planetary disc is fully within the stellar disc. Two different transit geometries are shown, corresponding to two different impact parameters and inclinations. R, Rp and ΔF/F are also noted. Image adapted from Seager & Mallen-Ornelas (2003). 6

15 To obtain a unique solution for these parameters, it is necessary to state the assumptions, which need to be made: 1. The target stars are observed from interstellar distances. 2. The orbits are circular. 3. M p M, and the candidate is much darker than the host. 4. The stellar mass and radius are known (here taken from the Ecliptic Plane Input Catalog (EPIC) 6 available online). 5. Effects of limb darkening are negligible or can be estimated. 6. The source of the light comes from a single host. All but the last of the above are reasonable assumptions. Number 6 can be inaccurate in cases of multiple hosts, i.e. an eclipsing binary (hereafter EB), another star orbiting a binary host, or in case of a chance alignment between the observed host and a fore- or background star (Seager & Mallen-Ornelas, 2003) (more on this in Section 2.3.). The probability for this to be the case is dependent on the stellar type. Lada (2006) inferred that the fraction of stellar systems in the G-M range without a companion of stellar type, or the Single Star Fraction (SSF), is dependent on the star s spectral type. The SSF is highest for brown drawfs (~85%) and lowest for G-type stars (~43%). M-stars have been estimated to have an SSF of ~74%. Thus, with M-stars being the dominant stellar type in the Galactic disc, it was deduced that two thirds of all main sequence primary stars are single stars. Quintana & Lissauer (2007) simulate different stellar binary scenarios and the case for terrestrial planet formation in circumbinary, or P-type orbits (planets orbiting both stars), and S-type orbits (planets orbiting one of the stars). They found that a larger separation between the two stars allows for a formation of larger terrestrial systems. They state that ~40-50% of binaries are wide enough for the stable existence of Earth-like planets in S-type orbits, while 10% have separation small enough for terrestrial planets to form in P-type orbits Governing Equations The amount of light periodically blocked when an opaque object passes in front of a star, or the ratio of flux observed during a transit (ΔF) to the flux with no transit (F), allows for the R 2 2 p /R ratio to be estimated: F F = R 2 p (1) R2 Assumption 1 is particularly relevant here, as planetary transits within our Solar system have a slightly different form. For conciseness and simplicity, the fraction ΔF/F will be denoted solely by ΔF. The semi-major axis of the planetary candidate s orbit can be determined via Kepler s third law: a 3 P 2 = G(M + M P ) 4π 2 (2) where G is the universal gravitational constant and P is the period. 6 This is specific to K2 targets. Kepler targets were predefined and a catalogue with KOIs (Kepler Object of Interest) was created. A few words about EPIC and problems associated with it are found in a later section. 7

16 Invoking assumption 3 to Equation (2) and rewriting for a, gives: a (GM ( P 2π ) 2) 1/3 (3) The full transit duration is: t 14 = P (R p + R ) 2 b 2 π sin 1 a ( ) (4) where the impact parameter, b, is a dimensionless quantity given by the relation: b = a cos i R (5) Equation (4) can be further simplified if a R R p and the orbit is circular (assumption 2), thus becoming: t 14 = P π ( R a ) 2 cos 2 i (6) From Equation (5) the orbital inclination can be calculated to be: Another useful equation is the stellar mass-radius relation: i = cos 1 (b R a ) (7) R = km x (8) with k being a constant coefficient dependent on the part of the stellar sequence the star is on, and x describes the power law of the sequence (for a F K star on the main sequence k = 1 and x 0.8). It can be shown that the stellar mass, radius and mean density are related in the following way: R = k ( M x ) = (k 1/x ρ x/(1 3x) ) R ʘ M ʘ ρ ʘ (9) The stellar density is an important parameter because it gives the location of the star with respect to the main sequence and, assuming that t 14 π/p 1, together with Kepler s third law and Equation (1), can be determined using the following relation: ρ = 32 Gπ P F 3/4 (t 2 14 t 2 (10) 23 ) 3/2 8

17 Errors The above relations are very useful in obtaining an initial estimate for the transit parameters. However, an evaluation of the effects of the errors in some of these crude estimates needs to be made. The impact parameter, b, is mostly dependent on the transit duration, t14, and the sizes of planet and star, R p and R. To a lesser extent, it also depends on the period, P. ρ, as seen in Equation (10), is dependent on R p, R and b. R depends on the transit duration and is constrained by Kepler s third law as it defines the mass contained inside a planet s orbit. Seager & Mallen-Ornelas (2003) perform a simulation to estimate the errors in the planetary and stellar parameters caused by limitations in photometric precision and time sampling. They performed a χ 2 minimisation to 1000 simulated two-transit light curves with added Gaussian noise, ignoring limb darkening effects. Two different impact parameter models were considered as the errors are very sensitive to changes in the impact parameter. For large t23 / t14 ratios (steep flanks, box-shaped transit), a small change in the value of this ratio corresponds to a large change in b. This makes it difficult to determine b accurately for such transits. The opposite case of less steep, more oval or V-shaped transits (small t23 / t14) even a larger change in t23 / t14 has a much less prominent effect on b. This non-linear relation between b and t23 / t14 means that noisy data will cause b to be underestimated because a symmetric error in t23 / t14 leads to an asymmetric error in b. As b, ρ, R p and R are interrelated, this effect on b will cause an overestimate in ρ and an underestimate in R p and R (in such cases when R is not known or is believed to be inaccurate). Seager & Mallen-Ornelas s analysis emphasises the importance of high photometric precision and frequent time sampling to better define the ingress/egress times and transit shape. A poorly constrained transit shape resulting in the above effects may lead to M-dwarf stars being wrongly classified as planetsized. Ignoring the effects of limb darkening can also bring about incorrect parameter estimates because of underestimated uncertainties. More on limb darkening in the next section Geometric Probability of Transit Detection A transit can only be seen by an observer if the orbital geometry of the system is favourable. A partial or grazing eclipse is visible from the penumbra (the partially shaded outer region) and a full eclipse is visible from the antumbra (the region where the planet is fully contained within the disc of the star). The geometry is shown on Figure 4 (Winn, 2010). 9

18 Figure 4: The favourable geometry for a transit to be visible. Left: to witness a transit, the observer must be located within the shaded region. Right: a close-up of this geometry. The grazing region, enclosed by the thick lines is the penumbra and the full region, subtended by the thin lines, corresponds to the antumbra, where a full transit would be visible. Image from Winn (2010). At inferior conjunction (the point where the occulting body is closest to the observer), the phase, φ, of the orbit is defined to be 0. For a circular orbit, the distance between the two objects at φ = 0 is d = a cos i. The condition which needs to be met for a transit to occur is then: a cos i R + R p (11) The condition for a grazing transit is R R p < a cos i < R + R p. The probability of a planet being detected by an observer from any random direction is the probability for which a random inclination satisfies Equation (11). Thus, the geometric transit probability, ptra, is: p tra = number of all orbits transiting all orbits = R + R p a R a (12) The right side of Equation (12) applies for cases when R R p. A quick analysis shows that ptra is highest for close-in planets with large radii and large host stars (Haswell, 2010; Cameron, 2016). The probability for a secondary eclipse is given by the same expression, because for circular orbits the two go together, which is not the case for eccentric orbits (one can be visible and not the other) (Winn, 2010) Stellar characteristics Apart from the density, mass and radius of the host star, other stellar parameters also become important in exoplanet investigations. These are the stellar metallicity, surface gravity and effective temperature. A trend of planets forming around high-metallicity, i.e. metal-rich 7 stars, has been observed. The metallicity of a star is determined by comparing the ratios of iron (Fe) to 7 The term metal in this context does not have the traditional meaning. Rather, when it comes to stellar structure, it means elements different from hydrogen and helium. 10

19 hydrogen (H) in relation to the Sun. If NFe and NH are the number of iron and hydrogen atoms, the metallicity ratio is given by: [Fe/H] log 10 [ (N Fe/N H ) star (N Fe /N H ) ] (13) Relative to the Sun, metal-poor stars have [Fe/H] < 0, with lowest values in our galaxy of about 5.4, and for relatively metal-rich stars [Fe/H] > 0, with highest values ~0.6. The most commonly used unit is the dex, short for decimal exponent (Carroll & Ostlie, 2014). Surface gravity represents the photospheric pressure of the stellar atmosphere and, logarithmically, is given by the stellar mass and radius. log g = log M 2 log R (14) Like [Fe/H], the unit is dex. Assuming a blackbody, the stellar effective temperature, Teff, is the temperature calculated based on the star s emitted radiation, in units of Kelvin (K). From Stefan- Boltzmann s law, it is related to the radiant power at the stellar surface, the stellar luminosity and radius by Equation (15) (Smalley, 2005). 4 σt eff = F = L 4πR 2 σ (15) 2.3. Limb Darkening Limb darkening is the phenomenon which causes stars to appear brighter towards the middle of the disc and darker (redder) towards the edge (limb). When an observer looks straight on towards the centre of a stellar disc, the angle of the line of sight, θ, with respect to the disc is 0. However, looking towards the limb of a star, this angle increases, and the observer is only able to see into less shallow layers of the stellar atmosphere, as compared to looking at the centre. Because deeper layers are generally hotter than shallow layers, photons from the limb appear redder and photons originating closer to the centre appear bluer. This effect occurs due to the optical depth, τλ, of the stellar atmosphere. One way to think of the optical depth is as the number of mean free paths 8 along a ray of light from the initial position to the surface. It is wavelength- and medium-dependent. For a given value, the line of sight has travelled further as θ increases (Figure 5) (Carroll & Ostlie, 2007). 8 In the context of stars, the mean free path is the average distance travelled by a photon before colliding with other particles. 11

20 Figure 5: A schematic of the effect of limb darkening. For the same optical depth (here τ λ=2/3), an observer is able to see into deeper, hotter layers at the centre, than at the edge of the disk as the angle, θ, increases. r is the radial distance from the star s centre. Image from (Carroll & Ostlie, 2007). Limb darkening is also the reason why transits have rounded bottoms and smooth edges around contact points 2 and 3. As previously noted, the effect of limb darkening was neglected in the equations above. However, when performing a full transit analysis, this effect needs to be accounted for as it can change the transit shape and thus lead to incorrect interpretation of the results. The impact parameter, b, is the most affected because limb darkening makes t23 appear smaller and b is overestimated. This, in turn, has an effect on the other parameters as discussed in the previous section (Seager & Mallen-Ornelas, 2003; Winn, 2010). Different limb darkening laws exist, the simplest and computationally fastest of which is the linear one (Equation 16). However, it is also the least accurate. It is possible to choose a quadratic (Equation 17), square-root, logarithmic, exponential, four-parameter nonlinear, or another law but the choice is somewhat arbitrary. Each of these produces coefficient(s) (u1,,un), which govern the intensity reduction gradient from centre to limb. I(μ) = I 0 [1 u 1 (1 μ)] (16) I(μ) = I 0 [1 u 1 (1 μ) u 2 (1 μ) 2 ] (17) In the above equations μ = (1 x 2 ) 1/2, I0 is the stellar intensity at the centre and I(μ) is the surface brightness of the star, x is the radial distance from the centre of the stellar disc to the limb, i.e. 0 < x < 1. Although called laws, the above are more accurately referred to as fitting formulas. This is because limb darkening is not absolute and can differ substantially depending on the spectral type of the star. This makes limb darkening very hard to model and predict (Cameron, 2016; Kreidberg, 2015). 12

21 The effects of limb darkening are less prominent for longer wavelengths. In contrast, for shorter wavelengths, a transit appears so curved that it can be extremely difficult to make out the different contact points (Haswell, 2010; Winn, 2010) Transits, Occultations and False Positives A transit event is essentially an eclipse where the sizes of the two bodies differ greatly, with the smaller one passing in front of the larger one. An occultation, or a secondary eclipse, on the other hand, is the opposite event, i.e. when the smaller body passes behind the larger one (Figure 6) (Winn, 2010). Figure 6: A schematic of a transit and a secondary eclipse (occultation). The most intensity is missing during the transit. The greatest intensity is recorded just before the eclipsing object moves behind the target. This is because the day side of the eclipsing object is illuminated by the star, which adds to the total detected brightness. Image from Winn (2010). Larger eclipsing bodies produce larger occultations, which are identifiable in a light curve. The size and timing of the secondary dip they produce can provide important information about the eclipsing body. Similar to the depth of a transit, a light curve with a very prominent secondary dip often means that the eclipsing body is likely self-luminous, thus pointing to an EB scenario. It is not uncommon that the eclipsing object is not a planet. In fact, apart from planets, there are three main culprits which can cause a false positive detection: a blended EB (hereafter simply referred to as a blend ), a grazing EB with stars of ~equal mass, and transit by a planetsized star. Each is discussed below, as per Haswell (2010), and illustrated in Figure 7: It is usually easy to distinguish between a planet and a dimmer star as the transiting object, as stars produce a noticeably larger dip. The transit depth is related to the 13

22 squared ratio of the planet-star radii (Equation (1)). However, this relation assumes that the entire disc of the eclipsing object passes fully within the face of the stellar disc, between contact points 2 and 3. This may not always be the case. A binary system with two stars of similar mass, where the transits are grazing, will produce a small dip in brightness, which can be mistaken for a planetary transit. In such a scenario the occulted area would be much smaller than the area of the occulting object. Additionally, if one star is brighter than the other, the primary and secondary eclipses will have different depths. It is more complicated if both stars are on the main sequence and of similar spectral type, size and mass, which makes the two eclipses indistinguishable. Giant planets have a similar size to brown dwarfs and white dwarfs are closer to the Earth in size. Since Equation 1 only takes into account the size of the eclipsing objects, it cannot differentiate between planets, brown and white dwarfs. A blend occurs when the light of the constant star is blended with the light of an EB, thus causing the transit depth to appear shallower (~1%). In most cases the background star and the EB are not aligned and part of the same system, and the observed transit is rather due to a chance alignment. Figure 7: There are the four types of eclipsing systems. All, apart from the bottom right case, are considered false positive detections. Image from Cameron (2016). 3. Data Reduction and Light Curve Generation As mentioned earlier, it is a real challenge to make a positive transit detection and a correct judgement. This is, to a large extent, due to the precision required to detect a transit; the distance to the objects being observed and their often unpredictable variability; random problems from the spacecraft platform itself, onboard instruments and the local environment. Unfortunately, in the case of K2, systematic noise is the biggest problem. Light curves are generated from raw pixel data downloaded from the spacecraft. Removing the systematic noise is a particularly challenging task due to the complex corrections that need to be applied to the data to remove the artefacts resulting from the pointing adjustment motion of the spacecraft 14

23 The light curves are then processed using different techniques to extract any potential transit features. Since the photometric precision of the K2 pixel data is 3-4 times worse than Kepler s (Howell, et al, 2014), a correction method for the systematic noise produced as a result of the spacecraft motion is needed. Three of the most popular methods will be described here: the light curves developed by the Kepler team (hereafter referred to as Kepler light curves), the Vanderburg light curves, and EVEREST (EPIC Variability Extraction and Removal for Exoplanet Science Targets) light curves Kepler Data Processing Pipeline Once the raw pixel data is downloaded from the spacecraft, the Kepler team at NASA first works to produce the calibrated target and background pixels and their uncertainties. As a next step the sky background is removed, and basic aperture photometry is extracted. At this stage the centroid position of each target for each frame is estimated. Finally, sources of systematic error are removed from the light curves. The output data after each of these steps is made available to the public (Jenkins, et al, 2010). An important note should be made here. This procedure produces the light curves which were used to search for planets during the Kepler mission. As explained by Grziwa & Pätzold (2016), the Kepler Science Pipeline relies on weaker filtering and the Kepler team has developed their own algorithm for the purposes of detection. Still, as these light curves were available first, in this work, a first attempt for exoplanet detection was made. However, understandably better results (more and clearer detections) were achieved using the below-described Vanderburg light curves Vanderburg light curves One such method developed by Vanderburg & Johnson (2014) and further refined by Vanderburg, et al. (2016) will be described here, following in part the short description of Mayo, et al. (2018). Due to bandwidth restrictions, the downlinked data does not contain fullframe images, but small sub-images called postage stamps instead. An aperture mask was chosen for each of these stamps using two different approaches: selecting a circular region of pixels around the target star and another one formed by flux from the Pixel Response Function 9 (PRF) of the telescope. The one yielding the best photometric precision is chosen. The apertures are stationary as the motion of the star over the detector is too small to justify the use of moving apertures. The circular masks are made large to prevent flux from the target star from spilling out of the aperture as it drifts across the detector. On the other hand, the Kepler PRF-defined masks result in a tightly fitted aperture around the image of the star. While the circular masks tend to yield better precision for saturated stars and stars with a companion, the latter method gives better photometric precision for faint or background-limited stars as it prevents background light contamination. The background pixel median value (from pixels lying outside the aperture) is subtracted from the image and the pixels within the aperture are summed. The extracted photometry for each star is normalised by dividing by the median background brightness. The location of the star on the detector for each datapoint is determined by using two methods: a centre of flux centroid calculation and a Gaussian fit to the PSF. The approach exhibiting the least root mean square (RMS) residuals is chosen. The recurrent motion of the centroid across the detector resulting from the corrective thruster fires 9 As per Bryson, et al. (2010), the Kepler s PRF is the composite of Kepler's optical point spread function, integrated spacecraft pointing jitter during a nominal cadence and other systematic effects. 15

24 (approximately every 6 hours) is identified and removed. Data taken during these fires and other poor-quality data is disregarded. Low-frequency variations (>1.5 days) are removed using a basis spline and a piecewise function is fit between centroid position and flux (Mayo, et al, 2018; Vanderburg & Johnson, 2014, Vanderburg, et al, 2016). Following this procedure for each target star yields, raw and corrected (pre-processed) light curves such as in Figure 8. The long vertical lines in the raw light curve (blue) are due to the drifting of the spacecraft. Figure 8: Raw (blue) and corrected (yellow) light curve after applying the above-described procedure. The improvement is clear. The vertical lines in the raw light curve are due to the spacecraft drift and loss of pointing. A transit with duration of 2.37 days is visible in the corrected light curve (short vertical lines). The target star was observed during K2 Campaign 3. Image from Vanderburg, et al. (2016) EVEREST light curves EVEREST stands for EPIC Variability Extraction and Removal for Exoplanet Science Targets. The EVEREST method produces high quality data; however, it is a few campaigns behind and at the time of writing (June 2018) light curves from Campaign 15 have not yet been made publicly available. For this reason, I have not had the opportunity to test them and have only used Kepler and Vanderburg light curves. Essentially all K2 light curve pre-processing techniques rely on numerical methods to remove correlations between stellar position and changes in flux and make assumptions about correlations between the instrument variability and spacecraft motion. Furthermore, the process of determining the centroid position is a source of uncertainties and relies on more assumptions about the target Point Spread Function (PSF) 10. Luger, et al. (2016) use an alternative method, called Pixel Level Decorrelation (PLD). The main advantage of this technique is that correcting for motion-induced noise does not require knowledge of the star s position and this removes 10 In the case of an imaging system, such as telescope, the PSF is the system s response to a point source. 16

25 centroid determination as an uncertainty source. Instead, the primary data products of the photometry are used, i.e. the intensities in each pixel, which are then normalised by the total flux in the chosen aperture. This removes astrophysical effects from the basis set, making the PLD sensitive only to different signals across the aperture and minimising assumptions about instrument variability and correlations between spacecraft motion and intensity fluctuations. EVEREST light curves generally have less scatter and fewer outliers as compared to the Vanderburg (and other) light curves. The PLD method used in this application is, of course, not perfect. The two most significant limitations are manifested in cases of saturated stars and crowded apertures (significant contamination from other stars). Both scenarios lead to overfitting. The PLD is unable to detrend the light curve properly and as a result removes the transit feature altogether (Luger, et al., 2016). 4. Transit Detection and Transit Modelling Many software packages and programs have been developed throughout the years since the transit method became popular. Some of these focus on detection, others on transit modelling. In this section, I will present the algorithms I worked with to find promising candidates (EXOTRANS) and then follow-up some of the most interesting ones in more detail via transit modelling (Exotrending and Pyaneti) EXOTRANS For this work I used a MATLAB-based routine called EXOTRANS, developed by the Department of Planetary Research of the Rheinisches Institut für Umweltforschung (RIU-PF) at Cologne University to find promising transiting events. It has so far been used successfully in both the CoRoT and the Kepler/K2 missions. This algorithm can use light curves produced via any of the above described methods (see Section 3). However, none of these methods remove the intrinsic variability (periodic or non-periodic) of the target stars (rotation effects, stellar flares, star spots, pulsations), which is the main reason for missing small planets. The below summary of the algorithm is based on Grziwa & Pätzold (2016) and Grziwa, et al. (2012). More details and thorough descriptions can be found there. EXOTRANS applies to the pre-processed light curves two wavelet-based filtering subroutines (VARLET and PHALET) and a box-fitting least-squares (BLS) algorithm (Kovacs, et al, 2002). The VARLET filtering method is applied first to reconstruct the stellar variability and discontinuities of the signal without the white noise and the transit feature (residual). The reconstructed light curve is subtracted from the original, leaving only the residual. Since a transit event is a steep and shallow dip with a short duration, it appears boxlike. The BLS algorithm is then used to phasefold the light curve at a wide range of periods and extract the box-like event by fitting a square box to the phase range. The period which delivers the lowest χ 2 is selected as the correct one, binned to reduce noise, and the transit feature is amplified several times. EXOTRANS does not take into account the effects of limb darkening Wavelets and the Stationary Wavelet Transform Denoising The continuous Wavelet transform (WT) uses finite pulses (wavelets) to find powers of frequencies. It works by computing the convolutions of different wavelets with the data to be analysed by scaling and translation. This produces the coefficients for each frequency. The 17

26 magnitude of the coefficients can be used to measure the similarity between the chosen wavelet and the signal at a given time. The WT is a more sophisticated alternative method to the Fourier transform (FT) for time-frequency analysis. What makes the WT superior is the fact that it is localised in both time and frequency domain, while the FT is only defined in frequency domain. Due to the Sampling theorem, which states that one cannot have infinite frequency and time resolution at the same time, the frequency and time estimates in the WT are approximated, which means it is not as precise. However, it is due to this ability to decompose a signal in ways which allow the retention of high frequency resolution in one part and high time resolution in another (not simultaneously), together with the fact that a wavelet can be localised in time, be smooth (vanish towards high frequencies), and have fast vanishing moments (decay towards low frequencies), that makes the WT so suitable for analysis of complex, abruptlychanging signals. More specifically, the Stationary Wavelet Transform (SWT) is typically used for noise reduction in complex time series. Both the SWT and the Discrete Wavelet Transform (DWT) convolve the input signal with a high and a low pass filter, thus producing two vectors one for the high and one for the low pass part. However, the SWT is preferred in many applications because it does not downsample the signal, i.e. the coefficient vectors produced are of the same length as the input data. The DWT, in contrast, reduces it by half at each level. What is meant by each level is that the denoising works by calculating and applying a new noise threshold and a new wavelet resolution to filter and separate finer and finer parts of the noise until the optimal signal reconstruction is achieved by combining all the different levels. Finally, goodness of the filtered signal is dependent on the wavelet, the noise thresholds of choice, and the number of iterations. Both VARLET and PHALET filter routines used in the EXOTRANS pipeline are built using the SWT VARLET In this filter routine, the SWT reconstructs the signal the stellar variability and light curve discontinuities by using a series of scaled wavelets. The reconstruction, however, does not include the short duration and small amplitude of a transit. The actual transit is contained, together with the noise, in the residual signal. The residual is obtained by subtracting the reconstructed signal from the original. The transit search is done by the previously mentioned BLS algorithm because a transit appears box-like. It is important to note that while it is convenient to use the residual light curve for transit detection, it is not to be used for further analysis of the candidate because the VARLET routine affects (reduces) the depth of the transit, which is crucial in determining the size of the potential planet (Equation 1). Therefore, the characterisation following the detection is to be done using the original light curve PHALET While VARLET is used to remove the stellar variability and abrupt discontinuities in the pre-processed light curve, PHALET is designed to remove disturbances of known frequencies (and their harmonics). This is particularly useful in the case of orbital resonances or when a stellar binary or a planet of a dominant frequency conceals the existence of a(nother) planet. In such a scenario, the frequency of the known disturbance is removed, and the residual 18

27 light curve is searched again for any additional transits. The procedure followed by PHALET is described below: 1. After detrending each segment, the residual light curve is phasefolded at a period detected by the BLS algorithm. This amplifies the feature at that period and blurs out the rest. 2. Once the light curve is phasefolded, it is denoised using SWT and binned. 3. The resulting light curve is unfolded and subtracted from the input False Positives and Signal Detection Efficiency EBs and blends were discussed as sources of false positive detections in Section 2.3. However, it is any kind of disturbance that is not a planet that is considered a false positive, including strong stellar variability, discontinuities and other periodic variability in the light curve. Unlike the latter group, EB s are not removed after filtering and the BLS routine cannot differentiate between a stellar binary and a planetary transit due to their strong similarity. The Signal Detection Efficiency (SDE) is a metric employed in EXOTRANS, which is calculated for each light curve and used to describe the quality of the fit and the transit probability. The confidence level of the SDE noise threshold is 95% and it is light curves which pass this condition that are entered in for further analysis. At this point visual inspection of the light curves becomes important. This is the stage where all the candidates are considered and shortlisted for further study if one is possible. To illustrate, in Figure 9 I show four typical plots derived by EXOTRANS, for four different targets. Only one (bottom right) contains a discernible (possibly) planetary transit. Figure 9: Four typical plots for visual inspection derived by EXOTRANS. The light curves are phasefolded and binned. The vertical axes correspond to relative flux and horizontal axes represent the phase (0 to 1). Each red bar represents binned values, centred on their average. The height of the bars depends on how far apart the values in each bin are. The light curves belong to: EPIC (top left), EPIC (top right), EPIC (bottom left), EPIC (bottom right). Top and bottom left do not contain a discernible transit feature. The two dips in the top right (one shallower than the other) may signify an EB, as described in Section 19

28 2.3. The light curve in the bottom right corner contains a clear transit feature, consistent with a planetary transit. This type of candidate enters the next stage of analysis. The list for visual inspection can be further reduced by employing Transit-Timing Variations (TTV). TTVs occur when there are gravitational perturbations between different planets, which lead to changes in the orbital period. If the variations match the forcing frequency of the other planet, this indicates that the two planets are in the same system, sometimes in some form of orbital resonance (Fabrycky, et al, 2012). In other words, if a planet is not observed at a given position at the expected time, and instead has yet to reach or has just passed this position, this can be a sign that another planet is influencing its orbit. However, the presence of TTVs does not exclusively lead to this conclusion. Sometimes, they can be caused by starspots, which complicates the interpretations of measured TTVs. Simulations performed by Ioannidis, et al. (2015) show that TTV amplitudes larger than ~1% are unlikely to be caused by starspots. The most problematic cases are those of lowamplitude TTVs with statistically significant 11 periods found in high transit signal-to-noise ratio (SNR 15) light curves. A probability and/or a binary simulation algorithm can also be used to perform further vetting of candidates. Ultimately, it is only after RV measurements (discussed in further detail in a later section) or another type of validation technique has been performed, that more definite system parameters and conclusions can be derived Pyaneti Although the transit parameter equations in Section 2.1. are useful, they are crude at best. Their purpose is to obtain a general idea about the system by an order of magnitude estimation, rule out obvious false positives and decide on follow-up priority. For the transit modelling part of my project I used Pyaneti, developed by O. Barragán, D. Gandolfi and G. Antonicello at University of Turin and University of Padova, Italy. The below description of their software follows Barragán, et al. (in prep). In practice, to obtain more accurate physical parameters of the planetary candidate, a parametric model is constructed and compared with the light curve data. A Bayesian approach is often used to do this because it allows to estimate the probability of a certain model to be accurately representing the observations. This can be written as P (M D), i.e. the probability, P, of the model, M, given the data, D. This can be estimated using Bayes theorem: P(M D) = P(D M)P(M) P(D) (18) Here P (M D) is the joint probability distribution, P (D M) is called the likelihood of observing the data, D, given the model, M; P(M) is the prior probability, and the probability of the data, or the evidence, is P(D). The prior is the previous information known about a parameter. The most typical priors in use, also used in Pyaneti, are the Uniform and the Gaussian. The former is called weakly informative and is used when the only information available about a parameter is its range (e.g. the orbital eccentricity and the semi-parameter 11 Unlikely to be caused by chance. 20

29 both lie in the range [0:1]). If the parameter is equally likely to lie in the range between a and b, then its uniform prior is given by the expression: P = 1 (19) b a Otherwise, the prior is 0. The Gaussian prior is an informative prior and is used when there is a previous measurement, which is used to improve the fit. For a median, a, a standard deviation, b, and a parameter φ, the Gaussian prior is given by: P = 1 (φ a) 2 2πb 2 e 2b 2 (20) The likelihood, P (D M), that the model, M, describes the data, D, where Di is a given datapoint, and Mi is a given model point, can be written as: P(D i M i ) = ( 1 (D i M i ) 2 2π(σ 2 i + σ 2 j ) e 2(σ 2 i +σ 2 j ) ) (21) The correctness of Equation (21) is subject to some key assumptions: the data is normally distributed, with only uncorrelated noise, σi, present, and each data point, Di, is independent. The term, σj, becomes important when σi is underestimated, in which case it normalises the likelihood. The posterior distributions of two models can be used to compare how well they represent the data, with the following equation: P(M 2 D) P(M 1 D) = P(D M 2)P(M 2 ) P(D M 1 )P(M 1 ) (22) If this ratio exceeds 1, then P (M2 D) represents the data better. P (M1 D) is better if the ratio is less than 1. Pyaneti employs a Markov Chain Monte Carlo (MCMC) method. Markov chains are used to generate different models with a different starting point in the parameter space. The future state of the chain is dependent only on its current state. The method is MCMC if the connection between the chains is random. There are many sampling methods used in MCMC and they all ensure that the chains converge to the best solution. Pyaneti uses an advanced sampling method, which uses an affine invariant ensemble sampler. This method uses groups of Markov chains to sample the parameter space. Each chain starts at a point and evolves using the complementary chains. After N iterations and L chains, the user has N L samples with different probabilities for each parameter, from which the optimal solutions for all parameters can be obtained. For final parameter solutions to be obtained, the chains need to have converged. Pyaneti uses a popular convergence test developed by Gelman & Rubin (1992). This test employs a scaled factor R = B/W, where B is the variance between chains and W is the variance within a chain. The threshold value for R used in Pyaneti is

30 Calculations of likelihoods and orbital solutions are computationally demanding and so Pyaneti utilises the computational speed of FORTRAN (for the calculations) and the userfriendly Python (for plots and data preparation). Pyaneti is Python-controlled in the sense that all the input information necessary for the simulation is entered into an input file, allowing Pyaneti to be run with only one command line. Pyaneti, however, does not include TTVs, occultations and planetary contributions to the total flux, Rossiter-McLaughlin effect, gravitational perturbations between different planets, multi-band photometry and others. Apart from the input file mentioned above, Pyaneti uses a detrended light curve as input. The detrending is done using a routine called Exotrending, developed by the same authors. Both Pyaneti and Exotrending are freely available and can be found on GitHub Other Methods Other methods and models have also been used successfully: Petigura, et al. (2013) and Yu, et al. (2018) used TERRA, which identifies Threshold-Crossing-Events (TCEs), which, as with EXOTRANS, are then visually examined to remove obvious false-positives. Two popular alternatives to Pyaneti are batman, a Python-based software package (Kreidberg, 2015) and EXOFAST (Eastman, et al, 2013), which is written in IDL. Due to the large number of light curves to visually examine, human vetting is a timeconsuming process. The human element inevitably introduces a bias and uncertainties, which are difficult, if not impossible, to quantify. An interesting alternative to the visual inspection is the robovetter, a robotic vetting procedure, developed by Coughlin, et al. (2016). The robovetter achieves this by employing multiple threshold conditions and constraints: several metrics, designed to test the similarity of a TCE to a planetary transit, the statistical significance of a TCE, the detection significance (related to the SNR), among others. Among the most common and useful tests are the even-odd and secondary eclipse tests. The even/odd difference is caused by a variation in the depth between the primary and secondary eclipse of an EB. A significant variation could mean that the transiting object is not a genuine planet (Livingston, et al, 2018). EBs usually produce a secondary eclipse of significant depth, indicating a selfluminous object, while planets cause a very small to negligible secondary eclipse, which is generally not discernible in the light curve. For circular orbits, the secondary eclipse is found at half a phase shift from the primary transit. Coughlin, et al. (2016) state that the robovetter is overall comparable to, and in most cases superior to, the human vetting procedures. Since the amount of data, which needs to be processed and visually inspected is greatly increased for future missions (such as the recently launched TESS (Transiting Exoplanet Survey Satellite) mission and the upcoming PLATO (PLAnetary Transits and Oscillations) mission), it is worth investing effort in developing such an automatic procedure. 5. Results As a first step, the Vanderburg light curves (Section 3.1.) were downloaded from the Mikulski Archive for Space Telescopes (MAST) as soon as they were available and were fed through the EXOTRANS pipeline. When EXOTRANS finished its run on K2 Campaign 15 (C15 for short), I visually examined and flagged the plots which appeared to contain a transit- 22

31 like feature. The candidates which passed the visual inspection (Section , Figure 9) I then subjected to further vetting to identify the false positives among them. I did this by examining the difference between the even and odd transits and by checking for the presence of a secondary eclipse. EXOTRANS plots the even and odd transits so the user can check if such a variation exists. While EB transits are often V-shaped and deep (>1%), they are not exclusively so, and this factor cannot be the only reason for discarding a candidate as a false positive. It only reduces the likelihood of the candidate being a planet and reduces the priority for further analysis. A transit depth of over 1% can be easily produced by planets orbiting small M-dwarfs. Generally, transit depths of over ~3% I did not consider for follow-up. Additionally, the target star has to be on the main sequence. The 30-minute sampling of KST can also be the reason for a transit to appear V-shaped (Petigura, et al, 2017). In other words, in order for a transit event to be classified as a viable planetary candidate, this event has to be unique in depth, duration and period in the phasefolded light curve (Coughlin, et al, 2016). Following the above steps, I compiled a list of viable candidates and stellar parameters for the candidates hosts were obtained from the EPIC database. Next, I set some constraining parameters: maximum semi-major axis, maximum planet radius and transit depth, and I used them, along with parameters derived by EXOTRANS, to calculate which transit events cannot be of planetary origin. Overall, giant stars were excluded in this step and candidates, which were either too close to the star to be planets, or have a a < R*, i.e. the planet is inside the star. Such candidates I removed from the list, thus leaving only the strongest ones, which were then entered into the transit modelling stage. Firstly, I used the Exotrending routine to detrend each of the shortlisted candidates Vanderburg light curves and to find and phasefold the separate transits. I used the transit period (P), full duration (t14) and the time of first transit (T0), derived by EXOTRANS, as input parameters for the Exotrending input file. Similarly, I fed an input file for Pyaneti, together with the Exotrending detrended light curve file, into Pyaneti, to obtain the transit fit plot, estimated stellar and candidate parameters, chains distributions (to check for convergence), posterior distributions and correlations between the different parameters. The Pyaneti input file takes as input the following parameters: thin factor, number of iterations, number of chains, stellar parameters and their uncertainties, T0, transit period and their uncertainties, ranges of the semi-major axis and impact parameter, and limb darkening coefficients (u1 and u2). The latter are calculated using the quadratic limb darkening model, as per Mandel & Agol (2002) (Equation 17). A uniform prior can be used to fit for u1 and u2, but this works best for giant planets and not so well for smaller ones. Given the stellar metallicity [Fe/H], effective temperature (Teff), and surface gravity (log(g)), different values for the limb darkening coefficients were calculated using the online applet 12 by Eastman, et al, (2013). The fitted parameters are T0, P, b, a/r*, Rp/R*. The last two are dimensionless and give the semimajor axis over the stellar radius, and the scaled planetary radius, respectively. The Pyaneti derived parameters are: Rp; i; a; the insolation (amount of stellar flux reaching the planet); one value for ρ* from the input stellar parameters and one derived from the transit light curve; equivalent temperature (Teq) for the planet, assuming the albedo = 0; t23 (T_tot) and t14 (T_full). Although, the eccentricity and argument of periastron can be added to the model, for this study I assumed the orbits to be circular, i.e. e = 0 and ω = 90. I performed several test runs for each candidate (thin factor = 1, number of iterations = 500, number of independent Markov chains = 100) until the optimal parameter ranges were determined. Once this was done, for the final 12 Available at: 23

32 runs I had a thin factor of 10 (i.e. the chains state was saved every 10 iterations), 500 iterations and 500 independent chains. The EXOTRANS plots for 30 candidates can be seen in Appendix A. Table 1 13 below contains their parameters as detected by EXOTRANS, except for Teff values and uncertainties, which are taken from EPIC. T0 is given in Barycentric Julian Date (BJD). This is the Julian Date (JD) corrected for the position of the Earth relative to the Solar system s barycentre (centre of mass, which all bodies orbit around). EPIC ID P (days) ΔF t 14 (hours) T 0 (BJD) T eff (K) % % % % ± % % % % ± % % % ± % % % % % % % % % ± % % % ± % % % ± % % % % Table 1: Period, flux change, transit duration and time of transit for 30 planetary candidates (27 targets) as detected by EXOTRANS and Teff with uncertainties from EPIC. 13 This list is by no means exhaustive. Although they appear to be good candidates, some of them may be false positives. Also, there are almost certainly more good candidates in Campaign

33 6. Discussion 14 More thorough investigation of my transit modelling results for seven of the most promising targets (at least one of which hosting two or more candidates, totalling eight) are illustrated below and contain the transit events produced by Exotrending using the Vanderburg light curves (top figure) and the transit fit (marked by red dots) for each candidate, calculated by Pyaneti (bottom figure). The convention is that the first planet for a star is named after the star s designation with an added letter b in the end. Every other planet found in the same system takes the letter c, d, etc., in order of discovery. This does not necessarily follow the distance between the planets and their star. RE stands for Radius of Earth and all estimated planetary radii (RP) are in RE units. AU stands for Astronomical Unit and is the average distance from the Earth to the Sun a standard unit for distance, most suitable for celestial objects in the same system. The orbital periods of the candidates are shown, also where the transit events are found, are marked by red dotted vertical lines. Table 2 contains a few key parameters for each planetary candidate. The tables in Appendix A contain all parameters calculated by Pyaneti for each of the eight candidates and the joint posterior probability distributions for some of their parameters. EPIC Figure 10: Top panel: pre-processed light curve of EPIC with five prominent transits. The fifth transit event was removed because of a disturbance. Bottom panel: candidate is consistent with a warm Jupiter. P = days, t 14 = 3.5 hours, R P = 8.5 R E, a = 0.15 AU. 14 All spectral types were estimated approximately based on the effective temperatures for the stars found in EPIC from (Pecaut & Mamajek, 2013). The uncertainties can make a greater difference for late-type stars due to the smaller increments in T eff reduction. 25

34 The top figure shows five transit events produced by a single source. One transit is removed due to a disturbance in the data. The ΔF of this transit is about 0.6%. Pyaneti estimates that the candidate is in an 0.15 AU orbit with a period of nearly 15 days and a radius of ~8.5 RE, consistent with a warm Jupiter. The host star is consistent with a G2V-type star, i.e. solartype. EPIC Figure 11: Top panel: pre-processed light curve of EPIC Small dips corresponding to the transits are visible. Bottom panel: Pyaneti fit of the eight transits of EPIC b: P = days, t 14 = 3.5 hours, R P = 6.8 R E, a = 0.22 AU. The Vanderberg light curve shows eight transits of EPIC b. The individual transits are not as prominent due to the continuous reduction in stellar intensity. This single source produces a flux reduction of 0.1%. The semi-major axis of this orbit is 0.22 AU, the orbital period is 10.3 days, and the candidate has an estimated radius of 6.8 RE. The host star is of type G8V. EPIC

35 Figure 12: Top panel: Pre-processed light curve of EPIC The nine small transits are barely visible. The most noticeable feature are the two transit events of what could be a Jupiter planet with a period ~5 times larger than the candidate under investigation. Bottom panel: Transit fit corresponding to the multiple shallow transits in the preceding light curve. The candidate is consistent with a super-earth in a tight orbit. P = 9.21 days, t 14 = 4.86 hours, R P = 2.7 R E, a = 0.08 AU. EPIC is an interesting light curve because it shows two contributors. EXOTRANS only detected one candidate (EPIC b), however nine transits with a period of 9.2 days and a depth of ~0.5%. Pyaneti estimated a radius of 2.7 RE and a = 0.08 AU. This is quite small because, if the Teff from EPIC is accurate, the host star s spectral type is ~F8V, i.e. hotter and larger than the Sun. The Vanderburg light curve also shows a second contributor (EPIC c) with only two prominent transits, which could be a reason why EXOTRANS did not make this detection 15. For lack of input data to use for Pyaneti for this second candidate, I have not tried to fit a transit curve. Nevertheless, the transit of EPIC c is about five times deeper and its period about five times longer than EPIC b. Another interesting moment is the flatness of the transit fit. This means that the candidate is travelling across the face of the disk for a substantial amount of time, which is consistent with the relatively long transit time and the relatively small planet, orbiting a star larger than the Sun. EPIC In theory, two transits should be sufficient to make a detection and fit a transit curve through the data but this is not always as straightforward. 27

36 Figure 13: Top panel: Pre-processed light curve of EPIC Dips of two different candidates can be identified. The two potential planets are in a 2:1 orbital resonance. Middle panel: P = 9.95 days, t 14 = 3.88 hours, R P = 5.7 R E, a = 0.19 AU. Bottom panel: P = days, t 14 = 4.88 hours, R P = 7.8 R E, a = 0.31 AU. This is the light curve of EPIC , which also shows two contributors. Both the light curve and the periods of these two potential planets show that they are in a 2:1 orbital resonance (for two orbits of EPIC b, EPIC c completes one). EPIC b has produced nine transits during the observation period, and EPIC c is visible on five of those. EPIC b has produced a transit depth of 0.06%, has a period of almost ten days, a semi-major axis of ~0.2 AU and a radius of 5.7 RE. EPIC c has a period of just over 20 days and has produced a transit depth of 0.13%, with its 7.8 RE and 0.3 AU orbit around a K0.5V star. EPIC

37 Figure 14: Top panel: Pre-processed light curve of EPIC , a very active star, with intensity changes as high as ~1%. Nevertheless, the seven dips are visible. Bottom panel: Transit fit for a candidate the size of a small Neptune. P = days, t 14 = 3.88 hours, R P = 3.45 R E, a = 0.11 AU. The light curve of EPIC is highly irregular, showing strong intrinsic stellar variability with no apparent periodicity and intensity changes as high as ~1%. Seven transits of depth ~0.1% are identifiable but not prominent due to the large variability amplitude. The size of EPIC b is consistent with a small Neptune with a period of 12.3 days and orbiting a ~G4V star at 0.11 AU. EPIC Figure 15: Top panel: Pre-processed light curve of EPIC Bottom panel: Transit fit for a large Jupiter. P = days, t 14 = 3.56 hours, R P = 13 R E, a = 0.21 AU. The Vanderburg light curve of EPIC shows a singular source of five clear transits, with the beginnings of a sixth, but it is insufficient to add to the fit. The depth of these transits is almost 0.6% and the candidate has a period of nearly 15 days and the radius of a giant planet 13 RE. This large potential planet would produce a higher flux deficit around a 29

38 solar type star with it a of just over 0.2 AU. However, the effective temperature of this planet suggests a host star of spectral type F2V over 1.6 times larger than the Sun. EPIC Figure 16: Top panel: Pre-processed light curve of EPIC Bottom panel: Transit fit for another super- Earth in a close orbit. Eleven transits for a candidate with P = 7.82 days, t 14 = 2.82 hours, R P = 2.25 R E at a = 0.07 AU. Finally, the Vanderburg light curve of EPIC shows eleven transits with a depth of 0.1% from a candidate orbiting the coolest star in this sample a K2V. Unsurprisingly, this candidate the shortest transit duration of all of the above. The period of this candidate is ~7.8 days, it is 2.25 times larger than the Earth and orbits its late-type star the shortest distance so far 0.07 AU. EPIC ID P (days) t 14 (hours) R P (R E) a (AU) a/r * b ± ± ± b ± b ± ± ± b ± c ± , ± b ± ± ± b ± ± b ± ± Table 2: Summary of the parameters and uncertainties of the eight planetary candidates as calculated by Pyaneti. The period distribution of these candidates is expected to be Gaussian and if the model is correct, the true values will lie in the 68% credible interval, which is equivalent of 1-σ for Gaussian-shaped distributions. 30

39 Figure 17: Period distribution of the 30 EXOTRANS detections and the modelled transits. The above period distribution peaks at ~10 days. Although it was not expected to, this corresponds to the observation stated in Ricker, et al. (2014) that the peak of the Kepler planets period distribution is reached at ~10 days. Formation of planets has been correlated with high stellar metallicity. The metallicity distribution of the hosts of the 30 candidates is shown in Figure 18 below. It can be seen that the candidates are clearly related to stars if similar metallicity to the Sun. Figure 18: Metallicity distribution of the 30 EXOTRANS candidates. Figure 19 (left) shows the Teff distribution of the targets in Campaign 15, for which there was data in EPIC. There are targets in this figure, split into 200 bins. There are three clear peaks here at ~4000K, ~5000K and ~6200K. Figure 19 (right) shows the Teff distribution for the 27 stars hosting the 30 candidates in Table 1. The peak here is at K. This is in agreement with the metallicity distribution as metallicity tends to be higher for cooler stars. It is not surprising that the distribution on the right does not mimic the distribution on the left since the number of stars on the right is much lower. 31

40 Figure 19: Left: effective temperature distribution of targets from Campaign 15, for which there were values in EPIC. Three clear peaks are visible at ~4000K, ~5000K and ~6200K. Right: distribution of the 27 hosts in Table 1. Unsurprisingly, this distribution does not mimic the one on the left as the sample is much smaller. As mentioned previously, to make any remarks about characterisation regarding these candidates if they were confirmed to be of planetary origin, information of the planets masses is needed (see following section). While it is possible to gain such knowledge via means other than RV measurements, e.g. via probabilistic mass-radius relation models, such as the R-based code developed by Wolfgang, et al. (2016), this is beyond the scope of this work. 7. Validation and Follow-up Observation Due to the relatively large PSF of KST, its limited photometric precision and the different false positive scenarios described in Section 2.3., the false positive detection rate is high. Therefore, planetary candidates, once vetted manually, need to be confirmed via further detailed photometric studies or RV measurements (Collins, et al, 2018). It is also possible to perform statistical validation on the strongest candidates. This is done by high resolution imaging to rule out EBs and blends, and rigorous light curve shape analysis (Petigura, et al, 2017). However, RV measurements (also called Doppler spectroscopy) are not used to just confirm that the candidate is a genuine planet, but to also constrain its mass, through which the bulk density is determined. The density is needed to draw conclusions regarding the structure of the planet whether it is made of rock, gas, ice, etc. The working principle of the RV method involves the Doppler effect (the stellar motion-induced wavelength shift of the spectral lines). More specifically, it uses the fact that both the star and the planet orbit their common centre of mass (barycentre). Due to the large size of the star, the barycentre is usually located inside the star, causing the star to appear to wobble. This wobble is determined by measuring the radial (line-of-sight) component of the velocity around the barycentre. It goes without saying that to obtain reliable and useful measurements, the SNR and resolution of the instrument need to be optimal, but in fact the type of star being observed has a much greater influence on the gathered data. Hot stars have fewer spectral lines, which also tend to be shallow, due to their fast rotation rate. Finding the position of the centroid makes the detection of Doppler shifts in such lines challenging. In contrast, cooler stars are slower rotators 32

41 and produce many deep and narrow lines, which are much easier to study (Figure 20, left). A nominal value of 10 m/s is needed to detect a Jovian planet orbiting a Sun-like star at a distance of 5AU. Figure 20 (right) shows that the RV error (horizontal dashed line) is comfortably below this point for stars of spectral type around F6 and later. It, however, rapidly increases for hotter stars. Figure 20: Left: The spectrum of a B9 star (top). Only a single wide and shallow spectral line can be seen. The spectrum of a K5 star can be seen at the bottom. In this case, there are many lines, all deep and narrow, which makes it easier to detect a Doppler shift. The horizontal axis gives the wavelength in units of Ångström (1Å=0.1 nm) vs the relative flux on the vertical. Right: The RV error as a function of stellar spectral type. The dashed horizontal line gives the nominal 10 m/s needed for the detection of a Jupiter-sized planet at 5AU from a solartype star. The RV error is well below this nominal value for stars around F6 and later and increases fast for earlier types. Image adapted from One of two reasons for this is the rotational broadening of spectral lines for hotter stars. Stars outer convection zone is the driver for magnetic activity. Magnetic activity, in turn, is the main reason for slowing down stars rotational rate. The other reason is that the number of spectral lines decreases with increasing Teff. These effects explain why most exoplanet discoveries so far have been around stars later than F6 (Hatzes, 2016). G-K stars have been studied extensively. More and more attention is falling on cooler stars with lower mass late K, and especially M dwarfs, because lower mass planets will be easier to detect around them. Additionally, the habitable zones (HZ) of these cool, dim stars (where water can exist in liquid form) are much closer to their stars. Potential planets in the HZ of these stars will have their semi-major axes much closer to their host stars, making them easier to detect by RV measurements (Perryman, 2011). For slowly rotating stars, high-precision masses of planetary companions can be derived with high resolution and SNR at the level of 3-30 m/s. The RV for fast rotators with rotational speeds of ~40 km/s is often impossible to determine with sufficient precision. If the transit light curve suggests a planetary companion, for rotational speeds of up to ~200 km/s, it is possible to use the method of Doppler Tomography (DT) to confirm that the source of the transit signal is indeed a planet-sized body, ruling out a possible blend scenario (Collins, 2018). Apart from the problem with quantity and quality of stellar spectral lines, other possible obstacles may be that the target star is too faint, or the expected radial velocity amplitude too small to be measured by the available instrumentation. Other possible methods are performing transit modelling based on adaptive optics imaging and/or photometric observation at longer wavelengths. 33

42 Sometimes, however, the sheer number of candidates renders these options impractical. In such cases, one may adopt a Bayesian approach to determine the probability of the transit candidate being an actual planet or a false positive. One such method is PASTIS (Díaz, et al, 2014), which uses an MCMC approach to model the light curve with different scenarios genuine planet and the different false positive scenarios. Using the likelihood for each scenario, it estimates the probability that the genuine planet scenario exceeds the probabilities of the other scenarios combined (Cameron, 2016). 8. Error Sources and Uncertainties To interpret a light curve and classify a candidate correctly, the stellar parameters obtained from EPIC, need to be known to a good degree of accuracy. Unfortunately, this is not always the case as EPIC is made up of archival photometric, rather than spectroscopic measurements, and is thus more prone to systematic errors and offsets in the photometry, yielding untrustworthy results (Mayo, et al, 2018). Specifically, metallicity estimates for stars without spectroscopic observation are only statistical and are not to be used for in-depth investigation. The precision for Teff values is estimated to be 2-3%, 0.3 dex for surface gravity and metallicity. The stellar radii of 55-70% of subgiant and low-mass stars are also systematically underestimated as they are misclassified as dwarfs (Huber, et al, 2016; Yu, et al, 2018). For the purpose of this work, however, these estimates are sufficient. Sometimes the detrending of the light curve is not enough to make a good fit. It is often necessary to remove outliers and parts of the light curve, which have experienced a disturbance. Care has been taken to remove only strong outliers in the light curve (>5σ), however there are inevitable exceptions, which need to be treated on a case-by-case basis (e.g. EPIC ). In any case, due to the unstable pointing of the Kepler spacecraft, disturbances are common and K2 data needs heavy multi-level detrending to give acceptable precision. The detrending causes the appearance of artefacts, which can in turn deteriorate the data. The limb darkening approximation used in the MCMC models is far from perfect. Observing transits at longer wavelengths would reduce the uncertainty in the parameters because it would make the transit look more like a box, thus minimising the effect of inaccurate limb darkening modelling. Another assumption made in the modelling is that measurement errors are statistically independent. This is often not the case. Apart from intrinsic stellar variability, the variability in the light curve can be attributed to different sources: instrumental noise, extinction, disturbances in the local environment, or detrending artefacts. Such variability does not have to be restricted to a single point and is, in fact, often present in multiple data points, reducing the number of independent data points. While ignoring these effects leads to reduced precision, it is very difficult to account for them and the procedure can be computationally demanding (Winn, 2010). As mentioned previously, the presence of sunspots, TTVs, the Rossiter McLaughlin effect, among others, are not accounted for in the methods and models used in this work. Since it is humans that write the programs, choose the models, perform the visual inspections and make the judgement calls, the uncertainties are inevitable and often unpredictable: bias regarding past experience and familiarity with a method or model, personal preference and conviction, as well as recent experiences can and do affect an investigator s decisions. Nothing will replace seeing the systems studied up close as no two stars are the same 34

43 and no model describes nature perfectly, but given the current technological state, it is worth appreciating how amazing it is that we are even able to make the deductions that we do. 9. Summary and Conclusion Exoplanet discovery and characterisation is one of the most exciting and fast-moving fields in astronomy in modern times. Learning more about the planets has pushed the boundaries of understanding in stellar astrophysics, stellar and planetary formation and migration scenarios with the increasingly technologically daring space- and ground-based telescopes and instruments. Many beliefs, like most stars being roughly as quiet as our Sun, planets around other stars being non-existent or extremely rare, most existing planets being hot gas giants in close orbits, and many others, have been shown to be wrong. A lot of the misconceptions were caused by our own human bias based on our own experience. Thanks to the ground-breaking discoveries by missions utilising high-precision photometry, such as KST and CoRoT, it is now known that most stars are active and variable, making the confident detection of a transit very difficult; the current census of the exoplanet population shows that planets around other stars are not only common, but that smaller planets, such as Super-Earths, are in fact the most common (Figure 2). Batalha (2014) made the observation that 85% of the detections made by Kepler have radii smaller then Neptune (~4RE). In this work, I have shown one possible way to create an exoplanet candidate list for follow-up investigation: first using a transit detection algorithm (EXOTRANS) on the preprocessed light curves (Vanderburg or other), followed by visual examination and fast calculation to rule out obvious false positives, and finishing by using a combination of a light curve detrending algorithm (Exotrending) with a transit modelling tool (Pyaneti) to derive the most plausible parameters of the system. This procedure using these software packages will also be used in future missions like TESS and Planetary Transits and Oscillations (PLATO). While this method is undoubtedly largely successful, KST is at the end of its life. Thankfully, with the four wide-field CCD optical cameras of TESS (Ricker, et al, 2014), launched in April 2018, looking at 200,000 nearby main-sequence stars in the search for transits, it is expected that over a thousand planets with radii smaller than Neptune will be found, with dozens in the Earth-size range. Additionally, the stars monitored by TESS are several times brighter than the ones monitored by Kepler, so they will be much easier to follow up by other telescopes. This is a required step if we are to move from demographics to characterisation of the continuously growing exoplanet census, because by observing in a single bandpass, transit photometry only yields the transit duration, planet radius, orbital period and semi-major axis. Future missions, such as the Characterising Exoplanets Satellite (CHEOPS) (Benz, et al, 2017) will provide preliminary characterisation of planets whose mass is already known, using ultrahigh precision photometry. CHEOPS is on track for a 2019 launch and will measure radii of planets in the size range between Earth and Neptune (~1-4 RE), thus providing the best targets for spectroscopic observation in search for planetary atmospheres by both ground- and space-based, current and future instruments. The upcoming and muchanticipated James Webb Space Telescope (JWST) (Beichman & Greene, 2017) will provide information on the composition of atmospheres planets discovered by KST, TESS and other transit surveys may have. After numerous delays, it is set for a 2021 launch and will use transit spectroscopy on short period planets, and coronographic imaging of planets orbiting their host stars at greater distances. JWST is expected to answer questions about the variability of 35

44 atmospheres with mass, radius, insolation and location; atmospheric dynamics and vertical structure; planetary formation mechanisms of close-in vs far-out planets, and more. Another upcoming mission with high hopes from the scientific community is PLATO (Rauer, et al, 2014, 2017) on track for a 2026 launch. As a transit survey mission, it will detect new planets around bright stars, with a focus on small planets in the HZ of Sun-like stars. It will provide radius estimates with unprecedented precision, as well as the age of these planets. The brightness of the stars will ensure that they can be followed up by radial velocity observation, thus yielding accurate mass estimates. Asteroseismology being one of PLATO s key objectives, will allow the precise characterisation of the host stars, thus ensuring the accurate determination of the planet parameters. PLATO will answer key questions about the properties of (terrestrial) planets in the HZ of their stars; the relation between stellar and planet parameters; the temporal evolution and typical architecture of planetary systems. PLATO will produce high-quality candidates for follow-up investigation. Finally, thanks to the accurate parallaxes (angular distances) by the Gaia spacecraft (Gaia collaboration, 2016), launched in 2013, PLATO (as well as current missions) will be able to measure the luminosities of the target stars and determine their location on the HR diagram. Providing an independent measurement on the distances to the most nearby stars, Gaia provides a critical constraint towards stellar modelling (Catala, C. & PLATO Consortium, 2009). 36

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48 Appendix A Class Ia-O Ia Ib II III IV V VI, sd D Type of Star Extreme, luminous supergiants Luminous supergiants Less luminous supergianats Bright giants Normal giants Subgiants Main sequence stars Subdwarfs White dwarfs Table 3A: The M-K system of spectral classification. Class Temperature (K) Apparent Colour O 30,000 blue B 10,000-30,000 blue-white A 7,500-10,000 white to blue-white F 6,000-7,500 white G 5,200-6,000 yellow-white K 3,700-5,200 yellow or orange M 3,700 orange or red Table 4A: The seven main spectral classes, their associated temperatures and colours. Appendix B EXOTRANS plots for the 30 candidates that were visually examined. EPIC b EPIC b EPIC b 40

49 EPIC b EPIC b EPIC b EPIC b EPIC b EPIC b EPIC b EPIC b EPIC b EPIC b EPIC b EPIC b EPIC b EPIC c EPIC b EPIC c EPIC b EPIC b EPIC b EPIC b EPIC b 41

50 EPIC b EPIC b EPIC c EPIC b EPIC b EPIC b Appendix C Posterior distributions and tables for the parameters of the eight planetary candidates from Section 6, as derived by Pyaneti b Fitted T0 = days P = days b = a/r* = rp/r* = Derived Rp = R_earth i = deg a = AU Insolation = F_Earth rho* = g/cm^3 (transit) rho* = g/cm^3 (stellar paramters) Teq = K (albedo=0) T_tot = hours T_full = hours 42

51 b Fitted T0 = days P = days b = ar = rpr = Derived Rp = R_earth i = deg a = AU Insolation = F_Earth rho = gcm^3 (transit) rho = gcm^3 (stellar paramters) Teq = K (albedo=0) T_tot = hours T_full = nan - nan + nan hours 43

52 b Fitted T0 = days P = days b = a/r* = rp/r* = Derived Rp = R_earth i = deg a = AU Insolation = F_Earth rho* = g/cm^3 (transit) rho* = g/cm^3 (stellar paramters) Teq = K (albedo=0) T_tot = hours T_full = hours 44

53 b Fitted T0 = days P = days b = a/r* = rp/r* = Derived Rp = R_earth i = deg a = AU Insolation = F_Earth rho* = g/cm^3 (transit) rho* = g/cm^3 (stellar paramters) Teq = K (albedo=0) T_tot = hours T_full = hours 45

54 c Fitted T0 = days P = days b = a/r* = rp/r* = Derived Rp = R_earth i = deg a = AU Insolation = F_Earth rho* = g/cm^3 (transit) rho* = g/cm^3 (stellar paramters) Teq = K (albedo=0) T_tot = hours T_full = hours 46

55 b Fitted T0 = days P = days b = ar = rpr = Derived Rp = R_earth i = deg a = AU Insolation = F_Earth rho = gcm^3 (transit) rho = gcm^3 (stellar paramters) Teq = K (albedo=0) T_tot = hours T_full = hours 47

56 b Fitted T0 = days P = days b = a/r* = rp/r* = Derived Rp = R_earth i = deg a = AU Insolation = F_Earth rho* = g/cm^3 (transit) rho* = g/cm^3 (stellar paramters) Teq = K (albedo=0) T_tot = hours T_full = hours 48

57 b Fitted T0 = days P = days b = a/r* = rp/r* = Derived Rp = R_earth i = deg a = AU Insolation = F_Earth rho* = g/cm^3 (transit) rho* = g/cm^3 (stellar paramters) Teq = K (albedo=0) T_tot = hours T_full = hours 49

58 50