Searching for Exoplanets in K2 Data

(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)

I

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.

(3)

II 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.

(4)

III

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. ... 2 FIGURE 2:THE EXOPLANET POPULATION DISTRIBUTION BY PLANET RADIUS, EXPRESSED IN TERMS OF JUPITER RADII (RJ).THE HIGHEST

PEAK CORRESPONDS TO ~10^-0.9RJ, OR ~1.38EARTH RADII (RE).THIS MEANS THAT THE MOST COMMON PLANETS DISCOVERED SO FAR ARE IN THE SUPER-EARTH RANGE. ... 4 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 ∗,R^P 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). ... 10 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). ... 12 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). ... 13 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). ... 14 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 K2CAMPAIGN 3.IMAGE FROM VANDERBURG, ET AL.(2016). ... 16 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:EPIC204313328(TOP LEFT),EPIC249140665(TOP RIGHT), EPIC249166632(BOTTOM LEFT),EPIC249780293(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. ... 19 FIGURE 10:TOP PANEL: PRE-PROCESSED LIGHT CURVE OF EPIC249451861 WITH FIVE PROMINENT TRANSITS.THE FIFTH TRANSIT

EVENT WAS REMOVED BECAUSE OF A DISTURBANCE.BOTTOM PANEL: CANDIDATE IS CONSISTENT WITH A WARM JUPITER.P= 14.89 DAYS, T14=3.5 HOURS,RP=8.5RE, A =0.15AU. ... 25 FIGURE 11:TOP PANEL: PRE-PROCESSED LIGHT CURVE OF EPIC249703125.SMALL DIPS CORRESPONDING TO THE TRANSITS ARE

VISIBLE.BOTTOM PANEL:PYANETI FIT OF THE EIGHT TRANSITS OF EPIC249703125B:P=10.31 DAYS, T14=3.5 HOURS,RP

=6.8RE, A =0.22AU. ... 26 FIGURE 12:TOP PANEL:PRE-PROCESSED LIGHT CURVE OF EPIC249624646.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=2.7RE, A =0.08AU... 27

(6)

V

FIGURE 13:TOP PANEL:PRE-PROCESSED LIGHT CURVE OF EPIC249675967.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=5.7RE, A =0.19AU.BOTTOM PANEL:P=20.23 DAYS, T14=4.88 HOURS,RP=7.8RE, A =0.31AU. ... 28 FIGURE 14:TOP PANEL:PRE-PROCESSED LIGHT CURVE OF EPIC249476992, 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=12.33 DAYS, T14=3.88 HOURS,RP=3.45RE, A =0.11AU. ... 29 FIGURE 15:TOP PANEL:PRE-PROCESSED LIGHT CURVE OF EPIC249378363.BOTTOM PANEL:TRANSIT FIT FOR A LARGE JUPITER.P

=14.87 DAYS, T14=3.56 HOURS,RP=13RE, A =0.21AU... 29 FIGURE 16:TOP PANEL:PRE-PROCESSED LIGHT CURVE OF EPIC249559552.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=2.25RE AT A = 0.07AU. ... 30 FIGURE 17:PERIOD DISTRIBUTION OF THE 30EXOTRANS DETECTIONS AND THE MODELLED TRANSITS. ... 31 FIGURE 18:METALLICITY DISTRIBUTION OF THE 30EXOTRANS CANDIDATES. ... 31 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. .. 32 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 ... 33

(7)

VI 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... 24 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. ... 40 TABLE 4A:THE SEVEN MAIN SPECTRAL CLASSES, THEIR ASSOCIATED TEMPERATURES AND COLOURS. ... 40

(8)

VII 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

(9)

1 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.

1.1. 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 (late- type) 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,

(10)

2 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 larger¹. 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: http://www.sociostudies.org/journal/files/seh/2005_1/seh_4_1_c_fig_2.JPG

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/𝑇𝑒𝑓𝑓√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 5.67×10^-8 W/m²K⁴.

(11)

3 of 20Rʘ to 0.1Rʘ. On the other hand, the giant stars at the top right have ranges of 10-100Rʘ

(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 – 700-1000Rʘ, 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.

1.2. 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 2³.

2 The relations between spectral type, luminosity, temperature and other parameters, can be found at:

http://www.pas.rochester.edu/~emamajek/EEM_dwarf_UBVIJHK_colors_Teff.txt. 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:

https://exoplanetarchive.ipac.caltech.edu/cgi-bin/IcePlotter/nph-icePlotInit?mode=demo&set=confirmed.

(12)

4

Figure 2: The exoplanet population distribution by planet radius, expressed in terms of Jupiter radii (RJ). The highest peak corresponds to ~10^-0.9 RJ, or ~1.38 Earth radii (RE). 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.

1.3. 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 exoplanets⁴. 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 (1.25 – 2 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 2014. 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) + 476 (K2) await confirmation. Numbers as per the NASA Exoplanet Archive:

https://exoplanetarchive.ipac.caltech.edu/docs/counts_detail.html.

5 The Super-Earth range is as per Ricker, et al. (2014).

(13)

5 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.

1.4. 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

(14)

6 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. 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).

(15)

7 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.

2.1.1. 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_𝑝²/𝑅_∗² ratio to be estimated:

∆𝐹 𝐹 = 𝑅_𝑝²

𝑅_∗² (1)

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:

𝑎³

𝑃² =𝐺(𝑀_∗+ 𝑀_𝑃)

4𝜋² (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.

(16)

8 Invoking assumption 3 to Equation (2) and rewriting for a, gives:

𝑎 ≈ (𝐺𝑀_∗(𝑃 2𝜋)

2

)

1/3

(3) The full transit duration is:

𝑡₁₄= 𝑃 𝜋𝑠𝑖𝑛⁻¹

(

√(𝑅_𝑝+ 𝑅_∗)²− 𝑏² 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:

𝑡₁₄= 𝑃 𝜋√(𝑅_∗

𝑎)

2

− 𝑐𝑜𝑠²𝑖 (6)

From Equation (5) the orbital inclination can be calculated to be:

𝑖 = 𝑐𝑜𝑠⁻¹(𝑏𝑅_∗

𝑎) (7)

Another useful equation is the stellar mass-radius relation:

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_∗

𝑅_ʘ = 𝑘 (𝑀_∗ 𝑀_ʘ)

𝑥

= (𝑘^1/𝑥 𝜌_∗

𝜌_ʘ)^{𝑥/(1−3𝑥)} (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 𝑡₁₄𝜋/𝑃 ≪ 1, together with Kepler’s third law and Equation (1), can be determined using the following relation:

𝜌_∗ = 32

𝐺𝜋𝑃 ∆𝐹^3/4

(𝑡₁₄² − 𝑡₂₃² )^3/2 (10)

(17)

9 2.1.2. 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_𝑝and 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 χ² 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 thatnoisy 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_𝑝and 𝑅_∗ are interrelated, this effect on b will cause an overestimate in 𝜌_∗ and an underestimate in 𝑅_𝑝and 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 planet- sized.

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.

2.1.3. 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).

(18)

10

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).

2.2. 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-rich⁷ 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.

(19)

11 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] ≡ 𝑙𝑜𝑔₁₀[(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 + 4.437 (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).

𝜎𝑇_𝑒𝑓𝑓⁴ = 𝐹_⨀ = 𝐿

4𝜋𝑅²𝜎 (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 paths⁸ 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.

(20)

12

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).

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) = 𝐼₀[1 − 𝑢₁(1 − 𝜇)] (16)

𝐼(𝜇) = 𝐼₀[1 − 𝑢₁(1 − 𝜇) − 𝑢₂(1 − 𝜇)²] (17) In the above equations 𝜇 = (1 − 𝑥²)^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).

(21)

13 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).

2.4. 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 planet- sized 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

(22)

14 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

Searching for Exoplanets in K2 Data