CALIFORNIA STATE UNIVERSITY, NORTHRIDGE a Unique Data

Home , Aleksander Wolszczan, NASA Exoplanet Archive

CALIFORNIA STATE UNIVERSITY, NORTHRIDGE

A Unique Data Reduction Methodology to Obtain Full Resolution Spectra of Exoplanet

Atmospheres Observed Using the Hubble Space Telescope in Stare Mode

A thesis submitted in partial fulfillment of the requirements For the degree of Master of Science in Physics

Amanda L. Rowen

August 2018

ã Copyright by Amanda L. Rowen 2018

The thesis of Amanda L. Rowen is approved:

______Dr. Mark R. Swain Date

______Dr. Gael M. Roudier Date

______Dr. Say-Peng Lim Date

______Dr. Damian J. Christian, Chair Date

California State University, Northridge

iii Dedication

To my Parents for their patience and continuous belief in me

iv Acknowledgements

I would like to express my utmost gratitude to my group at JPL, especially Dr. Mark Swain and Dr. Gael Roudier for their immense support and guidance throughout my research and for constantly pushing me to be a better scientist. I would also like to thank Dr. Damian Christian and Dr. Say-Peng Lim for their patience and support - especially during hard times - during my years at CSUN. If it were not for the continued support and guidance from each one of them, this thesis would not be possible.

This research is based on observations made with the NASA/ESA Hubble Space Telescope, obtained from the Data Archive at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555.

This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program.

Some/all of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555.

v Table of Contents

Copyright ...... ii Signature Page ...... iii Dedication ...... iv Acknowledgements ...... v List of Figures ...... vii Abstract ...... ix

1 Introduction ...... 1 1.1 Detecting Exoplanets ...... 1 1.1.1 Pulsar Timing ...... 2 1.1.2 Astrometry ...... 3 1.1.3 Radial Velocity ...... 3 1.1.4 Gravitational Microlensing ...... 3 1.1.5 Direct Imaging ...... 4 1.1.6 Transits ...... 4

2 Characterizing Exoplanet Atmospheres ...... 5 2.1 Transmission Spectroscopy ...... 5 2.2 Hubble Space Telescope: Stare vs. Scan ...... 9

3 Methods ...... 11 3.1 Data Reduction and Background Subtraction ...... 11 3.2 Building an Instrument Model ...... 14 3.3 Bayesian Statistics: The MCMC Method ...... 15

4 Results and Analysis ...... 18

5 Discussions ...... 22

6 Conclusions ...... 24

References ...... 25

vi List of Figures

1.1 Mass – Period relationship of all confirmed exoplanets, to date, color coded by their detection method. Image Credit: NExScI Exoplanet Archive, March 26, 2018, Akeson et al. 2013...... 2

2.1 A transmission spectrum from Wakeford et al. 2017 of WASP-39 showing enriched band amplitudes at varying wavelengths due to chemical abundances causing atmospheric opaqueness...... 6 2.2 An illustration of an exoplanet transit and it’s corresponding light curve. The solid blue line shows what the transit would look like due to limb darkening and the red dotted line shows what the transit would look like if there was no limb darkening...... 6

2.3 Comparing performances of three different limb darkening laws for the star GJ 436. From top to bottom: A Linear, Quadratic, and Non-Linear Fit to the Phoenix grid model. The three sets of semi-empirical models for each graph were attributed errors coming from the uncertainties in temperature, metallicity, and logg of GJ 436. Three models were chosen per graph at different wavelengths to account for the wavelength dependence of the limb darkening laws...... 8

2.4 GJ 436 limb darkening spectrum. The four coefficients (and their errors) of the non-linear limb darkening law...... 9

2.5 A spatially scanned image of GJ1214 compared to a staring mode image (red insert) of the star. The 0th and 1st orders are labeled in red. Image Credit: P. McCullough & J. MacKenty, May 02, 2012...... 10

3.1 Calibration and data reduction steps for post processing the STScI data...... 11

3.2 A Linearity Plot showing the signal of WASP-12 data versus time for Nsamp 0, 1, and 2...... 11

3.3 The standard deviation of the data with varying box size. The solid blue line shows the background subtracted data, while the dotted red line shows the data before background subtraction...... 12

3.4 Spectrum of WASP-12 during each main step of the data reduction and background subtraction steps. The steps in these examples (clockwise from top left) are the original image pre-reduction, the trimmed image after removing dark current and artificial negative pixels, the background selection region, the selected spectrum from the background...... 13

vii

3.5 The data timing calibration sequence for WASP-12b. (Top Left): A diagram showing the orbital phase angle of the planet with respect to its host star and the observer. (Top Right): The orbital phase sampling for each frame. (Bottom Left): The orbit sampling to determine the orbit number for each frame. (Bottom Right): The visit sampling showing the total number of orbital visits...... 14

3.6 The White Light Curves of WASP-12b (Top) and WASP-17b (Bottom). The original data are represented as black circles. The black dotted line shows the theoretical model. The red triangles are the corrected data points...... 15

4.1 Comparing my stare spectrum of WASP-12b to Kreidberg et al. 2015 spectrum shot in scan mode, and Swain et al. 2013 spectrum shot in stare mode. All spectra have been mean shifted so as to be directly compared. The top images aren’t binned while the bottom images have been binned down so as to match the format of the published spectra...... 18

4.2 The spectrum of WASP-12b over plotted with previously published spectra from Swain and Kreidberg showing a mean shift in average transit depth...... 19 4.3 The retrieval of WASP-12 for both sets of data are plotted in order to show the mean shift in posteriors and the associated errors. From top to bottom: Average transit depth, inclination, and mid-transit time. Collins and Southworth parameters are over plotted along with their relative errors to show how WFC3 localizes each parameter due to the sensitivity of the instrument when run through the EXCALIBUR pipeline...... 21

viii Abstract

A Unique Data Reduction Methodology to Obtain Full Resolution Spectra of Exoplanet

Atmospheres Observed Using the Hubble Space Telescope in Stare Mode

By Amanda L. Rowen Master of Science in Physics

Transit spectroscopy was the method by which the existence of exoplanet atmospheres was definitely established (Charboneau et al. 2002) and the molecular composition first probed (Tinetti et al. 2007, Swain et al. 2008). In the last decade, transit spectroscopy has grown rapidly and revolutionized our ability to probe atmospheric composition and conditions in a wide range of exoplanet atmospheres (Sing et al. 2016, Iyer et al. 2016, Kreidberg et al. 2014). However, interpretations of spectral transit light curves are heterogeneous in the field with multiple teams reducing their spectra in unique ways. A homogeneous extraction process is needed to produce spectra from multiple Hubble instruments (COS, STIS, WFC3) in order to combine them for more wavelength coverage, giving a more complete picture of exoplanet atmospheres. I report a tool to extract and produce exoplanet spectra from Hubble Space Telescope’s Wide Field Camera 3 (WFC3) instrument, shot in Stare mode. A transmission spectrum of WASP-12b is extracted in the near-infrared from 1.1–1.7 µm and compared to previously obtained spectra shot in stare and scan mode.

ix Chapter 1

Introduction With the implementation of increasingly precise technology and instruments, our knowledge of exoplanets has been broadening exponentially. Understanding multiple aspects such as the formation processes, host star properties, orbital parameters, and atmospheric composition are all critical in our full understanding of exoplanets. Over the past two decades, multiple missions have been tasked with discovering and characterizing exoplanets in their own unique way. On April 24, 1990, NASA’s Hubble Space Telescope launched on what would become one of the longest ongoing missions in history, discovering and characterizing multiple exoplanets through varying methods such as radial velocity, direct imaging, and the transit method. With the rising number of confirmed exoplanets each day, the task of characterizing them in terms of density, insulation, and atmospheric composition has become a necessary step in fully understanding exoplanets and our place in the universe. Chapter 1 will discuss the most common methods for detecting exoplanets as well as noteworthy discoveries made throughout history using each. Chapter 2 will explain the concepts of transmission spectroscopy as well as go on to compare the two observing modes of the Hubble Space Telescope in terms of their significance to this research. Chapter 3 will list the specific data reduction and calibration steps used to extract the spectrum of WASP-12b and how that spectrum can be used to probe atmospheric characteristics of the planet. Markov chain Monte Carlo sampling methods and Bayesian Statistics will be discussed pertaining to their importance in converging on the most likely solution for instrumental and science parameters. Chapter 4 will reveal the results of the spectral retrieval of WASP-12b as well as compare it to previously published spectra shot in scan and stare mode. Chapter 5 will discuss the implications of these results as well as their significance. Finally, Chapter 6 will detail future work that is made possible by this new calibration tool as well as conclude scientific findings involving the calibration of WASP-12b spectrum via stare mode.

1.1 Detecting Exoplanets An exoplanet is a planet that orbits a star outside of our solar system. According to NASA’s exoplanet archive, as of July 2018, there are 3,772 confirmed exoplanets and 2,720 candidates. There are a range of detection methods used to find and confirm exoplanets. Each

1 method has a preferred category of planet they discover due to the nature of the method. For instance, the transit method has a bias towards higher radius, short-period planets because the larger the planet, the easier it is to detect a dip it induces in the light curve as it passes in front of its star. With the same reasoning, it is also biased towards smaller stars. Also, since the transit method needs to measure multiple transit events in order to confirm the existence of an exoplanet, short-period candidates get confirmed quicker. The preference of this method is shown on figure 1.1 as a surplus of transit detections tend to be towards the upper left region of the figure.

Figure 1.1: Mass – Period relationship of all confirmed exoplanets, to date, color coded by their detection method. Image Credit: NExScI Exoplanet Archive, March 26, 2018, Akeson et al. 2013.

1.1.1 Pulsar Timing In 1992, the first two exoplanets were detected by Aleksander Wolszczan and Dale Frail orbiting PSR B1257+12, a pulsar in the constellation Virgo, using a method known as pulsar timing (Wolszczan & Frail 1992). A pulsar is a rapidly spinning neutron star that emits beams of extreme electromagnetic radiation that, when viewed from Earth, appear in periodic regular pulses. The pulses are timed so precisely that they are more accurate than an atomic clock (Guinot & Petit 1991). When an exoplanet is orbiting a pulsar, it causes the center of gravity to shift outside of the center of the pulsar which in turn causes the pulsar to wobble. An observer viewing this phenomenon would measure a periodic variation in the pulsar’s beam of radiation being emitted and be able to determine a presence of a planetary companion. Through this time variation method, exoplanets around pulsars can be detected.

2 1.1.2 Astrometry As discussed previously, an orbiting exoplanet shifts the center of mass of the planetary system and in turn causes its host star to wobble. In extreme cases, this wobble may be able to be detected via a method known as Astrometry, the method in which a star’s relative position in comparison to nearby field stars appears to shift. Through very precise measurements this wobble may be physically observed and measured in the detection of exoplanets. A research team led by Johannes Sahlman took astrometric measurements over the course of two years to reveal the orbital motion of a nearby ultracool brown dwarf DENIS-P-J082303.1-491201 caused by a companion (Sahlman et al. 2013). This companion was found to be an exoplanet of 28 ± 2 Jupiter masses in a highly eccentric orbit and became the first confirmed exoplanet discovered via astrometry.

1.1.3 Radial Velocity Another way to view a star’s wobble is through the method of radial velocity. If the inclination of a planetary system is nearly parallel to the line of sight of an observer, one may be able to measure a star’s wobble through the use of Doppler shift. Doppler shift is the measurement of how much a wave is stretched or compressed when reflected back to an observer due to the relative movement of the star. When the star is wobbling towards the observer, the wavelengths become compressed towards the bluer end of the spectrum and when a star is wobbling away from the observer the wavelengths become stretched towards the red end of the spectrum. This allows one to measure the effects that an exoplanet has on its host star and in turn determine properties of the planet such as size, period, and distance from the star. Pegasi 51, a solar-type star, was found to have a Jupiter-mass companion with an orbital period of 4 days (Mayor & Queloz 1995). This planet, presently known as Pegasi 51b, was discovered using regular periodic variations in the radial velocity measurements of its host star.

1.1.4 Gravitational Microlensing When an exoplanet system passes between an observer and a background star, the background star’s light is bent towards the observer due to the gravitational pull from the exoplanet system bending the fabric of space-time, causing light to take the shortest possible path through space. This phenomenon causes the brightness of the background star to increase or ‘magnify’ as the planetary system passes in front. As the planet passes in front of the star,

3 a spike in brightness may be measured in the light curve for a shorter period of time than the actual full lensing event allowing for an exoplanet detection. A sub-Neptune planet of 5.5 earth masses, named OGLE-2005-BLG-390Lb, was discovered orbiting 2.6 astronomical units from a 0.22 solar mass M-dwarf star through a microlensing event (Beaulieu et al. 2006). Recently, it has even been suggested that quasar microlensing can be used to probe extragalactic planets within a lens galaxy (Dai & Guerras 2018).

1.1.5 Direct Imaging An exoplanet can be seen directly when imaged very carefully with precise instruments. This process, known as direct imaging, is done through the use of coronagraphs (commonly called ‘masks’) which are used to physically block out the light coming from an exoplanetary system’s host star into the telescope. This allows the image to detect the fainter light coming from the exoplanet itself, rather than the host star’s light saturating the field of view in the image. The first exoplanet was directly imaged by a team of researchers led by Paul Kalas of the University of California, Berkeley in which they imaged the planet Fomalhaut b, located 25 lightyears from Earth (Kalas et al. 2008).

1.1.6 Transits Encompassing the majority of exoplanet discoveries, the transit method is responsible for roughly 78% of all confirmed exoplanets (e.g., Charbonneau et al., 2000; Knutson et al. 2014; Sing et al. 2016). On April 1, 2008, the planet WASP-12b was discovered via the transit method by the ground-based SuperWASP observatory (Pollacco et al. 2006). The planet, with a radius of R = 1.79 ± 0.09 Jupiter Radius (RJ), and mass of M = 1.41 ± 0.1 Jupiter Masses

(MJ) (Hebb et al. 2009), has become an ideal candidate for exoplanet atmospheric studies (Swain et al. 2013). When a planet passes between an observer and its host star, the planet blocks some of the star’s light, therefore a decrease in brightness is measured. The time dependence of the light curve can tell many properties of the system such as planetary radius, inclination, semi-major axis, etc. By plotting its fractional planet to star radius versus wavelength (exoplanet spectrum) one can extract the properties of the planet’s terminator region atmosphere (Tinetti et al. 2010).

4 Chapter 2

Characterizing Exoplanet Atmospheres When a planet transits its host star during the primary eclipse (transit), a small fraction of the star’s light passes through the planetary atmosphere. Because the light from the star will be absorbed differently when passing through the exoplanet atmosphere, depending on its composition, the radius of the planet will change depending on the wavelength. The modulation in the planetary radius will follow the absorption lines of each element constituting the exoplanet atmosphere. Therefore, a transmission spectrum is obtained by plotting the fractional radius of the planet to that of the star with respect to varying wavelength. The spectral retrieval can give insight into chemical abundances and physical processes that comprise the planet’s atmosphere such as clouds, hazes, and (non) equilibrium chemistry allowing an in-depth study of exoplanetary atmospheres. NASA’s Hubble Space Telescope has the capability to probe exoplanetary atmospheres thanks to its on-board spectrograph (WFC3+grisms). In this chapter, the fundamentals of transmission spectroscopy are discussed focusing on the influence of stellar limb darkening. The two data-taking modes of Hubble Space Telescope are then discussed to demonstrate their relative performances in obtaining exoplanet transit spectra.

2.1 Transmission Spectroscopy Atmospheres are more or less opaque at different wavelengths, with the amount of opaqueness depending on the chemical composition of the atmosphere. Depending on the wavelength probed, we can learn many different things about an exoplanet’s atmosphere through transmission spectroscopy. At ultraviolet wavelengths, Lymann Alpha and ionized metals can be measured which can provide insight as to atmospheric mass loss. At visible wavelengths, sodium and potassium can be measured which can provide insight as to the physical composition of the atmosphere such as clouds, hazes, or alternatively a clear atmosphere (Knutson 2012). Probing at infrared wavelengths can determine the chemical composition of the atmosphere to determine if the relative abundances of H2O, CH4, CO, and

CO2 are in thermochemical equilibrium (Burrows et al. 2006).

5 Figure 2.1: A transmission spectrum from Wakeford et al. 2017 of WASP-39 showing enriched band amplitudes at varying wavelengths due to chemical abundances causing atmospheric opaqueness.

In order to properly understand a planet’s transmission spectrum, the limb darkening of its host star must be well known and represented. Limb darkening is an optical phenomenon which appears as the decrease in brightness across the visual disk of a star from its center to its edge or limb, which is responsible for light curves appearing rounded in shape rather than sharp linear dips as shown in figure 2.2. To accurately model the light curve, a limb darkening law must be chosen that can produce intensities over the entire stellar disk, while preserving the flux. Figure 2.3 compares the performances of three different limb darkening laws for the

Figure 2.2: An illustration of an exoplanet transit and it’s corresponding light curve. The solid blue line shows what the transit would look like due to limb darkening and the red dotted line shows what the transit would look like if there was no limb darkening.

6 star GJ 436. The best fit results show that unlike linear and quadratic, a non-linear limb darkening law is best suited to match the Phoenix grid model over the entire disk, therefore, a four-parameter non-linear limb darkening law is chosen for this study. GJ 436 is an M dwarf star that stress tests the limb darkening laws to their limits (Maness et al. 2007). If the limb darkening laws are well suited for this type of star it will work for all of the stellar candidates within the pipeline.

Assuming spherical symmetry, a non-linear limb-darkened source can be represented by equation 2.1.

= 1 − � 1 − � − � 1 − � − � 1 − � − � 1 − � (2.1) where I(1) is the specific intensity at the center of the disk, µ is the distance from the edge of the stellar disk to the center (0 being the edge; 1 being the center), and an are the limb darkening coefficients, which can be derived and expanded from the general law shown in equation 2.2 (Claret 2000).

= 1 − � 1 − � (2.2)

Similarly, linear and quadratic limb-darkened sources are represented by equations 2.3 and 2.4.

= 1 − � 1 − � (2.3) = 1 − � 1 − � − � 1 − � (2.4)

Limb darkening laws have spectral variations and in order to not confuse them into the exoplanets spectrum it is very important to have the limb darkening law as accurate as possible. The four coefficients of the limb darkening spectrum for GJ 436 are shown in figure 2.4.

Figure 2.3: Comparing performances of three different limb darkening laws for the star GJ 436. From top to bottom: A Linear, Quadratic, and Non-Linear Fit to the Phoenix grid model. The three sets of semi-empirical models for each graph were attributed errors coming from the uncertainties in temperature, metallicity, and logg of GJ 436. Three models were chosen per graph at different wavelengths to account for the wavelength dependence of the limb darkening laws.

Figure 2.4: GJ 436 limb darkening spectrum. The four coefficients (and their errors) of the non-linear limb darkening law.

2.2 Hubble Space Telescope: Stare vs. Scan This research is based off of observations taken from Hubble Space Telescope using the Wide Field Camera 3 (WFC3) detector and G141 grism filter in the infrared channel. The wavelength range for this setup is 1075-1700 nm, probing in particular the change in the water vapor absorption coefficient at 1400 nm. There are two modes that Hubble can obtain data for this setup: Scan Mode and Stare Mode. Scan mode spatially scans a wider area without looking directly at the target in order to avoid saturation for bright objects (which is beneficial when looking at bright stars), and also leads to increased scientific efficiency and improved spectrophotometry due to larger photon collection per Hubble orbit (Dressel, 2018). Unlike scan mode, stare mode points directly at the star and it remains fixed in the same area of the detector. The benefits of staring mode are that background sources are less likely to overlap due to a crowded field of view thus reducing the need to decouple overlapping spectra. Pixel gain variations are minimized due to lack of scanning over a larger range of pixels. Exoplanet

9 transit data taken with other detectors on Hubble such as the Space Telescope Imaging Spectrograph (STIS) and the Cosmic Origins Spectrograph (COS) are shot in stare mode and therefore stare data from WFC3 can be combined with that of STIS and COS for a more complete analysis of exoplanetary atmospheres with greater wavelength coverage. Figure 2.5 shows a comparison of Scan Mode versus Stare Mode for the star GJ1214 (McCullough & MacKenty 2012).

Figure 2.5: A spatially scanned image of GJ1214 compared to a staring mode image (red insert) of the star. The 0th and 1st orders are labeled in red. Image Credit: P. McCullough & J. MacKenty, May 02, 2012.

10 Chapter 3

Methods In this section I will discuss the methods used in extracting a spectrum of WASP-12b. The steps involved in the background subtraction and spectrum selection are described. The calibration methods such as wavelength calibration are provided. The concepts of Bayesian statistics and their use in developing an instrument model for fitting the data are discussed. These calibration steps are outlined in figure 3.1.

Figure 3.1: Calibration and data reduction steps for post processing the STScI data. 3.1 Data Reduction and Background Subtraction Since observations can easily become oversaturated as a result of using stare mode, it was necessary to first verify that WASP-12 was not saturating the detector. For each non- destructive readout (Nsamp), the signal was plotted verses the integration time to check for non-linearity which would demonstrate saturation being reached (figure 3.2). The detector

Figure 3.2: A Linearity Plot showing the signal of WASP-12 data versus time for Nsamp 0, 1, and 2.

11 never becomes saturated for WASP-12 as demonstrated by the linear fit of the non-destructive reads. The max signal achieved is during Nsamp=2, with a signal of 12,454 e-. This can be converted into data numbers (DN) by multiplying by the gain of the filter (2.5). From this we find that WASP-12 reaches a value of 31,135 DN which is well below the saturation level of the detector; 65,534 DN (~78,000 e-) (Dressel. 2018). During the data reduction phase of WASP-12b, Nsamp=0 .ima frame data were selected because the standard deviation of the out of transit data of the white light curve was minimal compared to Nsamp=1 and Nsamp=2. Dark pixels (the upper and lower negative pixel rows in the upper left quadrant of figure 3.3) were manually removed by trimming the edges of the frame by 20 pixels. In order to automatically select the spectrum from the background noise, two thresholds were

Figure 3.3: The standard deviation of the data with varying box size. The solid blue line shows the background subtracted data, while the dotted red line shows the data before background subtraction. implemented: The first threshold set the spectrum selection to values which were of at least 10 standard deviations above the background noise level. The second threshold was to only select data that fell within the 98th percentile of the data distribution, which was chosen in order to assure with nearly 3 sigma confidence that the data selected was that of the spectrum, and not the background noise.

12 In order to minimize the standard deviation of the dataset, a box size of 15 pixels was drawn, centered on the spectrum, with the outside background values set to NAN. This box size was chosen by plotting the standard deviation of the out of transit data of the white light curve vs. varying box sizes as shown in Figure 3.3, and choosing that which minimized the

Figure 3.4: Spectrum of WASP-12 during each main step of the data reduction and background subtraction steps. The steps in these examples (clockwise from top left) are the original image pre-reduction, the trimmed image after removing dark current and artificial negative pixels, the background selection region, the selected spectrum from the background. standard deviation. The last, and final step was manually subtracting the mean background value. A 50-pixel height box was chosen at the top of the image, spanning the entirety of the image to represent the background of the data (shown in the bottom right quadrant of Figure 3.4). The mean value of the box was subtracted from the value of the spectrum in order to reduce the background level and achieve the final background subtracted image (shown in the bottom left quadrant of Figure 3.4).

13 3.2. Building an Instrument Model The instrument model is time dependent and depends on the Hubble orbits around Earth as well as the visits. The first thing needed is to find the number of visits and then split each visit into individual Hubble orbits. This is done through orbital phase sampling of the system. The frame number for each image is compared to the orbital phase of WASP-12b in order to determine the orbit numbering per frame number. This is done to account for Hubble’s orbit around Earth every 96 minutes. The threshold for determining these separate orbits is set to six and found by plotting discontinuities in the data. Larger discontinuities in the orbital phase data are marked to determine how many separate visits of WASP-12 there are. These three visits of WASP-12 are represented in the bottom of Figure 3.5. The first and third visit are omitted for this data because the transit is only present for the second visit. For the instrument

Figure 3.5: The data timing calibration sequence for WASP-12b. (Top Left): A diagram showing the orbital phase angle of the planet with respect to its host star and the observer. (Top Right): The orbital phase sampling for each frame. (Bottom Left): The orbit sampling to determine the orbit number for each frame. (Bottom Right): The visit sampling showing the total number of orbital visits.

14 model retrieval, the same ramp systematic is fit to all orbits. Due to the difference in the HST ramp parameters, the common practice of omitting the first orbit due to the steep ramp systematic is carried out (Deming et al. 2013, Stevenson et al. 2014).

Figure 3.6: The White Light Curves of WASP-12b (Top) and WASP-17b (Bottom). The original data are represented as black circles. The black dotted line shows the theoretical model. The red triangles are the corrected data points.

3.3 Bayesian Statistics: The MCMC Method In order to clearly explain the methods used in extracting a best fit model, I will explain the concepts of Bayesian Statistics and discuss the strengths of the Bayesian method over the Frequentist method in determining the posterior distribution of my data.

15 The frequentist method tackles the question of “Does an event occur?”, thus calculating the probability of an event occurring over the long run by sampling distributions of fixed size and iterations. The limitation in the frequentist method arises in that the result of the event depends on the number of iterations and the initial conditions (initial guess of each parameter to fit), therefore the best estimate of the parameter can change if the initial condition changes. This means that if two people were to sample a dataset for an absolute minimum and start with different inferences and iterate over different timespans, they might end up with different results. This is where Bayesian statistics becomes useful in that it accounts for newly learned data and can update beliefs, as well as properly sampling multimodal solutions if they exist. The Bayesian method looks for the best estimate called conditional probability in which the likelihood of one event occurring takes into account the belief of that event to be relevant. Bayes theorem as shown in equation 3.1 expresses this conditional probability:

� �|� � � � � � = (3.1) � � � � �(��) where q is the state of nature, and t is the outcome. The left-hand side of the equation is the posterior probability of q, and the quantity P(q), the prior probability of q (Cornfield 1967).

Bayesian statistics, in short, calculates a likelihood function and combines that with a prior distribution to determine a posterior distribution of data. Markov Chains are events or states where the next state of the process only depends on the previous state, sampling highly the region of interest and poorly the region where the probability of the model to describe the data is minimal. The easiest example of a Markov Chain is one with two states. The chain has four possible behaviors since it can transition between separate states (2) or transition into the same state (2, one for each state), with the probability of its transition depending on the current state it is in. The more probable the state, the less likely for a state to depart from it significantly. In other words, if the likelihood is high (model is close to the data), the sampler will stay in the region of interest, transiting from very close states to others. The Markov Chain Monte Carlo (MCMC) method is a general class of algorithms which essentially is a Monte Carlo integration using Markov Chains (Gilks et al. 1996). This means that random samples are chosen from a distribution to determine properties of the distribution and each random

16 sample is used as a stepping stone to generate the next random sample (van Ravenzwaaij et al. 2018).

Through the use of MCMC algorithms, a linear recovery is performed to the entire visit to account for the general shape of the data. Separate linear fits are then applied to each Hubble orbit combined with an exponential term in order to capture the ramp effect and curve of the data for each orbit in order to fit the instrument model, represented in equation 3.1.

� � �� = e � + � × b � + � × 1 − �� (3.1)

where e is the linear slope of the visit, g is the intercept of the visit, tv is the reference time of the visit (tMJD – tcenter of visit), b is the linear slope for each orbit, to is the reference time of each orbit (tMJD – tcenter of orbit), a is the intercept of the orbit, t is the time constant, and z is the delay.

The priors are set via a normal distribution around a pre-estimated value conditioned by the out of transit data for each orbital prior. The MCMC model generates a best fit overall model (instrument model and science model) to the white light curve which is shown in Figure 3.6. The averaged data over the entire bandpass (white data) are used to constrain the orbital parameters in a first step because the orbital solution is a common mode amongst wavelengths.

Free parameters are the fractional planet to star radius Rp/R★, inclination i, mid-transit time T0, and instrument parameters.

In a second step, the orbital solution is fixed, and individual wavelength dependent fits are performed, adjusting instrumental parameters and planetary radius in each wavelength bin, recovering the fractional planet to star radius vs. wavelength to achieve a full resolution spectrum.

17 Chapter 4

Results and Analysis In this chapter I show my spectrum of WASP-12b shot in stare mode, as well as compare it to those obtained by Swain et al. 2013 and Kreidberg et al. 2015.

Figure 4.1: Comparing my stare spectrum of WASP-12b to Kreidberg et al. 2015 spectrum shot in scan mode, and Swain et al. 2013 spectrum shot in stare mode. All spectra have been mean shifted so as to be directly compared. The top images aren’t binned while the bottom images have been binned down so as to match the format of the published spectra.

18 Figure 4.2: The spectrum of WASP-12b over plotted with previously published spectra from Swain and Kreidberg showing a mean shift in average transit depth.

The mean shift in the average transit depth of WASP-12b represented in Figure 4.2 could be caused by the orbital solution used in the retrieval. Accurate interpretation of a spectral transit light curve typically requires four stellar and six planetary priors; Stellar

Radius, R*, Stellar Temperature, T*, Metallicity, FeH, Surface Gravity, Logg, Eccentricity, e,

Inclination, i, Semi Major Axis, a, Periodicity, p, Mid Transit Time, t0, and Planet Radius, Rp. An important question is the extent to which results are sensitive to the solution adopted. The impact of priors on average transit depth results for a solar type star are considered. The parameters of Southworth 2012 and Collins et al 2016 are considered for periodicity. The difference between these orbital solutions is 13.571s. After an initial run on the EXCALIBUR pipeline, Southworth’s model would produce a fractional transit depth of 1.29982% while Collins data produces 1.32945% as shown in Figure 4.3. The uncertainty in these results is significant with a difference of 15s. The main difference is in the difference in periodicity of 13.571s. Given the sampling of the light curve, the periodicity is highly correlated with inclination and average transit depth which leads to two very different modes in the orbital

19 solution. Due to this mean shift, the effect of orbital solution, especially the orbital period of the system, leaking into the average transit depth is not negligible and therefore careful consideration must be taken when choosing a dataset.

Table 4.1: Orbital solutions of this study and previously published data from Swain et al. 2013 and Kreidberg et al 2015. Star Parameter Description This Study Swain et al. 2013 Kreidberg et al. 2015

R★ Star Radius [AU] 0.0073922±0.0008369 0.0072992±0.0003254 0.0072992±0.0003254

T★ Star Temperature [K] 6300±150 6300±200 6300±200 FeH Star Metallicity 0.30 null null Log g Star Surface Gravity 4.38±0.10 null null Planet Description This Study Swaint et al. 2013 Kreidberg et al. 2015 Parameter e Eccentricity 0.05±0.01 null 0.049±0.015 i Inclination [degrees] 82.50±0.75 null 83.1±1.3 Semi Major Axis a null null 0.0229±0.004 [AU] p Periodicity [Days] 1.09142±0.0 1.09142162±0.00000021 1.091423±0.000003

rp Planet Radius [AU] 0.00086977±0.00010228 0.000855207 0.00085545±0.0000418

Table 4.2: Both sets of orbital solutions from Collins and Southworth. The sigma difference in values is computed in the final column to show how much each prior varies between datasets. Difference Star Parameter Description Collins Southworth s

R★ Star Radius [AU] 0.00770582879±0.00020927115 0.0075291±0.00036738713 0.845

T★ Star Temperature [K] 6360.0±135.0 6250.0±100.0 1.1 FeH Star Metallicity 0.33±0.155 0.32±0.12 0.0833 Log g Star Surface Gravity 4.157±0.0125 4.159±0.024 0.159 Planet Difference Description Collins Southworth Parameter s e Eccentricity 0.0 0.0 0.0 i Inclination [degrees] 83.37±0.68 83.3±1.1 0.103 Semi Major Axis a 0.0234±0.00053 0.02309±0.00101 0.585 [AU] p Periodicity [Days] 1.09142030±0.00000014 1.0914222±0.0000011 13.571 Mid Transit Time t 56176.168258±0.00034 56176.168258±0.00034 0.0 0 [MJD]

rp Planet Radius [AU] 0.00090798663±0.00002674 0.00087214505±0.0000451096 1.34

20 Figure 4.3: The retrieval of WASP-12 for both sets of data are plotted in order to show the mean shift in posteriors and the associated errors. From top to bottom: Average transit depth, inclination, and mid-transit time. Collins and Southworth parameters are over plotted along with their relative errors to show how WFC3 localizes each parameter due to the sensitivity of the instrument when run through the EXCALIBUR pipeline.

21 Chapter 5

Discussions In the Excalibur pipeline, there are 42 targets with scan mode data that use one specific instrument model to produce spectra. A uniform reduction method is needed for the extraction of spectra for the 11 targets within the pipeline that have stare mode data, therefore, the same instrument model was used to produce a spectrum for WASP-12 for a homogeneous extraction method. The spectrum of WASP-12b was comparable to the spectrum achieved by Swain et al. (2013) as demonstrated by the data falling within the error bars of each spectral bin, with the exception of the lack of water feature at ~ 1.4 µm. The spectrum was much different than it’s scan mode counterpart represented by Kreidberg et al. (2015) with the data falling well beyond the error bars of each dataset. The possible sources of discrepancy that could be accountable for this difference can stem from the calibration level, the instrument model, physical phenomena, or a combination of each.

The spectrum of WASP-12b for this study had a mean fractional planet to star radius value of 1.35%, which was a 0.10% shift from the mean value of 1.45% found by Swain and Kreidberg. Since the scan and stare data were taken at different times over a period of years, a physical phenomenon that could be causing the data to not fit Kreidberg’s spectrum at a spectral modulation level is stellar activity. WASP-12 could have large sunspots, flares, coronal mass ejections, and other stellar activity during stare mode observations that make the data not fit the data of those previously taken. Since the model is highly dependent on the fractional planet to star radius, flares and coronal mass ejections would cause an increase in observed stellar flux and in turn cause the fractional planet to star radius to appear lower than expected, causing a mean shift in the spectrum as observed in this study. Another cause of this mean shift could be due to the difference in limb darkening laws used for each study. As discussed previously, linear and quadratic limb darkening laws do not accurately constrain the flux near the limbs of the star and therefore could lead to an overestimation of the stellar flux towards its edge. This would lead to a larger fractional planet-to-star radius value as opposed to that obtained from using a non-linear limb darkening law such as this study. This could be the reason why the spectrum obtained by Kreidberg has a higher mean value.

22 The stellar priors from Swain et al. 2013’s spectrum were taken from Hebb et al. 2009 and the systems ephemeris were taken from Maciejewski et al. 2011. My stellar priors and system ephemeris were taken from Stassun et al. 2017. The difference between these orbital priors are 0.2857s for stellar radius, and 0.889s for planetary radius and therefore a mean shift in resulting average transit depth would be expected after an initial run through the pipeline similar to that shown in Figure 4.3 comparing datasets from Southworth and Collins.

23 Chapter 6

Conclusions A transmission spectrum of WASP-12b was extracted and compared to previously obtained spectra shot in stare and scan mode. The spectrum was comparable to that achieved by Swain et al. (2013) (which was also shot in stare mode) and was different than the scan mode data of Kreidberg et al. (2015) with the data falling well beyond the error bars of each dataset. The spectrum of WASP-12b for this study had a mean transit depth of 1.35%, which was a 0.10% shift from the mean value of 1.45% found by Swain and Kreidberg. The importance in choosing an orbital solution is critical in understanding the posterior distributions and mean shift of a spectral retrieval. Discrepancies in the data calibration and instrument model must be accounted for in order to fully automate the stare mode extraction for the remaining targets in the pipeline.

The calibration of the spectrum could require more rigorous steps such as accounting for tilted spectra. A tilted spectrum could throw off the calculation of where the mean value of the spectrum is for each pixel column and therefore cause the summed values over each wavelength bin to be skewed. Also, the Nsamp selection is not automated and therefore would need to be implemented in a way to test for saturation and standard deviation of the out of transit white light data. Lastly, the background selection region is a fixed box that could contain another background source or spectrum depending on the target imaged in the pipeline. Therefore, a more careful background selection needs to be implemented in the future.

The instrument model was not successful in fitting the data of WASP-17, likely due to the b term in equation 3.1 not accurately fitting for the sub-orbits present in the orbital phase sampling, thus leading to the spread in the data as shown in Figure 3.5. A higher threshold was attempted in order to try and capture these sub-orbits but was still unsuccessful in capturing the ramp. A sub-orbit term needs to be added to the instrument model in order to accurately fit the data of WASP-17. Other targets within the pipeline could encounter the same issues when trying to fit the same instrument model due to the vast differences in the stare data from target to target. It is clear that a unique instrument model must be implemented in order to fit the uniqueness of stare data.

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