Statistical Correlations Hidden in Phase Variations of Exoplanet Data
AS A thesis presented to the faculty of San Francisco State University 3 fc In partial fulfilment of The Requirements for pH VS -S 5 C. Master of Science In Physics: Astronomy
by
Charles G. Showley
San Francisco, California
August 2015 Copyright by Charles G. Showley 2015 CERTIFICATION OF APPROVAL
I certify that I have read Statistical Correlations Hidden in Phase Varia tions of Exoplanet Data by Charles G. Showley and that in my opinion this work meets the criteria for approving a thesis submitted in par tial fulfillment of the requirements for the degree: Master of Science in
Physics: Astronomy at San Francisco State University.
Dr. Stephen Kane, Ph.D. Astrophysics Assistant Professor of Physics
C ^ ... Dr./Joseph Barranco, Ph.D.'Astrophysics Associate Professor of Physics
p . Dr. Ron Marzke, Ph.D. Astronomy Professor of Physics Statistical Correlations Hidden in Phase Variations of Exoplanet Data
Charles G. Showley San Francisco State University 2015
A method is developed to process the entire catalog of known exoplanets up through those confirmed by the Kepler telescope and produce phase plots for all star systems
based on the planets’ properties. We generate correlation plots from the calculated
data to identify patterns with the hope of improving planetary detection methods
for future missions and to create better plots that identify groupings of types of
planets. Ultimately all equations to simulate multiple planets’ orbits around host
stars is fully automated within a Python script from pulling the raw exoplanets data
from the Internet to returning a unique phase curve for 947 exoplanetary systems.
Based on case studies the two properties of planets affecting their flux the most is
their size and distance to host star.
I certify that the Abstract is a correct representation of the content of this thesis.
*?/// //.c Chair, Thesis Committee Date ACKNOWLEDGMENTS
Thank you to my research advisor, Stephen Kane, and the entire SFSU
Physics & Astronomy Department. TABLE OF CONTENTS
1 Introduction...... 1
1.1 The Kepler M ission...... 1
1.2 Exoplanet Discoveries...... 4
1.3 Exoplanetary Atmospheres...... 4
1.4 Reflected Light from Exoplanets...... 6
2 Theory of Exoplanet Phase C u r v e s ...... 9
2.1 Basic Formulation...... 9
2.2 The Phase Function...... 12
2.3 A lb e d o ...... 14
2.4 Effects of Keplerian O rbits...... 15
3 Application to Known Exoplanets...... 17
3.1 Importing the Exoplanet D a ta ...... 17
3.2 Solving Kepler’s E q u a tion ...... 19
3.3 Writing the Phase Variation Function...... 19
3.4 Light Curves of Exoplanetary S y ste m s ...... 21
4 Correlations...... 24
4.1 Motivation for Plotting Calculated D a ta ...... 24
4.2 Data to Compare...... 25
4.3 Results from Data Analysis ...... 27 5 Case Studies...... 32
5.1 Selected S y ste m s...... 32
5.1.1 K epler-9...... 34
5.1.2 GJ 5 8 1 ...... 35
5.1.3 GJ 876 ...... 36
5.1.4 Kepler-341...... 37
5.1.5 HD 4 1 1 3 ...... 39
5.1.6 KIC 11442793 ...... 41
5.2 Implications of Behaviors...... 42
6 Conclusions...... 44
7 Appendices...... 46
vii LIST OF TABLES
Table Page
7.1 Brightest 50 planets from exoplanet da t a ...... 59
7.2 Brightest 50 stars from exoplanet da t a ...... 62
7.3 Case study system s...... 65
7.4 Physical p roperties...... 67
viii LIST OF FIGURES
Figure Page 1.1 Kepler candidates...... 3
1.2 Planet reflectance under different m od els...... 6
1.3 Phases of V enus...... 7
2.1 Solar system phase p lo t...... 10
2.2 Geometry of an orbit ...... 11
3.1 Power law relation ...... 18
3.2 Examples of calculated phasepl o t s ...... 22
4.1 Scatter plot data cle a n in g ...... 26
4.2 Histogram data clea n in g...... 27
4.3 Peak flux ratio v. semi-major a x is...... 28
4.4 Semi-major axis v. planetary radius...... 29
4.5 Albedo plots...... 30
4.6 V-magnitude v. peak f l u x ...... 31
5.1 Kepler-9 ...... 34
5.2 GJ 5 8 1 ...... 35
5.3 GJ 876 ...... 36
5.4 Kepler-341...... 38
5.5 HD 4 1 1 3 ...... 39
5.6 KIC 11442793 ...... 41
ix 1
Chapter 1
Introduction
1.1 The Kepler Mission
To compile evidence of the existence of planets outside our solar system requires
extraordinarily precise equipment capable of filtering out noise that would block
such evidence. Where before it was difficult to distinguish anything within the light
output by another star directly the astronomy community has entered a new age
of photometric collecting power such that detection strength can identify photons
coming from individual planets. The Kepler telescope, launched in March 2009,
was the most recent such mission to empirically determine the fraction of Sun-like
stars in our galaxy that have Earth-sized planets; in only a few successful years of operation it’s been confirmed that most, if not all, stars host multiple planets where before the frequency of planets orbiting stars were only suspected but unverifiable, along with the discoveries of some Earth-like planets in their host stars’ habitable 2
zones (Borucki et al. 2010). Imaging power only allowed for data on giant planets
so formation models predicted there to be many more terrestrial planets but it
couldn’t be confirmed before the inception of this mission. Results have shown the
three most frequently occurring planets easily seen by Kepler are gas giants, ice
giants, and hot super-Earths (see Fig. 1.1). The ideal planet to identify would be
an Earth size (0.5-2 Earth-radii) planet within its star’s habitable zone, a region in
which temperatures are just right to potentially allow liquid water to form on its
surface given sufficient atmospheric pressure.
Kepler consists of a 0.95 m photometer oriented towards the Lyra and Cygnus
constellations (approximately 19h20min right ascension, 45° declination), simul
taneously and continuously monitoring more than 150,000 target stars within its
100 square degrees field of view (0.25% of the sky). By comparison the Hubble
Space Telescope has a field of view of only 10 square arcminutes. Its fixed orien tation away from the ecliptic plane prevents light from the Sun being included in
its measurements. It has an Earth-trailing orbit with a period of 372.5 days to
minimize occultation, stray light, and gravitational perturbation from Earth. As
a space-based telescope it is free from day-night and seasonal cycles, atmospheric
interference, and extinction that ground-based telescopes must account for. The
initial method of detection (until Kepler began experiencing mechanical trouble) was planetary transits, the periodic change in a star’s light due to a planet passing
in front of it. The likelihood of an occurrence is equal to the radius of the target 3
Period [days]
Figure 1.1: In this plot of radii of exoplanets (in Earth-radii) versus orbital period the bulk of planets discovered are between 1-4 Earth-radii. The colors correspond to the date of release of a new set of discovered exoplanets: blue for June 2010, red for February 2011 and yellow for February 2012. Most yellow points occur around 1 Re where more time to discern the signal of smaller planets would allow for their discoveries. The blue dots indicate larger planets (than Earth) which would be the easiest to identify with the transit method used by Kepler; a larger planet will induce more dimming in starlight when crossing in front of it (Batalha et al., 2012). star divided by the mean radius of the planet’s orbit, e.g. an observer watching our solar system would have only a 0.46% chance of seeing Earth transit. Many target stars dramatically increases the chance of Kepler observing transiting planets. 4
1.2 Exoplanet Discoveries
To date Kepler has found more than 1,000 exoplanets thanks to its photometry data alone. In 2012 one of its four reaction wheels malfunctioned and in 2013 a second wheel began to experience elevated levels of friction during operation. As a result Kepler is no longer able to easily search for new planets as it has been but there still remain a backlog of data for Keplerian Objects of Interest (KOIs) to study and determine their statuses as planets or not.
Despite the malfunctions the Kepler missions is considered a huge success for the exoplanetary field. Milestone discoveries include Kepler’s first series of confirmed planetary systems containing five hot Jupiters (rather unimpressively named Kepler-
4b, Kepler-5b, Kepler-6b, Kepler-7b, and Kepler-8b) which are planets with radii comparable to Jupiter and short period orbits among 156,000 observed stars (Koch et al. 2010) and Kepler-186f, the first Earth-like planet discovered in its host star’s habitable zone (Quintana et al. 2014).
1.3 Exoplanetary Atmospheres
Discovery of planets is entirely dependent on the amount of light reflecting off of the atmosphere of a planet and eventually being detected by our instruments; a planet with very low reflectance, or a very small planet is much harder to detect that a large, bright planet (e.g. Jupiter). To describe how much light a planet reflects is 5
to describe a planet’s albedo.
An albedo is in part determined by its surface and atmospheric composition.
Albedo can be affected by the following, but is certainly not limited to: clouds, temperature, water, changing pressure with depth, and gases in the atmosphere like carbon monoxide or nitrogen. With such variability present and given the lack of comprehensive spectroscopic data on exoplanetary atmospheres at the time of writing we have chosen to simplify our models of planets’ reflectivity to one similar to that of a sphere exhibiting Lambertian reflectance, that is, where the intensity of reflected light is constant regardless of the angle at which the object is viewed. Outlined in Madhusudhan & Burrows (2012) are several models for exoplanet reflectivity, including Lambertian reflectivity. Their derivations plotted in Fig. 1.2 show that approximating planets as having surfaces following Lambertian reflectance reflect the most light as a function of phase angle, an ideal solution for our model given that we are plotting a flux ratio of planets’ brightnesses with their much brighter host stars. For continued discussion on albedo, see §2.3.
A small correction is made with the Hilton phase function (see §2.2, Eqn. 2.1) to allow for non-isotropic scattering due to clouds that may appear in exoplanets’ atmospheres. 6
Figure 1.2: According to different scattering models the Lambertian scattering model (black line) is independent of scattering albedo. The Rayleigh models and Isotropic scattering models are plotted for varying values of scattering albedo be tween 0 and 1 (Madhusudhan &: Burrows, 2012).
1.4 Reflected Light from Exoplanets
To better understand how light reflecting off of planets is calculated it is instruc tive to think of planets going through phases during their orbits the same way our moon goes through phases when orbiting the Earth. For a stationary observer on
Earth looking at the moon a different percentage of its surface being illuminated will be visible depending on its positioning relative to the Earth and sun, e.g. a quarter phase when the angle subtended by the moon and the sun is 90°. Similarly, 7
the observer on Earth watching a planet orbit a star will see different phases of the planet dependent on that planet’s position in orbit. In our solar system phases of
Venus have been studied at length as the planet is a good candidate because of its proximity and high albedo (Fig. 1.3).
Figure 1.3: With Earth at a fixed point, Venus appears in different phases depending on where it is with respect to the sun and the Earth. It goes through the same principle phases as the moon (full, new, quarter, crescent, gibbous) and is a useful model for visualizing how exoplanet phases will progress.
Using the high-precision data collected by Kepler and other exoplanet missions we can search for correlations in exoplanetary characteristics and identify systems exhibiting unusual behavior for further study. We use properties of confirmed plan ets (inferred directly from data about their orbits) including semi-major axis, in- clination angle, radius, eccentricity, mass and period in our method which involves plotting a ratio of the theoretical phase variations in the constant luminosity of host stars and the sum of the variable luminosities of planets orbiting them as a function of the phase of the longest-period planet. A plot is generated for every star that hosts one or more planets where Python code calculates the flux ratio for n number of increments of the orbital phase of the system’s outermost planet; there are 947 phase variation plots created via this method.
Correlation plots are generated to look for patterns in the characteristics of the exoplanets. Do planets above a certain eccentricity tend to be larger or smaller than
Earth? What is the prevalence of hot Jupiters with respect to other gas and ice giants farther away from host stars? What can be inferred about exoplanets that can be applied to KOIs? 9
Chapter 2
Theory of Exoplanet Phase Curves
2.1 Basic Formulation
We can think of planets going through phases the same way the moon progresses through phases in its approximate one month period. This effect has been well documented within our solar system where Jupiter and Venus dominate its phase curve (Kane & Gelino, 2013). With data taken from the confirmed exoplanets we can calculate a theoretical phase curve of any solar system given each planets’ masses, radii, et cetera based solely on the geometry of their orbits about their host stars.
The most important element in making such a calculation is ensuring ubiquity in the positioning of the system by setting a t = 0 point, or the point in time at which the calculation begins. For a circular orbit the maximum amount of light reflected by a planet is when it is directly opposite its host star relative to the observer’s line of sight. We choose this point in time (for the planet with the largest period in a 10
system) as t = 0 and calculate the total reflected light of every planet in the time it takes the outermost planet to complete one orbit, to eventually combine into one curve. In Fig. 2.1 is an example of a phase curve from our solar system.
Orbitol Phcse
Figure 2.1: The phase curve for our solar system is plotted over one phase of Jupiter’s orbit. The dashed line represents Jupiter’s reflected light while the solid line is the total light from all planets; the periodic spikes come from Venus’ change in phases. Venus’ high albedo is attributable to its reflective atmosphere while Jupiter dominates on account of it having the largest radius of any planet in the solar system. (Kane & Gelino, 2013).
The orientation of a planet’s orbit relative to an observer influences the calculated flux. This is known as the argument of periastron (a>), measured from 0 — 2n radians. A planet with u = 0 rad would be oriented such that the position of its
periastron would form a left-handed right angle with the observer’s line of sight,
with u increasing counter-clockwise about the host star, e.g. cj = | rad means the planet’s periastron occurs directly on an observer’s line of sight before intersecting the host star and u = ^ rad would place periastron on the opposite side of the star. The angles to be taken into account when calculating the flux of a planet at 11
any given point in time are known as the true anomaly ( /) and eccentric anomaly
(E) shown in Fig. 2.2.
Figure 2.2: The true and eccentric anomalies track the position of a planet (point P ) in its orbit, where / (here represented by 6) is the angle subtended between periastron and the planet with respect to the host star (point F) and E is subtended between periastron and the corresponding point on the orbit’s auxiliary circle (point P', a circle whose center is at the ellipse’s centroid (point C ) with radius equal to the ellipse’s semi-major axis, with respect to the centroid of the ellipse.
In order to orient the outermost planet of a system correctly such that it will begin its orbit opposite its host star relative to the line of sight (where the planet’s phase angle
(a) is zero radians) it is necessary to solve for the minimum of an array where, given n steps of an orbit and the computed / for each step, each element of the array is equal to ^ — (ui+f) rad. The phase angle is a number from 0—27r radians and defines the current phase the target object is in, e.g. a = 0 rad is where the target object 12
is in its full phase, a = n rad is where the target object is in its new phase. The element closest to zero in the array then takes the correct number of steps starting from periastron to position the planet at such a point where a = 0 rad; all other planets in the system (if they exist) take the same number of steps through their orbits to start at randomly distributed positions relative to the outermost planet.
It is important to note that a phase angle of zero is not necessarily correlated with maximum flux, as the flux is highly dependent on star-planet separation, especially in systems with high eccentricities (Kane & Gelino, 2010).
2.2 The Phase Function
When constructing a model of planet reflectivity we use a phase function found to be more effective than the model for an isotropically reflecting Lambert sphere
(§1.3) because it accounts for the backscattering in planet atmospheres due to clouds.
To solve for the flux ratio of the planet/star pair requires a solution to the phase function at every position of the planet in its orbit (Kane & Gelino, 2011). This
Hilton phase function (a polynomial fit determined by Hilton in 1992 from empirical data) is written as
g(a) = l0-°-4Am(“), (2.1) where A m(a) is the visual magnitude of an object given by 13
A m(a) = 0 .0 9 -? - + 2 .3 9 ( - ^ - ) 2 - o W - ^ - ) 3. (2.2) v ’ 100° V100°J VlOO0/ v ’
In this expression a is the phase angle of the object whose solution depends on
Kepler’s Equation
E = M + e • sin(E), (2.3) a transcendental equation that can only be solved numerically where e is the eccen tricity of the orbit and M is the mean anomaly of a planet; mean anomaly is defined as a parameter that is proportional to the area swept out by a planet in a length of time and increases uniformly from 0 — 2tt. Solving Kepler’s Equation (§3.2) yields sin(/) and cos(/), where
/ = 27T — cos_1cos(/) if sin(/) < 0, (2.4) or
/ = cos-1cos(/) if sin(/) > 0. (2.5)
The true anomaly is related to the phase angle by the equation
cos(a) = sin(u/ + / + 7r)sin(i), (2,6) 14
so it is trivial to solve for a (i is the inclination angle of the planet). Thus, Eqn.
2.1 can be solved for any value of a, bring us one step closer to solving for the flux
ratio of a planet-star system.
2.3 Albedo
We determine the geometric albedo for planets in our data and do not take into
account the inherent reflectivity of a planet’s atmosphere which is typically thought
of as a planet’s albedo. The atmospheric albedo is a number from 0 — 1 in which an
object reflecting 100% of the light incident on its atmosphere/surface has an albedo
of 1. Conversely something with an albedo of 0 reflects no light. 1
At the time of writing there is little data on any exoplanetary atmospheres as data on them must be inferred from how scattering properties of the atmospheres affect optical phase curves, and certainly not at the scale of our experiment (Gelino
& Kane, 2014) so we rely on the percentage of light visible during any phase of a planet given by the geometric albedo equation
er-i _ gl-r 3 ^ = 5(e'-> + ei-) + TO’ (2'7) where r is the star-planet separation which is dependent on the orbit’s eccentricity
xOne might believe an object like a black hole would have an albedo of 0 but in actuality quantum effects at the event horizon can cause photons incident on it to be reflected, with a probability proportional to the black hole’s temperature (Kuchiev 2003). 15
and semi-major axis (a), written as
a( 1 — e2) r = r ^ i ( 7 ) ■ (28)
N.B. The e that appears in Eqn. 2.7 is the exponential function and not the
eccentricity of a planet.
2.4 Effects of Keplerian Orbits
From Eqn. 2.8 there is a time-dependence introduced in the form of the true
anomaly reappearing in the denominator of the expression. When solving Kepler’s
Equation in §3.2 the position every point a planet occupies takes up the known
quantities r and e for the planet as well as the to-be-determined array for true
anomaly, the time-dependent variable. The shape of the orbital path is affected by
its eccentricity which, according to Kepler’s Laws, changes the rate at which the
planet moves through its orbit. It sweeps out equal areas with respect to its host
star in equal times so it travels faster when closer to its star, and slower as it moves
farther away from the star. Depending on the planet’s argument of periastron this
will affect the shape of the resultant phase curve of the system because of the time-
dependance of the brightness due to the Keplerian orbit. A highly eccentric orbit
can dominate the changes in flux ratio because of the rapidly changing velocity of the planet; such an effect increases with systems approaching an edge-on orientation 16
relative to observers on Earth. The Kepler mission sought out transiting planets which, by definition, are in systems with inclination angles very close to 90°. With a face-on system (i = 0°) the planet would always appear in half full phase so the distance between it and its star becomes the only time-dependent factor. 17
Chapter 3
Application to Known Exoplanets
3.1 Importing the Exoplanet Data
To calculate the flux ratio of planets and host stars the accumulated data have to be imported to a local machine. A repository of exoplanet data on all confirmed planets and KOIs is held on exoplanets.org and can be processed with Python code
(refer to Appendix I for the full script) although it contains many more properties of planets than are needed to find the flux ratio, specifically, 339 of them; for our purposes we only need to know a planet’s system, mass, period, semi-major axis, radius, eccentricity, argument of periastron, and orbital inclination. Some data taken from exoplanets.org are missing in which case we make the assumptions of an orbital inclination of 90°, an eccentricity of 0, an argument of periastron of 90°, and in the case of an unknown radius either Rp = Rj or is solved with the power law relation (see Fig. 3.1) depending on if Mp > 0.3 Mj or Mp < 0.3 Mj, respectively. 18
Mass (Mj)
Figure 3.1: Planetary radii are calculated from the power law equation for planets with masses less than 0.3 Mj. The fit of the plot is expressed using the equation Rp = 1.91286Mp 0513388 (Kane & Gelino, 2012).
In the case where the period and semi-major axis are unknown then that planet is ignored.
With planets’ properties successfully imported into the function they are orga nized into a dictionary where every dictionary key is the name of the host star, containing an inner dictionary where each imported planet characteristic functions as its own key, as well as creating additional keys for the characteristics to be solved for (e.g. albedo, flux ratio, etc.). It is at this point, with all the initial conditions 19
ready to process, that the flux ratios for each system may be solved.
3.2 Solving Kepler’s Equation
Now the script must solve Eqn. 2.3 by numerically iterating until it converges to a solution for eccentric anomaly with a function calculateStartTime(). The motivation for this is twofold: first, for a given system it needs to start in a position such that the outermost planet starts at a position where its phase angle is 0 rad i.e. solving
Eqn. 2.3 will return the start time for the entire system. Second, with the start time known Eqn. 2.3 must be solved for again, this time taking into account the previously solved for start time in the function keplerQ to find an n-length array of values for true anomaly, where n is the number of steps the planets will take around the host star; the code is designed such that after n steps the outermost planet will complete one full orbit and all other planets (following Kepler’s Third Law) will complete > 1 orbits.
3.3 Writing the Phase Variation Function
The script loops through every system, producing a flux ratio plot for each before iterating to the next system. Inputs for the function phaseVariation() include arrays for planet radii, semi-major axes, eccentricities, arguments of periastron, and orbital inclinations (all taken from exoplanet data). We choose n — 1000 as the resolution 20
of the plots and evaluate the variable days to be the period of the outermost planet.
There are some systems in which a planet’s period is as much as two orders of magnitude greater than that of any other planet of that system; calculating the phase variation of such a system for one complete phase of that planet produces a plot that resembles noise only because, in one planet’s orbit, all other planets orbit several hundred times. These systems are a minority of the total number of Kepler systems and altering the code would bias our observations toward them so this issue can be ignored. We use a step size of create a zeroes matrix (size n x m , where m is the number of planets in the system), and evaluate the function for the current system.
As explained in §3.2 the kepler() function runs m times to return an n-length array of true anomaly values for each planet of a system. Ultimately the flux ratio is dependent on geometric albedo, Hilton phase function, and star-planet separation, or Eqns. 2.7, 2.1, and 2.8 respectively, as well as planetary radius. The flux ratio is written as
£ = A M a )% , (3-D where fv and /* are the fluxes of the planet and star and Rp is the radius of the planet. This is a simplified equation from the two equations for each flux:
U = A,g(a)Ft( A ) S (3.2) 21
and
'• - L-3 ' ^
We have Fj(A) = > the incident flux at wavelength A on a planet, L*(A) as the luminosity of a star at the specific wavelength, and d as the distance from observers on Earth to the target star. Dividing Eqns. 3.2 and 3.3 results in Eqn. 3.1.
Within phase Variation() a loop fills the n x m matrix for each planet’s flux ratio at each n number of points. With all planets’ flux ratios of a system computed the n elements of each coulmn of the matrix are summed to arrive at the total flux ratio of the points in the orbital phase of the outermost planet’s.
3.4 Light Curves of Exoplanetary Systems
It is a simple matter, once the n points of total flux have been evaluated for a system, to plot them as a function of the phase angle of the system’s outermost planet. Plots are generated from these arrays, with strings containing the name of the system and its number of planets as the plot’s title, six examples of which are shown in Fig. 3.2. 22
14 Andr 1 planet(||
04 a* Orbital phase of outermost planet
a tyn, J. taltHI 'rr. II UMi, I pl»nft(s) .i »*..* i#, i- «| I Id M31 E I 5 i al mm *8« n r «4 - a# .... «r^ Orbital phase of outermost planet Orbital phase of outermost planet
11 Com, 1 planet (s)
ptt ',,u if i&T~~“ Figure 3.2: Orbital phase is calculated over 1000 steps using • 2tv where j goes from 0-999. Hundreds of plots were generated and, due to space constraints, a select few are shown here as representatives. Refer to §5.1 for case study systems. 23 A problem that arose in the plotting of the data originated from the calculation of the exoplanet data, manifesting as some erroneous points within the kepler() function. Our initial guess for eccentric anomaly (E in the script) was n (originally derived in Charles & Tatum, 1998 who found the Kepler equation to converge fastest using 7r as the initial guess) but for an unknown reason in approximately 10% of the resultant plots the loop to solve Eqn. 2.3 never converges for a small number of points. To alleviate this we choose a new guess for eccentric anomaly where E = 1.5. This change fixes the problem for nearly all of the final phase plots which, again, a solution to why a difference in guesses affects the rate of convergence for Eqn. 2.3 we did not find. 24 Chapter 4 Correlations 4.1 Motivation for Plotting Calculated Data Over 900 flux ratio plots have been generated and relevant data for every system calculated and stored in a file called planetData.csv. We’ve reached the point where we want to extract some patterns from the systems or look for some emerging structure from the planetData file when certain characteristics are plotted against each other. There is a chance some plots may answer (fully or in part) the questions posed in the introduction, or exhibit an intrinsic quality shared by all planets in all systems, or the inverse of that where conditions are shown in which all planets, or planets of a specific type (e.g. terrestrial planets) will never exist in such a position. 25 4.2 Data to Compare We choose to visualize our data using two kinds of mapping techniques: scatter plots and histograms. Computed data for every confirmed planet from Chapter 3 and saved in planetData.csv is as follows: • Period • Semi-major Axis • Radius • Eccentricity • Argument of Periastron • Orbital Inclination • Mean Albedo • Albedo at Semi-major Axis • Mean Flux • Peak Flux • Total Flux of a System • Host Star Mass 26 • V-Magnitude Before creating scatter plots, in an effort to clean our data we remove planets in which characteristics have been assigned to them in cases where no data exists in Kepler’s database, e.g. a planet with no documented inclination angle is registered with i = 90°. A significant percentage of planets have at least one of the above values missing, a negligible effect when calculating phase plots but necessary to eliminate in these cases. In Fig. 4.1 are shown two such scatter plots, one with planets whose eccentricities and radii have been assigned values of e = 0 and Rp = 1 Rj (Jupiter radius) are included in the data, the other where the same planets have been omitted for clarity. Planetary Radius y. Eccentricity Planetary Radius v. Eccentricity Figure 4.1: The left figure has a distinct horizontal line at y = l, with another, slightly less clear, vertical line at x=0. A large portion of exoplanet data were missing values of eccentricity and radius and including the points that the script assigned to the blank spots by default would make the correlation plots appear to have very strong patterns at those values. The right figure shows that, after the removal of the planets whose data were not observationally determined, the shapes formed by the data that were once there are present no longer. 27 A similar problem arises in the creation of histograms (example Fig. 4.2); a saturation of a single value drastically overshadows the rest of the data which ne cessitates the same filtering method. Eccentricity Eccentricity _ 10’ 10J ■ i M ...... 0 0 0.? 0 4 0 6 0,8 1,0 0.6 0,8 10 Eccentricity Eccentricity Figure 4.2: Any observationally determined values for eccentricity are dwarfed in a histogram by the number of planets whose eccentricities have been assigned 0 in the left figure. Such a model is effective for approximations in our phase plots but renders such correlation plots unusable. With the e = 0 values filtered out the remainder of the planets’ eccentricities exhibit a much clearer pattern, seen in the right figure. 4.3 Results from Data Analysis There are multiple plots in which underlying structure emerges from the com parison of data points once the accumulated data have been cleaned. A majority of the comparisons may return plots with no coherency but, as stated, some inter esting shapes (Figs. 4.3-4.6) take form. Of special note, in Figs. 4.4, 4.5, and 4.6 28 there are some red data points accompanying the blue ones; these points signify planets within the top 50 list of brightest peak fluxes (for the complete list, refer to Appendix II, Table 6.1) and will be referred to as Peak Flux Planets (PFPs) below for brevity. Semi-major Axis (AU) Figure 4.3: There are three groupings of points that stand out. The largest cluster in the lower-left are terrestrial planets; the color of the points correlates with the size of the planet such that bluer points are smaller and redder points are larger (in units of Jupiter-radii). The cluster in the lower-right contains gas giants close to Jupiter’s size. The final cluster in the upper-left contains hot Jupiters, that is to say very large gas giants that are close to their host stars. Conspicuously absent are a large number of Neptune-sized planets which would appear in the gap between the lower-left cluster and the other two clusters. 29 Semi-major Axis v. Planetary Radius 101 10° ? i/i I o 10l 2oT: E COO) 10‘2 10'3 10'2 101 10° 101 Radius (Jupiter-radii) Figure 4.4: As with Fig. 4.3 there is a gap in points where Neptune-size planets would lie. It is an indication of a bias against Neptunes during planetary formation, and that any Neptunes that do form are unlikely to migrate (as hot Jupiters do, i.e. the smaller cluster in the lower-right of large planets with small semi-major axes). The positioning of the two clumps relative to each other is interesting in that the larger, closer planets appear lower on the plot than the others. This can be explained by planetary formation in how gas giants form outside the frost line of a given host star and then migrate inwards. Smaller terrestrial planets generally remain wherever they form because their masses are too small to interact in any meaningful way with their stars through gravitational interaction. All PFPs appear within the hot Jupiter group suggesting that the intrinsic size to a planet is a greater factor in determining its total reflected light than its distance to its host star, e.g. of the planets appearing in the lower-left of the plot (with small radii and close to host star) none of them are among the 50 PFPs. 30 Figure 4.5: Here albedo of planets is plotted first against a planet’s semi-major axis and then against a planet’s period. With the first figure a smooth line is formed from the points since albedo is directly dependent on semi-major axis (see §2.3, Eqns. 2.7 and 2.8). But with the second figure, period is only proportional to semi-major axis (Kepler’s Third Law: P 2 Peak Flux Figure 4.6: This plot illustrates ideal target planets, that is, planets with low mag nitude host stars (i.e. the brightest stars) and themselves having high peak fluxes (whether due to size, eccentricity, or a high albedo). Such a planet would appear in the lower-right of the plot and though there are no obvious outliers in that area there are some that don’t lie within the two loose groupings present. The lower-left cluster represents terrestrial planets with low peak fluxes relative to other planets and the upper-right cluster is composed of the larger gas giants. It should be obvi ous that the rightmost data should be the PFPs in a plot where the x-axis measures peak flux. 32 Chapter 5 Case Studies 5.1 Selected Systems The following figures are six systems intended to display the differing phase curves that typically occur, including single planet systems of varying eccentricities and multi-planet systems with periodic or chaotic shapes. Solid black lines denote the total flux ratio for all planets and different colored dashed lines each represent a single planet’s flux ratio that contributes to the total. Also worth commenting on are the contributing factors towards the total flux from each planet. Prom the flux ratio equation (§3.3, Eqn. 3.1) the radius of a planet and its distance to the host star affect the greatest change in the result, with geometric albedo contributing relatively little. Both radius and distance are squared values with albedo only being linear, and even then only varying between 0 — 1. The most significant contribution a planet’s albedo can make to the flux ratio 33 would be one very close to 0; the resultant flux ratio would be proportionally closer to zero regardless of a planet’s size or proximity to its host star because it would be reflecting very little light. We attempt to find whether a planet’s size or its distance between it and its star predominantly determine its contribution to the total flux ratio. All relevant data on case study systems is contained in Appendix II, Table 6.3. Naming conventions for the planets follow alphabetically (starting with b) for every new planet discovered within a given system. Planets that are later redacted due to a misinterpretation of the data do not cause previously named planets to change, e.g. a system with planets b, c, and d with planet c later discovered to be nothing more than noise in observational data becomes a system with planets b and d. This should alleviate any confusion that may arise when reading planets’ names contained below. 34 5.1.1 Kepler-9 Figure 5.1: An example of a periodic multi-planet system. All three planets’ periods are clearly visible within the continuous curve representing the sum of all the plan ets’ flux ratios with their host star. The planets, represented by dashed lines, are: Kepler-9d (red), Kepler-9b (green), and Kepler-9c (blue). The most apparent fluctu ations in the total curve are attributable to Kepler-9d as its period is much shorter than the other two planets. The three peaks forming the rough w-shape whose points occur at phases of approximately 0.0, 0.5, and 1.0 originate from Kepler-9b’s three peaks at identical locations. The two highest points of the curve (at phase ~0.0 and 1.0 come from Kepler-9c’s peak every 38.91 days. From Fig. 5.1 Kepler-9b has the greatest peak flux but the phase curve is dominated by Kepler-9d which is much closer to its star. Kepler-9c is almost the size of b but much farther away, consequently its contribution to the total phase 35 curve is minimal. This is an example of how the effects of albedo are overshadowed by radius and distance as Kepler-9c has the greatest albedo at semi-major axis (Ag — 0.17) but has the lowest peak flux of the three planets. 5.1.2 GJ 581 Figure 5.2: A system with three planets but no evident periodicity. The orbits of GJ 581c (green), GJ 581b (red), and GJ 581e (blue) never appear to sync in such a way as to create a very large peak, thus, the total phase curve is dominated by GJ 581e, whose peak flux is more than twice that of the second-brightest planet. Albedos of b, c, and e have little variation with respect to each other (Ag = 0.152 ± 0.002) so contributions to total flux ratio are dominated by radius and 36 distance. GJ 581b is larger than GJ 581e at Rb = 0.41 R j v. Re = 0.14 R j but has a larger periastron of rmj„ = 0.039 AU v. rmjn = 0.019 AU. GJ 581e, despite its smaller radius, shines more brightly and GJ 581c with a periastron of rmjn = 0.067 AU has almost no visible contribution to the total. 5.1.3 GJ 876 10 le-5______GJ 876, 4 planet(s) Orbital phase of outermost planet Figure 5.3: A system where a fourth planet is introduced and the resultant phase curve becomes dominated by two planets: GJ 876d (red) and GJ 876c (blue). The other two planets GJ 876b (teal) and GJ 876e (green) do not contribute in any meaningful way because of their long orbital periods and low peak fluxes compared to the d and c planets. 37 The peak flux of GJ 876d is much higher than that of the next brightest planet, GJ 876c on account of its closeness to its host star (rmin = 0.016 AU v. rmin — 0.096 AU at their respective periastrons). GJ 876d has a radius approximately four times smaller but it is so much closer it dominates the phase curve. The individual shape of GJ 876c’s phase curve can be seen when the two curves constructively interfere but GJ 876b has very little measurable effect and GJ 876e essentially contributes nothing from the very bottom of the plot. Albedos of d and c are similar with c reflecting slightly more light with Ag = 0.160 and d with Ag = 1.50. 5.1.4 Kepler-341 Periastrons for the pair of interior planets are less than half of periastron of the third-closest planet. Planetary radii aren’t dramatically different and all albedos exist within a range of Ag = 0.16 ± 0.01 so it is the proximity to the host star that leads planets c and b to dominate the phase curve. 38 Figure 5.4: A four-planet system with an interesting resonance effect resultant from two pairs of planets with similar semi-major axes. Kepler-341e (teal) and Kepler- 341d (blue) make up the outer pair indicated by their long periods. Kepler-341c (green) and Kepler-341b (red) have shorter periods (each less than 10 days). The inner planets’ brighter peak fluxes are attributable to their proximity to their host star; the outer planets have larger radii but contribute little to the phase curve. The three maxima occur when planets c and b peak simultaneously, with periodic local maxima occurring in groups of three when they peak out of sync. Little contribution to the phase plot comes from the two outer planets. 5.1.5 HD 4113 The argument of periastron is u = 317.7° for HD 4113b, explaining why the peak is so close to phases of 0.0 and 1.0 (oj = 270° would place the maximum exactly at those phases). The semi-major axis for this planet is a = 1.273 AU but with its 39 extreme eccentricity its periastron is less than a tenth of that (rmin = 0.123 AU). It also has the greatest albedo of any planets in the case study systems at Ag = 0.35. This is a time-dependent albedo in an orbit whose star-planet separation varies wildly so the mean albedo is even greater (Ag = 0.41) than its albedo at its semi major axis. le-6 HD 4113, 1 planet(s) 2.0 1.5 0.5 0.0 0.0 0.2 0.4 0.6 0.8 10 Orbital phase of outermost planet Figure 5.5: HD 4113b is among the most eccentric planets discovered to date with an eccentricity calculated to be 0.903. The peak in its phase curve occurs when it is at superior conjunction (directly opposite its host star with respect to Earth) so the time it takes to transit, despite a 526.62 day period, is only on the order of hours. The inner plot is a closeup of the left side of the phase curve; despite the short transit time of HD 4113b there is still a recognizable peak, albeit for small timescales (on the order of one one hundredth of the total phase). 40 5.1.6 KIC 11442793 , . ie-7 KIC 11442793, 7 planet(s) Z.D ------1------1------1------ Orbital phase of outermost planet Figure 5.6: This seven-planet system is among the most complex systems in the exoplanet database. The color designations for the planets, ordered by descending peak flux, are: b (teal), h (blue), g (green), c (black), d (yellow), e (purple), / (red). The clear primary contributor to the total phase curve is b with the long period of h affecting either side of the plot and synchronized peaks of g, d, e, and / at close to an orbital phase of 0.45 producing the highest peak. Table 6.3 covers all relevant properties of the seven planets as it would be tedious to describe them at length here. Consider that all eccentricities have been assumed to be 0 and inclination angles are within two tenths of a degree of each other (i = 89.6° ± 0.2°). What’s also interesting is all the planets are within 1 AU of their star making it a very compact system. KIC 11442793b is the second-smallest planet 41 but the brightest on account of its closeness to its star, with an orbital period of only seven days with the second brightest planet being the one furthest out h, being a Jupiter-sized planet (Rp — l.OlRj) with the highest albedo in the system (Ag = 0.299). The planets increase in size as they get further away from the host star which explains why their individual brightnesses have some contribution to the total and are not for all intents and purposes invisible (e.g. GJ 876e in Fig. 5.3). 5.2 Implications of Behaviors Of a planet’s albedo, radius, and distance to its host star the planet’s periastron and its intrinsic size are the two most dominant factors in affecting its contribution to a system’s total flux ratio (see §5.1 for breakdown of individual contributions of Ag/Rp/r to the flux ratio); hot Jupiters are, naturally, going to be the brightest of any planet because of their radii and proximity to their host stars, along with their short orbital periods allowing for multiple transit detection opportunities in a short time frame. If we take ratios of planets’ semi-major axes and radii, we can determine whether their peak fluxes will be similar. By way of example, with Kepler-9b and Kepler-9d, we find ^ = 5.22, = 5.77, (5.1) aa Rd where a and Rp are the semi-major axis and planetary radius, respectively, and 42 fluxes would be equal if the ratios were equal. Since -§&■ > —, we would expect ■K’d Q'd the flux of Kepler-9b to be greater, which we can confirm from Fig. 5.1. Without even having to directly calculate peak flux ratios, the planets with the greatest contributions can be inferred from this simple ratio alone. With Figs. 5.3 and 5.4 we see how very long period planets contribute virtually nothing to the resultant phase plot; their radii simply aren’t large enough to account for their vast distance from their host stars. For planets with periods bordering on > 600 days the transiting method employed by Kepler may not catch them. It isn’t even a planet’s low flux that may escape Kepler, but the time required for multiple transits to confirm that a planet is, in reality, a planet and not errant noise. Planets with low flux that take more than 18 months to orbit their star are easily overshadowed by planets that transit every 8 days at 100 times the brightness. 43 Chapter 6 Conclusions Writing a code to automate the calculation for the amount of light reflected off of exoplanets’ surfaces based on the shapes of their orbits doesn’t just produce several hundred phase plots. The data calculated can be used to create correlation plots that begin to visualize the relationships between different types of planets and their frequency of occurrence. Atmospheric models for planets can be compared against such plots to verify the validity of planet reflectivity. Comparisons with observational data are necessary because the flux ratio equation is highly dependent on planetary radius. Uncertainties in measurements can affect how reflective a planet appears to be which has to be negated with more precise measurement instruments. Beyond phase plot calculation, code that can automate an algorithm for the aggregate exoplanet data is capable of performing any calculation once the data is available for processing. In the coming years as more exoplanets are discovered with higher precision telescopes the number of systems to study will increase, and the 44 importance of automating the analysis of systems will take on more significance. With upcoming exoplanet survey missions such as TESS (est. launch date 2017) or JWST (est. launch date 2018) the potential use for the tools developed in this thesis grows. The expansion of the catalog of known exoplanets and the greater degree of accuracy and precision in measuring properties of planets can modify the results presented here to an equally higher degree of accuracy and precision, calculate additional phase plots for planets to fill in the correlation plots, or the script can be retooled for new calculations beyond phase amplitudes since its structure is designed to extract raw exoplanet data for ease of manipulation. 45 Chapter 7 Appendices Appendix I: Code # Plot phase variations for all confirmed exoplanets ’ systems # Works by pulling a spreadsheet of all the raw data from the internet then manipulates and reorganizes it , outputting a streamlined spreadsheet and a plot for each system (947 plots) import numpy as np import matplotlib . pyplot as pit import math import csv from collections import defaultdict from itertools import chain import os import re from sortedcontainers import SortedDict 46 import wget # Define the functions required for flux ratio calculation def calculateSt artTime (T, e ,w, stepSize ) : # Find t_o (starting point) for Kepler equation solution x = int (1) if w > 270.0: w = w — 360.0 w = math . radians (w) findStartDate = [] if T < 1: T = 1.0 for i in range(10**x * int(T)): M = (2 * np.pi / T) * (stepSize + 10**( — x) * i) M = M%(2*np . p i) — 2*np.pi Ei = np . pi for j in range (0 ,100) : # Arbitrary number of steps chosen; it should find a solution for E after only a few iterations E = Ei — (Ei — e*np.sin(Ei) — M) / (1 — e*np.cos(Ei)) if np . abs ( (E-Ei) /E i) < le —4: break else : Ei = E cos_f = (np.cos(E) — e) / (1.0 — e * np.cos(E)) sin.f = (np.sqrt(1.0 — e**2) * (np.sin(E) / (1.0 — e * np.cos(E 47 )))) if sin_f < 0: f = 2*np.pi — np. arccos ( cos_f) findStartDate . append ( abs (math. radians (270.0) — (w-ff))) else : f = np . arccos ( cos.f) findStartDate . append ( abs (math. radians (270.0) — (w-ff))) for i in range ( len ( findStartDate )) : if findStartDate [ i ] = min( findStartDate ) : t_o = float(i) / len (findStartDate) * T break return t_o def kepler(T, t_o , e , n ,w,stepSize): # Solve for true anomaly true.anomaly = [ for i in range(0 ,n ) : M= (2 * np.pi / T) * ((i) * stepSize + t_o) M = M%(2*np . p i) — 2*np.pi E = 1.5 # Initial guess of pi doesn’ t work but 1.5 has empirically been determined to work for all but three systems ; reason unknown for j in range(0,100): Ei = E E = Ei — (Ei — e*np.sin(Ei) — M) /(1.0 — e*np.cos(Ei)) if np . abs ((E—Ei)/Ei) < le—4: break 48 else : continue cos_f = (np.cos(E) — e) / (1.0 — e * np.cos(E)) sin_f = (np.sqrt(1.0 — e**2) * (np.sin(E) / (1.0 — e * np.cos(E )))) if sin_f < 0: f = 2*np.pi — np . arccos ( cos_f) true_anomaly . append ( f ) else : f = np.arccos(cos_f) true-anomaly . append ( f ) return true_anomaly def phaseVariation( days , n ,T, t_o , R_E ,R, e,w,a,I,stepSize , name) : T = np . array (T) R = R_E * np. array (R) # Convert R to km e = np.array(e) w = np . array (w) a = np.array(a) I = np.array(I) fluxRatio = np . zeros ( shape=(len (R) ,n)) # Create matrix for planets’ fluxes A = [] # Albedo smA = [] # Albedo at semi—major axis flux = [] # Flux (of individual planets) sps = [] # Star—planet separation 49 timeRange = [] for i in range(n): s = stepSize*i timeRange . append (s ) # Calculate flux ratio for i in range( len(R)): f = np.array(kepler(T[i],t_o,e[i],n,w[i],stepSize)) # True anomaly f = f . t olist () alpha = [ for j in range( len( f )): angle = np . arccos (np . sin (np . radians (w[ i ]) + f [ j ] + np.pi) * np.sin (np.radians(I [i ]))) alpha . append ( angle ) r = a [ i ] * ( 1 — (e [ i ]) ** 2) / (1 + e[i] * n p.cos(f)) # Star — planet separation sps.append( r ) albedo = (np.exp(r—1) — np . exp(l —r ) ) /(5 * (np.exp(r—1) + np . exp (1 —r))) H- 3.0/10 # Calculate geometric albedo smA. append (( np . exp (a [ i ] —1) — np . exp(l — a [ i ]) ) /(5 * (np.exp(a[i ]—1) + np . exp(l — a [ i ]) )) + 3.0/10) A.append(albedo) ma = .09 * ( alpha/np . radians (100) ) + 2.39 * ( alpha/np . radians (100)) ** 2 — .65 * ( alpha/np. radians (100)) ** 3 # Hilton phase function g = 10.0 ** ( — .4*ma) 50 fluxRatio[i] = albedo * g * (R [ i ] / ( r * 1.496 e8)) **2 # Flux ratio flux . append ( flux Ratio [ i ]) totalFlux = np.zeros(n) for i in range( len(R)): totalFlux += fluxRatio[i] pit . figure () pit . plot (timeRange/max(T) , totalFlux , ’ k ’ , marker size =2) pit . ticklabel_format ( style = ’ sci ’ , axis = ’y ’ , scilimits =(0,0)) pit . xlim ([0 , timeRange [ — 1] /max(T) ]) pit . xlabel ( ’ Orbital phase of outermost planet’ ) pit . ylabel ( ’ Flux r a tio ’ ) pit . title (name+’ , ’+str ( len (R)) + ’ planet (s) ’) return A,smA, flux , totalFlux , sps # Pull constants from Kepler data # Read in the EOD column: if EOD = 1,add it to dictionary. If EOD = 0 (the candidates), skip it # If no value for inclination , use I = 90 # Read in a column for planet mass # If no value for planetary radius, use equation sent inKane’ s email: if planet mass > .3, use R = 1 jupiter radius. If planet mass < .3, use the power law relation # If no eccentricity , use e = 0 # If no argument of periastron , use w = 90 51 # If no period AND semi—major axis , ignore exoDict = {} columns = defaultdict ( list ) if not os . path . exists ( ’ exoplanets . csv’) : url = ’http://exoplanets.org/csv-files/exoplanets.csv’ filename = wget. download ( url) with open (” exoplanets . csv rU” ) as data: reader = csv . DictReader (data , delimiter = ’ , ’) for row in reader: for (h,v) in row.itemsQ: columns [h] . append(v) i = 0 for name in columns [ ’NAME’ ] : if columns [ ’EOD’ ][ columns [ ’NAME’ ]. index (name) ] = ’1’: # if not re . search ( ’KOI’ ,name) : planetName = name[ —1] name = name [0:—2] if exoDict . has.key (name) : if columns [ ’ I ’][i] = columns [ ’ I ’ ][ i ] — 90.0 if columns [ ’R ’ ][ i ] = ’ ’ : if columns [ ’MASS’ ][ i ] = ’ ’ : columns [’R’][ i ] = 0.001 elif columns [ ’MASS’ ][ i ] >= ’0.3’: columns [ ’R ’ ][ i ] = 1.0 52 else : columns [’R’][ i ] = 1.91286 * float ( columns [ ’MASS’ ][ i ]) ** 0.513388 i f columns [ ECC’ ] [ i] = columns ’ECC’ ] [ i] = 0.0 i f columns [ OM’] [ i] = ’ columns ’OM’ ] [ i] = 90.0 i f columns [ MSTAR’ ] [ i ] = ” : columns ’MSTAR’ ] [ i] = 0.0 if columns [ V’ ] [ i ] = columns ’V’] [ i] = 30.0 i f columns [ A’lfil = ” and columns [ ’PER’ 1 [ i i = i + 1 continue exoDict [name] [ ’ Planet ’ ] . append (planetName) exoDict [name] [ ’ Period ’ ] . append (columns [ ’PER’ ] [ i ]) exoDict [name] [ ’ Semi—major Axis ’ ] . append (columns [ ’ A ’ ] [ i ]) exoDict [name] [ ’ Planetary Radius ’ ] . append (columns [ ’R ’ ] [ i ]) exoDict [name] [ ’ Eccentricity ’ ]. append (columns [ ’ECC’ ] [ i ]) exoDict [name ] [ ’ Argument of Periastron ’ ]. append ( columns [ ’OM ’ ] [ i ]) exoDict [name] [’Orbital Inclination ’ ]. append (columns [ ’ I ’ ][ i ]) exoDict [name] [ ’ Mass ’ ] . append (columns [ ’MASS’ ] [ i ]) exoDict [name] [ ’ Star Mass ’ ] . append (columns [ ’MSTAR’ ] [ i ]) exoDict [name] [ ’ V—Mag ’ ] . append (columns [ ’ V’ ] [ i ]) 53 i = i + 1 else : if columns [ ’ I ’ ] [ i ] = columns [ ’ I ’ ][ i ] = 90.0 if columns [’R’][i] = if columns [ ’MASS’ ] [ i ] = ’ ’ : columns [’R’][ i ] = 0.001 elif columns [ ’MASS’ ][ i ] >= ’0.3’: columns [ ’R ’ ][ i ] = 1.0 else : columns [’R’][ i ] = 1.91286 * float (columns [ ’MASS’ ][ i ]) ** 0.513388 if columns [ ’ECC’ ] [ i ] = 5 columns [ ’ECC’ ][ i ] = 0.0 if columns [ ’OM’ ] [ i] = columns [ ’OM’ ][ i ] = 90.0 if columns [ ’MSTAR’ ] [ i ] = ” : columns [ ’MSTAR’ ] [ i ] = 0.0 if columns [ ’V ’ ] [ i ] = ’ columns [’V’][ i ] = 30.0 if columns [’A’][ i ] = ’’ and columns [ ’PER’ ][ i ] = ’ ’ : i = i + 1 continue exoDict [name] = { ’ Planet ’ :[ planetName ] , ’ Period ’ :[ columns [ ’ PER’ ] [ i ] ] , ’ Semi—major Axis ’ : [ columns [ ’A ’ ] [ i ] ] , ’ Star — Planet Separation ’ : [] , ’ Planetary Radius ’ : [ columns [ ’R ’ ] | 54 i ] ] , ’ Eccentricity ’ : [ columns [ ’ECC’ ] [ i ] ] , ’ Argument of Periastron ’ :[ columns [ ’OM’ ] [ i]] , ’ Orbital Inclination ’ : [ columns [ ’ I ’ ] [ i ] ] , ’ Mass ’ : [ columns [ ’MASS ’ ] [ i ] ] , ’ Star Mass [columns [ ’MSTAR’ ] [ i ] ] , ’ V-Mag’ :[ columns [’V’][ i ] ] , ’ Mean Albedo ’ : [ ] , ’ Semi—major Albedo ’ : [] , ’ Mean Flux ’ : [ ] , ’ Peak Flux ’ : [] , ’ Total Flux’:[]} i = i -h 1 else : i = i + 1 ” ”” # This section contains the Kepler candidate planets; for now they can be ignored else : if exoDict . has_key (name) : exoDict [name] [ ’ Planet ’ ] . append ( ’N/A’ ) exoDict [name] [ ’ Period ’ ] . append (columns [ ’PER’ ] [ i ]) exoDict [name] [ ’ Semi—major Axis ’ ] . append (columns [ ’ A ’ ] [ i ]) exoDict [name] [ ’ Planetary Radius ’ ] . append (columns [ ’R ’ ] [ i ]) exoDict [name ] [ ’ Eccentricity ’ ]. append (columns [ ’ECC’ ] [ i ]) exoDict [name] [ ’ Argument of Periastron ’ ]. append (columns [ ’OM exoDict [name] [ ’ Orbital Inclination ’ ]. append (columns [ ’ I ’ ][ i ]) i = i + 1 else : exoDict [name] = { ’ Planet ’ : [ ’N/A’ ] , ’ Period ’ : [ columns [ ’PER’ ] [ i ] ] , ’ Semi—major Axis ’ : [ columns [ ’ A’ ] [ i ] ] , ’ Star—Planet 55 Separation ’ : [] , ’ Planetary Radius ’ : [ columns [ ’R ’ ] [ i ] ] , ’ Eccentricity ’ :[ columns [ ’ECC’ ] [ i ] ] , ’ Argument of Periastron ’ :[ columns [ ’OM’ ][i]] , ’ Orbital Inclination ’ :[ columns [ ’ I ’ ] [ i ] ] , ’ Mass ’ : [ columns [ ’MASS ’ ] [ i ] ] , ’ Star Mass ’ : [ columns [ ’MSTAR’ ] [ i ] ] , ’ V—Mag ’ : [ columns [ ’ V’ ] [ i ] ] , ’ Mean Albedo Semi—major Albedo Mean Flux Peak Flux ’ : [] , ’ Total Flux ’ : [] } i = i + 1 55 5? # Run code to create phase curves for Kepler systems if not os . path . exists ( ’ Plots ’ ) : os . makedirs ( ’ Plots ’ ) for name in exoDict : print name T = [] R = [] e = [ w = [ 1 = [] a = [ ] for j in range ( len ( exoDict [name] [ ’ Planet ’])) : T. append ( float ( exoDict [name] [ ’ Period ’ ] [ j ]) ) R. append ( float ( exoDict [name] [ ’ Planetary Radius ’ ] [ j ]) ) e . append ( float ( exoDict [name] [ ’ Eccentricity ’ ] [ j ]) ) 56 w. append ( flo at ( exoDict [name ] [ ’ Argument of Periastron ’ ][ j ]) ) I . append ( float ( exoDict [name] [ ’ Orbital Inclination ’][j])) a . append ( flo at ( exoDict [name] [ ’ Semi—major Axis ’ ] [ j ]) ) n = 10000 R_E = 71492.0 # Jupiter—radius (km) maxPeriod = T. index (max(T)) # The number of elements into the array the maximum period occurs; use to calculate the start date days = max(T) stepSize = days/n # Dividing outermost planet’ s period into n equal parts t_o = calculateStartTime (T[ maxPeriod ] , e [ maxPeriod ] ,w[ maxPeriod ] , stepSize) A,smA, flux,tot alFlux,sps = phaseVariation( days , n ,T, t_o , R_E ,R, e , w, a , I , stepSize ,name) for i in range ( len ( exoDict [name ] [ ’ Planet ’])) : exoDict name ] [ ’ Mean Albedo ’ ] . append (np . mean (A [ i ])) exoDict name ] [ ’ Semi--major Albedo ’ ] . append (smA [ i ]) exoDict name ] [ ’ Mean Flux ’ ] . append (np . mean( flux [ i])) exoDict name ] [ ’ Peak Flux ’ ] . append (max( flux [ i ]) ) exoDict name ] [ ’ Total Flux ’ ] . append (max( totalFlux)) exoDict name ] [ ’ Star--Planet Separation ’ ] . append (min( sps pit . savefig(os.path .join (” P lots/” ,name)) pit . close () exoDict = SortedDict ( exoDict) with open (” planetData . csv ” ,”w” ) as data: 57 data. write(’% s ’ % ’System’ ’%s ’ % ’ Planet’ ’%s ’ % ’ Period’ ’% s’ % ’Semi-major Axis’ 4-” ,” + ’%s ’ % ’Minimum Star—Planet Separation’ ’%s ’ % ’Planetary Radius’ + ’%s ’ % ’ E ccentricity’ ’%s ’ % ’Argument of Periastron’ ’% s’ % ’Orbital Inclination’ + ’%s ’ % ’Albedo’ + ’%s ’ % ’ Albedo at Semi-Major Axis’ ’%s ’ % ’Flux’ 4-”,”+ ’% s ’ % ’Peak Flux’ ’%s ’ % ’ Total Flux’ ’%s ’ % ’ Star Mass’ ’%s\n ’ % ’V-Magnitude ’ ) for names in exoDict : for i in range ( len ( exoDict [ names ] [ ’ Planet ’])) : with open (” planetData . csv ” a” ) as data: data . write (’%s ’ % names ’%s ’ % exoDict [names] [ ’ Planet ’ ] [ i ] ’% s’ % exoDict [names] [ ’ Period ’][i] ’% s’ % exoDict [names] [ ’ Semi—major Axis ’ ] [ i ] ’%s ’ % exoDict [ names ] [ ’ Star—Planet Separation ’ ][ i ] +” ’%s ’ % exoDict [names] [ ’ Planetary Radius ’ ][ i ] ’%s ’ % exoDict [names] [ ’ Eccentricity ’ ][ i ] ’%s ’ % exoDict [ names] [ ’ Argument of Periastron ’ ][ i ] ’%s ’ % exoDict [names] [ ’ Orbital Inclination ’][i] ’%s ’ % exoDict [names ] [ ’ Mean Albedo ’ ][ i ] ’%s ’ % exoDict [ names ] [ ’ Semi—major Albedo ’ ][ i ] ’%s ’ % exoDict [ names] [ ’ Mean Flux ’ ] [ i ] ’%s ’ % exoDict [ names ][ ’ Peak Flux ’ ] [ i ] ’%s ’ % exoDict [names] [ ’ Total Flux ’][i] ’%s ’ % exoDict [names ][’Star Mass ’ ] [ i ] + ’%s\n’ % exoDict [names ] [ ’V—Mag’ ][ i ]) Appendix II: Tables Table 7.1: Brightest 50 planets from exoplanet data System Planet Peak Flux Ratio WASP-19 b 2.38E-04 WASP-12 d 2.33E-04 WASP-103 c 2.01E-04 WASP-43 b 1.43E-04 WASP-18 b 1.34E-04 CoRoT-1 b 1.17E-04 WASP-33 b 1.17E-04 TrES-3 b 1.16E-04 WASP-4 b 1.16E-04 OGLE-TR-56 b 1.07E-04 HAT-P-23 d 9.94E-05 Qatar-2 c 9.64E-05 WASP-46 e 9.61E-05 HAT-P-32 b 9.37E-05 CoRoT-2 b 9.06E-05 HATS-2 b 8.79E-05 WASP-77 A b 8.65E-05 59 HAT-P-36 b 8.47E-05 Kepler-17 b 8.41E-05 Qatar-1 c 8.31E-05 TrES-5 b 8.23E-05 WASP-36 b 7.88E-05 WASP-48 b 7.83E-05 WASP-64 b 7.82E-05 OGLE-TR-113 b 7.77E-05 WASP-78 b 7.50E-05 WASP-52 b 7.44E-05 Kepler-412 b 6.66E-05 CoRoT-18 b 6.20E-05 WASP-5 b 5.98E-05 KOI-13 b 5.76E-05 WASP-3 b 5.71E-05 CoRoT-14 b 5.44E-05 HAT-P-41 b 5.40E-05 WASP-1 b 5.33E-05 OGLE-TR-132 b 5.29E-05 WASP-17 b 5.17E-05 WASP-50 b 5.12E-05 60 OGLE2-TR-L9 b 5.08E-05 WASP-14 b 4.96E-05 WASP-100 b 4.65E-05 Kepler-76 b 4.64E-05 HD 189733 b 4.59E-05 HAT-P-7 b 4.37E-05 WASP-95 b 4.32E-05 WASP-24 b 4.31E-05 Kepler-45 b 4.26E-05 IVES-4 b 4.19E-05 WASP-2 b 4.03E-05 TrES-2 b 4.02E-05 61 Table 7 .2: Brightest 50 stars from exoplanet data System V-Magnitude beta Gem 1.15 alpha Cen B 1.33 alpha Ari 1.996 gamma Leo A 2.12 gamma Cep 3.21 iota Dra 3.29 omicron UMa 3.362 epsilon Tau 3.53 epsilon Eri 3.72 beta Pic 3.861 7 CMa 3.95 upsilon And 4.1 epsilon CrB 4.13 HD 20794 4.26 epsilon Ret 4.44 HD 60532 4.45 tau Boo 4.5 HD 11977 4.695 xi Aql 4.71 62 11 Com 4.783 kappa CrB 4.79 42 Dra 4.833 HD 114613 4.85 61 Vir 4.866 HD 102365 4.89 70 Vir 4.97 11 UMi 5.024 47 UMa 5.03 HD 19994 5.07 HD 33564 5.08 mu Ara 5.12 14 And 5.22 omega Ser 5.23 75 Cet 5.364 HD 147513 5.37 rho CrB 5.39 HD 81688 5.4 iota Hor 5.4 51 Peg 5.45 Kepler-439 5.46 63 18 Del 5.51 omicron CrB 5.519 HD 10647 5.52 81 Cet 5.65 HD 39091 5.65 HD 142 5.7 HD 190360 5.73 HD 192310 5.73 HD 89744 5.73 HD 30562 5.77 Table 7.3: Properties of case study systems fp System Planet Period (d) ^min (A?/) Radius (R j) e u (°) Ag f* max Kepler-9 b 19.24 0.129 0.842 0.151 18.6 0.16 1.28E-06 Kepler-9 c 38.91 0.195 0.823 0.133 348.7 0.17 5.70E-07 Kepler-9 d 1.59 0.027 0.146 0 90 0.15 1.03E-06 GJ 581 b 5.37 0.039 0.411 0.031 251 0.15 3.74E-06 GJ 581 c 12.92 0.068 0.235 0.07 235 0.15 4.10E-07 GJ 581 e 3.15 0.019 0.140 0.32 236 0.15 1.66E-06 GJ 876 b 61.12 0.201 1 0.032 50.3 0.17 8.48E-07 GJ 876 c 30.09 0.096 1 0.256 48.78 0.16 1.74E-06 GJ 876 d 1.94 0.016 0.246 0.207 234 0.15 7.14E-06 GJ 876 e 124.26 0.315 0.363 0.055 239 0.18 5.41E-08 Kepler-341 b 5.20 0.059 0.105 0 90 0.15 1.12E-07 Kepler-341 c 8.01 0.078 0.152 0 90 0.15 1.32E-07 Kepler-341 d 27.67 0.179 0.165 0 90 0.16 3.20E-08 Kepler-341 e 42.47 0.238 0.178 0 90 0.17 2.18E-08 HD 4113 b 526.62 0.123 1 0.903 317.7 0.35 1.94E-06 KIC 11442793 b 7.01 0.076 0.117 0 90 0.15 8.32E-08 KIC 11442793 c 8.72 0.088 0.106 0 90 0.16 5.14E-08 KIC 11442793 d 56.74 0.307 0.256 0 90 0.18 2.86E-08 KIC 11442793 e 91.94 0.423 0.237 0 90 0.20 1.40E-08 KIC 11442793 f 124.91 0.520 0.26 0 90 0.21 1.20E-08 KIC 11442793 g 210.61 0.736 0.723 0 90 0.25 5.47E-08 KIC 11442793 h 331.60 0.996 1.01 0 90 0.30 7.02E-08 N.B. 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