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A Study of the Detection, Observation, Analysis and Mod-elling of Transiting Exoplanets

Figure 1: Artists conception of wasp17 [1]

By: Benjamin John CookStudent No.: 488622

Project unit: (M301)Supervisor: (Dr Michael McCabe)Mentors: (David Harris, Chris Priest, Steve Futcher)

Date: March 26, 2013

Abstract

This Project begins with an introduction into what are Transiting Exoplanetary Systems,then Project will then go into the methods for detecting Transiting Exoplanetary Systemsdescribing the mathematical and scientific basis these methods are founded on. Resultsobtained during observations at Clanfield Observatory will be analysed as will data fromonline databases such as The Exoplanetary Transit Database website (http://var2.astro.cz/ETD/). We will then research into the methods for modelling the data analysing howeffects such as limb darkening have on the transit light curve. We will then finish with athree dimensional visual representation of the exoplanetary system.

http://var2.astro.cz/ETD/

http://var2.astro.cz/ETD/

Student Number: 488622, Unit Code: M301

Contents

1 Exoplanetary Systems 31.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 What makes a planet habitable? . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Stellar Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.2 Planetary Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Habitable Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Exoplanet Detection Methods 82.1 Direct Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1 Coronagraphy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Astrometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Radial Velocity Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3.1 The Stellar Reflex orbit with a single planet . . . . . . . . . . . . . . . . 102.3.2 Reflex radial velocity of multiple non-interacting planets . . . . . . . . 14

2.4 Transits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4.1 Geometric Probability of a Transit . . . . . . . . . . . . . . . . . . . . . 16

2.5 Gravitational Microlensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Observations and Construction of the Transit Light Curve 193.1 Clanfield Observatory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Transit Candidates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3 Making Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.4 Image manipulation using AIP4WIN . . . . . . . . . . . . . . . . . . . . . . . . 243.5 Light Curve Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.5.1 Excel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.5.2 Minitab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 Analysis of Exoplanetary Systems 344.1 Characteristics of Host Star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2 Orbital Period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.3 Exoplanet Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.4 Calculation of Semi-major Axis . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.5 Orbital Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.6 Impact Parameter and Transit Duration . . . . . . . . . . . . . . . . . . . . . . 394.7 Mass of Exoplanet and Eccentricity of Orbit . . . . . . . . . . . . . . . . . . . . 44

5 Limb Darkening 455.1 Geometry of the Transit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.2 Light lost during the Transit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.3 Laws for Limb Darkening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

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6 Modelling of the Transiting Exoplanetary System 546.1 Exoplanetary System Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.2 Maple Model factoring in Limb Darkening . . . . . . . . . . . . . . . . . . . . 556.3 NAAP Transit Simulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606.4 Autodesk Maya . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

7 Conclusion and Recommendations 687.1 Detection Methods and Observations . . . . . . . . . . . . . . . . . . . . . . . 687.2 Light Curve Construction and Limb Darkening . . . . . . . . . . . . . . . . . . 697.3 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697.4 Recommendations for future investigation . . . . . . . . . . . . . . . . . . . . 69

8 Appendices 728.1 Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728.2 Appendix B - Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798.3 Appendix C - Excel Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798.4 Appendix D - Maple Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 798.5 Appendix E - Autodesk Maya Model . . . . . . . . . . . . . . . . . . . . . . . . 79

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Chapter 1

Exoplanetary Systems

1.1 Introduction

From our perspective standing on our planet it looks of colossal size. Moving further andfurther into the vacuum of space you can begin to see how small it is compared to the uni-verse as a whole. It also makes us think how lucky we are to live on a planet that can sustainlife. This then begs the question, is there other intelligent life out there in the universe? Acouple of years ago, theoretical physicist Stephen Hawkings told the world that the exis-tence of Extraterrestrials is almost certain. Stephen Hawking said that his conclusion was”unusually simple”. By knowing that the universe contains roughly 100 billion galaxies andeach galaxy contains around 100 million stars then there is a high probability that our planetEarth is not the only planet in our Universe that has life on it that has evolved [6]. Aroundthe world astronomers are now competing against one another to be the first to discover ahabitable planet.

For hundreds of years scientist, astronomers and philosophers have predicted thatplanets outside our own solar system must exist; these were called Exoplanets and weredefined as ”planets orbiting stars other than our own” [2]. It was not until 1995 with thediscovery of the first exoplanet that orbited a star much like our own sun that astronomersaround the world came together in a global effort to discover more exoplanets [2].

Figure 1.1: Artists conception of the planet with 4suns [7]

As of January 15th 2013 there havebeen 859 exoplanets discovered[5], yet thediscovery of a habitable planet still eludesus (based on our knowledge of carbonbased life forms requiring oxygen and wa-ter to survive). While our main goal hasnot yet been achieved, other fascinating dis-coveries has been made. One such recentdiscovery is the planet with four suns, dis-covered by volunteers using http://www.planethunters.org/. The planet they dis-covered is assumed to be a gas giant orbit-ing a binary star system along with anotherpair of stars orbiting the binary star system. All four stars have a gravitational pull on thisone planet which baffles scientist as the planet itself is in a stable orbit, which makes it andamazing and unexpected discovery [7].

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http://www.planethunters.org/

http://www.planethunters.org/


1.2 What makes a planet habitable?

Planetary habitability is defined as ”the measure of a planet’s or a natural satellite’s potentialto develop and sustain life” [8].

Figure 1.2: Hertzsprung-Russel diagram [10]

Figure 1.3: Diagram showing the range of the hab-itable zone is compared to the size of differentstars [11]

When looking for indicators of whether aplanet is habitable or not astronomers lookfor various key characteristics of the se-lected star system.

1.2.1 Stellar Characteristics

The key stellar characteristics required forplanetary habitability are:

• Spectral Type of Star - Through us-ing spectroscopy to find the spectraltype of stars, astronomers are able todetect the temperature of the photo-sphere (”The photosphere of an as-tronomical object is the region fromwhich externally received light origi-nates” [9]). The temperature of thephotosphere is related to the total massof the star, this relation is only forstars on the main sequence as shown inthe Hertzsprung-Russel diagram Fig-ure 1.2. Astronomers suggest thatthe best spectral types for habitablestars are F, G and to the middle of K.This correlates to stars on the main se-quence with a photospheric tempera-ture of 4000 degrees Kelvin to 7000 de-grees Kelvin.

Stars in this section of the main se-quence are more suitable candidatesbecause:

1. The stars have longer life timescompared with more luminousstars that burn their fuel morequickly due their massive size.These very luminous stars live fora few millions years, while starslike our sun live for a few billion years.

2. The stars emit high frequency UV (ultra-violet) radiation, this is required for at-mospheric conditions such as ozone layer formation.

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3. Liquid water may exist on the surface of planets orbiting them at a specific distancethat does not induce tidal locking [8] (captured rotation/synchronus rotation).

• Stability of Habitable Zone - The habitability zone also know as the ”circumstellar hab-itable zone” or ”Goldilocks zone” is the scientifice terminology used to describe theregion around a star in which it is possible for a planet to have a stable atmosphere,enough atmospheric pressure and liquid water to make it habitable [11]. This zone canbe calculated by using the size of the star as seen in Figure 1.3.

• Low Stellar Variation - Stars experience small fluxations in their luminosity, a minutenumber of stars experience a significant change in their luminosity (variable stars).This significant change in luminosity is likely to be combined with large amounts ofradiation such as gamma rays and x-rays. Planets around stars that experience thisphenomena are not habitable because of severe temperature change would make lifeunable to survive, along with the large amounts of radiation would make it impossiblefor life to sustain itself or even exist.

• High Metallicity - Using Spectroanalysis of stars astronomers have seen that there isa correlation between high metallicity in stars and the chances of an exoplanet beingfound. Astronomers theorise that the high metallicity corresponds to the amount ofheavy elements available in the protoplanetary disc, so a high metallicity in the starwould mean there is a high chance that an exoplanet would be orbiting it.

1.2.2 Planetary Characteristics

Figure 1.4: Diagram showing the sizes of planetswith various compositions [12]

The key planetary characteristics requiredfor planet habitability are:

• Mass of Planet - The mass of a planetcan effect the habitability of a planetin different ways; the mass effects thegravitational force of a planet, thiswould mean that lower mass plan-ets would have a lower gravitationalforce. The planets ability to retainan atmosphere is dependent upon theplanets gravitational force and there-fore also its size. This means that as-tronomers are having to look for plan-ets that are the size of the earth orlarger. Astronomers also theorise thatlarge planets are likely to have iron cores thus generating a magnetic field, this wouldprotect the planet from stellar winds and cosmic radiation [8].

• Planetary Composition (Geochemistry) - Astrobiologists theorise that life may exist onplanets that have the same primal elements as earth (carbon, oxygen, nitrogen and hy-drogen) that are crucial for life as we know. These elements combined produce aminoacids which in turn make up proteins, the proteins are needed to form DNA and RNA.

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• Orbital and Rotational Properties of Planet - The Orbital eccentricity of a planet ef-fects its surface temperature; for example if a planets orbital eccentricity was large, thiswould lead to drastic rises and falls in temperature. Therefore the planet would be un-able to support life. The Axis tilt of a planet effects the stasis of the biosphere. No axistilt or very little axis tilt leads to no occurrence of seasons, therefore a main catalyst inthe dynamics of the biosphere would not exist. Too much tilt would lead to extremeseasons which would not allow the biosphere to remain stable to sustain life [8].

1.3 Habitable Zone

The habitable zone as mentioned earlier is the scientific terminology used to describe theregion around a star that a planet would have to orbit in to have the neccessary conditionsto sustain life.

Teq =12

((1−A)L∗

σπa2

) 14

(1.1)

• Teq - The equilibrium temperature. As the planet orbits its star we assume it stays at aconstant temperature so the temperature on the surface of the planet, so it be charac-terised by Teq.

• σ - The Steffan-Boltzmann constant which is 5.671×10−8Jm−2K−4s−1.

• a - The semi major axis (longest diameter of an ellipse).

• A - The albedo (reflection coefficient)

• L∗ - The luminosity of the star in units of our suns luminosity (L� = 3.8939×1026Js−1).

Rearranging equation 1.1 we can get a to be the subject to work out the boundaries of thehabitable zone knowing the required equilibrium temperature.

a =1

(2Teq)2

((1−A)L∗

σπ

) 12

(1.2)

Using our own solar system as an example we know that:

• The Luminosity of the Sun L� = 3.839×1026Js−1.

• The albedo (reflection coefficient) of earth A⊕ = 0.3. [2]

• The Steffan-Boltzmann constant σ = 5.671×10−8Jm−2K−4s−1.

• The inner boundary temperature Teq(inner) = 273K (freezing point of water)

• The outer boundary temperature Teq(outer) = 373K (boiling point of water)

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Substituting into equation 1.2 we get

a(inner) =1

(2×373K)2

((1−0.3)×3.839×1026Js−1

π×5.671×10−8Jm−2K−4s−1

) 12

=1

(746)2

(0.7×3.839×1026m2

π×5.671×10−8

) 12

= 6.9787×1010m

(1.3)

a(outer) =1

(2×273K)2

((1−0.3)×3.839×1026Js−1

π×5.671×10−8Jm−2K−4s−1

) 12

=1

(546)2

(0.7×3.839×1026m2

π×5.671×10−8

) 12

= 1.30×1011m

(1.4)

The Earths semi major axis is a⊕ = 1.496×1011m so by this calculation our planet is outsidethe theoretical habitable zone; but we are making the assumption that earth has a uniformtemperature and that it is also a perfect black body. Calculating the Earths equilibriumtemperature, using equation 1.1 we can see that

Teq =12

((1−0.3)×3.839×1026Js−1

π×5.671×10−8Jm−2K−4s−1 × (1.496×1011m)2

) 14

=12

((0.7)×3.839×1026K4

π×5.671×10−8 × (1.496×1011)2

) 14

= 255K

(1.5)

So looking at equation 1.5 we can see that Earths Teq = 255K =−19◦C. Earth is not a perfectblack body and the temperature of our planet is not uniform, therefore it is capable to sustainlife at the distance it is from the sun even though it is outside the theoretical habitable zone.

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Chapter 2

Exoplanet Detection Methods

2.1 Direct Imaging

Compared to stars planets are extremely faint and most of the light that they emit is lost dueto the glare of light emitted from their host star. Detecting planets using direct imaging iscan be extremely difficult if the planet you are trying to detect is some light years away fromour own solar system, which is why it mainly works best to detect objects nearby our ownplanet. Therefore scientists and astronomers have to use other methods to try and detectplanets outside our solar system.

2.1.1 Coronagraphy

Figure 2.1: Diagram showing the Fomalhuat Sys-tem as seen using Coronagraphy [2]

In 1930 french astronomer Bernard Lyot in-vented the coronagraph [13]. This telescopicinstrument allowed the astronomer to blockout direct light emitted by the sun; there-fore he was able to observe objects orbit-ting the sun that would not have previ-ously been seen because of the suns glare.This coronagraph was originally designedto study our suns solar corona but hasbeen adapted by astronomers into the stel-lar coronagraph to detect exoplanetary sys-tems.The stellar coronagraph stops the de-tector being flooded with light from the dis-tant star allowing astronomers to observe ifthere are objects such as exoplanets orbitingthe star.

Fomalhuat is roughly 25 light years fromEarth and is roughly twice the mass of oursun, due to its mass it is therefore more lu-minous than our sun; therefore from earth itis one of the brightest stars in the night sky. Astronomers observed Fomalhuat and discov-

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ered that surrounding was a ring of dust, this dust meant that it was possible that Fomalhuathad a exoplanet orbitting it. Due to the luminosity of the star, direct observation was use-less in detecting the existence of an orbiting planet. It was not until 2008 that the HubbleTelescopes combined images of the star taken in 2004 and 2006 using a stellar coronagraphcould the exoplanet be detected clearly. Figure 3.1 shows the pictures the Hubble telescopetook combined together. From the images astronomers we able to calculate that Fomalhuatb is roughly 109 times fainter than its host star and was orbiting at a distance of 100 timesthe distance earth is from our sun (100AU). The use of a stellar coronagraph in detecting Fo-malhuat b allowed astronomers to directly detect exoplanetary systems in a new way whilebefore they were only detectable in exceptional circumstances.

2.2 Astrometry

’Astrometry is the science of accurately measuring the position of stars’ [2]. Astrometry isone of the oldest methods used to detect extrasolar planets indirectly. It uses the fact thatplanets do not orbit their host star but the barycentre (centre of mass of the solar system),therefore all bodies in the star system orbit the barycentre including the star itself. If thereare planets in the star system then the sun performs a reflex orbit around the barycentre tokeep the centre of mass at the barycentre. Many astronomers make observations of variousstars to detect whether they have a reflex orbit, it is from this they can predict the presenceand parameters of an exoplanet. Using Figure 3.2 it can be seen that the motion of a star inits reflex orbit has a semi-major axis a∗; the planets orbit around the barycentre has a semi-major axis aP, therefore using figure 3.2 it can be shown that the sum of the semi-major axesis a = a∗+aP.

Figure 2.2: Diagram based on Figure 1.12, page27 [2][17]

Kepler’s Laws

1. The orbit of every planet is an ellipse with the Sun at one of the two foci [19].

2. A line joining a planet and the Sun sweeps out equal areas during equal intervals oftime [19].

3. The square of the orbital period of a planet is directly proportional to the cube of thesemi-major axis of its orbit (P2 ∝ a3) [19].

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Generalizing Kepler’s third law it can be shown that:

a3

P2 =G(M∗+MP)

4π2 (2.1)

Where a is the sum of the semi-major axes and P is the orbital period.

Since the reflex orbit of a star has a semi-major axis a∗ it can be detected in as an angulardisplacement β from an observer at a distance d. Therefore the angular displacement orastrometric wobble as it is known can be show as proportional to the semi-major axis of thestars reflex orbit a∗:

β∝a∗ ∴ β = a∗d

Since we know that

a∗ =MP

M∗aP (2.2)

We can see that

β =MPaP

M∗d(2.3)

From equation 2.3 you can see that as MP and aP increases so does β therefore we see moreangular displacement (astrometric wobble) with large mass Exoplanets that have a largeobit aswell. From the equations we can see that Astrometry is useful in detecting large massExoplanets such as gas giants.

2.3 Radial Velocity Measurements

The Radial Velocity method shares similarities with the astrometric method in that it alsouses the reflex orbit of the star. Instead of using the change in position of the star to detectExoplanets it uses the change in radial velocity of the star’s reflex orbit. Since the star ismoving towards and away from the observer the radial velocity can measured using theDoppler shift of the light emitted by the star.

2.3.1 The Stellar Reflex orbit with a single planet

When analyzing the motion of a planet it is most simple in doing so when it is in the restframe of its host star, this is also known as the astrocentric frame. Figure 3.3 shows anexample of the astrocentric frame of a planet. The planet performs an orbit around its host

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Figure 2.3: Diagram based on Figure 1.14, page31 [2][17]

star with a semi-major axis, a, with an orbital period, P, and an orbital eccentricity (howmuch an orbit around a body deviates from a perfect circle), e.

The point at which the orbiting planet is closest to its host star is defined as the pericentreand the star is positioned at the focal point of the elliptical orbit. Using the cartesian co-ordinate system we center it on the focal point of the system which in this case is the star,figure 3.3 shows an example of this. We define the angle between the orbital plane and thesky plane as the inclination, i. We position the x axis along the line between the 2 points ofintersection between the orbital plane and the sky plane as seen by the observer. Positivevalues of x mean that the planet is moving towards the observer, we define γ as the pointin which the positive x axis intersect the orbital plane. We define θ as the angle known asthe true anomaly, θ measures the distance the planet has moved along its orbital from thepericentre. The orientation of the pericentre is defined by the angle ωOP which is measuredwith respect to γ.

Using these definitions we can define the velocity which has components in the x, y and zdirections. These can be shown as:

vx =− 2πa

P√

1− e2(sin(θ+ωOP)+ esinωOP)

vy =− 2πacos i

P√

1− e2(cos(θ+ωOP)+ ecosωOP)

vz =− 2πasin i

P√


(2.4)

Astronomers using this method do not observe the planet as they simply cannot, instead

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they observe the planets host star. Therefore they require analogus equations for the hoststars reflex orbit.

Only looking at the astrocentric frame means that the planet is the only object moving inthe system, while in fact the star is also moving. The star performs a reflex orbit around thebarycentre as shown in figure 3.2. Therefore the velocity v of the planet in its astrocentricframe(fig 3.2) is given by

v = vP,bary −v∗ (2.5)

In equation 2.3 the astrocentric velocity is defined as v, the velocity of the star performingits reflex orbit around the barycentre is defined as v∗. Using equation 2.2 we can see that:

MPaP = M∗a∗

Using vector notation and knowing that the position of the barycentre is fixed; then the onlyvariable is the distance the star and the planet are from the barycentre, which are propor-tionate to each other. We can then deduce that:

M∗r∗ =−MPrP

We have made one side of the equation negative due to the fact that the planet and the starare in opposite directions to the barycentre.

The by differentiating M∗r∗ =−MPrP with respect to time we get:

M∗v∗ =−MPvP,bary (2.6)

Now we have the relationship of the velocities of the planet and the star in the barycentricframe. Using equation 2.6 we can see that the orbit of the planet and the reflex orbit of thehost star around the barycentre are the same shape (ellipse), the only difference betweenthem is the size of their elliptical orbits.

We now rearrange equation 2.6 to get vP,bary as the subject of the equation.

vP,bary =−M∗v∗MP

(2.7)

We now substitute equation 2.7 into equation 2.5 to get:

v∗ =MP

MP +M∗v (2.8)

Looking at equation 2.8, v is defined as the astrocentric orbit velocity of the planet, as shownin equation 2.4. From the view of the observer the barycentre of the transiting exoplanetary

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system will have velocity that is non-zero, we define the velocity of the barycentre as V0.V0 is dependent on time, therefore as time progresses the velocity of the barycentre changesaswell. The change in velocity takes hundreds of millions of years and therefore is chosento be a constant due to the fact the change in V0 takes more time than the orbital period.

From the point of view of the observer, we can define the velocity of the stars reflex orbit as:

V = v∗+V0 (2.9)

Hence for the velocity of the stars reflex orbit factoring in the motion of a planet orbiting itwe get the components:

Vx =V0,x +2πaMP

(MP +M∗)P√

1− e2(sin(θ+ωOP)+ esinωOP)

Vy =V0,y +2πaMP cos i

(MP +M∗)P√


Vz =V0,z +2πaMP sin i

(MP +M∗)P√


(2.10)

The observed radial velocity is given by:

Vz =V0,z +2πaMP sin i

(MP +M∗)P√

1− e2(cos(θ+ωOP)+ ecosωOP) (2.11)

Part of this equation is a variable, and is shown as:

2πaMP sin i

(MP +M∗)P√

1− e2cos(θ(t)+ωOP) (2.12)

Since we know that cos(θ(t)+ωOP) can only go between −1 and 1, we can define the ampli-tude of the radial velocity variations as:

ARV =2πaMP sin i

(MP +M∗)P√

1− e2(2.13)

Using the Doppler shift we can measure the radial velocities by knowing what the specificfeatures that appear the in stars stellar spectrum are. We define ∆λ as the change in thewavelength from the effect of the Doppler shift, and we define c as the speed of light. Therelationship between the velocities, the change in wavelength and the speed of light is givenby:

∆λλ

=Vc

(2.14)

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2.3.2 Reflex radial velocity of multiple non-interacting planets

In the previous section we only considered a system involving a single planet and its hoststar, this is not always the case. To make an approximation of the observed radial velocitiesof a system with multiple planets we must assume that the planets do not alter each otherselliptical orbits. Hence combining all of the elliptical reflex orbits around the barycentre theapproximation to the observed radial velocity is as follows:

V =V0,z +n

∑k=1

Ak(cos(θk +ωOP,k)+ ek cosωOP,k) (2.15)

Here we define n as the number of the planets in the system, Mk as the mass of each planetand ak, ek, θk and ωOP,k as the instantaneous orbital parameters. We also define Ak as:

Ak =2πakMk sin i

MtotalPk

√1− e2

k

(2.16)

Ak includes Mtotal which is the mass of the entire system, so the mass of the sun added withthe mass of all the planets orbiting it. Therefore we have the amplitude of the reflex orbit ofthe star with multiple planets, ARV = Ak

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2.4 Transits

The Transit method is the main the main method of detection used in this project. The tech-nique is relatively simple as it uses the idea that when a planet crosses infront of its hoststar it absorbs the light being emitted. Therefore the method does not require a high levelof precision, hence the data can be collected using telescopes on the ground in observato-ries. A transiting giant planet such as a gas giant like Jupiter will show a dip of 1% of thelight emitted by its host star; terrestrial planets however cause a dip of 10−2% therefore thephotometric precision required to detect terrestrial planets is 10−4 [2], because of the preci-sion required it is impossible for observatories to detect terrestrial planets. Using groundbased telescopes Astronomers observe stars to detect the dip in the light being emitted, thenthe astronomers can deduce that an object is orbiting the star if the dip in the light occursperiodically.

Figure 2.4: Diagram of transit light curve as an onject crosses infront of its host star [20]

As we can see from figure 3.4 the intensity of light decreases as the object moves betweenthe observer and the star. The light curve can be separated into different sections of time, t:

• t < t1: Pre-Ingress - The planet is orbiting its host star but has not begun its transit.

• t1 � t � t2: Ingress - The planet has begun its transit and has moved passed the leftoutter edge to the inside of the star’s disc.

• t2 � t � t3: Ingress and Egress - The planet is in the middle of its transit and is movingpassed the centre of the star’s disc.

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• t3 � t � t4: Egress - The planet is coming to the final phase of its transit and is startingto pass across the right outer edge of the star’s disc.

• t4 < t: Post-Egress - The transit has ended and the planet has passed the right outeredge of the star’s disc.

Using the size of the dip in the star’s brightness we can estimate the fraction of star’s discbeing covered by the exoplanet.

∆FF

=R2

PR2∗

(2.17)

We define F as the Flux of the star and therefore we define ∆F as the change in the Flux

measured by an observer. The right hand side of the equation R2P

R2∗

is the ratio between theareas of the exoplanet and its host star. Therefore Equation 2.17 allows us to calculate andestimation of the planets size in terms of the size of the host star the planet is orbiting.

2.4.1 Geometric Probability of a Transit

A transiting exoplanet can be detected when its orbital plane is sufficiently close the the lineof sight of the observer. To be seen the disc of the exoplanet must cross infront of the disc ofits host star. The closest approach of the centre of the exoplanets disc and the centre of itshost star’s disc occurs at the inferior junction, this phase is when the planet is closet to theobserver [2]. The inferior junction is defined with φ = 0.0 and the distance between the twodiscs can be calculated using the smei-major axis, a, and the inclination, i. Hence:

d(φ = 0.0) = acos i (2.18)

Figure 2.5: Diagram of the geometry of a transit viewed from above and from the view of the observer[2]

The orbital inclination must satisfy equation 2.19 for the planet’s disc to occult its host star[2].

acos i � R∗+RP (2.19)

16


The Projection of the unit vector normal to the orbital plane onto the skyplane is defined ascos i. cos i can take a random value between 0 and 1, we can replace cos i with x. Thereforewe can work out the geometric transit probability:

geometric transit probability =number of orbits transiting

all orbits

=

∫ (R∗+RP)a

0 dx∫ 1

0 dx

(2.20)

Hence

geometric transit probability =R∗+RP

a≈ R∗

a(2.21)

From the equation we can see transits are more probable when the planet has a small orbitand a large host star.

2.5 Gravitational Microlensing

Gravitational Microlensing uses the lensing effect of the general relativistic curvature ofspacetime to detect exoplanets. This method is used to detect planets in regions of wherethere exists a dense cluster of stars. These dense clusters of stars are chosen because mi-crolensing requires the alignment of stars from the point of view of the observer. Thereforea dense cluster of stars provides a higher probability of the occurrence of star alignment. Thebest case study is the galactic bulge which is the dense region of stars at the centre of ourgalaxy, the Milky Way. Another bulge that has be subjected to this method is in the spiralgalaxy known as M31. Over 2000 microlensing events have been observed and only a few ofthem have yielded results of exoplanet detection around the foreground of the lensing star[2].

Figure 2.6: Diagram of showing the focusing of light passing through a cluster of stars [22]

17


Figure 2.7: Diagram of showing the method of detection through Microlensing [23]

The two diagrams show what occurs during a Gravitational Microlensing observation. Thefirst figure shows the how the light is focused as it passes through a cluster of stars. Thesecond figure shows the magnification curve we obtain through the microlensing technique,we observe an extra peak indicating that a planet is present.

18


Chapter 3

Observations and Construction of theTransit Light Curve

3.1 Clanfield Observatory

As part of my final year project I was required to make observations to collect transit pho-tometry data with the help of my mentors David Harris, Chris Priest and Steve Futcher.Since my mentors are members of the Hampshire Astronomical Society I was permitted touse the 24 inch telescope at Clanfield Observatory to collect data. The University has a closerelationship with the Hampshire Astronomical Society for the past years; this relationshiphas brought with it success for students requiring to make observations to obtain data fortheir projects. My mentors utilised their experience and knowledge to teach me the meth-ods required for identifying a suitable transit candidate and for using the 24 inch telescopeto obtain data to analyse.

Figure 3.1: Photo of the Clanfield Observatory site [?]

My initial aim was to organise andarrange a date and time to make anobservation at Clanfield Observa-tory to gain first hand experiencein the use of the telescope. Duringthe window of opportunity I wasable to make one observation withthe 24 inch telescope though thedata did not show any discernibledip in the transit light curve; how-ever the methods used to choose asuitable transit candidate and howto use the 24 inch telescope are ex-plained in this chapter.

The data used up to the AIP4Winsection is of the Transiting Exoplanet Qatar1b. Since previous students have already studiedQatar1b, for the purpose of this project I decided to use data of the Transiting ExoplanetCorot12 from the Exoplanet Transit Database (ETD) to construct a light curve using varioussoftware.

19


3.2 Transit Candidates

Before observations can begin a suitable candidate must be chosen to achieve good dis-cernible data. There are many factors that need to be addressed when choosing a transit, tofind a suitable exoplanet we must use the Exoplanet Transit Database.

Figure 3.2: ETD website showing the predictions of transits[24]

Figure 3.3: ETD website showing the predictions of transitsbut magnified [24]

Using the Exoplanet Transit Databasewe can obtain all the data requiredto find a predicted transit. The Pro-cess to do so is as follows:

1. Using the Hampshire Astro-nomical Society’s website[26]we obtain the latitude andlongitude of Clanfield Ob-servatory, which is longitude359◦ and latitude 51◦. We in-sert these coordinates on theETD website shown in figure4.1.

2. A main factor that can effectthe observation is weather, ifthe weather on a certain dayis cloudy an observation isout of the question, thereforereviewing the weather fore-cast is key. The optimumweather to make an observa-tion is clear skies, after look-ing at the weather forecasta date can be chosen whenthe skies are clear. Using theETD website you can click onthe most appropriate date tomake an observation to to ob-tain a list of all predicted tran-sits for that day.

3. When you have your chosendate you are then given a ta-ble of transits predicted forthat day. The table containsdata that will need to be anal-ysed to choose the best candi-date for observation.

4. Using figure 4.2 we can anal-yse the table of data. Themagnitude of star is indicated with the column marked V(MAG). For a transit to bedetected by the 24 inch telescope the magnitude of the star requires to be 11 magni-tudes or higher, anything lower and the transit is too dim to detect.

20


5. The Element Coordinates gives us the Right Ascension (RA) and the Declination (DE).For a transit to be detected by the 24 inch telescope it needs a declination value of 30◦

above the horizon from start to finish, this is so that atmospheric conditions and lightpollution do not corrupt the data. The position and phase of the Moon has to be takeninto consideration; if there is a full Moon then the data might become obscured due tothe light from the Moon. If the Moon moves across the area of sky being observed thedata will also become obscured. Therefore the right ascension must be analysed so thatan exoplanet can be chosen that when observed will not be affected by the Moon.

6. The column marked DEPTH(MAG) indicates the dip in the magnitude of the star fromthe transit. For a transit to be detected by the 24 inch telescope the DEPTH is requiredto be at least 15 millimagnitudes (0.015 MAG), this is due to the precision of groundbased telescopes which is detailed in chapter 2.

7. Looking at each row we can see the time each transit begins and ends. Each transitmust be observed 30 minutes before it begins and observed 30 minutes after it ends.The time it would take to travel to Clanfield Observatory and set up all the equipmentwould have to be calculated as well. Therefore we can calculate the time to start theobservation and end the observation, this would be subject to the availability of thementors.

After going through this process, hopefully a transit has been chosen to observe. Now the fi-nal step is to communicate your chosen transit along with the date and time to your mentorsand perform the observation if they are available and the weather stays clear.

21


3.3 Making Observations

Figure 3.4: Photo of an SBIG STL-1000E CCD Cam-era [27]

To make and observation of the suitabletransit candidate one needs three pieces ofvery important equipment, these are:

• CCD Camera: For this project we useda SBIG STL-1000E with the followingspecifications[27]:

1. Focal Length at 1 arcsecond perpixel: 195 inches

2. Total Pixels: 1.0 million3. Array 1024×1024 pixels4. Pixel Size: 24 micrometres5. Full Well Capacity: 200000e-6. Cooling: 2 stage thermoelectric,

water circulation, −40◦C belowambient with uncooled water, reg-ulated to +/- 0.1 degree (roughly32◦C air only). Further coolingmay be achieved by using watercooled below ambient and abovethe dew point.

• Telescope: For this project we used the24 inch telescope. This allows the exposure time to be longer which is optimal forobserving a transit over a long time.

• Laptop

When setting up all the equipment with my mentors, the process needs to be done with care.Therefore a series of steps can be created to ensure your observation does not encounter anydifficulties.

1. The clamps attached to the dome need to be undone so that the dome is free to rotate.

2. The cover on the telescope is required to be removed. This cover prevents condensa-tion accumulating on the telescope; if the telescope was not covered it would result inobservations that would be obscure and possibly damage the telescope in temperaturesbelow freezing in which the condensation turns to ice.

3. The clamps attached to the telescope are then undone to allow the telescope to movefreely.

4. The dome and the telescope are attached to stepper motors[17], this allows the processof following the transit to be automated. By simply using the the the Right Ascension(Qatar1b RA = 20h13m32s) and Declination (Qatar1b DE = +65◦9m43s) and inputtinginto the point targeting system console as shown in figure 4.6 the telescope will auto-matically position itself to observe that point of the sky.

22


Figure 3.5: Picture of the 24 inch telescope used for this project [26]

Figure 3.6: Photo showing the point targeting system console for the 24 inch telescope [26]

5. Then the CCD Camera is attached to the 24 inch telescope, this is then connected to alaptop which will store the images of the transit. The laptop also serves the regulatorfor the CCD Camera’s temperature. If the Camera’s temperature is too high, the pixelscan become saturated and this in turn will ruin any transit data collected. The noise inthe image is also affected by the temperature, the colder the temperature the less noisein the image. The laptop also allows us to control the exposure time, the length of timefor each exposure depends on the period of the transit you are observing.

6. The mirror is now uncovered.

7. The dome is then rotated to line up correctly with the 24 inch telescope, this is doneusing the dome’s motor control.

8. Every step is completed and we are ready to start taking images of the chosen transit.

23


3.4 Image manipulation using AIP4WIN

The data collected from the observation of the transit at Clanfield Observatory must beanalysed to obtain the data required to construct a transit light curve. To do this we mustuse AIP4Win which will convert all of the images we collected into quantifiable data. Thefollowing steps show how the images obtained by Thomas Stephens [17] were used withAIP4Win.

1. Once AIP4Win has loaded up we can begin to convert the images to the data we re-quire. Looking at the top toolbar, click on Measure, mouse down to Photometry andclick Multiple Images. Figure 4.7 shows this step clearly.

Figure 3.7: Screenshot of step 1 using AIP4Win [17]

2. A window labeled multiple image photometry should now have opened. There shouldbe a box next to Auto Calibrate, click on the box so it is ticked. Then click on select filesand a browser box should have opened. Search through the folders until you find theimages of your transit, these should be .FIT files. Then while holding the shift key clickon the first image and then the last image, this should select all the images. Once allthe images are selected, click the open button.


24


3. Now the first image should be displayed in a small window. Each of the images will beslightly blurred, this has been done deliberately during the observation phase so thatthe results from the star are averaged to produce improved readings. Now using thewindowed image we need to determine which star is the transit we observed. Usingan image of the target field we can compare it with the observational image to identifyour transit star, this can be done by simply looking for particular star clusters aroundthe star we want on the target field image. Then by looking for these star clusters onthe observational image we can determine our transit star. Once found right click themiddle of the transit star, now the transit star has been selected. Now two more starsmust be chosen so our transit star can be compared to them. The comparison stars musthave the same magnitude (brightness) as the transit star that was observed. Choosingthe comparison stars can be a difficult process and will require some experimentation toget it just right. Clicking on the settings tab of the Multiple Image Photometry windowyou can adjust the radii so that the aperture captures all the star light and the annulusexcludes starlight.


4. Once the previous steps are complete and the transit star has been found on the obser-vational image we can begin the image analysis. To start this process click the executebutton in the multiple image photometry window. When AIP4Win has gone throughall the images two graphs should appear like in figure 4.10. The V-C1 graph shows thedip in magnitude of the star we observed compared to the comparison star. The C1-C2graph shows the difference in magnitude between the two comparison stars. Whenlooking at the C1-C2 graph it is best to choose two comparison stars whose variation inmagnitude is on the green line, if this is not the case then new comparison stars needto be chosen.

25



5. Now that images have been turned into data we can use it has to transfered to anotherprogram to create the light curve. Firstly we click ’the save to file’ tab to save the datalog, save it as a text file. Then open excel and using excel open the folder containingthe data log, choosing all files allows us to see our text file. Loading the text file willbring up the ’Text Import Wizard’ window. Click on the option for the fixed width andclick next, now you can create columns in your data which will be imported into excell.When you have finished all the data will be on your excel spreadsheet with hopefullyall your data in their intended columns.


26


3.5 Light Curve Construction

Now that the data has been extrapolated from the images obtained from the observationalvisits to Clanfield Observatory, the construction of the light curve can begin. In this sectionI shall be using data collected from the Exoplanet Transit Database (ETD) since my observa-tional data did not provide any conclusive results.

3.5.1 Excel

Using the data for Corot2b as shown in the Appendix we can begin to construct the lightcurve using excel. The data on the ETD website is ranked on its quality, rank 1 is the best dataand will give us a light curve with a definitive dip in the magnitude. As the rank numberincreases the quality in the data worsens and the dip in the light curve is less discernible.For my project the data used has a quality rank of 1 to give more accurate results.

Figure 3.12: Screenshot of the Corot2b data from the ETD website [25]

Using the ETD website we simply choose data that has a quality rank of 1 as shown inFigure 4.12. Clicking on the data quality rank number a new tab should open with all thedata. After selecting all the data, copy it and paste it in a text file; once that is done save thetext file and follow the process in the section 4.4 to import it into Microsoft Excel. Once itis in Excel we can delete columns that are not needed to produce the light curve. Now weshould be left with 2 columns, the Julian Dates and the magnitude of light from the observedstar.

27


Figure 3.13: Screenshot of the formula used in Excel

1. We begin by zeroing the Julian Dates, this is done so when the data is plotted the graphwill begin at zero. As you can see from figure 4.13 we use absolute cell referencing totakeaway the beginning Julian Date from each of the Julian Dates.

2. Now that the Julian Dates start from zero they can now be converted from days tohours. This can be simply done by multiplying each cell in column D by 24.

3. We now need to create the characteristic dip in the flux, we firstly average out the first20-30 results in column F to calculate the average magnitude of the star. The equationfor the average magnitude of Corot2a is shown in cell I3 of figure 4.13.

4. Using the average magnitude we subtract it from each magnitude in column F, theequation can be seen in column G of figure 4.13.

5. Now the final phase is to get the flux, to do this we must use equation 3.2 where B1B is

the flux and ∆m is the change in magnitude. Figure 4.13 shows the equation for the fluxfor Excel.

∆m =−2.5log10B1

B(3.1)

To do step 5 we rearrange equation 3.1 to get:

B1

B= 10−

∆m2.5 (3.2)

Now we are ready to plot the light curve, using the time in hours on the x-axis and the fluxon the y-axis we get:

28


Figure 3.14: Excel graph showing the transit light curve for Corot2b

Now that the graph has been plotted we can experiment with Excel to try and fit a line ofbest fit. Using Excel we can create a graph with adjoining lines between each point. Thelight dip of the transit light curve can be seen clearly with this new graph which can be seenin figure 4.15 .

Figure 3.15: Excel graph showing the transit light curve for Corot2b with adjoining lines

Figure 4.16 adds a trend line to the graph, the trend line uses a polynomial of sixth degree.The trend line shows the dip in the flux of the transit light curve more definitively, the onlydownside with the trend line is that the ends of the polynomial curves downwards. Thisis not very representative of the transit light curve as the flux should level out and have aconstant magnitude before the transit passes in front of the star and after the transit passesin front of the star. From this graph we are however able to approximate the period of thetransit and the radius of the exoplanet which will be covered in chapter 5.

29


Figure 3.16: Excel graph showing the transit light curve for Corot2b with 5th polynomial trendline

Figure 4.17 shows the transit light curve but with a trend line added that averages the datafor every five points. Figure 4.18 shows a similar graph of the transit light curve but with atrendline that averages the data for every ten points. These two graphs do not significantlyhelp with the analysis of the data but add to the visual illustration of the transit light curve.

Figure 3.17: Excel graph showing the transit light curve for Corot2b with a trend line averaging thedata at every 5 points

30


Figure 3.18: Excel graph showing the transit light curve for Corot2b with a trend line averaging thedata at every 10 points

3.5.2 Minitab

Using minitab we can also achieve a line of best fit for our transit light curve.

1. After starting up minitab we open up our Excel worksheet. This is done by clicking on’File’ and then clicking on ’Open Worksheet’.

Figure 3.19: Step 1 using minitab

2. Now a new window should have opened, change the ’Files of type’ to Excel (* xls;*xlsx). Then find the folder with your excel spreadsheet and click on the file and openit.

31



3. Now we can begin to plot the data and produce a line of best fit. Click on ’Graph’ thenclick ’Scatterplot’. Select ’Simple’ scatterplot and click ’ok’. Click on ’X Variables’ rowone and click on the time of the transit measured in hours and click ’select’. This setthe X variable for the graph as the time, now doing the same for the first row for ’Yvariables’ but this time selecting the ’Flux’.


4. Click on ’Data View’ and then click the ’Smoother’ tab and click the circle marked’Lowess’. Fitting a lowess smoother to the scatterplot allows us to see the relationshipbetween the time and the flux without having to fit a specific model. Altering thedegree of smoothing changes the fraction of the total number of points used to calculatethe fitted values at each x-value. Altering the number of steps changes the number ofiterations of smoothing to limit the influence of outliers. To get the line of best fit I

32


chose 10 steps with a degree of smoothing of 0.2 and we can see the result in Figure4.22.

Figure 3.22: Transit Light Curve produced using minitab

33


Chapter 4

Analysis of Exoplanetary Systems

This chapter will discuss the various characteristics that can be found from the informationtaken from the transit light curve. The methods used will be explained and a running ex-ample using the Corot2b data from the ETD website we be shown. It must be noted that notevery single characteristic of the transit can be found using the transit photometry observa-tion method. We begin by analysing the characteristics of the star.

4.1 Characteristics of Host Star

The star for this project is Corot2a, it has the following characteristics:

• Spectral Type: G7V

• Mass (M∗): 0.97±0.06M�

• Radius: 0.902±0.018R�

• Metallicity Fe/H: 0±0.1

• Right Acension: 19h27m0.6496s

• Declination: +01◦23′01.38′′

• Magnitude V: 12.57

4.2 Orbital Period

The Orbital Period of a transit can be calculated using two different techniques. The first isby using the radial velocity method and the other is to use the time that has passed betweenobservations of your chosen transit. We define the time elapsed between observations as T,

34


the number of transits that occur a certain time period as N and the Orbital Period P; whichhas the following relation.

P =Telapsed

Ncycles(4.1)

Using the transit light curve we can determine orbital period transit by comparing the mid-point of each transit light curve from a set of continuous transit observations. The midpointof the transit can be estimated using the transit light curves we have already produced usingexcel and minitab.

Figure 4.1: Transit Light Curve produced using excel showing the estimated mid and end points oftransit [25]

Figure 4.2: Transit Light Curve produced using minitab showing the estimated mid and end pointsof transit

Now comparing the estimated mid point from our transit light curve with another transitlight curve using data of the same transit we can determine the orbital period knowing the

35


number of cycles that occurred between the two observations. We can do this by using dataobtained from the Exoplanet Transit Database.

Figure 4.3: Observation Data collected from the ETD website

Using the column labeled HJD(mid) we can obtain the Julian Date of which the midpoint ofthe transit occurred. We can also calculate the number of transit cycles that have occurredby using the column labeled epoch.

Ncycles = E poch45−E poch43= 1090−1086= 4cycles

(4.2)

Telapsed = HJD45−HJD43= 56137.40209−56130.43098= 6.97111days

(4.3)

P =Telapsed

Ncycles

=6.97111days

4cycles= 1.7427775days

(4.4)

Comparing the answer from our calculation with the actual orbital period stated on the ETDwebsite (P = 1.7429935days) there is an error of 0.000216days. To get a more accurate resultwe would have to compare more observations.

4.3 Exoplanet Dimensions

We derived a relation between the change in flux and the ratio between the radius of theexoplanet (RP) and the radius of its host star (R∗) in the second chapter. It is as follows:

∆FF

=R2

PR2∗

(4.5)

36


Using the Transit Light Curves produced by excel and minitab we can approximate thechange in the flux caused by Corot2b transiting its host star.

Figure 4.4: Transit Light Curve produced using excel showing the estimated change in flux

Using the Transit Light Curve of Corot2b I estimate the change in flux to be roughly 0.0155.The maximum and minimum points of the light curve can however be deduced since thetrend line is a sixth degree polynomial.

y(x) =−7×10−5x5 −0.004x4 +0.0345x3 −0.0795x2 +0.0399x+0.9968dydx

=−00035x4 −0.016x3 +0.1035x2 −0.1590x+0.0399(4.6)

Using dydx and setting it equal zero we can use maple to solve the equation to find the values

of x when y(x) is either a minimum or maximum point. Then using these x-values we canobtain the maximum value of flux and the minimum value of flux and therefore the changein the flux.

Figure 4.5: Calculations using maple16 to determine the change in flux of the transit

As we can see the change in flux according to maple is roughly 0.0342107818mag, using thedata from the ETD website we can actually see that the change in flux is 0.0322mag [25]

37


so we can see that the accuracy from modeling with excel quite accurate. Now using thetransit depth (0.0322mag) and the radius (0.902R� = 627341000m) of Corot2a [28] we cannow calculate the radius of the exoplanet.

∆FF

=R2

PR2∗

0.0332 =R2

P

6273410002

(4.7)

Therefore,

RP =√

0.0322×6273410002

= 1.125723177×105km= 1.61RJ

(4.8)

The actual radius of Corot2b is in fact 1.429± 0.047RJ [25]. The result I have calculated isclearly larger than that obtained from the ETD website; one possible explanation for thisresult is that Corot2b may have an enlarged atmosphere which absorbs light from the hoststar causing the area occulted to be larger [2].

4.4 Calculation of Semi-major Axis

Using Kepler’s third law we can derive the semi-major axis of our transiting exoplanetarysystem. This can be done by using the following relation:

a3

P2 =G(M∗+MP)

4π2 (4.9)

Where a is the semi-major axis, P is the orbital period, G is the gravitational constant, M∗ isthe mass of the star and MP is the mass of the planet. We can therefore rearrange to make athe subject:

a =

(G(M∗+MP)P2

4π2

) 13

(4.10)

We can neglect the mass of the planet since it only makes up a small fraction of the mass ofthe entire transiting exoplanetary system, therefore we can obtain an estimation of the semi-major axis by just using the mass of Corot2a. Using spectrography analysis and comparing

38


the spectral type of the star with the main sequence on the Hertzsprung-Russel Diagram,thethe mass of the star can be deduced. So we no have the following:

M∗ = 1.928893500×1030

G = 6.67×10−11

P = 1.7427775days = 1.505759760×105seconds

(4.11)

Now inputting these values into equation 4.10 we obtain,

a =

(6.67×10−11(1.928893500×1030)(1.505759760×105)

2

4π2

) 13

=

(2.917060638×1030

4×π2

) 13

= 4.196255338×109m= 0.0280AU

(4.12)

Using the ETD website we can see that the actual Semi-Major axis is 0.0281AU so the estima-tion we calculated is quite accurate.

4.5 Orbital Speed

The eccentricty of Corot2b’s orbit is 0 and we can simply use Kepler’s second law to calculatethe orbital speed of our exoplanet. We define v as the orbital speed, a = 4.196255338×109mas the semi-major axis and P = 1.505759760× 105s as the orbital period. We can now inputthese values into the following equation.

v =2πaP

=2π×4.196255338×109

1.505759760×105

= 1.750999767×105ms−1

(4.13)

4.6 Impact Parameter and Transit Duration

Using the previous data we have obtained we can now begin to calculate an approximationto the impact parameter. In figure 5.6 the impact parameter is noted with the letter b, and isdefined as ’the shortest distance from the centre of the disc to the locus of the planet’[2].

39


Figure 4.6: Diagram based on on figure 2.6 pg57 [2] [17]

Figure 4.7: Diagram based on on figure 2.5 pg56 where V and W are the points of intersection betweenthe parallel light and the planet’s orbit[2] [17]

To calculate the duration of a transit defines as Tdur, with an impact parametre value of b=0,we can use the following relationship:

Tdur = P× length of arc from V to W2πa

≈ P×2R∗2πa

≈ P×R∗πa

(4.14)

40


Using the values from the ETD website P, a and R∗ we can approximate the transit duration.

Tdur ≈41.8318440hr×6.273410000×108m

π×4.203760000×109m≈ 1.987114512hr≈ 119.2268707mins

(4.15)

From the ETD website we can see that the transit duration is in fact 136.8 minutes, there-fore our calculation gave a rough approximation to the actual value. We can also find theduration of the transit analysing the transit light curve from chapter 3. From now on wewill use the value of Tdur from the ETD website for better accuracy. Comparing the two Tdurvalues indicates that our assumption that setting the impact parameter b=0 was incorrect.Figure 5.8 shows the relation between the impact parameter, b, orbital inclination, i and thesemi-major axis, a.

Figure 4.8: Diagram based on on figure 3.2 pg93 showing the geometrical representation of the impactparameter[2] [17]

Figure 4.9: Diagram based on on figure 3.3 pg93 showing a geometrical representation to allow us toexpress the length, l, in terms of the impact parameter using Pythagoras’s Theorem[2] [17]

41


As you can see from figure 5.9 we can express the impact parameter in the following way:

b = acos i (4.16)

It can also be clearly seen that the hypotenuse, h, is equal to RP +R∗. We can therefore usePythagoras’s theorem to calculate the length, l:

l2 = (RP +R∗)2 − (acos i)2

l =√(RP +R∗)

2 −a2cos2i(4.17)

Figure 4.10: Diagram based on on figure 3.4 pg94 showing a geometrical representation of the exo-planet going from point A to point B, to give a triangular shape.[2] [17]

Using figure 5.10 we can observe that the distance along the straight line between points Aand B is equal to 2l. As the planet orbits the star it covers a distance of 2πa therefore as itmoves along its orbit between A and B it covers an arc length of αa where the angle α is inradians. Using the triangle formed by the centre of the star and the points A and B we candeduce that the length, l, has the following relation:

sin(α

2

)=

la

(4.18)

Since α is measured in radians we can simply write the Tdur as the following:

Tdur = Pα2π

=Pπ

sin−1(

la

)(4.19)

42


Using equation 4.17 we can substitute the value for l in to equation ?? so that we now havethe inclination, i, included in the equation. Then we can input the values for P, RP, R∗, Tdurand a.

Tdur =Pπ

sin−1

√(RP +R∗)

2 −a2cos2i

a

8208s =1.5059463841×105s

π

× sin−1

√(9.990281900×107m+6.273410000×108m)

2 − (4.203760000×109m)2cos2i

4.203760000×109m

8208s = 47935.76220sin−1

(√5.288835723×1017 −1.767159814×1019cos2i

4.203760000×109

)

0.1712291538 = sin−1

(√5.288835723×1017 −1.767159814×1019cos2i

4.203760000×109

)

0.1703936562 =

√5.288835723×1017 −1.767159814×1019cos2i

4.203760000×109

7.162940362×108 =√

5.288835723×1017 −1.767159814×1019cos2i

5.130771463×1017 = 5.288835723×1017 −1.767159814×1019cos2i

−1.580642600×1016 =−1.767159814×1019cos2i

0.0008944536807 = cos2i0.02990741849 = cos i1.540884448rad = i

1.540884448× 180π

= i

88.28617555◦ = i(4.20)

Looking at the ETD website we can see that the inclination of the orbit is 87.84◦ which is veryclose to the value we calculated. Now using the actual inclination we can now use equation4.16 to work out the impact parameter, b.

b = acos i

b = 4.203760000×109 × cos(87.84π

180)

b = 1.584404821×108mb = 0.2525587872R∗

(4.21)

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4.7 Mass of Exoplanet and Eccentricity of Orbit

Using the radial velocity method we can determine an accurate value of the mass of theexoplanet. Now we can use equation 2.13 from chapter two to calculate the amplitude ofthe radial velocity using the mass of the exoplanet, MP, the mass of the star, M∗, the periodof the transit, P, the eccentricity of the orbit, e which is assumed to be 0, and the inclinationof the orbit, i. We can now insert these values into the following equation:

ARV =2πaMP sin i

(MP +M∗)P√

1− e2

ARV =2π×4.203760000×109 ×6.283273137×1027 sin1.533097215

(6.283273137×1027 +1.928893500×1030)1.505946384×105√

1−02

ARV ≈ 569.0689375ms−1

(4.22)

Figure 4.11: Diagram showing the radial velocity of Corot2b corresponding to its phase[29]

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Chapter 5

Limb Darkening

5.1 Geometry of the Transit

We can picture the geometry of the transit as two intersecting discs, the exoplanetary discand the stellar disc. The area of the stellar disc is occulted during the transit of the exo-planetary disc. Using the geometry of figure 6.1 we can calculate the area of the stellar discocculted by the exoplanetary disc during its egress and ingress.

Figure 5.1: Diagram showing the geometry of the transit using partially overlapping discs of the starand the exoplanet, based on figure 3.13 pg105[2][17]

Now using figure 6.1 we can calculate the eclipsed area, Ae which is the area of the intersec-tion of the two discs. Looking at the diagram we can see that Ae is equal to twice the areaof shape (e), we can then calculate the transit depth by taking Ae away from the area of the

45


stellar disc.

Ae = 2× [(d)− (c)]= 2× [(d)− [(a)− (b)]]= 2× [(d)− (a)+(b)]

(5.1)

We can define the ratio between the radius of the stellar disc and the radius of the planetarydisc as equal to p, so we have

p =RP

R∗(5.2)

Now we can calculate the area of each of the shapes using equation 5.2 to get the following:

(d) = πR2P ×

α1

2π=

p2R2∗α1

2

(b) = πR2∗ ×

α2

2π=

R2∗α2

2

(a) =R∗ ×ξR∗ sinα2

2=

ξR2∗

2sinα2

(5.3)

Using (a) from the diagram we can evaluate the angles α1 and α2 by using the cosine rule.

cosα1 =p2 +ξ2 −1

2ξp

cosα2 =1+ξ2 − p2

2ξ

(5.4)

We can then obtain the following using Pythagoras’s Theorem

sinα2 =

√4ξ2 − (1+ξ2 − p2)

2

2ξ(5.5)

We can now express equation 5.1 using these results to get the following

Ae = 2×

p2R2

∗α1

2+

R2∗α2

2−

ξR2∗

√4ξ2 − (1+ξ2 − p2)

2

2ξ

Ae = R2∗

p2α1 +α2 −

√4ξ2 − (1+ξ2 − p2)

2

2

(5.6)

46


The area of intersection between the two discs is in terms of ξ which is defined as the param-eterised distance from the centre of the star to the planet. The distance between the centreof the star and the planet is defined as s(t) and is in terms of the stellar radius R∗ and ξ.

s(t) = ξR∗ (5.7)

We can calculate s(t) using time, t, the orbital parameters, and the orbital angular speed, ω.ω is defined in terms of the orbital period, P, as shown in equation 5.8.

ω =2πP

(5.8)

Figure 5.2: Diagram showing the transit geometry face on and from the viewpoint of the observer,based on figure 3.12 pg104[2][17]

Using figure 6.2 we can see how the exoplanet moves around its host star in a circular orbit.It can be seen that from the observers perspective the orbit looks elliptical due to the angularinclination of the orbit. Since the component of the displacement s(t) is foreshortened theactual observed displacement is given by acos icosωt. Therefore to calculate s(t) we needto use the actual observed displacement along with Pythagoras’s theorem, in doing so weobtain

s2(t) = (asinωt)2 +(acos(i)cos(ωt)2

s(t) = a[sin2ωt + cos2(i)cos2(ωt)]12

(5.9)

To generalise Ae we must express it in its two cases in terms of our variables ξ and p:

1. Planetary Disc falls withing the stellar disc.

2. Planetary Disc falls outside the stellar disc.

47


Using equations 5.7 and 5.9 we can express ξ in terms of time to obtain Ae(t). For case 1where the planetary disc falls outside the stellar disc, if the distance between each of thedisc centres exceeds the sum of their radii the we get the following

Ae =

0 if R∗+RP < s,0 if R∗(1+ p)< ξR∗,0 if 1+ p < ξ

(5.10)

For case 2 where the planetary disc falls inside the stellar disc, if the distance between eachof the discs centres is less than the difference of their radii then we get the following

Ae =

πR2P if R∗ −RP ≥ s,

πp2R2∗ if R∗(1− p)≥ ξR∗,

πp2R2∗ if 1− p ≥ ξ

(5.11)

Bringing case 1 and case 2 together we can create a generalised form

Ae =

0 if 1+ p < ξ,

R2∗

(p2α1 +α2 −

√4ξ2−(1+ξ2−p2)

2

2

)if 1− p < ξ ≥ 1+ p,

πp2R2∗ if 1− p ≥ ξ

(5.12)

Therefore Ae also known as the eclipsed area is a function of R∗, p and ξ(t).

5.2 Light lost during the Transit

The total flux emitted by the stellar disc is related to the intensity, I. The integral of theintensity over the surface area of the star is equal to the total flux; this is achievable whenthe intensity of the star is equally distributed. For our analysis we need to restrict it to axiallysymmetric intensity distributions [2], therefore I = I(r′), where r′ is the measurement fromthe centre of the stellar disc as shown in figure 5.3.

Therefore it can be shown that

F =∫

discI(r′)dA =

∫ r′=R∗

r′=0I(r′)2πr′dr′ (5.13)

48


Figure 5.3: Diagram showing the intensity of light distributed across the stellar disc, based on figure3.14 pg109[2][17]

Using a stellar disc with uniform brightness we can say that the intensity therefore is con-stant, I = I0 for the entire disc. Therefore we can simplify equation 5.13 into the following

F =∫ r′=R∗

r′=0I02πr′dr′

= 2πI0

∫ r′=R∗

r′=0r′dr′

= 2πI0

[r′2

2

]r′=R∗

r′=0

= πI0R2∗

(5.14)

We can now write an expression for ∆F in terms of time. ∆F is the flux that is lost when thestellar disc is occulted by the exoplanet during its transit phase. Therefore to obtain ∆F wemust integrate the intensity, I(r′), over the occulted area to get the following equation

∆F =∫

occulted areaI(r′)dA (5.15)

Since the brightness is uniform we can once again let I = I0, therefore we get

∆F = I0

∫

occulted areadA (5.16)

Looking back at the previous section we can see that the integration is in fact equal to the Ae

49


(ecclipsed area), hence our equation becomes

∆F = I0Ae (5.17)

So for the simple case where the intensity is uniformly distributed across the star the cor-responding change in flux is I0Ae. On the other hand we need to take into account thenormalised axial coordinate, r, this is used for the case where the distribution of intensity isnon-uniform or the limb darkened case. We therefore define r so that the centre of the stellardisc starts at zero and the limb is at 1, hence we get the following

r =r′

R∗(5.18)

Figure 5.4: Diagram showing the eclipsed area Ae made of series of partial annuli, based on figure3.15 pg111[2][17]

Figure 6.4 clearly shows us that the eclipsed area within the stellar disc of radius rR∗ is equalto the same area that would be eclipsed if the star was radius rR∗ than radius R∗. Hencewe can use the equation we derived early for the eclipsed area, Ae, but with a change ofvariables so that we can integrate over the axially symmetric disc. The shaded area in figure6.4 shows the additional eclipsed area when increasing r by an amount dr, we define thisadditional area as dA(r). Therefore the shaded area can be expressed as the following

dA(r) =dAe

dr′dr′ (5.19)

Now we can write the expression using the star in figure 6.4 with radius rR∗. Therefore wewill need to change our variables for the function Ae which previously used p and ξ. Sincep = RP

R∗changing the radius changes p to p

r . ξ has a similar change as p, ξ changes to ξr .

50


Therefore using equation 5.12 the eclipsed area of the star is as follows

∫ r′=rR∗

0

dAe

dr′dr′ = Ae

(rR∗,

pr,ξr

)

= r2Ae

(R∗,

pr,ξr

) (5.20)

Comparing equations 5.19 and 5.20 we can see that the integrand on the left hand side ofequation 5.19 is equal to that of the right hand side of equation 5.20. Hence we get

dA(r) =ddr

[r2Ae

(R∗,

pr,ξr

)]dr (5.21)

Therefore combining equation 5.21 with equation 5.15 we can get a general expression for∆F Hence for the limb darkened axially symmetric case we get

∆F =∫ r=1

r=0I(r)A(r)

=∫ r=1

r=0I(r)

ddr

[r2Ae

(R∗,

pr,ξr

)]dr

(5.22)

Therefore using equation 5.22 we can calculate the exact shape of a transit light curve usingthe limb darkening law, sizes of star and planet along with the orbital parameters P, a and i.

5.3 Laws for Limb Darkening

The surface of stars along the main sequence are comprised of super hot gases also knownas plasma.The light that is emitted from the star comes from various levels in the stars at-mosphere. The probability that the photon emitted can escape within a particular layer isdependent on the optical depth of that particular layer. For a given frequency, ν we cancalculate the optical depth, τν, as being equal to the integral of the opacity, κν, multiplied bythe density, ρ(s) along the path taken by the photon. Hence we have

τν =∫ ∞

Xρ(s)κνds (5.23)

The opacity and the optical depth are both dependent on the frequency of the radiation.Therefore particular physical depths will have different optical depths which depend on thefrequency of the radiation. The probability that the photon will not be absorbed or scatteredas it travels alongs its path is e−τν , hence the ratio between the emitted intensity, Iemitted , andthe emergent intensity, Iemergent , is as follows

IIemitted

= e−τν (5.24)

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Figure 5.5: Diagram showing the cross section of a star with depths and directions of emergingphotons, based on figure 3.8 pg99[2][17]

As you can see on figure 6.5, a photon that is emerging from the stellar disc travels at anangle γ from the the radius vector. The two photons in figure 6.5 have been emitted atthe same depth in the stellar atmosphere. For the photon emitted in the limb of the stellaratmosphere to reach the observer it must pass through a larger path length, s, of the stellaratmosphere. Therefore for photons emitted at a depth of h in the stellar atmosphere, thelength of the path can be calculated using the following equation, where µ = cosγ.

s ≈ hcosγ

=hµ

(5.25)

Since the optical depth for a particular physical depth grows larger towards the limb ofthe star, a small fraction of the photons emitted from the stars limb will escape and reachthe observer. Therefore as you look at the limb of a star it appears to be dimmer than thecentre of the stellar disc. Photons that are emitted radially outwards mean they travel ashorter path. This also means that the probability of photons escaping from deeper in thestellar atmosphere towards the centre of the stellar disc is higher. Hence as you observe thestar, the edges appear redder than the centre. This is because photons from deeper layersapproximate a bluer black body spectrum due to the being hotter.

The effect of the limb of the star being redder and dimmer is known as limb darkening. Thisis dependent on opacity and emissivity at each depth within the stellar atmosphere whichin turn is dependent on the wavelength and the thermodynamic properties of the stellaratmosphere at each point. Hence limb darkening is dependent on the spectral type and

52


composition of the star[2].

The following laws are models of what happens at the limb of a star, since the only star wecan observe with great detail is our sun these laws are not precise. Using these laws we cansimulate the data to obtain the finest representation.

1. The Linear Limb Darkening Relationship: We define u as the limb darkening coefficientwhich governs the the gradient of the intensity drop between the centre and limb of thedisc.

I(µ)I(1)

= 1−u(1−µ) (5.26)

2. The Logarithmic Law: similar to linear law but uses two limb darkening coefficients u1and νl .

I(µ)I(1)

= 1−u1(1−µ)−νlµln(µ) (5.27)

3. The Quadratic Law:

I(µ)I(1)

= 1−uq(1−µ)−νq(1−µ)2 (5.28)

4. The Cubic Law:

I(µ)I(1)

= 1−uc(1−µ)−νc(1−µ)3 (5.29)

Laws 2, 3 and 4 are more complex relationships but give more flexibility to allow empircaldata to be fitted more closely. The drawback with these three laws is having two coefficientsthat need to be determined or arbitrarily fixed.

53


Chapter 6

Modelling of the Transiting ExoplanetarySystem

6.1 Exoplanetary System Properties

The star, Corot2a, has the following characteristics:

• Spectral Type: G7V

• Mass (M∗): 0.97±0.06M�

• Radius: 0.902±0.018R�

• Metallicity Fe/H: 0±0.1

• Right Acension: 19h27m0.6496s

• Declination: +01◦23′01.38′′

• Magnitude V: 12.57

The exoplanet, Corot2b has the following properties:

• P = P = 1.7429935days

• RP = 1.429RJ

• a = 0.0281AU

• v = 1.750999767×105ms−1

• Tdur = 136.8mins

• i = 1.533097215rad = 87.84◦

• b = 0.2525587872R∗

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6.2 Maple Model factoring in Limb Darkening

Using the characteristics of the exoplanet and its host star that we collected we can nowbegin to produce a visual representation of the transiting exoplanetary system. Using Maplewe aim to generate a theoretical light curve presented over the data retrieved from the ETDwebsite for Corot2b. After generating the theoretical light curve we can then implement thelimb darkening laws that were derived in chapter 5 to produce theoretical light curves thatfactor in limb darkening.

To begin our maple program we must first use the ’restart’ command to reset all the variablesand implement the ’plottools’ and ’plots’ packages.

Now we scale down the radius of the star (Corot1a) so that it is equal to one, then we scalethe radius of the planet so it is in units of R∗. We then input and define these variables alongwith the impact parameter, b, in ourMaple program. Then we calculate x which is defined asthe the starting point of the exoplanet on the x-axis, which is calculated using Pythagoras’stheorem.

Next we begin to create a basic visual representation of the transiting exoplanetary systemusing Maple. We set the centre of the star at [0,0] with radius Rstar = 1 and make the linearound the circumference of the star thicker and colour it yellow. Then we set the centre ofthe exoplanet at [−x,b] = [−1.131401810,0.2525587872] with radius rplanet = 0.1592480310and make the line around the circumference of the planet thicker and colour it green.

No we can begin to create an animation of the transit using Maple. We start by calculatingthe distances between the centre of each of the discs, the maximum distance between themwe define as dmax where the exoplanet shall begin its transit. Then we calculate the maxi-

55


mum distance from the centre of the exoplanet and the y-axis suing Pythagoras’s theoremand define this as ymax. Then we can begin to set up the animation. We begin by settingA as the varying x coordinate using the a sequence that involves ymax and the variablei = 0,1,2, ...,20. The sequence will begin at ymax and then will increase in sizes of 1

10ymax,therefore setting the number of steps to 20 means that the exoplanet will complete its ani-mation and end up on the opposite side of the y-axis.

Now we can proceed to define a function for the theoretical light curve which we can use toproduce a light curve for the case where there is no limb darkening. Therefore we requirethe radius of the exoplanet and star as well as a variable which we define as y. For thisformula we require several other variables to evaluate the transit light curve which we thendefine for this function using local.

We define rmax as the maximum distance of d when the exoplanet disc is about to intersectwith the stellar disc, hence rmin is the minimum radius. We define dmax similarly to whatwe had defined previously but we change it so it is in terms of rmax.

Using an ’if’ command we can describe the value of dmin which is the minimum distance

56


between the centre of each disc. If y (impact parameter) is less than rmin (the minimumdistance between the two centre points) then we calculate dmin using Pythagoras’s theorem,if this is not true then dmin = 0.

We can now calculate A which is the eclipsed area produced by the two intersecting discsby using the formula for the circular segment [30].

The variation in the observed light we would see by the area of the stellar disc minus theeclipsed area we define as B.

Now we can input the values of the exoplanet and the stars radius into our equation B whichwe define as B1. This process leaves us with the function we need to plot our transit lightcurve.

The following command calls on the function of for our transit light curve using the vari-ables we have obtained.

Now that we have our light curve we can now call on our data collected from the ETDwebsite and plot the light curve over the data. Using our excel file we copy the ’time inhours’ and ’flux’ columns into a .txt file. This can then be called on by Maple by using the’readdata’ command by entering the file name and the number of columns.

We then need to scale and translate the data from our excel file to fit our transiting lightcurve. We begin by translating the our data so that the mid point of the transit is at x = 0.Now we have to scale the data to fit the light curve more accurately. Then we can alter theaxis values since to fit in with the beginning and end of the transit.

57


Now we begin to try and implement the limb darkening laws using Maple. We begin bydefining a new function similar to the one used to produce the previous transit light curve.Therefore we start by defining our variables that our function requires by using the ’local’command.

We then begin to define all the parameters and values that were in the previous code, hencewe define ds as previously stated before.

Looking back at the previous chapter the limb darkening law requires a coefficient, u. Wecan then input this coefficient into our linear limb darkening law which, we define this lawas i which requires the value ds.

To calculate the total intensity of the stellar disc we must integrate over the stellar disc, whichwe define as F. We the calculate I(1) as shown in the previous chapter on limb darkening.Unlike before we do not integrate over the entire eclipsed area but a midpoint in the eclipsedarea is used.

We once again define rmax, rmin, dmax and dmin as we did previously for our transit lightcurve function.

58


The eclipsed area, A, is calculated similarly as for the transit light curve function. We alsodefine A1 as we did before inputting the radius of the exoplanet and star.

The difference in intensity of the star is defined as DF1, which is the eclipsed area multipliedby the linear limb darkening mid point I(1)

We can now proceed to construct the plots that are required to build our model factoring inthe limb darkening of the light curve. We define p1L as the plot showing the characteristicdip in the light curve. We set the light curve to begin and end with a y-value of 1, theminimum point of the dip is set from −dmin to dmin which is defined by p5L where themaximum eclipsed area is equal to the area of the planets disc.

We define Ptheory to call on the function of the light curve using the values of the exoplanetand star radius and the impact parameter.

We once again import our data from our excel file by using the .txt file we created earlier.We then scale and translate the data to fit the light curve, for this we use the same values weused previously.

59


The new graph now shows the expected transit light curve along with a series of plots withlimb darkening values to best represent our data.

6.3 NAAP Transit Simulator

Using the characteristics of our transiting exoplanetary system we can use the NAAP onlinesimulator to construct the theoretical light curve we would observe not factoring in limbdarkening. The simulator can then process the data and produce a simulation model of the

transiting exoplanetary system as viewed from the side.

Figure 6.1

From the simulation we can see the transit depth simulated without the effects of limb dark-ening. We can also observe the effects that noise has on the transit light curve. Also weare able to alter the sizes of the planet and star and observe the changes in the light curvefrom these adjustments. From altering the mass of the planet we can see that this has aneffect on the eclipsing time and the orbital period. Therefore if the mass of the exoplanet

60


was reduced we would see a orbital period would increase hence therefore increasing themass would decrease the orbital period. It can also be noted that making adjustments to theradius of the planet has an effect on the transit depth, we have already seen this in chapter4, hence large planets will result in a larger transit depth and smaller planets will have asmaller transit depth. Altering the eccentricity of the exoplanets orbit effects the length ofthe transit period as well as the time the planet spends in at mid-transit. By changing thesize of the host star leads to similar effects as that of altering the planet’s size.

Figure 6.2

We can see from figure 7.2 the effect of noise on the transit light curve. From this we cancompare if we had any original data to assess its accuracy. I have chosen to set the noiselevel as 0.0014 to approximate the data on Corot2b retrieved from the ETD website.

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Figure 6.3

We can see from figure 7.3 the representation of the radial velocity curve for Corot2b. Tocalculate the curve we require various planetary and stellar characteristics as we did for thetransit light curve simulator. From the radial velocity curve we can gain a glance at the orbitwe would expect the exoplanet to have and the effect that the exoplanet has on its host staras they both orbit the barycentre of the exoplanetary system. We can also see a geometricalvisualisation of the transiting exoplanetary system from different perspectives.

6.4 Autodesk Maya

The following is a walkthrough I have created for the creation of the three dimensionalmodel of the Corot2 planetary system. After opening Autodesk Maya please follow theseinstructions, if required there are online tutorials for guidance on sites such as ’Youtube’.

62


63


The end result is the following model which can be seen in the following five figures aswell as the CD that contains video footage of the model performing the transit from variousangles.

64


Figure 6.4: Exoplanetary Sytem as viewed from Corot2b

Figure 6.5: Exoplanetary Sytem as viewed from Corot2a

65


Figure 6.6: Exoplanetary Sytem as viewed from above

Figure 6.7: Exoplanetary Sytem as viewed from the side with exoplanet moving behind the star as itfollows its orbit

66


Figure 6.8: Exoplanetary Sytem as viewed from the side as the exoplanet performs its transit

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Chapter 7

Conclusion and Recommendations

7.1 Detection Methods and Observations

Analysing the various detection methods that can be implemented to discover new exo-planetary systems revealed that the methods are all on equal standing. It can be seen thatthe different methods each had better accuracy for detecting specific types of exoplanet, forexample the transit method that was implemented in this project is best used to find largeexoplanets orbiting small stars. The dip that we see from the transit light curve producedby this method is deep enough to counteract the noise created by atmospheric conditions.We can also see that just using the transit method alone we are unable to obtain all the char-acteristics of the exoplanetary system that we require for modeling. Researching into theradial velocity method we can see that it has the ability to find multiple planets orbiting asingle star, though it does have its limitations. The radial velocity method requires that thestar in the exoplanetary system be bright enough that you can measure the Doppler shift inthe stars light, this means that the star requires to be closer to the observer to increase bright-ness or be of significant size and mass. Using both the transit method and the radial velocitymethod together we are then able to determine the characteristics of the exoplanetary sys-tem. Using the astrometry requires a high level of precision and the constant monitoringof the star you are observing. The advantage of this method is that it can too detect mul-tiple planets orbiting around a single star. This method does require however sufficientlylarge enough planets to exert a pull on the star, therefore terrestrial planets are unlikely tobe discovered by this method. The use of direct imaging has progressed further and hasbeen found to be very useful in studying our own solar system and its development. Thismethod however is incapable of detecting planets beyond our solar system therefore it is notimplemented in searcing for terrestrial planets. Due to Microlensing’s occurence factor it isquite unreliable. Therefore at this point in time we can not conclude that there is one sin-gle method of detection that can determine all characteristics of an exoplanetary system butrequires the combination of methods such as the transit and radial velocity method. Usingboth of these methods combined can give us the full picture of the exoplanetary system andthe basis for a model.

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7.2 Light Curve Construction and Limb Darkening

This project has shown the methods of observation and data analysis. Making my own ob-servations has been quite unsuccessful due to many factors such as weather, hence I haveused data from the Exoplanet Transit Database to supplement this. Using the data obtainedfrom the ETD website we were able to generate a transit light curve on excel, minitab andmaple. We were also able to use the data collected and the transit light curves to determinethe exoplanetary systems characteristics to a good degree of accuracy. Our research intolimb darkening showed that the expected light curve is not always what we obtain fromobservations. We also discovered that the intensity of the star is not distributed across theentire surface equally, this leads to the limbs of the star to appear darker and dimmer. There-fore the bottom of the light curve becomes rounder dure to limb darkening. Using geometrywe were also able to show how we can calculate an approximation to limb darkening usingthe limb darkening laws which could then be applied to out transit light curve to model thelimb darkening effect.

7.3 Modelling

Using maple we were able to produce an expected light curve which could then be plottedagainst the data we collected. We were also able to generate a simplistic two dimensionalmodel of the transit for Corot2b, this gave a visual representation of our exoplanetary systemas viewed from the side. Using a simplified version of the limb darkening laws we were ableto implement them to create a set of light curves that factored in limb darkening. We plottedthese against our data and could see that they fitted the data more closely. Using AutodeskMaya we were able to generate a three dimensional model that no previous student has everdone before. It has produced a superb visual representation of the exoplanetary system,though some of the characteristics are not completely accurate due to having to be scaleddown.

7.4 Recommendations for future investigation

Future students could look at taking multiple sets of images or data from the ETD websiteto compile them together to create a better representation of the light curve with more datapoints. This would then lead to obtaining better characteristics for the transiting exoplane-tary system they are studying. Future student could also look into furthering the work withthe Maple program by improving the limb darkening laws by using algorithms to producea better light curve factoring in limb darkening. Future students could also try and calculatethe orbital period directly by using the transit predictions from the ETD website. Repre-senting the error of the light curve can be quite problematic but is an area of research thata future student could undertake. The theoretical habitable zone calculations have recentlybeen looked into due to that they do not always give an accurate representation of the habit-able zone as we can see in chapter 1. Therefore a student could look into the habitable zonecalculations and how they could be improved to provide a more accurate answer. Futurestudents could look into improving the Autodesk Maya model that I have created, a stu-dent could try and model limb darkening of the star or try and add in orbital rotation of theplanet for example.

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Bibliography

[1] wasp17. Nasa/Hubble (2012). Huge new planet tells of game of planetary billiards. Re-trieved January 22, 2013, from http://www.stfc.ac.uk/News+and+Events/10582.aspx

[2] Haswell, C. A.(2010). Transiting Exoplanets. Cambridge: University Press.

[3] Norton, A. J. (2004). Observing the Universe: A Guide to Observational Astronomy andPlanetary Science. Cambridge: University Press.

[4] Bruce, G. (2007). Exoplanet Observing for Amateurs http://brucegary.net/book_EOA/EOA.pdf. Hereford, Arizona, USA: Reductionist Publications.

[5] Wikipedia (2013). Extrasolar Planet. Available: http://en.wikipedia.org/wiki/Extrasolar_planet. Last accessed 22th January 2013.

[6] Leake, J. (2010, April 26). Extraterrestrials are almost certain to exist saysBritish theoretical physicist Stephen Hawking. Daily Telegraph. RetrievedDecember 10, 2012 from http://www.dailytelegraph.com.au/news/weird/extraterrestrials-are-almost-certain-to-exist-says-british-theoretical-physicist-stephen-hawking/story-e6frev20-1225858062554.

[7] Rincon, P. (2012, October, 15). Planet with four suns discovered by volunteers.BBC NEWS science and environment. Retrieved from http://www.bbc.co.uk/news/science-environment-19950923

[8] Wikipedia (2013). Planetary Habitability. Available: http://en.wikipedia.org/wiki/Planetary_habitability. Last accessed 25th January 2013.

[9] Wikipedia (2013). Photosphere. Available: http://en.wikipedia.org/wiki/Photosphere. Last accessed 25th January 2013.

[10] Wikipedia (2013). Main Sequence. Available: http://en.wikipedia.org/wiki/Main_sequence#Formation. Last accessed 25th January 2013.

[11] Wikipedia (2013). Habitable Zone. Available: http://en.wikipedia.org/wiki/Habitable_zone#Circumstellar_habitable_zone. Last accessed 26th January 2013.

[12] Wikipedia (2013). Super Earth. Available: http://en.wikipedia.org/wiki/Super-Earth. Last accessed 26th January 2013.

[13] Wikipedia (2013). Coronagraph. Available: http://en.wikipedia.org/wiki/Coronagraph. Last accessed 28th January 2013.

[14] Wikipedia (2013). Fomalhuat. Available: http://en.wikipedia.org/wiki/Fomalhaut.Last accessed 28th January 2013.

70

http://en.wikipedia.org/wiki/Fomalhaut

http://en.wikipedia.org/wiki/Coronagraph

http://en.wikipedia.org/wiki/Coronagraph

http://en.wikipedia.org/wiki/Super-Earth

http://en.wikipedia.org/wiki/Super-Earth

http://en.wikipedia.org/wiki/Habitable_zone#Circumstellar_habitable_zone

http://en.wikipedia.org/wiki/Habitable_zone#Circumstellar_habitable_zone

http://en.wikipedia.org/wiki/Main_sequence#Formation

http://en.wikipedia.org/wiki/Main_sequence#Formation

http://en.wikipedia.org/wiki/Photosphere

http://en.wikipedia.org/wiki/Photosphere

http://en.wikipedia.org/wiki/Planetary_habitability

http://en.wikipedia.org/wiki/Planetary_habitability

http://www.bbc.co.uk/news/science-environment-19950923

http://www.bbc.co.uk/news/science-environment-19950923

http://www.dailytelegraph.com.au/news/weird/extraterrestrials-are-almost-certain-to-exist-says-british-theoretical-physicist-stephen-hawking/story-e6frev20-1225858062554



http://en.wikipedia.org/wiki/Extrasolar_planet

http://en.wikipedia.org/wiki/Extrasolar_planet

http://brucegary.net/book_EOA/EOA.pdf

http://brucegary.net/book_EOA/EOA.pdf

http://www.stfc.ac.uk/News+and+Events/10582.aspx


[15] Wikipedia (2013). Methods of detecting extrasolar planets. Available: http://en.wikipedia.org/wiki/Methods_of_detecting_extrasolar_planets#Astrometry. Lastaccessed 30th January 2013.

[16] Wikipedia (2013). Astrometry. Available: http://en.wikipedia.org/wiki/Astrometry.Last accessed 30th January 2013.

[17] Stephens, T. (2012). Observation and Modelling Study of Exoplanet Qatar1b. RetrievedJanuary 22, 2013, from Final Year Student Projects 2012: http://mccabeme.myweb.port.ac.uk/projects2012/ThomasStephens.pdf.

[18] Miller, P. (2012). An in-depth Study of the Detection, Observation and Modelling ofTransiting Exoplanetary Systems. Retrieved January 22, 2013, from Final Year StudentProjects 2012: http://mccabeme.myweb.port.ac.uk/projects2012/PeterMiller.pdf.

[19] Wikipedia (2013). Kepler’s Laws of planetary motion. Available: http://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion. Last accessed 30th Jan-uary 2013.

[20] Wallentinsen, D. (1985). Title. (Observation and Analysis of Eclipsing Binary Stars).Retrieved January 31st, 2013, from the Wallentinsen.com: http://www.wallentinsen.com/binary/intro.htm.

[21] Wikipedia (2013). Gravitational Microlensing. Available: http://en.wikipedia.org/wiki/Gravitational_microlensing. Last accessed 2nd February 2013.

[22] Pictures of Microlensing. Accessed 02/02/13.http://nexsci.caltech.edu/workshop/2011/

[23] Space news (2012). 100 billion planets in the Milky Way. Available: http://www.sen.com/news/100-billion-planets-in-the-milky-way.html. Last accessed 2nd February2013.

[24] Exoplanet Transit Database (ETD) Transit Predictions. Accessed 25/02/2013. http://var2.astro.cz/ETD/predictions.php?delka=359&submit=submit&sirka=51

[25] Exoplanet Transit Database (ETD) Corot2b Data. Accessed 28/02/2013. http://var2.astro.cz/ETD/etd.php?STARNAME=CoRoT-2&PLANET=b

[26] Hampshire astronomical group website, details on 24 telescope. Accessed 25/02/2013.http://www.hantsastro.org.uk/index.php

[27] SBIG Astronomical Instruments website, details on SBIG STL-1001E Specifications.Accessed 4th February 2013. http://www.astro.sunysb.edu/metchev/PHY517_AST443/STL1001E_specs_7.11.11.pdf

[28] Wikipedia (2013). Corot-2. Available: http://en.wikipedia.org/wiki/COROT-2. Lastaccessed 1st March 2013.

[29] Wikipedia (2013). Corot-2b. Available: http://en.wikipedia.org/wiki/COROT-2b. Lastaccessed 1st March 2013.

[30] Circle-circle intersection theory. Accessed 3rd March 2013. http://mathworld.wolfram.com/Circle-CircleIntersection.html

[31] Jupiter Texture. Accessed 4th March 2013. http://www.celestiamotherlode.net/.

71

http://www.celestiamotherlode.net/

http://mathworld.wolfram.com/Circle-CircleIntersection.html

http://mathworld.wolfram.com/Circle-CircleIntersection.html

http://en.wikipedia.org/wiki/COROT-2b

http://en.wikipedia.org/wiki/COROT-2

http://www.astro.sunysb.edu/metchev/PHY517_AST443/STL1001E_specs_7.11.11.pdf

http://www.astro.sunysb.edu/metchev/PHY517_AST443/STL1001E_specs_7.11.11.pdf

http://www.hantsastro.org.uk/index.php

http://var2.astro.cz/ETD/etd.php?STARNAME=CoRoT-2&PLANET=b

http://var2.astro.cz/ETD/etd.php?STARNAME=CoRoT-2&PLANET=b

http://var2.astro.cz/ETD/predictions.php?delka=359&submit=submit&sirka=51

http://var2.astro.cz/ETD/predictions.php?delka=359&submit=submit&sirka=51

http://www.sen.com/news/100-billion-planets-in-the-milky-way.html

http://www.sen.com/news/100-billion-planets-in-the-milky-way.html

http://nexsci.caltech.edu/workshop/2011/

http://nexsci.caltech.edu/workshop/2011/

http://en.wikipedia.org/wiki/Gravitational_microlensing

http://en.wikipedia.org/wiki/Gravitational_microlensing

http://www.wallentinsen.com/binary/intro.htm

http://www.wallentinsen.com/binary/intro.htm

http://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion

http://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion

http://mccabeme.myweb.port.ac.uk/projects2012/PeterMiller.pdf

http://mccabeme.myweb.port.ac.uk/projects2012/ThomasStephens.pdf

http://mccabeme.myweb.port.ac.uk/projects2012/ThomasStephens.pdf

http://en.wikipedia.org/wiki/Astrometry

http://en.wikipedia.org/wiki/Methods_of_detecting_extrasolar_planets#Astrometry

http://en.wikipedia.org/wiki/Methods_of_detecting_extrasolar_planets#Astrometry


Chapter 8

Appendices

8.1 Appendix A

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3 Overall Approach

3.1 Strategy and/or Methodology

• Generate Project Plan

• Undertake Research

• Analyse the Data

• Model the Data

• Generate first draft of Project Report

• Submit first draft to Supervisor and Mentors for review and comment

• Compile final version of Project Report

3.2 Important Issues to be Addressed

• Weather Patterns : To make any observations at Clanfield Observatory a clear night with no cloudcoverage is optimum. It is not possible to get weather predictions weeks in advance. Thereforethe chance of making Observations have to be calculated each week. Supplemental data will berequired due to the limited amount of data that can be collected personally by making observationsat the Observatory.

• Position and Phase of the Moon : To take observations of an Exoplanetary System the position andphase of the moon has to be taken into account. The Transiting Exoplanet would need to be thesufficient Right Acension (RA) away from the position of the moon throughout the Transit Period.Depending on the phase of the moon the RA will need to be adjusted.

• Supervisor and Mentor Availability : During different phases of the project, help will be required,e.g. to undertake observations. The supervisor and mentors are not available 24/7, so organisingmeetings and trips to the Observatory will be required.

3.3 Scope

The project will cover:

• The gathering of data from my own observations and from online databases.

• The problems of retrieving data.

• The analysis of the data collected.

• The modelling of analysed data.

• The review of various techniques in improving the modelling of the data.

• The utilisation of various techniques to improve modelling of the data.

The project will not cover:

• The search for new Exoplanetary Systems.

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3.4 Critical Success Factors

The success of the project will be determined by the following:

1. Favourable weather conditions at Clanfield Observatory to make observations.

2. Favourable lunar phase and positioning to make observations.

3. The availability of the Observatory and Mentors.

4. Access and availability to Computer 1 in the Technology Learning Centre (TLC) to process obser-vations using AIP4WIN.

5. Access and availability to a Computer with Excell, Maple 16 and Matlab to analyse and modelprocessed observations.

6. Access to resource material e.g. the Exoplanet Transit Database, Wikipedia, previous studentsprojects, etc.

7. The availability of my Supervisor and mentors to discuss the project.

4 Project Outputs

My Project will deliver

• A Report Consiting of:

1. Contents

2. Abstract

3. Chapter 1 (Introduction into Exoplanetary Systems) - An enlightening prelude into thediscovery of Exoplanetary Systems along with the aims and objectives of the project.

4. Chapter 2 ( Review of previous projects/research) - An evaluation of previous studentswork into the study of Exoplanets along with a review of their recommendations for futureprojects.

5. Chapter 3 (Exoplanet detection methods) - An investigation into the methods of detectingTransiting Exoplanetary Systems and the mathematics behind them.

6. Chapter 4 (Observations and Construction of the Transit Light Curve) - An overviewof the method used to make the observations along with an excerpt from the Observation Log.Aswell as a detailed description of how the observational data is processed by AIP4WIN; andan in-depth method on how the data from AIP4WIN is converted into a Transit Light Curve.

7. Chapter 5 (How can the Transit Light Curve be affected?) - An investigation intothe various problems that can affect the Transit Light Curve e.g. Limb Darkening. Also ananalysis into the methods to overcome such problems followed by applying such methods tothe previous Transit Light Curves.

8. Chapter 6 (Analysis of Exoplanetary Systems) - An analysis of the Transit Light Curve todetermine various aspects of the Exoplanetary System e.g. The Semi-Major axis, The OrbitalSpeed, etc.

9. Chapter 7 (Modelling of the Transiting Exoplanetary System) - An overview of theprocess to develop models of the different Exoplanetary Systems using mathematical software.

10. Chapter 8 (Conclusions and Recomendations) - A discussion of the project results andrecomendations to further the work of this project in the future.

11. Bibliography

12. Appendix A - Project Plan

13. Appendix B - Observation Schedule

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14. Appendix C - Observation Data

15. Appendix D - Observation Log

• A ten minute presentation

5 Project Outcomes

At the end of the project I expect:

• To have a greater understanding of Exoplanetary Systems, e.g. How they are detected, the problemsdetecting them, etc.

• To become proficient in using AIP4WIN to convert images into data that can be used to form alight curve.

• To become competent in making observations using a telescope with a Charged Coupled Device(CCD) camera.

6 Risk Analysis

The following table on the next page shows the risks identified for the project at this time.

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Risk LogRisk ID

Risk Probability

Severity

Score

Mitigation Plan Proximity Risk Response

Status

1 Unable to make Observations

3 5 15 Use data from the online Exoplanet Transit

Database

No later than 23rd

DecemberContingency Open

2 Unable to get time on Computer 1 at TLC

2 5 10 Try and secure a copy of AIP4WIN to run on a separate computer

No later than 6th

Jan 2013Transfer Open

3 Losing Data 1 5 5 Back up data on a separate memory stick /

HDD

Up to the 23rd

April 2013Contingency Open

4 Availability of Supervisor and/or

Mentors

2 4 8 Organise specific times to meet up and discuss project in advance.

Up to the 23rd

April 2013Reduce Open

5 Availability of Mentors

3 4 12 Ensure that I have more than one mentor

Up to the 23rd

April 2013Avoidance Open

6 Availability of Resources

2 4 8 Ensure that resources are obtained/booked in

advance

Up to 23rd April2013

Accept Open

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7 Technical Development

I intend to:

• Develop new models for Transit Light Curves.

• Develop new models for the Exoplanetary Systems.

8 Project partners

The Project is undertaken in partnership with Clanfield Observatory.

9 Workpackages

The Project consists of the following work packages:

1. Generate Project Plan

2. Submit Project Plan

3. Research Previous Projects

4. Evaluate Previous Projects

5. Define Strategy and/or Methodology

6. Make Observations and Record Results

7. Analyse Data

8. Model Data

9. Evaluation

10. Conclusion

11. Write up first draft of project report

12. Submit first draft of project report to supervisor and mentors

13. Write up final draft of Project Report

14. Submit Project Report

15. Generate Presentation

16. Perform Presentation

The following table on the next page shows a gantt chart of the amount of time required for eachworkpackage.

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8.2 Appendix B - Images

Images taken of the Exoplanet Qatar1b at Clanfield Observatory are included on a CD, thesewere used during the conversion of images to data using AIP4WIN.

8.3 Appendix C - Excel Data

The Analysed Data of Corot2b including the text file with the Corot2b data downloadedfrom the ETD website are included on the CD.

8.4 Appendix D - Maple Files

The text file with the data from excel aswell as the Maple modelling file created by supervi-sor Dr Michael McCabe endited for Corot2b which are included on the CD.

8.5 Appendix E - Autodesk Maya Model

File for the Autodesk Maya Model aswell as rendered pictures and a video of the transitwhich are included on the CD.

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