A Study of the Detection, Observation, Analysis and Modelling of Transiting Exoplanets

Transcription

1 A Study of the Detection, Observation, Analysis and Modelling of Transiting Exoplanets Figure 1: Artists conception of wasp17 [1] By: Benjamin John Cook Student No.: Project unit: (M301) Supervisor: (Dr Michael McCabe) Mentors: (David Harris, Chris Priest, Steve Futcher) Date: March 26, 2013

2 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 Systems describing the mathematical and scientific basis these methods are founded on. Results obtained during observations at Clanfield Observatory will be analysed as will data from online databases such as The Exoplanetary Transit Database website ( cz/etd/). We will then research into the methods for modelling the data analysing how effects such as limb darkening have on the transit light curve. We will then finish with a three dimensional visual representation of the exoplanetary system.

3 Contents 1 Exoplanetary Systems Introduction What makes a planet habitable? Stellar Characteristics Planetary Characteristics Habitable Zone Exoplanet Detection Methods Direct Imaging Coronagraphy Astrometry Radial Velocity Measurements The Stellar Reflex orbit with a single planet Reflex radial velocity of multiple non-interacting planets Transits Geometric Probability of a Transit Gravitational Microlensing Observations and Construction of the Transit Light Curve Clanfield Observatory Transit Candidates Making Observations Image manipulation using AIP4WIN Light Curve Construction Excel Minitab Analysis of Exoplanetary Systems Characteristics of Host Star Orbital Period Exoplanet Dimensions Calculation of Semi-major Axis Orbital Speed Impact Parameter and Transit Duration Mass of Exoplanet and Eccentricity of Orbit Limb Darkening Geometry of the Transit Light lost during the Transit Laws for Limb Darkening

4 6 Modelling of the Transiting Exoplanetary System Exoplanetary System Properties Maple Model factoring in Limb Darkening NAAP Transit Simulator Autodesk Maya Conclusion and Recommendations Detection Methods and Observations Light Curve Construction and Limb Darkening Modelling Recommendations for future investigation Appendices Appendix A Appendix B - Images Appendix C - Excel Data Appendix D - Maple Files Appendix E - Autodesk Maya Model

5 Chapter 1 Exoplanetary Systems 1.1 Introduction From our perspective standing on our planet it looks of colossal size. Moving further and further into the vacuum of space you can begin to see how small it is compared to the universe as a whole. It also makes us think how lucky we are to live on a planet that can sustain life. This then begs the question, is there other intelligent life out there in the universe? A couple of years ago, theoretical physicist Stephen Hawkings told the world that the existence of Extraterrestrials is almost certain. Stephen Hawking said that his conclusion was unusually simple. By knowing that the universe contains roughly 100 billion galaxies and each galaxy contains around 100 million stars then there is a high probability that our planet Earth is not the only planet in our Universe that has life on it that has evolved [6]. Around the world astronomers are now competing against one another to be the first to discover a habitable planet. For hundreds of years scientist, astronomers and philosophers have predicted that planets outside our own solar system must exist; these were called Exoplanets and were defined as planets orbiting stars other than our own [2]. It was not until 1995 with the discovery of the first exoplanet that orbited a star much like our own sun that astronomers around the world came together in a global effort to discover more exoplanets [2]. As of January 15th 2013 there have been 859 exoplanets discovered[5], yet the discovery of a habitable planet still eludes us (based on our knowledge of carbon based life forms requiring oxygen and water to survive). While our main goal has not yet been achieved, other fascinating discoveries has been made. One such recent discovery is the planet with four suns, discovered by volunteers using planethunters.org/. The planet they discovered is assumed to be a gas giant orbiting a binary star system along with another Figure 1.1: Artists conception of the planet with 4 suns [7] pair of stars orbiting the binary star system. All four stars have a gravitational pull on this one planet which baffles scientist as the planet itself is in a stable orbit, which makes it and amazing and unexpected discovery [7]. 3

6 1.2 What makes a planet habitable? Planetary habitability is defined as the measure of a planet s or a natural satellite s potential to develop and sustain life [8]. When looking for indicators of whether a planet is habitable or not astronomers look for various key characteristics of the selected star system Stellar Characteristics The key stellar characteristics required for planetary habitability are: Spectral Type of Star - Through using spectroscopy to find the spectral type of stars, astronomers are able to detect the temperature of the photosphere ( The photosphere of an astronomical object is the region from which externally received light originates [9]). The temperature of the photosphere is related to the total mass of the star, this relation is only for stars on the main sequence as shown in the Hertzsprung-Russel diagram Figure 1.2. Astronomers suggest that the best spectral types for habitable stars are F, G and to the middle of K. This correlates to stars on the main sequence with a photospheric temperature of 4000 degrees Kelvin to 7000 degrees Kelvin. Stars in this section of the main sequence are more suitable candidates because: Figure 1.2: Hertzsprung-Russel diagram [10] Figure 1.3: Diagram showing the range of the habitable zone is compared to the size of different stars [11] 1. The stars have longer life times compared with more luminous stars that burn their fuel more quickly due their massive size. These very luminous stars live for a few millions years, while stars like our sun live for a few billion years. 2. The stars emit high frequency UV (ultra-violet) radiation, this is required for atmospheric conditions such as ozone layer formation. 4

7 3. Liquid water may exist on the surface of planets orbiting them at a specific distance that does not induce tidal locking [8] (captured rotation/synchronus rotation). Stability of Habitable Zone - The habitability zone also know as the circumstellar habitable zone or Goldilocks zone is the scientifice terminology used to describe the region 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 can be 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 minute number of stars experience a significant change in their luminosity (variable stars). This significant change in luminosity is likely to be combined with large amounts of radiation such as gamma rays and x-rays. Planets around stars that experience this phenomena are not habitable because of severe temperature change would make life unable to survive, along with the large amounts of radiation would make it impossible for life to sustain itself or even exist. High Metallicity - Using Spectroanalysis of stars astronomers have seen that there is a correlation between high metallicity in stars and the chances of an exoplanet being found. Astronomers theorise that the high metallicity corresponds to the amount of heavy elements available in the protoplanetary disc, so a high metallicity in the star would mean there is a high chance that an exoplanet would be orbiting it Planetary Characteristics The key planetary characteristics required for planet habitability are: Mass of Planet - The mass of a planet can effect the habitability of a planet in different ways; the mass effects the gravitational force of a planet, this would mean that lower mass planets would have a lower gravitational force. The planets ability to retain an atmosphere is dependent upon the planets gravitational force and therefore also its size. This means that astronomers are having to look for planets that are the size of the earth or larger. Astronomers also theorise that Figure 1.4: Diagram showing the sizes of planets with various compositions [12] large planets are likely to have iron cores thus generating a magnetic field, this would protect the planet from stellar winds and cosmic radiation [8]. Planetary Composition (Geochemistry) - Astrobiologists theorise that life may exist on planets that have the same primal elements as earth (carbon, oxygen, nitrogen and hydrogen) that are crucial for life as we know. These elements combined produce amino acids which in turn make up proteins, the proteins are needed to form DNA and RNA. 5

8 Orbital and Rotational Properties of Planet - The Orbital eccentricity of a planet effects its surface temperature; for example if a planets orbital eccentricity was large, this would lead to drastic rises and falls in temperature. Therefore the planet would be unable to support life. The Axis tilt of a planet effects the stasis of the biosphere. No axis tilt or very little axis tilt leads to no occurrence of seasons, therefore a main catalyst in the dynamics of the biosphere would not exist. Too much tilt would lead to extreme seasons 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 the region around a star that a planet would have to orbit in to have the neccessary conditions to sustain life. ( ) 1 (1 A)L 4 (1.1) T eq = 1 2 σπa 2 T eq - The equilibrium temperature. As the planet orbits its star we assume it stays at a constant temperature so the temperature on the surface of the planet, so it be characterised by T eq. σ - The Steffan-Boltzmann constant which is Jm 2 K 4 s 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 = Js 1 ). Rearranging equation 1.1 we can get a to be the subject to work out the boundaries of the habitable zone knowing the required equilibrium temperature. a = 1 ( ) 1 (1 A)L 2 (2T eq ) 2 (1.2) σπ Using our own solar system as an example we know that: The Luminosity of the Sun L = Js 1. The albedo (reflection coefficient) of earth A = 0.3. [2] The Steffan-Boltzmann constant σ = Jm 2 K 4 s 1. The inner boundary temperature T eq (inner) = 273K (freezing point of water) The outer boundary temperature T eq (outer) = 373K (boiling point of water) 6

9 Substituting into equation 1.2 we get ( 1 (1 0.3) Js 1 a(inner) = (2 373K) 2 π Jm 2 K 4 s 1 = 1 ( m 2 ) 1 2 (746) 2 π = m ) 1 2 (1.3) ( 1 (1 0.3) Js 1 a(outer) = (2 273K) 2 π Jm 2 K 4 s 1 = 1 ( m 2 ) 1 2 (546) 2 π = m ) 1 2 (1.4) The Earths semi major axis is a = m so by this calculation our planet is outside the theoretical habitable zone; but we are making the assumption that earth has a uniform temperature and that it is also a perfect black body. Calculating the Earths equilibrium temperature, using equation 1.1 we can see that ( T eq = 1 (1 0.3) Js 1 2 π Jm 2 K 4 s 1 ( m) 2 ( ) = 1 (0.7) K π ( ) 2 = 255K ) 1 4 (1.5) So looking at equation 1.5 we can see that Earths T eq = 255K = 19 C. Earth is not a perfect black body and the temperature of our planet is not uniform, therefore it is capable to sustain life at the distance it is from the sun even though it is outside the theoretical habitable zone. 7

10 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 due to the glare of light emitted from their host star. Detecting planets using direct imaging is can be extremely difficult if the planet you are trying to detect is some light years away from our own solar system, which is why it mainly works best to detect objects nearby our own planet. Therefore scientists and astronomers have to use other methods to try and detect planets outside our solar system Coronagraphy In 1930 french astronomer Bernard Lyot invented the coronagraph [13]. This telescopic instrument allowed the astronomer to block out direct light emitted by the sun; therefore he was able to observe objects orbitting the sun that would not have previously been seen because of the suns glare. This coronagraph was originally designed to study our suns solar corona but has been adapted by astronomers into the stellar coronagraph to detect exoplanetary systems.the stellar coronagraph stops the detector being flooded with light from the distant star allowing astronomers to observe if there are objects such as exoplanets orbiting the star. Fomalhuat is roughly 25 light years from Figure 2.1: Diagram showing the Fomalhuat System as seen using Coronagraphy [2] Earth and is roughly twice the mass of our sun, due to its mass it is therefore more luminous than our sun; therefore from earth it is one of the brightest stars in the night sky. Astronomers observed Fomalhuat and discov- 8

11 ered that surrounding was a ring of dust, this dust meant that it was possible that Fomalhuat had a exoplanet orbitting it. Due to the luminosity of the star, direct observation was useless in detecting the existence of an orbiting planet. It was not until 2008 that the Hubble Telescopes combined images of the star taken in 2004 and 2006 using a stellar coronagraph could the exoplanet be detected clearly. Figure 3.1 shows the pictures the Hubble telescope took combined together. From the images astronomers we able to calculate that Fomalhuat b is roughly 10 9 times fainter than its host star and was orbiting at a distance of 100 times the distance earth is from our sun (100AU). The use of a stellar coronagraph in detecting Fomalhuat b allowed astronomers to directly detect exoplanetary systems in a new way while before they were only detectable in exceptional circumstances. 2.2 Astrometry Astrometry is the science of accurately measuring the position of stars [2]. Astrometry is one of the oldest methods used to detect extrasolar planets indirectly. It uses the fact that planets 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 there are planets in the star system then the sun performs a reflex orbit around the barycentre to keep the centre of mass at the barycentre. Many astronomers make observations of various stars to detect whether they have a reflex orbit, it is from this they can predict the presence and parameters of an exoplanet. Using Figure 3.2 it can be seen that the motion of a star in its reflex orbit has a semi-major axis a ; the planets orbit around the barycentre has a semimajor axis a P, therefore using figure 3.2 it can be shown that the sum of the semi-major axes is a = a + a P. Kepler s Laws Figure 2.2: Diagram based on Figure 1.12, page27 [2][17] 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 of time [19]. 3. The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit (P 2 a 3 ) [19]. 9

12 Generalizing Kepler s third law it can be shown that: a 3 P 2 = G(M + M P ) 4π 2 (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 angular displacement β from an observer at a distance d. Therefore the angular displacement or astrometric wobble as it is known can be show as proportional to the semi-major axis of the stars reflex orbit a : β a β = a d Since we know that We can see that a = M P M a P (2.2) β = M Pa P M d (2.3) From equation 2.3 you can see that as M P and a P increases so does β therefore we see more angular displacement (astrometric wobble) with large mass Exoplanets that have a large obit aswell. From the equations we can see that Astrometry is useful in detecting large mass Exoplanets such as gas giants. 2.3 Radial Velocity Measurements The Radial Velocity method shares similarities with the astrometric method in that it also uses the reflex orbit of the star. Instead of using the change in position of the star to detect Exoplanets it uses the change in radial velocity of the star s reflex orbit. Since the star is moving towards and away from the observer the radial velocity can measured using the Doppler shift of the light emitted by the star 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 rest frame of its host star, this is also known as the astrocentric frame. Figure 3.3 shows an example of the astrocentric frame of a planet. The planet performs an orbit around its host 10

13 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 (how much 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 pericentre and the star is positioned at the focal point of the elliptical orbit. Using the cartesian coordinate 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 the sky plane as the inclination, i. We position the x axis along the line between the 2 points of intersection between the orbital plane and the sky plane as seen by the observer. Positive values of x mean that the planet is moving towards the observer, we define γ as the point in which the positive x axis intersect the orbital plane. We define θ as the angle known as the true anomaly, θ measures the distance the planet has moved along its orbital from the pericentre. The orientation of the pericentre is defined by the angle ω OP which is measured with respect to γ. Using these definitions we can define the velocity which has components in the x, y and z directions. These can be shown as: v x = 2πa P 1 e (sin(θ + ω OP) + esinω OP ) 2 v y = 2πacosi P 1 e (cos(θ + ω OP) + ecosω OP ) 2 v z = 2πasini P 1 e (cos(θ + ω OP) + ecosω OP ) 2 (2.4) Astronomers using this method do not observe the planet as they simply cannot, instead 11

14 they observe the planets host star. Therefore they require analogus equations for the host stars reflex orbit. Only looking at the astrocentric frame means that the planet is the only object moving in the system, while in fact the star is also moving. The star performs a reflex orbit around the barycentre as shown in figure 3.2. Therefore the velocity v of the planet in its astrocentric frame(fig 3.2) is given by v = v P,bary v (2.5) In equation 2.3 the astrocentric velocity is defined as v, the velocity of the star performing its reflex orbit around the barycentre is defined as v. Using equation 2.2 we can see that: M P a P = M a Using vector notation and knowing that the position of the barycentre is fixed; then the only variable is the distance the star and the planet are from the barycentre, which are proportionate to each other. We can then deduce that: M r = M P r P We have made one side of the equation negative due to the fact that the planet and the star are in opposite directions to the barycentre. The by differentiating M r = M P r P with respect to time we get: M v = M P v P,bary (2.6) Now we have the relationship of the velocities of the planet and the star in the barycentric frame. Using equation 2.6 we can see that the orbit of the planet and the reflex orbit of the host star around the barycentre are the same shape (ellipse), the only difference between them is the size of their elliptical orbits. We now rearrange equation 2.6 to get v P,bary as the subject of the equation. We now substitute equation 2.7 into equation 2.5 to get: v P,bary = M v M P (2.7) v = M P M P + M v (2.8) Looking at equation 2.8, v is defined as the astrocentric orbit velocity of the planet, as shown in equation 2.4. From the view of the observer the barycentre of the transiting exoplanetary 12

15 system will have velocity that is non-zero, we define the velocity of the barycentre as V 0. V 0 is dependent on time, therefore as time progresses the velocity of the barycentre changes aswell. The change in velocity takes hundreds of millions of years and therefore is chosen to be a constant due to the fact the change in V 0 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 + V 0 (2.9) Hence for the velocity of the stars reflex orbit factoring in the motion of a planet orbiting it we get the components: 2πaM P V x = V 0,x + (M P + M )P 1 e (sin(θ + ω OP) + esinω OP ) 2 2πaM P cosi V y = V 0,y + (M P + M )P 1 e (cos(θ + ω OP) + ecosω OP ) 2 2πaM P sini V z = V 0,z + (M P + M )P 1 e (cos(θ + ω OP) + ecosω OP ) 2 (2.10) The observed radial velocity is given by: V z = V 0,z + Part of this equation is a variable, and is shown as: 2πaM P sini (M P + M )P 1 e 2 (cos(θ + ω OP) + ecosω OP ) (2.11) 2πaM P sini (M P + M )P 1 e 2 cos(θ(t) + ω OP) (2.12) Since we know that cos(θ(t) + ω OP ) can only go between 1 and 1, we can define the amplitude of the radial velocity variations as: A RV = 2πaM P sini (M P + M )P 1 e 2 (2.13) Using the Doppler shift we can measure the radial velocities by knowing what the specific features that appear the in stars stellar spectrum are. We define λ as the change in the wavelength from the effect of the Doppler shift, and we define c as the speed of light. The relationship between the velocities, the change in wavelength and the speed of light is given by: λ λ = V c (2.14) 13

16 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 host star, this is not always the case. To make an approximation of the observed radial velocities of a system with multiple planets we must assume that the planets do not alter each others elliptical orbits. Hence combining all of the elliptical reflex orbits around the barycentre the approximation to the observed radial velocity is as follows: V = V 0,z + n k=1 A k (cos(θ k + ω OP,k ) + e k cosω OP,k ) (2.15) Here we define n as the number of the planets in the system, M k as the mass of each planet and a k, e k, θ k and ω OP,k as the instantaneous orbital parameters. We also define A k as: A k = 2πa km k sini M total P k 1 e 2 k (2.16) A k includes M total which is the mass of the entire system, so the mass of the sun added with the mass of all the planets orbiting it. Therefore we have the amplitude of the reflex orbit of the star with multiple planets, A RV = A k 14

17 2.4 Transits The Transit method is the main the main method of detection used in this project. The technique is relatively simple as it uses the idea that when a planet crosses infront of its host star it absorbs the light being emitted. Therefore the method does not require a high level of precision, hence the data can be collected using telescopes on the ground in observatories. A transiting giant planet such as a gas giant like Jupiter will show a dip of 1% of the light emitted by its host star; terrestrial planets however cause a dip of 10 2 % therefore the photometric precision required to detect terrestrial planets is 10 4 [2], because of the precision required it is impossible for observatories to detect terrestrial planets. Using ground based telescopes Astronomers observe stars to detect the dip in the light being emitted, then the astronomers can deduce that an object is orbiting the star if the dip in the light occurs periodically. 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 between the observer and the star. The light curve can be separated into different sections of time, t: t < t 1 : Pre-Ingress - The planet is orbiting its host star but has not begun its transit. t 1 t t 2 : Ingress - The planet has begun its transit and has moved passed the left outter edge to the inside of the star s disc. t 2 t t 3 : Ingress and Egress - The planet is in the middle of its transit and is moving passed the centre of the star s disc. 15

18 t 3 t t 4 : Egress - The planet is coming to the final phase of its transit and is starting to pass across the right outer edge of the star s disc. t 4 < t: Post-Egress - The transit has ended and the planet has passed the right outer edge of the star s disc. Using the size of the dip in the star s brightness we can estimate the fraction of star s disc being covered by the exoplanet. F F = R2 P R 2 (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 R2 P is the ratio between the R 2 areas of the exoplanet and its host star. Therefore Equation 2.17 allows us to calculate and estimation of the planets size in terms of the size of the host star the planet is orbiting Geometric Probability of a Transit A transiting exoplanet can be detected when its orbital plane is sufficiently close the the line of sight of the observer. To be seen the disc of the exoplanet must cross infront of the disc of its host star. The closest approach of the centre of the exoplanets disc and the centre of its host star s disc occurs at the inferior junction, this phase is when the planet is closet to the observer [2]. The inferior junction is defined with φ = 0.0 and the distance between the two discs can be calculated using the smei-major axis, a, and the inclination, i. Hence: d(φ = 0.0) = acosi (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]. acosi R + R P (2.19) 16

19 The Projection of the unit vector normal to the orbital plane onto the skyplane is defined as cosi. cosi can take a random value between 0 and 1, we can replace cosi with x. Therefore we can work out the geometric transit probability: number of orbits transiting geometric transit probability = all orbits (R +R P ) a 0 dx = 10 dx (2.20) Hence geometric transit probability = R + R P a R a (2.21) From the equation we can see transits are more probable when the planet has a small orbit and a large host star. 2.5 Gravitational Microlensing Gravitational Microlensing uses the lensing effect of the general relativistic curvature of spacetime to detect exoplanets. This method is used to detect planets in regions of where there exists a dense cluster of stars. These dense clusters of stars are chosen because microlensing requires the alignment of stars from the point of view of the observer. Therefore a dense cluster of stars provides a higher probability of the occurrence of star alignment. The best case study is the galactic bulge which is the dense region of stars at the centre of our galaxy, the Milky Way. Another bulge that has be subjected to this method is in the spiral galaxy known as M31. Over 2000 microlensing events have been observed and only a few of them 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

20 Figure 2.7: Diagram of showing the method of detection through Microlensing [23] The two diagrams show what occurs during a Gravitational Microlensing observation. The first figure shows the how the light is focused as it passes through a cluster of stars. The second figure shows the magnification curve we obtain through the microlensing technique, we observe an extra peak indicating that a planet is present. 18

21 Chapter 3 Observations and Construction of the Transit Light Curve 3.1 Clanfield Observatory As part of my final year project I was required to make observations to collect transit photometry 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 to use the 24 inch telescope at Clanfield Observatory to collect data. The University has a close relationship with the Hampshire Astronomical Society for the past years; this relationship has brought with it success for students requiring to make observations to obtain data for their projects. My mentors utilised their experience and knowledge to teach me the methods required for identifying a suitable transit candidate and for using the 24 inch telescope to obtain data to analyse. My initial aim was to organise and arrange a date and time to make an observation at Clanfield Observatory to gain first hand experience in the use of the telescope. During the window of opportunity I was able to make one observation with the 24 inch telescope though the data did not show any discernible dip in the transit light curve; however the methods used to choose a suitable transit candidate and how to use the 24 inch telescope are explained in this chapter. Figure 3.1: Photo of the Clanfield Observatory site [?] The data used up to the AIP4Win section is of the Transiting Exoplanet Qatar1b. Since previous students have already studied Qatar1b, for the purpose of this project I decided to use data of the Transiting Exoplanet Corot12 from the Exoplanet Transit Database (ETD) to construct a light curve using various software. 19

22 3.2 Transit Candidates Before observations can begin a suitable candidate must be chosen to achieve good discernible data. There are many factors that need to be addressed when choosing a transit, to find a suitable exoplanet we must use the Exoplanet Transit Database. Using the Exoplanet Transit Database we can obtain all the data required to find a predicted transit. The Process to do so is as follows: 1. Using the Hampshire Astronomical Society s website[26] we obtain the latitude and longitude of Clanfield Observatory, which is longitude 359 and latitude 51. We insert these coordinates on the ETD website shown in figure A main factor that can effect the observation is weather, if the weather on a certain day is cloudy an observation is out of the question, therefore reviewing the weather forecast is key. The optimum weather to make an observation is clear skies, after looking at the weather forecast a date can be chosen when the skies are clear. Using the ETD website you can click on the most appropriate date to make an observation to to obtain a list of all predicted transits for that day. 3. When you have your chosen date you are then given a table of transits predicted for that day. The table contains data that will need to be analysed to choose the best candidate for observation. Figure 3.2: ETD website showing the predictions of transits [24] Figure 3.3: ETD website showing the predictions of transits but magnified [24] 4. Using figure 4.2 we can analyse the table of data. The magnitude of star is indicated with the column marked V(MAG). For a transit to be detected by the 24 inch telescope the magnitude of the star requires to be 11 magnitudes or higher, anything lower and the transit is too dim to detect. 20

23 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 light pollution do not corrupt the data. The position and phase of the Moon has to be taken into consideration; if there is a full Moon then the data might become obscured due to the light from the Moon. If the Moon moves across the area of sky being observed the data will also become obscured. Therefore the right ascension must be analysed so that an 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 from the transit. For a transit to be detected by the 24 inch telescope the DEPTH is required to be at least 15 millimagnitudes (0.015 MAG), this is due to the precision of ground based telescopes which is detailed in chapter Looking at each row we can see the time each transit begins and ends. Each transit must 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 equipment would have to be calculated as well. Therefore we can calculate the time to start the observation and end the observation, this would be subject to the availability of the mentors. After going through this process, hopefully a transit has been chosen to observe. Now the final step is to communicate your chosen transit along with the date and time to your mentors and perform the observation if they are available and the weather stays clear. 21

24 3.3 Making Observations To make and observation of the suitable transit candidate one needs three pieces of very important equipment, these are: CCD Camera: For this project we used a SBIG STL-1000E with the following specifications[27]: 1. Focal Length at 1 arcsecond per pixel: 195 inches 2. Total Pixels: 1.0 million 3. Array pixels 4. Pixel Size: 24 micrometres 5. Full Well Capacity: e- 6. Cooling: 2 stage thermoelectric, water circulation, 40 C below ambient with uncooled water, regulated to +/- 0.1 degree (roughly 32 C air only). Further cooling may be achieved by using water cooled below ambient and above the dew point. Figure 3.4: Photo of an SBIG STL-1000E CCD Camera [27] Telescope: For this project we used the 24 inch telescope. This allows the exposure time to be longer which is optimal for observing 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 any difficulties. 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 condensation accumulating on the telescope; if the telescope was not covered it would result in observations that would be obscure and possibly damage the telescope in temperatures below freezing in which the condensation turns to ice. 3. The clamps attached to the telescope are then undone to allow the telescope to move freely. 4. The dome and the telescope are attached to stepper motors[17], this allows the process of following the transit to be automated. By simply using the the the Right Ascension (Qatar1b RA = 20h13m32s) and Declination (Qatar1b DE = +65 9m43s) and inputting into the point targeting system console as shown in figure 4.6 the telescope will automatically position itself to observe that point of the sky. 22

25 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 a laptop which will store the images of the transit. The laptop also serves the regulator for the CCD Camera s temperature. If the Camera s temperature is too high, the pixels can become saturated and this in turn will ruin any transit data collected. The noise in the image is also affected by the temperature, the colder the temperature the less noise in the image. The laptop also allows us to control the exposure time, the length of time for 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 done using the dome s motor control. 8. Every step is completed and we are ready to start taking images of the chosen transit. 23

26 3.4 Image manipulation using AIP4WIN The data collected from the observation of the transit at Clanfield Observatory must be analysed to obtain the data required to construct a transit light curve. To do this we must use AIP4Win which will convert all of the images we collected into quantifiable data. The following steps show how the images obtained by Thomas Stephens [17] were used with AIP4Win. 1. Once AIP4Win has loaded up we can begin to convert the images to the data we require. Looking at the top toolbar, click on Measure, mouse down to Photometry and click 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 should be a box next to Auto Calibrate, click on the box so it is ticked. Then click on select files and a browser box should have opened. Search through the folders until you find the images of your transit, these should be.fit files. Then while holding the shift key click on the first image and then the last image, this should select all the images. Once all the images are selected, click the open button. Figure 3.8: Screenshot of step 2 using AIP4Win [17] 24

27 3. Now the first image should be displayed in a small window. Each of the images will be slightly blurred, this has been done deliberately during the observation phase so that the results from the star are averaged to produce improved readings. Now using the windowed image we need to determine which star is the transit we observed. Using an image of the target field we can compare it with the observational image to identify our transit star, this can be done by simply looking for particular star clusters around the star we want on the target field image. Then by looking for these star clusters on the observational image we can determine our transit star. Once found right click the middle of the transit star, now the transit star has been selected. Now two more stars must be chosen so our transit star can be compared to them. The comparison stars must have the same magnitude (brightness) as the transit star that was observed. Choosing the comparison stars can be a difficult process and will require some experimentation to get it just right. Clicking on the settings tab of the Multiple Image Photometry window you can adjust the radii so that the aperture captures all the star light and the annulus excludes starlight. Figure 3.9: Screenshot of step 3 using AIP4Win [17] 4. Once the previous steps are complete and the transit star has been found on the observational image we can begin the image analysis. To start this process click the execute button in the multiple image photometry window. When AIP4Win has gone through all the images two graphs should appear like in figure The V-C1 graph shows the dip in magnitude of the star we observed compared to the comparison star. The C1-C2 graph shows the difference in magnitude between the two comparison stars. When looking at the C1-C2 graph it is best to choose two comparison stars whose variation in magnitude is on the green line, if this is not the case then new comparison stars need to be chosen. 25

28 Figure 3.10: Screenshot of step 4 using AIP4Win [17] 5. Now that images have been turned into data we can use it has to transfered to another program to create the light curve. Firstly we click the save to file tab to save the data log, save it as a text file. Then open excel and using excel open the folder containing the data log, choosing all files allows us to see our text file. Loading the text file will bring up the Text Import Wizard window. Click on the option for the fixed width and click 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 hopefully all your data in their intended columns. Figure 3.11: Screenshot of step 5 using AIP4Win [17] 26

29 3.5 Light Curve Construction Now that the data has been extrapolated from the images obtained from the observational visits to Clanfield Observatory, the construction of the light curve can begin. In this section I shall be using data collected from the Exoplanet Transit Database (ETD) since my observational data did not provide any conclusive results Excel Using the data for Corot2b as shown in the Appendix we can begin to construct the light curve using excel. The data on the ETD website is ranked on its quality, rank 1 is the best data and will give us a light curve with a definitive dip in the magnitude. As the rank number increases 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 in Figure Clicking on the data quality rank number a new tab should open with all the data. After selecting all the data, copy it and paste it in a text file; once that is done save the text file and follow the process in the section 4.4 to import it into Microsoft Excel. Once it is in Excel we can delete columns that are not needed to produce the light curve. Now we should be left with 2 columns, the Julian Dates and the magnitude of light from the observed star. 27

30 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 graph will begin at zero. As you can see from figure 4.13 we use absolute cell referencing to takeaway 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 to hours. This can be simply done by multiplying each cell in column D by We now need to create the characteristic dip in the flux, we firstly average out the first results in column F to calculate the average magnitude of the star. The equation for the average magnitude of Corot2a is shown in cell I3 of figure Using the average magnitude we subtract it from each magnitude in column F, the equation can be seen in column G of figure Now the final phase is to get the flux, to do this we must use equation 3.2 where B 1 B is the flux and m is the change in magnitude. Figure 4.13 shows the equation for the flux for Excel. m = 2.5log 10 B 1 B (3.1) To do step 5 we rearrange equation 3.1 to get: B 1 B = 10 m 2.5 (3.2) Now we are ready to plot the light curve, using the time in hours on the x-axis and the flux on the y-axis we get: 28

31 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 of best fit. Using Excel we can create a graph with adjoining lines between each point. The light dip of the transit light curve can be seen clearly with this new graph which can be seen in figure 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 only downside with the trend line is that the ends of the polynomial curves downwards. This is not very representative of the transit light curve as the flux should level out and have a constant magnitude before the transit passes in front of the star and after the transit passes in front of the star. From this graph we are however able to approximate the period of the transit and the radius of the exoplanet which will be covered in chapter 5. 29

32 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 data for every five points. Figure 4.18 shows a similar graph of the transit light curve but with a trendline that averages the data for every ten points. These two graphs do not significantly help 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 the data at every 5 points 30

33 Figure 3.18: Excel graph showing the transit light curve for Corot2b with a trend line averaging the data at every 10 points 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 open it. 31

34 Figure 3.20: Step 2 using minitab 3. Now we can begin to plot the data and produce a line of best fit. Click on Graph then click Scatterplot. Select Simple scatterplot and click ok. Click on X Variables row one and click on the time of the transit measured in hours and click select. This set the X variable for the graph as the time, now doing the same for the first row for Y variables but this time selecting the Flux. Figure 3.21: Step 3 using minitab 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 relationship between the time and the flux without having to fit a specific model. Altering the degree of smoothing changes the fraction of the total number of points used to calculate the fitted values at each x-value. Altering the number of steps changes the number of iterations of smoothing to limit the influence of outliers. To get the line of best fit I 32

35 chose 10 steps with a degree of smoothing of 0.2 and we can see the result in Figure Figure 3.22: Transit Light Curve produced using minitab 33

36 Chapter 4 Analysis of Exoplanetary Systems This chapter will discuss the various characteristics that can be found from the information taken from the transit light curve. The methods used will be explained and a running example using the Corot2b data from the ETD website we be shown. It must be noted that not every single characteristic of the transit can be found using the transit photometry observation 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: ± 0.018R Metallicity Fe/H: 0 ± 0.1 Right Acension: 19h27m0.6496s Declination: Magnitude V: Orbital Period The Orbital Period of a transit can be calculated using two different techniques. The first is by using the radial velocity method and the other is to use the time that has passed between observations of your chosen transit. We define the time elapsed between observations as T, 34

37 the number of transits that occur a certain time period as N and the Orbital Period P; which has the following relation. P = T elapsed N cycles (4.1) Using the transit light curve we can determine orbital period transit by comparing the midpoint of each transit light curve from a set of continuous transit observations. The midpoint of the transit can be estimated using the transit light curves we have already produced using excel and minitab. Figure 4.1: Transit Light Curve produced using excel showing the estimated mid and end points of transit [25] Figure 4.2: Transit Light Curve produced using minitab showing the estimated mid and end points of transit Now comparing the estimated mid point from our transit light curve with another transit light curve using data of the same transit we can determine the orbital period knowing the 35

38 number of cycles that occurred between the two observations. We can do this by using data obtained 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 of the transit occurred. We can also calculate the number of transit cycles that have occurred by using the column labeled epoch. N cycles = E poch45 E poch43 = = 4cycles (4.2) T elapsed = HJD45 HJD43 = = days (4.3) P = T elapsed N cycles = days 4cycles = days (4.4) Comparing the answer from our calculation with the actual orbital period stated on the ETD website (P = days) there is an error of days. To get a more accurate result we 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 the exoplanet (R P ) and the radius of its host star (R ) in the second chapter. It is as follows: F F = R2 P R 2 (4.5) 36

39 Using the Transit Light Curves produced by excel and minitab we can approximate the change 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 The maximum and minimum points of the light curve can however be deduced since the trend line is a sixth degree polynomial. y(x) = x x x x x dy dx = 00035x x x x (4.6) Using dy dx 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 can obtain the maximum value of flux and the minimum value of flux and therefore the change in 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 mag, using the data from the ETD website we can actually see that the change in flux is mag [25] 37

40 so we can see that the accuracy from modeling with excel quite accurate. Now using the transit depth (0.0322mag) and the radius (0.902R = m) of Corot2a [28] we can now calculate the radius of the exoplanet. F F = R2 P R 2 R 2 P = (4.7) Therefore, R P = = km (4.8) = 1.61R J The actual radius of Corot2b is in fact ± 0.047R J [25]. The result I have calculated is clearly larger than that obtained from the ETD website; one possible explanation for this result is that Corot2b may have an enlarged atmosphere which absorbs light from the host star 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 exoplanetary system. This can be done by using the following relation: a 3 P 2 = G(M + M P ) 4π 2 (4.9) Where a is the semi-major axis, P is the orbital period, G is the gravitational constant, M is the mass of the star and M P is the mass of the planet. We can therefore rearrange to make a the subject: ( G(M + M P )P 2 ) 1 3 a = 4π 2 (4.10) We can neglect the mass of the planet since it only makes up a small fraction of the mass of the entire transiting exoplanetary system, therefore we can obtain an estimation of the semimajor axis by just using the mass of Corot2a. Using spectrography analysis and comparing 38

41 the spectral type of the star with the main sequence on the Hertzsprung-Russel Diagram,the the mass of the star can be deduced. So we no have the following: M = G = P = days = seconds (4.11) Now inputting these values into equation 4.10 we obtain, ( ( )( ) 2 a = 4π 2 ) 1 3 ( = 4 π 2 = m = AU ) 1 3 (4.12) Using the ETD website we can see that the actual Semi-Major axis is AU so the estimation 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 calculate the orbital speed of our exoplanet. We define v as the orbital speed, a = m as the semi-major axis and P = s as the orbital period. We can now input these values into the following equation. v = 2πa P 2π = = ms 1 (4.13) 4.6 Impact Parameter and Transit Duration Using the previous data we have obtained we can now begin to calculate an approximation to the impact parameter. In figure 5.6 the impact parameter is noted with the letter b, and is defined as the shortest distance from the centre of the disc to the locus of the planet [2]. 39

42 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 between the parallel light and the planet s orbit[2] [17] To calculate the duration of a transit defines as T dur, with an impact parametre value of b=0, we can use the following relationship: length of arc from V to W T dur = P 2πa P 2R 2πa P R πa (4.14) 40

43 Using the values from the ETD website P, a and R we can approximate the transit duration. T dur hr m π m hr mins (4.15) From the ETD website we can see that the transit duration is in fact minutes, therefore our calculation gave a rough approximation to the actual value. We can also find the duration of the transit analysing the transit light curve from chapter 3. From now on we will use the value of T dur from the ETD website for better accuracy. Comparing the two T dur values 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 the semi-major axis, a. Figure 4.8: Diagram based on on figure 3.2 pg93 showing the geometrical representation of the impact parameter[2] [17] Figure 4.9: Diagram based on on figure 3.3 pg93 showing a geometrical representation to allow us to express the length, l, in terms of the impact parameter using Pythagoras s Theorem[2] [17] 41

44 As you can see from figure 5.9 we can express the impact parameter in the following way: b = acosi (4.16) It can also be clearly seen that the hypotenuse, h, is equal to R P + R. We can therefore use Pythagoras s theorem to calculate the length, l: l 2 = (R P + R ) 2 (acosi) 2 l = (R P + R ) 2 a 2 cos 2 i (4.17) Figure 4.10: Diagram based on on figure 3.4 pg94 showing a geometrical representation of the exoplanet 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 A and B is equal to 2l. As the planet orbits the star it covers a distance of 2πa therefore as it moves along its orbit between A and B it covers an arc length of αa where the angle α is in radians. Using the triangle formed by the centre of the star and the points A and B we can deduce that the length, l, has the following relation: ( α ) sin = l 2 a (4.18) Since α is measured in radians we can simply write the T dur as the following: T dur = P α 2π = P π sin 1 ( l a ) (4.19) 42

45 Using equation 4.17 we can substitute the value for l in to equation?? so that we now have the inclination, i, included in the equation. Then we can input the values for P, R P, R, T dur and a. T dur = P (R P + R ) 2 a π sin 1 2 cos 2 i a 8208s = s π sin 1 ( m m) 2 ( m) 2 cos 2 i m ( ) 8208s = sin cos 2 i ( ) = sin cos 2 i cos = 2 i = cos 2 i = cos 2 i = cos 2 i = cos 2 i = cos i rad = i π = i = i (4.20) Looking at the ETD website we can see that the inclination of the orbit is which is very close to the value we calculated. Now using the actual inclination we can now use equation 4.16 to work out the impact parameter, b. b = acosi b = cos( 87.84π 180 ) b = m b = R (4.21) 43

46 4.7 Mass of Exoplanet and Eccentricity of Orbit Using the radial velocity method we can determine an accurate value of the mass of the exoplanet. Now we can use equation 2.13 from chapter two to calculate the amplitude of the radial velocity using the mass of the exoplanet, M P, the mass of the star, M, the period of the transit, P, the eccentricity of the orbit, e which is assumed to be 0, and the inclination of the orbit, i. We can now insert these values into the following equation: 2πaM P sini A RV = (M P + M )P 1 e 2 2π sin A RV = ( ) A RV ms 1 (4.22) Figure 4.11: Diagram showing the radial velocity of Corot2b corresponding to its phase[29] 44

47 Chapter 5 Limb Darkening 5.1 Geometry of the Transit We can picture the geometry of the transit as two intersecting discs, the exoplanetary disc and the stellar disc. The area of the stellar disc is occulted during the transit of the exoplanetary disc. Using the geometry of figure 6.1 we can calculate the area of the stellar disc occulted 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 star and the exoplanet, based on figure 3.13 pg105[2][17] Now using figure 6.1 we can calculate the eclipsed area, A e which is the area of the intersection of the two discs. Looking at the diagram we can see that A e is equal to twice the area of shape (e), we can then calculate the transit depth by taking A e away from the area of the 45

48 stellar disc. A e = 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 planetary disc as equal to p, so we have p = R P R (5.2) Now we can calculate the area of each of the shapes using equation 5.2 to get the following: (d) = πr 2 P α 1 2π = p2 R 2 α 1 2 (b) = πr 2 α 2 2π = R2 α 2 2 (a) = R ξr sinα 2 2 = ξr2 2 sinα 2 (5.3) Using (a) from the diagram we can evaluate the angles α 1 and α 2 by using the cosine rule. cosα 1 = p2 + ξ 2 1 2ξp cosα 2 = 1 + ξ2 p 2 2ξ (5.4) We can then obtain the following using Pythagoras s Theorem sinα 2 = 4ξ 2 (1 + ξ 2 p 2 ) 2 2ξ (5.5) We can now express equation 5.1 using these results to get the following A e = 2 p2 R 2 α 1 + R2 α 2 ξr2 4ξ 2 (1 + ξ 2 p 2 ) ξ A e = R 2 p 2 4ξ 2 (1 + ξ 2 p 2 ) 2 α 1 + α 2 2 (5.6) 46

49 The area of intersection between the two discs is in terms of ξ which is defined as the parameterised distance from the centre of the star to the planet. The distance between the centre of 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 = 2π P (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 angular inclination of the orbit. Since the component of the displacement s(t) is foreshortened the actual observed displacement is given by a cos i cos ωt. Therefore to calculate s(t) we need to use the actual observed displacement along with Pythagoras s theorem, in doing so we obtain s 2 (t) = (asinωt) 2 + (acos(i)cos(ωt) 2 s(t) = a[sin 2 ωt + cos 2 (i)cos 2 (ωt)] 1 2 (5.9) To generalise A e 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

50 Using equations 5.7 and 5.9 we can express ξ in terms of time to obtain A e (t). For case 1 where the planetary disc falls outside the stellar disc, if the distance between each of the disc centres exceeds the sum of their radii the we get the following 0 if R + R P < s, A e = 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 each of the discs centres is less than the difference of their radii then we get the following πr 2 P if R R P s, A e = πp 2 R 2 if R (1 p) ξr, πp 2 R 2 if 1 p ξ (5.11) Bringing case 1 and case 2 together we can create a generalised form A e = R 2 ( p 2 α 1 + α 2 4ξ 2 (1+ξ 2 p 2 ) 2 2 ) 0 if 1 + p < ξ, if 1 p < ξ 1 + p, πp 2 R 2 if 1 p ξ (5.12) Therefore A e 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 the intensity over the surface area of the star is equal to the total flux; this is achievable when the intensity of the star is equally distributed. For our analysis we need to restrict it to axially symmetric intensity distributions [2], therefore I = I(r ), where r is the measurement from the centre of the stellar disc as shown in figure 5.3. Therefore it can be shown that F = disc I(r )da = r =R r =0 I(r )2πr dr (5.13) 48

51 Figure 5.3: Diagram showing the intensity of light distributed across the stellar disc, based on figure 3.14 pg109[2][17] Using a stellar disc with uniform brightness we can say that the intensity therefore is constant, I = I 0 for the entire disc. Therefore we can simplify equation 5.13 into the following F = r =R r =0 I 0 2πr dr r =R = 2πI 0 r dr = 2πI 0 [ = πi 0 R 2 r =0 ] r =R r 2 2 r =0 (5.14) We can now write an expression for F in terms of time. F is the flux that is lost when the stellar disc is occulted by the exoplanet during its transit phase. Therefore to obtain F we must integrate the intensity, I(r ), over the occulted area to get the following equation F = occulted area I(r )da (5.15) Since the brightness is uniform we can once again let I = I 0, therefore we get F = I 0 occulted area da (5.16) Looking back at the previous section we can see that the integration is in fact equal to the A e 49

52 (ecclipsed area), hence our equation becomes F = I 0 A e (5.17) So for the simple case where the intensity is uniformly distributed across the star the corresponding change in flux is I 0 A e. On the other hand we need to take into account the normalised axial coordinate, r, this is used for the case where the distribution of intensity is non-uniform or the limb darkened case. We therefore define r so that the centre of the stellar disc 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 A e made of series of partial annuli, based on figure 3.15 pg111[2][17] Figure 6.4 clearly shows us that the eclipsed area within the stellar disc of radius rr is equal to the same area that would be eclipsed if the star was radius rr than radius R. Hence we can use the equation we derived early for the eclipsed area, A e, but with a change of variables so that we can integrate over the axially symmetric disc. The shaded area in figure 6.4 shows the additional eclipsed area when increasing r by an amount dr, we define this additional area as da(r). Therefore the shaded area can be expressed as the following da(r) = da e dr dr (5.19) Now we can write the expression using the star in figure 6.4 with radius rr. Therefore we will need to change our variables for the function A e which previously used p and ξ. Since p = R P R changing the radius changes p to p r. ξ has a similar change as p, ξ changes to ξ r. 50

53 Therefore using equation 5.12 the eclipsed area of the star is as follows r =rr 0 ( da e dr dr = A e rr, p r, ξ ) r ( = r 2 A e R, p r, ξ ) (5.20) r Comparing equations 5.19 and 5.20 we can see that the integrand on the left hand side of equation 5.19 is equal to that of the right hand side of equation Hence we get da(r) = d ( [r 2 A e R, p dr r, ξ )] dr (5.21) r 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 r=1 F = I(r)A(r) r=0 r=1 = r=0 I(r) d dr ( [r 2 A e R, p r, ξ )] dr r (5.22) Therefore using equation 5.22 we can calculate the exact shape of a transit light curve using the 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 known as plasma.the light that is emitted from the star comes from various levels in the stars atmosphere. The probability that the photon emitted can escape within a particular layer is dependent on the optical depth of that particular layer. For a given frequency, ν we can calculate the optical depth, τ ν, as being equal to the integral of the opacity, κ ν, multiplied by the density, ρ(s) along the path taken by the photon. Hence we have τ ν = X ρ(s)κ ν ds (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 the frequency of the radiation. The probability that the photon will not be absorbed or scattered as it travels alongs its path is e τ ν, hence the ratio between the emitted intensity, I emitted, and the emergent intensity, I emergent, is as follows I I emitted = e τ ν (5.24) 51

54 Figure 5.5: Diagram showing the cross section of a star with depths and directions of emerging photons, 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 an angle γ from the the radius vector. The two photons in figure 6.5 have been emitted at the same depth in the stellar atmosphere. For the photon emitted in the limb of the stellar atmosphere to reach the observer it must pass through a larger path length, s, of the stellar atmosphere. Therefore for photons emitted at a depth of h in the stellar atmosphere, the length of the path can be calculated using the following equation, where µ = cosγ. s h cosγ = h µ (5.25) Since the optical depth for a particular physical depth grows larger towards the limb of the star, a small fraction of the photons emitted from the stars limb will escape and reach the observer. Therefore as you look at the limb of a star it appears to be dimmer than the centre of the stellar disc. Photons that are emitted radially outwards mean they travel a shorter path. This also means that the probability of photons escaping from deeper in the stellar atmosphere towards the centre of the stellar disc is higher. Hence as you observe the star, the edges appear redder than the centre. This is because photons from deeper layers approximate 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. This is dependent on opacity and emissivity at each depth within the stellar atmosphere which in turn is dependent on the wavelength and the thermodynamic properties of the stellar atmosphere at each point. Hence limb darkening is dependent on the spectral type and 52

55 composition of the star[2]. The following laws are models of what happens at the limb of a star, since the only star we can observe with great detail is our sun these laws are not precise. Using these laws we can simulate the data to obtain the finest representation. 1. The Linear Limb Darkening Relationship: We define u as the limb darkening coefficient which governs the the gradient of the intensity drop between the centre and limb of the disc. I(µ) = 1 u(1 µ) (5.26) I(1) 2. The Logarithmic Law: similar to linear law but uses two limb darkening coefficients u 1 and ν l. I(µ) I(1) = 1 u 1(1 µ) ν l µln(µ) (5.27) 3. The Quadratic Law: I(µ) I(1) = 1 u q(1 µ) ν q (1 µ) 2 (5.28) 4. The Cubic Law: I(µ) I(1) = 1 u c(1 µ) ν c (1 µ) 3 (5.29) Laws 2, 3 and 4 are more complex relationships but give more flexibility to allow empircal data to be fitted more closely. The drawback with these three laws is having two coefficients that need to be determined or arbitrarily fixed. 53

56 Chapter 6 Modelling of the Transiting Exoplanetary System 6.1 Exoplanetary System Properties The star, Corot2a, has the following characteristics: Spectral Type: G7V Mass (M ): 0.97 ± 0.06M Radius: ± 0.018R Metallicity Fe/H: 0 ± 0.1 Right Acension: 19h27m0.6496s Declination: Magnitude V: The exoplanet, Corot2b has the following properties: P = P = days R P = 1.429R J a = AU v = ms 1 T dur = 136.8mins i = rad = b = R 54

57 6.2 Maple Model factoring in Limb Darkening Using the characteristics of the exoplanet and its host star that we collected we can now begin to produce a visual representation of the transiting exoplanetary system. Using Maple we aim to generate a theoretical light curve presented over the data retrieved from the ETD website for Corot2b. After generating the theoretical light curve we can then implement the limb darkening laws that were derived in chapter 5 to produce theoretical light curves that factor in limb darkening. To begin our maple program we must first use the restart command to reset all the variables and 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 scale the radius of the planet so it is in units of R. We then input and define these variables along with the impact parameter, b, in ourmaple program. Then we calculate x which is defined as the the starting point of the exoplanet on the x-axis, which is calculated using Pythagoras s theorem. Next we begin to create a basic visual representation of the transiting exoplanetary system using Maple. We set the centre of the star at [0,0] with radius Rstar = 1 and make the line around the circumference of the star thicker and colour it yellow. Then we set the centre of the exoplanet at [ x, b] = [ , ] with radius rplanet = and 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 calculating the distances between the centre of each of the discs, the maximum distance between them we define as dmax where the exoplanet shall begin its transit. Then we calculate the maxi- 55

58 mum distance from the centre of the exoplanet and the y-axis suing Pythagoras s theorem and define this as ymax. Then we can begin to set up the animation. We begin by setting A as the varying x coordinate using the a sequence that involves ymax and the variable i = 0,1,2,...,20. The sequence will begin at ymax and then will increase in sizes of 10 1 ymax, therefore setting the number of steps to 20 means that the exoplanet will complete its animation 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 to produce a light curve for the case where there is no limb darkening. Therefore we require the radius of the exoplanet and star as well as a variable which we define as y. For this formula we require several other variables to evaluate the transit light curve which we then define for this function using local. We define rmax as the maximum distance of d when the exoplanet disc is about to intersect with the stellar disc, hence rmin is the minimum radius. We define dmax similarly to what we 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

59 between the centre of each disc. If y (impact parameter) is less than rmin (the minimum distance 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 discs by 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 the eclipsed area we define as B. Now we can input the values of the exoplanet and the stars radius into our equation B which we define as B1. This process leaves us with the function we need to plot our transit light curve. The following command calls on the function of for our transit light curve using the variables we have obtained. Now that we have our light curve we can now call on our data collected from the ETD website and plot the light curve over the data. Using our excel file we copy the time in hours 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 light curve. 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 the axis values since to fit in with the beginning and end of the transit. 57

60 Now we begin to try and implement the limb darkening laws using Maple. We begin by defining 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, hence we define ds as previously stated before. Looking back at the previous chapter the limb darkening law requires a coefficient, u. We can then input this coefficient into our linear limb darkening law which, we define this law as i which requires the value ds. To calculate the total intensity of the stellar disc we must integrate over the stellar disc, which we 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 eclipsed area is used. We once again define rmax, rmin, dmax and dmin as we did previously for our transit light curve function. 58

61 The eclipsed area, A, is calculated similarly as for the transit light curve function. We also define 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 multiplied by the linear limb darkening mid point I(1) We can now proceed to construct the plots that are required to build our model factoring in the limb darkening of the light curve. We define p1l as the plot showing the characteristic dip in the light curve. We set the light curve to begin and end with a y-value of 1, the minimum point of the dip is set from dmin to dmin which is defined by p5l where the maximum 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 exoplanet and 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 we used previously. 59

62 Figure Student Number: , Unit Code: M301 The new graph now shows the expected transit light curve along with a series of plots with limb 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 online simulator to construct the theoretical light curve we would observe not factoring in limb darkening. The simulator can then process the data and produce a simulation model of the transiting exoplanetary system as viewed from the side. From the simulation we can see the transit depth simulated without the effects of limb darkening. We can also observe the effects that noise has on the transit light curve. Also we are able to alter the sizes of the planet and star and observe the changes in the light curve from these adjustments. From altering the mass of the planet we can see that this has an effect on the eclipsing time and the orbital period. Therefore if the mass of the exoplanet 60

63 was reduced we would see a orbital period would increase hence therefore increasing the mass would decrease the orbital period. It can also be noted that making adjustments to the radius of the planet has an effect on the transit depth, we have already seen this in chapter 4, hence large planets will result in a larger transit depth and smaller planets will have a smaller transit depth. Altering the eccentricity of the exoplanets orbit effects the length of the transit period as well as the time the planet spends in at mid-transit. By changing the size 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 can compare if we had any original data to assess its accuracy. I have chosen to set the noise level as to approximate the data on Corot2b retrieved from the ETD website. 61

64 Figure 6.3 We can see from figure 7.3 the representation of the radial velocity curve for Corot2b. To calculate the curve we require various planetary and stellar characteristics as we did for the transit light curve simulator. From the radial velocity curve we can gain a glance at the orbit we would expect the exoplanet to have and the effect that the exoplanet has on its host star as they both orbit the barycentre of the exoplanetary system. We can also see a geometrical visualisation 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 dimensional model of the Corot2 planetary system. After opening Autodesk Maya please follow these instructions, if required there are online tutorials for guidance on sites such as Youtube. 62

65 63

66 The end result is the following model which can be seen in the following five figures as well as the CD that contains video footage of the model performing the transit from various angles. 64

67 Figure 6.4: Exoplanetary Sytem as viewed from Corot2b Figure 6.5: Exoplanetary Sytem as viewed from Corot2a 65

68 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 it follows its orbit 66

69 Figure 6.8: Exoplanetary Sytem as viewed from the side as the exoplanet performs its transit 67

70 Chapter 7 Conclusion and Recommendations 7.1 Detection Methods and Observations Analysing the various detection methods that can be implemented to discover new exoplanetary systems revealed that the methods are all on equal standing. It can be seen that the different methods each had better accuracy for detecting specific types of exoplanet, for example the transit method that was implemented in this project is best used to find large exoplanets orbiting small stars. The dip that we see from the transit light curve produced by 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 characteristics of the exoplanetary system that we require for modeling. Researching into the radial velocity method we can see that it has the ability to find multiple planets orbiting a single star, though it does have its limitations. The radial velocity method requires that the star in the exoplanetary system be bright enough that you can measure the Doppler shift in the stars light, this means that the star requires to be closer to the observer to increase brightness or be of significant size and mass. Using both the transit method and the radial velocity method together we are then able to determine the characteristics of the exoplanetary system. Using the astrometry requires a high level of precision and the constant monitoring of the star you are observing. The advantage of this method is that it can too detect multiple planets orbiting around a single star. This method does require however sufficiently large enough planets to exert a pull on the star, therefore terrestrial planets are unlikely to be discovered by this method. The use of direct imaging has progressed further and has been found to be very useful in studying our own solar system and its development. This method however is incapable of detecting planets beyond our solar system therefore it is not implemented in searcing for terrestrial planets. Due to Microlensing s occurence factor it is quite unreliable. Therefore at this point in time we can not conclude that there is one single method of detection that can determine all characteristics of an exoplanetary system but requires the combination of methods such as the transit and radial velocity method. Using both of these methods combined can give us the full picture of the exoplanetary system and the basis for a model. 68

71 7.2 Light Curve Construction and Limb Darkening This project has shown the methods of observation and data analysis. Making my own observations has been quite unsuccessful due to many factors such as weather, hence I have used data from the Exoplanet Transit Database to supplement this. Using the data obtained from the ETD website we were able to generate a transit light curve on excel, minitab and maple. We were also able to use the data collected and the transit light curves to determine the exoplanetary systems characteristics to a good degree of accuracy. Our research into limb darkening showed that the expected light curve is not always what we obtain from observations. We also discovered that the intensity of the star is not distributed across the entire surface equally, this leads to the limbs of the star to appear darker and dimmer. Therefore the bottom of the light curve becomes rounder dure to limb darkening. Using geometry we were also able to show how we can calculate an approximation to limb darkening using the limb darkening laws which could then be applied to out transit light curve to model the limb darkening effect. 7.3 Modelling Using maple we were able to produce an expected light curve which could then be plotted against the data we collected. We were also able to generate a simplistic two dimensional model of the transit for Corot2b, this gave a visual representation of our exoplanetary system as viewed from the side. Using a simplified version of the limb darkening laws we were able to implement them to create a set of light curves that factored in limb darkening. We plotted these against our data and could see that they fitted the data more closely. Using Autodesk Maya we were able to generate a three dimensional model that no previous student has ever done 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 scaled down. 7.4 Recommendations for future investigation Future students could look at taking multiple sets of images or data from the ETD website to compile them together to create a better representation of the light curve with more data points. This would then lead to obtaining better characteristics for the transiting exoplanetary system they are studying. Future student could also look into furthering the work with the Maple program by improving the limb darkening laws by using algorithms to produce a better light curve factoring in limb darkening. Future students could also try and calculate the orbital period directly by using the transit predictions from the ETD website. Representing the error of the light curve can be quite problematic but is an area of research that a future student could undertake. The theoretical habitable zone calculations have recently been looked into due to that they do not always give an accurate representation of the habitable zone as we can see in chapter 1. Therefore a student could look into the habitable zone calculations and how they could be improved to provide a more accurate answer. Future students could look into improving the Autodesk Maya model that I have created, a student could try and model limb darkening of the star or try and add in orbital rotation of the planet for example. 69

72 Bibliography [1] wasp17. Nasa/Hubble (2012). Huge new planet tells of game of planetary billiards. Retrieved January 22, 2013, from [2] Haswell, C. A.(2010). Transiting Exoplanets. Cambridge: University Press. [3] Norton, A. J. (2004). Observing the Universe: A Guide to Observational Astronomy and Planetary Science. Cambridge: University Press. [4] Bruce, G. (2007). Exoplanet Observing for Amateurs EOA.pdf. Hereford, Arizona, USA: Reductionist Publications. [5] Wikipedia (2013). Extrasolar Planet. Available: Extrasolar_planet. Last accessed 22th January [6] Leake, J. (2010, April 26). Extraterrestrials are almost certain to exist says British theoretical physicist Stephen Hawking. Daily Telegraph. Retrieved December 10, 2012 from extraterrestrials-are-almost-certain-to-exist-says-british-theoretical-physicist-ste story-e6frev [7] Rincon, P. (2012, October, 15). Planet with four suns discovered by volunteers. BBC NEWS science and environment. Retrieved from science-environment [8] Wikipedia (2013). Planetary Habitability. Available: Planetary_habitability. Last accessed 25th January [9] Wikipedia (2013). Photosphere. Available: Photosphere. Last accessed 25th January [10] Wikipedia (2013). Main Sequence. Available: sequence#formation. Last accessed 25th January [11] Wikipedia (2013). Habitable Zone. Available: Habitable_zone#Circumstellar_habitable_zone. Last accessed 26th January [12] Wikipedia (2013). Super Earth. Available: Super-Earth. Last accessed 26th January [13] Wikipedia (2013). Coronagraph. Available: Coronagraph. Last accessed 28th January [14] Wikipedia (2013). Fomalhuat. Available: Last accessed 28th January

73 [15] Wikipedia (2013). Methods of detecting extrasolar planets. Available: wikipedia.org/wiki/methods_of_detecting_extrasolar_planets#astrometry. Last accessed 30th January [16] Wikipedia (2013). Astrometry. Available: Last accessed 30th January [17] Stephens, T. (2012). Observation and Modelling Study of Exoplanet Qatar1b. Retrieved January 22, 2013, from Final Year Student Projects 2012: ac.uk/projects2012/thomasstephens.pdf. [18] Miller, P. (2012). An in-depth Study of the Detection, Observation and Modelling of Transiting Exoplanetary Systems. Retrieved January 22, 2013, from Final Year Student Projects 2012: [19] Wikipedia (2013). Kepler s Laws of planetary motion. Available: wikipedia.org/wiki/kepler%27s_laws_of_planetary_motion. Last accessed 30th January [20] Wallentinsen, D. (1985). Title. (Observation and Analysis of Eclipsing Binary Stars). Retrieved January 31st, 2013, from the Wallentinsen.com: com/binary/intro.htm. [21] Wikipedia (2013). Gravitational Microlensing. Available: wiki/gravitational_microlensing. Last accessed 2nd February [22] Pictures of Microlensing. Accessed 02/02/ / [23] Space news (2012). 100 billion planets in the Milky Way. Available: com/news/100-billion-planets-in-the-milky-way.html. Last accessed 2nd February [24] Exoplanet Transit Database (ETD) Transit Predictions. Accessed 25/02/ var2.astro.cz/etd/predictions.php?delka=359&submit=submit&sirka=51 [25] Exoplanet Transit Database (ETD) Corot2b Data. Accessed 28/02/ astro.cz/etd/etd.php?starname=corot-2&planet=b [26] Hampshire astronomical group website, details on 24 telescope. Accessed 25/02/ [27] SBIG Astronomical Instruments website, details on SBIG STL-1001E Specifications. Accessed 4th February STL1001E_specs_ pdf [28] Wikipedia (2013). Corot-2. Available: Last accessed 1st March [29] Wikipedia (2013). Corot-2b. Available: Last accessed 1st March [30] Circle-circle intersection theory. Accessed 3rd March wolfram.com/circle-circleintersection.html [31] Jupiter Texture. Accessed 4th March

74 Chapter 8 Appendices 8.1 Appendix A 72

75 73

76 Student Number: , Unit Code: M301 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 cloud coverage is optimum. It is not possible to get weather predictions weeks in advance. Therefore the chance of making Observations have to be calculated each week. Supplemental data will be required due to the limited amount of data that can be collected personally by making observations at the Observatory. Position and Phase of the Moon : To take observations of an Exoplanetary System the position and phase of the moon has to be taken into account. The Transiting Exoplanet would need to be the sufficient 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 organising meetings 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. 2 74

77 Student Number: , Unit Code: M 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 observations using AIP4WIN. 5. Access and availability to a Computer with Excell, Maple 16 and Matlab to analyse and model processed observations. 6. Access to resource material e.g. the Exoplanet Transit Database, Wikipedia, previous students projects, 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 the discovery of Exoplanetary Systems along with the aims and objectives of the project. 4. Chapter 2 ( Review of previous projects/research) - An evaluation of previous students work into the study of Exoplanets along with a review of their recommendations for future projects. 5. Chapter 3 (Exoplanet detection methods) - An investigation into the methods of detecting Transiting Exoplanetary Systems and the mathematics behind them. 6. Chapter 4 (Observations and Construction of the Transit Light Curve) - An overview of 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; and an 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 into the various problems that can affect the Transit Light Curve e.g. Limb Darkening. Also an analysis into the methods to overcome such problems followed by applying such methods to the previous Transit Light Curves. 8. Chapter 6 (Analysis of Exoplanetary Systems) - An analysis of the Transit Light Curve to determine various aspects of the Exoplanetary System e.g. The Semi-Major axis, The Orbital Speed, etc. 9. Chapter 7 (Modelling of the Transiting Exoplanetary System) - An overview of the process to develop models of the different Exoplanetary Systems using mathematical software. 10. Chapter 8 (Conclusions and Recomendations) - A discussion of the project results and recomendations to further the work of this project in the future. 11. Bibliography 12. Appendix A - Project Plan 13. Appendix B - Observation Schedule 3 75

78 Student Number: , Unit Code: M 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 problems detecting them, etc. To become proficient in using AIP4WIN to convert images into data that can be used to form a light 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. 4 76

79 Risk Risk ID 1 Unable to make Observations 2 Unable to get time on Computer 1 at TLC Risk Log Probability Severity Score Mitigation Plan Proximity Risk Response Use data from the online No later than 23 rd Contingency Exoplanet Transit December Database Try and secure a copy of No later than 6 th Transfer AIP4WIN to run on a Jan 2013 separate computer 3 Losing Data Back up data on a separate memory stick / HDD 4 Availability of Supervisor and/or Mentors 5 Availability of Mentors 6 Availability of Resources Organise specific times to meet up and discuss project in advance Ensure that I have more than one mentor Ensure that resources are obtained/booked in advance Up to the 23 rd April 2013 Up to the 23 rd April 2013 Up to the 23 rd April 2013 Up to 23 rd April 2013 Contingency Reduce Avoidance Accept Status Open Open Open Open Open Open 77

80 Student Number: , Unit Code: M301 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 each workpackage. 6 78

81 8.2 Appendix B - Images Images taken of the Exoplanet Qatar1b at Clanfield Observatory are included on a CD, these were 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 downloaded from 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 supervisor 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 transit which are included on the CD. 79