An Observational Study and Mathematical Model of the Transiting Exoplanet HAT-P-25b

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2 Abstract This project concerns the detection, observation and modelling of planets outside of the Solar System and demonstrates the mathematical processes involved in this fast developing area of astrophysics. Information is included about the various detection methods and the merits and problems with each one have been highlighted. Data collected at the Clanfield Observatory on the transiting exoplanet HAT-P-25b has been used to construct a light curve using Microsoft Excel. Using this light curve, various parameters of the exoplanet including its radius and impact parameter were calculated, which were later used in a Maple code to construct a theoretical light curve. The Clanfield Observatory data was then imported into the Maple file to fit with the theoretical light curve, before being compared to data retrieved from the Exoplanet Transit Database. The effects of limb darkening on the theoretical light curve have been studied, to identify whether this model fits much better with the actual transit data. It is demonstrated that mathematics is an essential tool when using the data to calculate particular properties of the planet, and model the transit more accurately. The original aims of the project are then discussed and conclusions drawn from the data in order to improve future work undertaken in this area. Finally some information on current research is included to highlight the uses of transit light curves in the study of exoplanet atmospheres as well as the discovery of starspots and exomoons. 1

3 Contents Nomenclature 5 Chapter 1: Introduction The Universe Exoplanets Exoplanet Habitability Project Aims.. 9 Chapter 2: Detection Methods Astrometry Radial Velocity Microlensing Direct Imaging and Coronagraphy Transit Method. 15 Chapter 3: Observations and Light Curve Construction Candidates Equipment Twenty Four Inch Newtonian Telescope CCD Camera Clanfield Observatory..20 Chapter 4: Image Manipulation and Light Curve Construction Image Manipulation: AIP4Win Light Curve Construction: Microsoft Excel.25 Chapter 5: Modelling Characteristics of HAT-P

4 5.2 Maple Animation Model Theoretical Light Curve. 37 Chapter 6: Modelling with Limb Darkening Transit Geometry Light Reduction Due to Transit Limb Darkening Laws Maple with Limb Darkening...48 Chapter 7: Discussion of Results Original Aims Colour Filters Observations Light Curve Construction Modelling Conclusions Future Work...53 Chapter 8: Telescope Missions and Current Research COROT Kepler Current Research Exoplanet Atmospheres Exoplanet Moons Starspots.. 55 References Appendices Project Plan..60 3

5 9.2 HAT-P-25b Clanfield Images AIP4Win Data Report HAT-P-25b Excel Spreadsheet and Graph Maple File Conversions The Drake Equation 64 4

6 Nomenclature The following is a list of the mathematical symbols that I have used throughout this project with their meanings a = semimajor axis a = distance of the star from the centre of mass a p = distance of the planet from the centre of mass AU = Astronomical Unit (distance between Earth and the sun) β = angular displacement b = impact parameter d = distance G = gravitational constant F = flux ΔF = change in flux m = magnitude M = Solar mass M = mass of the host star M J = mass of Jupiter M p = mass of the planet π = pi P = orbital period R = Solar radius R = radius of the star R J = radius of Jupiter R P = radius of the planet T dur = duration of transit v = velocity 5

7 Introduction CHAPTER The Universe The Universe is extremely vast, containing billions of stars formed across billions of galaxies. The observable Universe is everything that can be detected from our viewing point on Earth, and scientists have calculated that 13.8 billion light years can be seen in each possible direction (Taylor Redd, 2013). This figure corresponds to the furthest distance that light has travelled to reach us since the Big Bang, but it is widely accepted that the Universe as a whole is infinite (Dinwiddie, Gater, Sparrow, & Stott, 2012, p. 9). The Universe is also in a state of expansion and is therefore becoming more disperse (Taylor Redd, 2013). The Earth in comparison is almost inconceivably small and so it is no surprise that throughout history, humans have speculated the existence of planets beyond our own solar system and even the possibility of extra-terrestrial life forms. The ancient Greeks were aware of the local planets, recognising them as objects that moved among the stars ( Greek Astronomy, 2013) and these planets have become points of interest over the centuries. Our Solar System is made up of eight planets orbiting a central star, the Sun. The Sun itself is not particularly uncommon in size, luminosity or temperature and so it is sensible to assume that other stars could be part of a similar system, that is, exoplanets orbiting a parent star Exoplanets An exoplanet is defined as any planet that orbits a star outside of the Solar System (Mason, 2008). It was not until 1995 that the first exoplanet orbiting a main sequence star, 51 Pegasi b, was detected by Mayor and Queloz with the radial velocity method (see section 2.2). This planet has an orbital distance of 0.05 AU and has a mass of around 0.46 M J (Mason, 2008, p. 1), meaning that it is much larger than the Earth and much closer to its host star. It has an orbital period of 4.2 days (Haswell, 2013, p. 47), which is much shorter than any of the planets in the Solar System. As of 2 nd January 2014, a total of 1055 extra Solar planets have been discovered across 801 planetary systems, including 175 multi planet systems according to the Extrasolar Planets Encyclopaedia ( Catalogue, 2014). There are several classifications of planets which are as follows: Terrestrial Planets Terrestrial planets are rocky planets comparable to Mercury, Venus, Earth and Mars. It is speculated that these are smaller exoplanets, as in our own Solar System. Terrestrial planets have a metal core, with iron in particular abundance, which has an outer layer of silicate rock. Surfaces of these planets can vary from mountains to craters, but all are solid. Like the terrestrial planets of our Solar System, exoplanet atmospheres can be dense like that of Venus or almost non-existent, similar to Mars (Cessna, 2010). 6

8 Gas Giants As their name suggests, gas giants are almost totally composed of gaseous elements. They are much larger than terrestrial planets, making them easier to detect outside of our Solar System. Therefore, many of the first exoplanets discovered were of this kind. These planets do not have a solid surface and are comparable to Saturn and Jupiter in our own Solar System. For this reason, they are often call Jovian planets. Gas giants have a core that is referred to as rocky ; however the specific compositions of the cores of planets such as Jupiter are a matter of conjecture (Cessna, 2009). Most of the mass of a gas giant planet is contained within a thick atmosphere composed of Hydrogen and Helium with traces of heavier elements including Oxygen and Carbon. Ice Giants Similarly to gas giants, ice giants are believed to have a rocky core. It is thought that this core can be surrounded by water, methane and ammonia in a liquid state. This type of planet will have a thick atmosphere similar to gas giant planets, containing hydrogen and helium. Ice giants are usually found further from their host star than the above types and are most similar to Neptune. 1.2 Habitability The discoveries that have been made so far in exoplanetry science have excited scientists, philosophers and amateur astronomers alike. However the question that has fascinated humanity for centuries has yet to be answered; Are we alone? As this field of study develops and the sensitivity and quality of instruments improves, astronomers are getting increasingly close to detecting possible habitable planets. The principle factors affecting whether a planet can be habitable are its location both in the Milky Way, and its orbital distance from the parent star. The galactic habitable zone is the area at optimum distance from the centre of the Galaxy. Stars close to the centre are subject to excessive radiation from a supermassive black hole and supernovae explosions which are more common due to the density of stars, while stars at too great a distance from the centre may not have the abundance of heavier elements required for life as we understand it to develop (Freedman, Geller, & Kaufmann, 2011, p. 756). The circumstellar habitable zone (sometimes termed the Goldilocks zone by the media) is defined as the distance from the parent star at which liquid water can exist and the planet can sustain a state of equilibrium ( The Habitable Zone, n.d.). This distance depends on the size, luminosity and temperature of the parent star and it is believed that the entire planet must orbit within the zone. The location of the planetary system is not the only factor that needs to be considered when searching for so called habitable planets. There exist different classifications of stars and many 7

9 astronomers believe that a planet s chance of habitability is connected to the characteristics of its host star. In 1992, previous to the Mayor and Queloz discovery of 51 Pegasi b, two planets were discovered by Wolszczan and Frail orbiting the pulsar PSR B It was confirmed that these planets could not be habitable as they are constantly being bombarded by radiation from the dead neutron star they orbit ( Planet Quest Historical Timeline, n.d.). Figure Hertzsprung-Russell diagram ( Spectral Type and Luminosity Class, 2013) Unlike pulsars (or dead neutron stars), main sequence stars fuse hydrogen into helium in their cores and are the most stable. These are very common with around 90% of all stars being described as main sequence (Taylor Redd, 2013), and are the most popular candidates for speculated habitable exoplanets. Figure shows a Hertzsprung - Russell diagram which illustrates the abundance of main sequence stars. Stars evolve and do not remain in one section of the above diagram indefinitely. However, most of their lives are spent on the main sequence, accounting for the large number of main sequence stars found in the Universe. There are many dedicated astronomers working in this area and research is constantly taking place. This is demonstrated in particular by the space based missions, Corot and Kepler (discussed in Chapter 8) which have been successful in the discovery of exoplanets. 8

10 It can also be seen from the volume of results on the online Exoplanet Transit Database which is frequently updated, that exoplanetary science is an important area of interest for the scientific community, much as the study of our own Solar System was in the time of the ancient Greeks. 1.3 Project Aims This project focusses on the observation and mathematical modelling of a transiting exoplanet. The main aims are to collect and analyse astronomical images and create a light curve in Microsoft Excel which can then be used to calculate various parameters of the exoplanet mathematically. The data should then be compared with a theoretical light curve before the effects of limb darkening (chapter 6) are explored. Overall, I want to demonstrate the importance of mathematics as a tool in the collection, analysis and modelling of data for exoplanet systems. 9

11 Detection Methods CHAPTER 2 To date, over 1000 extra solar planets have been detected in the past 22 years, and this figure is still increasing. Astronomers have even begun to discover extra galactic planets using a technique called Pixel-Lensing which is an adaptation of the Microlensing techniques discussed in section 2.3 (Atkinson, 2009). In this chapter, I will explain how exoplanet discoveries have been made, together with their respective advantages and disadvantages. 2.1 Astrometry Astrometry is the one of the oldest methods used by astronomers and involves accurately measuring the spatial coordinates of a particular star as it moves across the celestial sphere (Yaqoob, 2011, p ). It is often thought that planets orbit a host star, but in reality both the planet and star orbit a centre of mass located between the two bodies called the barycentre, shown in Figure As the star usually accounts for the majority of the system s mass, the barycentre can be located within the star. This means that the planet will appear to orbit the star and the star itself will appear to wobble as it orbits the same point (Haswell, 2010, p. 26). This method measures the angular position and reflex ( wobble ) of a star with respect to the stationary background stars. The amplitude of the reflex is: a = a pm p M Equation Where a is the distance of the star from the centre of mass, a p is the distance of the planet from the centre of mass and M and M p are the masses of the star and planet respectively. I have added lines for a p in red and a in green on figure (Mason, 2008, p. 4) a + a p = a a P a Figure Diagram showing a planet and its parent star orbiting a centre of mass ( What s a Barycentre?, 2011) 10

12 If a star is found to have a regular reflex movement, then it is possible that it has a planet or planetary system orbiting it. This method makes use of Kepler s three laws of motion which are as follows: 1) Planets have an elliptical orbit where one focus is the host star. 2) The area between lines from the planet to the star are equal when measured for an equal amount of time. 3) The cube of the average distance from the star (a) is proportional to the square of the orbital period of the planet (P). ( Kepler s Laws of Planetary Motion, n.d.) P 2 a 3 Kepler s third law is particularly useful with this method when finding the mass of the planet and star in the relationship. This variation is as follows: (Haswell, 2010, p. 27) a 3 P 2 = G(M + M p ) 4π 2 Equation The equation for the angular displacement of the reflex movement of a star (β) is: β = a d Where d is the distance that the star is from the viewing point. By substituting a from equation the following is obtained: Which can be rearranged to give: βd = a pm p M β = a pm p M d Equation (Haswell, 2010, p. 27) Equation shows that as the mass of the star and its distance from Earth increase, the angular displacement decreases. It can also be concluded that a higher mass planet with a larger distance from its parent star will cause the value for angular displacement to increase. Therefore this method is more sensitive to higher mass planets orbiting far from a small star that is close to Earth. Advantages Certain properties of the planet can be calculated including its orbital inclination and the upper limit of its mass (Mason, 2008, p. 4). 11

13 This method is sensitive to planets orbiting further from their parent star. The further the planet, the further the barycentre will be from the star causing a larger reflex, thus making the planet more detectable. Disadvantages A high level of accuracy is required for these measurements and therefore specialist equipment is needed to generate results that are sensitive enough to confirm the existence of a planet (Mason, 2008, p. 4). The planet must be relatively close to Earth as is required for angular measurement (Mason, 2008, p. 4). However, it is very difficult to use this technique on Earth due to atmospheric distortion so space based telescopes are necessary. 2.2 Radial Velocity The radial velocity method is similar to astrometry in that it uses the fact that the planet and star orbit their centre of mass. The observer measures the light coming from the star as it travels either towards or away from them. The light waves from a star become compressed when it travels towards the Earth, meaning the light has the shorter wavelength and appears to be more toward the blue end of the visible spectrum. Equally when moving away the waves are more spread out making the light appear slightly redder (Planet Quest, n.d.). Astronomers measure the red or blue shifts in the spectral lines of the light compared with the average wavelength to determine the star s movement. This technique is most successful when finding larger mass planets that are close to their parent star, as they have a greater gravitational pull. The very first exoplanet orbiting a main sequence star was detected by using this method. Geoffery W. Marcy and R Paul Butler (University of California) have demonstrated the measurement of radial velocity changes of just 3ms 1 by letting the light pass through iodine paper (Carroll & Ostlie, 2007, p ). The iodine absorption lines are used as reference points for zero velocity and the star absorption and emission lines are then compared with these to find the radial velocity to a high level of accuracy (Carroll & Ostlie, 2007, p. 196). Advantages This was the first effective detection method and has paved the way for a vast area of scientific research and discoveries. Radial velocity has been one of the most successful methods, with a large number of planets being discovered (Mason, 2008, p. 1). As well as being a detection method, radial velocity can also be used as a tool along with other methods (such as transit observation) to confirm planet candidates and learn more about them. No exoplanet has been confirmed without the use of this method (Haswell, 2010, p. 41). Disadvantages The method is less effective for smaller mass planets or planets with a large orbit, which are currently thought to be the ones that are most likely to be habitable. However, as the 12

14 sensitivity of instruments increases it is becoming more possible to detect planets with larger orbits (Mason, 2008, p. 1). 2.3 Microlensing The microlensing technique (also known as gravitational microlensing) involves the observation of two stars referred to as the source star and the lens star. With this method, the observer would see the light from a distant source star bend around an intermediate lens star as the lens star passes close to or in front of it from their perspective on Earth. Figure Diagram showing the effect of a planet on the observed light curve of the source star ( Exoplanets- Microlensing, 2013) The lens star acts as a magnification tool for the light coming from the source star as shown in Figure If there is a planet orbiting the lens star, this too will have a magnifying effect causing a spike in the graph. Advantages This method can be used to detect lower mass planets than some of the other methods. This is because it depends on the ratio of the mass of the planet to the mass of the star (q) and the probability scaling on the sensitivity curve is shallower (Mason, 2008, p ). 13

15 Microlensing can be used to detect the furthest planets from Earth and be used in the search for planets outside of the Milky Way (Haswell, 2010, p. 44). Multi planet systems and even planet moons can be detected using gravitational microlensing as it is more sensitive than other methods of detection (Yaqoob, 2011, p. 58). Disadvantages Planet detection relies on the lens star passing in front of the source star from our perspective on Earth a chance occurrence that cannot be repeated. Therefore the data collected for a particular event can t be verified as each event is unique (Yaqoob, 2011, p. 58). 2.4 Direct Imaging and Coronagraphy Direct imaging is a technique in which a picture of the planet is achieved, in the same way that all of the planets in our own solar system were detected (Haswell, 2010, p. 20). This method relies on the planet itself being bright enough to be seen, something that is very rare as the light coming from it is usually a reflexion of the light of the parent star, which is extremely bright in comparison. Therefore to obtain a direct image, the planet must be large and orbit at a distance so that its image is not obscured by the glare of the star. Another factor affecting whether direct imaging is possible is the distance of the system from Earth. The further the distance, the smaller the angular separation of the planet from the star, making it more difficult to detect. As the visible light from the star is often too great, it can be advantageous to take an infrared image as the planet itself will emit infrared light (Yaqoob, 2011, p. 60). It is generally the case that the star s light emission peaks in the visible spectrum, while the planet peaks in the infrared, making the ratio of light more favourable in this case (Haswell, 2010, p. 22). Figure Image of the sun taken by NASA with the SOHO coronagraph ( Depth and Perception in the Solar System, n.d.) 14

16 Another way of getting around this problem is to use a coronagraph which is an instrument that blocks the light coming from the parent star in order to give the observer a better chance of seeing the actual planet. A coronagraph simulates a solar eclipse by using a disk to block the star light, leaving only the corona visible (Haswell, 2010, p. 24) demonstrated in Figure Advantages Direct imaging is the study of the actual planet and therefore provides conclusive proof of its existence. Disadvantages It is extremely difficult to capture an image of a planet outside of our solar system due to the ratio of the star to planet light being unfavourable. Therefore this has only been done successfully on very few occasions. 2.5 Transit Method When any planet moves in front of its parent star, it blocks a percentage of its light. If the system is positioned suitably with respect to Earth, it s possible to observe this by analysing the star s light curve, where a dip in the curve would imply the existence of a planet. Figure shows a graphed light curve dipping when the planet transits the star. The dip is the change in flux ( F), which when divided by the flux (F) can allow us to determine the planet s radius using the following equation: (Haswell, 2010, p.39). 2 F F = R P R2 Equation Figure Diagram showing the effect of a transit on the light curve of a star ( Discovery of Exoplanet Systems part 2, 2013) 15

17 In order for a transit to be detectable, the planet must move between its host star and Earth. Therefore it must have an orbital inclination of almost 90 in order to get the best results (Mason, 2008, p. 4). This method will be used throughout this project and I will therefore discuss the transit method in more detail in the following chapters. Advantages Certain parameters of the planet can be obtained such as its radius and mass (when used with radial velocity). In some cases, the atmosphere can be determined by studying the absorption lines in the spectrum (Mason, 2008, p. 7). As transits are periodic they can be observed on more than one occasion, making the data easier to verify and more reliable. Disadvantages The geometrical probability of an observable transit occurring can be very low particularly outside of our solar system (Haswell, 2010, p. 42). This is because we must be able to see the system from as close to edge on as possible. While this method can be used to learn more about known exoplanets, it is challenging to detect new transits. Large portions of the sky with many stars must be observed to make effective use of this detection method (Yaqoob, 2011, p. 57). 16

18 Observations CHAPTER Candidates Despite the large number of exoplanets discovered to date, finding a suitable transiting candidate is still a difficult task. There are various conditions that need to be satisfied, such as the following: Weather conditions - cloud cover makes observing difficult and often impossible, the clearest possible conditions are required to obtain the most accurate results. Observation position the location of the observation point must be relatively free of light pollution. Distance from the Moon and Sun - the ideal scenario is that the observed star is 90 away from both the Moon and Sun. This is because these objects make the entire sky appear brighter, reducing the ratio of brightness between the star and the background. For my purposes, a transit taking place 60 from the moon will be sufficient. Distance from the horizon - the star must be at least 30 above the horizon. This is because light from celestial objects at lower elevations have to pass through more of the Earth s atmosphere which absorbs light. Therefore lower stars appear to lose more light. Magnitude of the star - for my purposes, the star must have a magnitude brighter than +14 to be visible. Lower magnitude stars require longer image exposures, meaning that there would be less images and points in the resulting light curve. Reduction in magnitude caused by the transit - there must be a sufficient dip in the light coming from the parent star in order for the transit to be detectable. Transit time - the time of the transit has to be reasonable (less than 4-5 hours), bearing in mind that an extra 30 minutes to an hour either side of the event will be required to set up the equipment and take images of the star to compare with the transit images. The above conditions significantly reduce the number of suitable candidates and the next step is to choose from this vastly reduced list. In this case, the transit is chosen using the Exoplanet Transit Database (ETD) website ( by the Czech Astronomical Society. The first step is to enter the geographical coordinates of the observation point in the Transit predictions section shown in Figure My data was collected at Clanfield Observatory which has latitude and longitude -1.02, around 359 east as required ( Where We Are, n.d.). Figure 3.1.1: Screenshot showing the coordinates of Clanfield Observatory on the Exoplanet Transit Database 17

19 The ETD website then uses these coordinates to compile a list of transit candidates suitable for that specific location. Figure shows that the time and coordinates of the transit event are also given and it is possible to input the date on which you wish to observe. Figure 3.1.2: Screenshot of the Exoplanet Transit Database showing possible transit candidates that could be observed at Clanfield If a candidate is chosen on the ETD, then it is definitely possible to view from Earth. The probability of being able to observe a transit from Earth can be estimated as follows: (Haswell, 2010, p. 42) geometric transit probability = R + R P a R a Equation Where R and R p are the radii of the star and planet respectively and a is the semi major axis. My chosen star is HAT-P-25 (outlined in red in Figure 3.1.2), which has a radius of 0.959R ( HAT-P- 25, n.d.) and its planet HAT-P-25b has a semi major axis of AU ( HAT-P-25, n.d.). The radius of HAT-P-25 is originally given in terms of the solar radius as this makes it easier to compare with the sun. However, in order to use Equation in a meaningful manner the radius and semi major axis must be in the same units to obtain a percentage for the probability. 1R = m and 1AU = m 18

20 (Freedman, Geller, & Kaufmann, 2011, Appendices 6 and 7) It follows that 1R AU Using this, I can calculate the probability of being able to observe HAT-P-25b using Equation as follows: So according to this equation, there is roughly a 9.6% chance that this particular transit could be viewed from Earth. Fortunately, I know that this transit has been viewed before as the results from previous transits can be seen on the ETD. 3.2 Equipment Along with various computer programmes and some elementary equipment, I needed the use of two specialist instruments; the 24 inch Newtonian Reflector Telescope at Clanfield Observatory and a Charge Coupled Device (CCD) camera Twenty Four Inch Newtonian Reflector Telescope Traditional refracting telescopes have a convex glass lens which light passes through. The light is then collected and refracted causing it to converge to a focal point (Freedman, Geller, & Kaufmann, 2011, p. 130). The major drawbacks of refracting telescopes are that the lens can only be secured from the edges meaning that heavy lenses can become deformed under gravity, and that the lens must be ground to optical perfection (Carroll & Ostlie, 2007, p. 155). Figure The 24 inch telescope at Clanfield Observatory ( 24 Dome, n.d.) For my observations, a twenty four inch Newtonian reflecting telescope was used (Figure 3.2.1). Reflecting telescopes use a concave mirror to project parallel light rays to a focal point (Freedman, Geller, & Kaufmann, 2011, p. 135). This design is preferable, because the mirror can be supported 19

21 from behind, meaning that only one side must be ground and that it s less likely to deform (Carroll & Ostlie, 2007, p. 156). In order to get to the focal point, the light is reflected back. Unfortunately this creates a problem because if the observer wishes to view the focal point they will block out a large percentage of the light. This was resolved by Isaac Newton who used a flat mirror to alter the location of the focal point, by placing it in front of the original in 1668 (Freedman, Geller, & Kaufmann, 2011, p. 136). This type of telescope is now known as a Newtonian telescope CCD Camera For my purpose in this project, it was important to collect images of the observed event to analyse later, meaning that the use of a camera was essential. These images were required to have a long exposure in order to capture details that otherwise could not be detected. Ordinary photographic equipment is not appropriate for this work as it is not sensitive enough. If very old film cameras are used, only 2% of the light that hits the film is captured because roughly 1 in 50 of the light s photons actually reacts with the film (Freedman, Geller, & Kaufmann, 2011, p. 142). Fortunately, technology is consistently improving with digital cameras replacing film. For my observations, a CCD camera was attached to the telescope, with a lead into a computer for image viewing. CCD cameras contain a silicon chip and any light that falls on this chip is converted by the device to digital data (Dinwiddie, Gater, Sparrow, & Stott, 2012, p. 56). These images are more detailed because the CCD camera contains thousands of pixels, with the charge in each one proportional to the amount of photons that come into contact with it. Pixel charges are then individually analysed by the computer resulting in a much higher 35 in 50 of the light s photons reacting (Freedman, Geller, & Kaufmann, 2011, p. 142). 3.3 Clanfield Observatory As part of my project I wanted to collect my own transit data from Clanfield Observatory. The University has links with the observatory and the Hampshire Astronomical Group, which meant that I was fortunate enough to have Steve Futcher, David Harris and Chris Priest as my mentors and was also given access to the observatory equipment. My original plan was to use the 24 inch telescope to capture a full transit that I could later analyse. Unfortunately due to consistent poor weather conditions, I was unable to do this despite visiting the observatory on numerous occasions. However, thanks to the previous work of David Harris, I still had results from a transit for analysis. The following is the process that I learned about and completed twice, which would have been used to capture an actual transit: 1) Remove the protective cover from the telescope, uncover the mirror and release the clamps allowing it to move freely. 2) Attach the power supply to the drive system, allowing the celestial coordinates of the parent star to be set. The telescope is then automatically directed towards the transit and is calibrated making it move east to west to counteract the Earth s rotation. 20

22 3) Connect the CCD camera to the telescope eyepiece and ensure that the safety strap is attached. The camera must have a 12V power supply and a USB cable to a computer which collects and stores the images. 4) Open the CCD program on the computer and connect this to the camera. The computer is an important tool as it can be used to control the temperature of the CCD camera (colder images are less likely to have unwanted noise ). 5) Take a dark frame with an exposure of around 60 seconds to compare the pixels with the transit and subtract the images. 6) Ensure that the chains are released on the dome and that the slit in the dome is aligned to the telescope (so that the telescope view is not obstructed by the dome). 7) Collect images every minute and rotate the motorised dome every 20 minutes to ensure that it does not block the view of the telescope. An image exposure time of around 60 seconds is sufficient. After the implementation of the techniques outlined above, I have 160 images of HAT-P-25b in transit. These images will be analysed to construct a light curve and create a mathematical model of the transit in subsequent chapters. 21

23 Image Manipulation and Light Curve Construction CHAPTER Image Manipulation: AIP4Win To analyse the images that David Harris had taken of the HAT-P-25b transit mathematically, I needed to convert them to numerical data. This was done using the AIP4Win (Astronomical Image Processing for Windows) software which gives quantitative results from the original images. Figure Screenshot of AIP4WIN showing how to set up image manipulation After opening the programme, the first step was to select the Multiple Image Option as indicated in Figure 4.1.1, because I had 160 images that needed to be analysed. This brought up a new window titled Multi-Image Photometry which is shown in Figure Using the Select Files button, I imported all of the images. The first of the selected images was then displayed on the screen in a new window with the same title as the image file Screenshot of AIP4WIN showing the Multi-Image Photometry 22

24 I then had to locate my target star (HAT-P-25) using a SkyMap image of the same part of the sky and compare the positions of the stars. It can be seen from Figures that this is not necessarily an easy task as the telescope image shows more objects than the SkyMap one, making the correct choice less clear. Figure The SkyMap image (left) and observatory image (right) The stars in the observatory image in Figure are deliberately defocused so that the light is spread out allowing for a more accurate measurement of the light intensity (Morris, n.d.). Once I had identified the correct star, I had to select it by clicking on it. This brings up three circles that surround the star, the sizes of which can then be modified using the Settings tab on the Multiple-Image Photometry window. As the selected star was quite close to another, it was important to set radii values that would not allow the unwanted star to appear in the circles. I chose the values 15.2, 16.5 and 20.0 respectively for the radii of the circles (Figure 4.1.4), which fit the requirements. Figure The Settings tab showing the radii 23

25 Following this, I had to select two comparison stars by clicking on them on the same image (an example can be seen in Figure 4.1.5). These stars had to be of similar or brighter magnitude than my target star and could not be variable stars (eg pulsar with variable brightness over time). This is because AIP4Win subtracts the magnitude of the first comparison star, C1, from that of the target star, V, to demonstrate the target star s dip in light due to the transit. The first comparison star s magnitude is subtracted from the second, C2, to ensure that it was an appropriate choice for comparison. Figure Screenshot showing the target star (V) and two comparison stars (C1 and C2) AIP4WIN then returns two curves; V C1 and C2 C1 and the curves for the comparison stars in Figure are displayed in Figure Figure Screenshot showing the graphs of star comparisons 24

26 The ideal situation would be that the bottom graph in Figure is as close to a straight line as possible. This would indicate that their magnitudes of C 1 and C 2 do not change with respect to each other, implying that they are not variable stars and suitable to be used to compare to the transit star. This image shows an adequate choice of comparison stars as the data is fairly uniform across the horizontal axis. AIP4Win also returns a report detailing the specific numerical data for each image, which must be saved as it will be used in Excel to create the light curve of the target star in transit. The full report is included in Appendix 9.2 and a screenshot is also shown in Figure Figure Screenshot showing the report given by AIP4Win. 4.2 Light Curve Construction: Microsoft Excel After processing the images and receiving the data report constructed by AIP4Win, I had to use Microsoft Excel to construct my own light curve. My first step was to import the data by clicking on the Data tab, then From Text and selecting the appropriate file. I then had to choose the Fixed Width option so that I could arrange the columns correctly. This is shown in Figure Figure Screenshot showing the data input process on Excel 25

27 Following this, I deleted the columns that were no longer needed leaving me with the image number, the Julian day and V C 1. From these values I calculated some extra columns needed to compute the light curve as shown in Figure Figure Screenshot of the data in Excel used to construct the light The extra columns are explained as follows: 1) Time (days) The time in days of the corresponding image starting from 0. This was calculated by subtracting each Julian day value from the first. Excel input (entered into C2 and dragged down): =B2-$B$2 2) Time (hours) The time in hours calculated by multiplying the time in days by 24. Excel input (entered into D2 and dragged down): =24*C2 3) DM To calculate this column, I first calculated an average of the first 14 values for V-C1 (the light from the star in transit minus the light from the comparison star). Excel input (entered into N1): =AVERAGE(D2:D15) In theory, this should be the average value of light emitted from the star before the transit begins. This is then subtracted from the V-C1 column to give the difference in magnitude during the transit. 26

28 Flux Excel input (entered into F2 and dragged down): =D2-$N$1 4) DM Average This column is the ten point moving average of the difference in magnitude (DM) column. This was done by averaging a particular value with the 5 values either side of it. Excel input (entered into G7 and dragged down to G156): =AVERAGE(F2:F12) 5) Flux Finally, I calculated the change in flux which is used to determine the difference in brightness of the star. Given two magnitudes m 1 and m 2 the following relationship can be used: m 1 m 2 = 2.5log 10 ( F 1 ) F 2 (Carroll & Ostlie, 2007, p. 61) Therefore, upon rearranging this and substituting m 1 = V, m 2 = C 1 and F 2 = 1, it can be shown that the difference in flux F is: F = 10 (V C ) This was the equation I used in the spreadsheet for column G. Excel input (entered into G2 and dragged down): =10^(-F2/2.5) I then constructed the flux graph by selecting the Time (hours) and Flux columns and selecting the scatter graph option on the Insert tab. The resulting graph can be seen in Figure and the full spreadsheet is included as Appendix Clanfield Data Flux Time (hours) Flux curve plotted with Excel 27

29 Flux Flux Flux It can be seen from the graph that the light from the star HAT-P-25 is significantly reduced during the transit. I then plotted a four point moving average trend line, which can be seen in figure This helped with visual representation but not the necessary mathematical analysis. I decided to plot a second graph using data from the ETD to compare my graph to, this is displayed in figure Clanfield Data Flux ETD Flux Time (hours) Time (hours) Figure Flux graph with a four point moving average trend line Figure ETD data flux graph with a four point moving average I added a polynomial line of best fit of fourth degree to create the graph in figure This was selected visually using a trial and error method, but much better lines of best fit are calculated in chapters 5 and 6 in the modelling sections. This line is a smoother curve, but isn t as useful as it may seem, as the flux does not average out as expected before and after the transit Clanfield Data Flux Time (hours) Figure Flux graph with a fourth degree polynomial line of best fit 28

30 Flux Figure shows the ETD data graph without a trend line. It can be seen from this that the gradient of the flux is around zero before and after the transit as expected. I have used both sets of data for the next section to model my curve in order to make comparisons between the two ETD Flux Time (hours) Figure ETD data Excel flux graph Parameters Affecting the Light Curve Ratio of planet to star radius - the larger the planet radius is in comparison to the star, the greater the dip in the light curve will be. Magnitude of the star - the brighter a star is, the easier the dip in magnitude is to detect due to a higher signal to noise ratio (Haswell, 2010, p. 57). Length of the transit - the longer the transit, the wider the dip in the light curve (when viewed on the same scale). Impact parameter - if the full planet does not pass in front of the star from our viewing point, the dip will appear to be smaller. Standard Deviation The standard deviation of the data was calculated on the Excel Spreadsheet in order to give the error approximations required when calculating the parameters of HAT-P-25b. By inspection of the graph, I estimated the points at which the transit begins and ends and to two separate averages for the in transit flux and out of transit flux. I then calculated the flux deviation by subtracting the mean values from the flux column, before taking the mean to the squared flux deviation. Finally, I calculated the square root of the squared flux deviation, to obtain the in transit and out of transit standard deviation values. 29

31 Modelling CHAPTER Characteristics of HAT-P-25 The parent star of the transit I am studying (HAT-P-25) has the following characteristics: Star Mass: 1.01M Star Radius: 0.959R Right Ascension: (measured in hours, minutes and seconds) Declination: Magnitude: ( Planet HAT-P-25b, 2010) Orbital Period The orbital period (P) is the time that it takes the planet to completely orbit its host star. To measure this, several independently observed transits of the same star are required. (Haswell, 2010, p.51) P = T elapsed N cycles In equation 5.1.1, T elapsed is the time between two transits of the same planet measured from the same point on the light curve (for example the beginning of the transit). N cycles is the number of full transit cycles that can be observed in that time. As my observations only covered one transit and this equation requires more for accuracy purposes, I have used the standard given orbital period. P 3.65 days ( Planet HAT-P-25b, 2010) Equation Semi-Major Axis The semi major axis is the point of orbit at which the planet it furthest away from the star. Planets rarely have a perfectly circular orbital path and this is an important concept when considering the properties of exoplanet systems. It can be calculated if the orbital period (P) is known using equation a (GM ( P π ) ) Equation (Haswell, 2010, p.91) To calculate the semi major axis of my planet, I need the following parameters: G = Nm 2 kg 2 (Gravitational constant) M = kg (Star mass in kilograms) P = s (Period in seconds) 30

32 Flux a ( ( ) ) 2π When converted to astronomical units this value comes out to be around , which is exactly the given value of a m ( Planet HAT-P-25b, 2010) a = AU Orbital Speed A planet s orbital speed varies depending on its position with respect to the host star. As orbits are often elliptical rather than circular, the planet will travel fastest at its minimum distance from the star and slowest when furthest away (deduced from Kepler s second law, chapter 2.1). The orbital speed (v) of a planet with a circular orbit is: (Haswell, 2010, p.91) v = 2πa P Equation Assuming a circular orbit and using the values for a and P from the previous sections, I can calculate the orbital speed of HAT-P-25b as follows: v = 2π m s Giving a value of ms 1 or around 136 ± 0.2 kms 1 (due to rounding). Planet Radius The radius of the exoplanet (R P ) can be calculated using the following equation: ΔF F = R 2 P Equation R2 ( The Transit Light Curve, n.d.) Where F represents the flux, ΔF is the change in flux or transit depth and R is the radius of the star. The change in flux is a value that can be obtained from the light curve constructed in chapter 4 by subtracting the minimum point (mid transit) from the flux value outside of the transit ETD Flux Time (hours) 31

33 Flux Using my light curve, I estimate the change in flux to be around = 0.02 and using equation 5.1.4: = R P ( ) 2 Giving R P 1.32 ± 0.17 R J The standard deviation for the out of transit flux is and the standard deviation for the in transit flux is This means that the flux ratio could be between and giving the lower limit radius of R J and an upper limit of R J The actual planet radius is: ( Planet HAT-P-25b, 2010) R P = 1.19R J Impact Parameter The impact parameter of a planet is the distance between the centres of the planet and star when their orbits are in alignment ( The Transit Light Curve, n.d.). To calculate this value for HAT-P-25b, I first need the transit duration (T dur ) which is the amount of time that the planet is actually in front of the star from our viewing point. This can be calculated using equation Therefore for HAT-P-25b: ( minutes) T dur P2R 2πa PR πa T dur 3.653days ( )AU AU π hours Equation Clanfield Data Flux Time (hours) The above graph is the one that I plotted using the data collected at Clanfield (figure 4.2.4). As the graph does not display a definite beginning and end to the transit, it is difficult to estimate the transit duration accurately from this. However, it can be seen from the red lines placed at he estimated beginning and end of the transit that the calculated value of 2.67 hours is a sensible value when compared to the actual data. 32

34 Flux The ETD data curve below gives a transit time of roughly = 2.7 hours which is a slightly better estimate when compared with the calculated value ETD Flux Time (hours) Figure shows a right angle triangle where a is the hypotenuse and b and i are an edge and angle respectively. Hypotenuse Adjacent Figure The impact parameter in terms of a right angle triangle ( The Transit Light Curve, Using trigonometry, specifically the rule cos (i) = adjacent hypotenuse, it can be shown that: (Haswell, p. 93, 2010) b = acos (i) Equation Using Pythagoras theorem, with R P + R as the hypotenuse (a) it can be seen that: l 2 = (R p + R ) 2 (acos (i)) 2 l = (R P + R ) 2 a 2 cos 2 (i) Equation

35 Where l is the distance between the centre of the planet and the centre of the star. The orbit of the planet is displayed in figure 5.1.3, which shows that a distance of 2l is observed. Viewing point Figure The orbit with respect to the observer ( The Transit Light Curve, n.d.) Again, by using trigonometry on the triangle I have outlined in red on figure 5.1.3, the following equation is obtained: ( The Transit Light Curve, n.d.) sin ( α 2 ) = l a Since the planet must travel 2πa from the observer s point of view to complete and orbit (where a is the radius of orbit), the duration on the transit in terms of α is: ( The Transit Light Curve, n.d.) Substituting for l from equation 5.1.7: T dur = P α 2π = P π sin 1 ( l a ) (Haswell, p. 94, 2010) T dur = P π sin 1 ( (R P + R ) 2 a 2 cos 2 i ) a Equation Since the value for T dur is known, along with all of the other require parameters, all that remains is to substitute these in and rearrange equation to get the angle of incidence (i). T dur = 9600s = s π sin 1 ( ( m m) 2 ( ) 2 m cos 2 i ) m 34

36 sin ( 9600π ) = ( m) 2 ( ) 2 m cos 2 (i) m m sin ( π ) = ( m) 2 ( ) 2 m cos 2 (i) m = m m cos 2 (i) m = m m cos 2 (i) m = m cos 2 (i) cos 2 (i) = cos(i) = = i = cos 1 ( ) = radians = This value is very close to the given value of 87.6 ( Planet HAT-P-25b, 2010) and considering the inevitable rounding errors when working with a large number of variables in different units, gives a good approximation. i = 87.6 = π radians Using equation it can be seen that the impact parameter b b = acosi = cos ( 73π 150 ) = AU This corresponds to a value of R 5.2 Maple Animation Model For this section I have used some of the parameters calculated in section 5.1 to create a model of the HAT-P-25b transit using a Maple code given by my supervisor Dr Michael McCabe. The first value I needed to use was the radius of the star which was set to the value of one for simplicity. This means that to calculate the radius of the planet I needed to calculate the radius of the planet as a proportion of the radius of the star. The planet radius is 1.19R J, so using the following conversions: 1R J = m 35

37 1R = m The ratio of the radius of the planet is: m m Finally I needed the impact parameter which I calculated above to be R Figure A screenshot showing the Maple code with my parameters Figure shows the Maple file with which I worked to create a mathematical model of the transit of HAT-P-25b. The first line implements the plotting tools that are needed in this program and the following lines are the parameters calculated in section 5.1. x is the distance from the centre of the planet and the centre of the star, calculated in figure by: x = (R + R P ) 2 b 2 This is just a variation of formula where l has been replaced by x and the impact parameter (b = acosi) is now known. The next part of the code adjusts how the graph will be displayed visually where c 1 is the star, centre (0,0) and radius 1, and c 3 is the planet, centre ( x, b) and radius rplanet. The star is represented by a yellow circle, while the planet is represented by a green one (figure 5.2.2). Figure A screenshot showing the Maple model with my parameters 36

38 5.2.2 Theoretical Light Curve The next step was to create a theoretical light curve and add my data to see how well it fits in the model. Figure shows a screenshot of the Maple code that was needed for this. Figure Maple screenshot of the theoretical light curve calculation It can be seen from figure that the theoretical light curve depends on the radius of the star and planet and the impact parameter. The steps are explained below: ds = d 2 + y 2 is the distance from the centre point of the planet disc to the centre point of the star disc rmax = R + R P is the maximum distance the centre of the star from the centre of the planet when the discs are about to intersect rmin = R R P is the minimum distance from the star centre to planet centre dmax = rmax 2 y 2 is the point where the transit starts if y < rmin then dmin = rmin 2 y 2 else dmin = 0 means that if the impact parameter is smaller than the minimum distance between the centre of the star and planet, then dmin is calculated using Pythagoras theorem, if not it is zero A is the area of intersection between the planet and the star when the planet is in transit (explained in depth in chapter 6) B is a constant value for when the planet and star discs are not intersecting B1 substitutes the parameters entered in the previous part of the code 37

39 p1 p5 plot the various sections of the theoretical light curve, p1 and p2 plot the egress and ingress respectively (p2 is a reflection of p1 since the theoretical light curve is symmetrical), p3 and p4 are the points before the ingress and after the egress respectively and p5 is the dip in the theoretical light curve Figure Maple screenshot showing the data input Figure shows that my own data for the HAT-P-25b transit was inputted into Maple as a text file to display on the same graph as the theoretical light curve. In this part of the code: pdata formats the points displayed from my data file HAT-P-25b for Maple.txt pdata2 = plottools[translate](pdata, 1.5, 0.003) translates the data points from the text file (move them left or right and up or down) pdata3 = plottools[scale](pdata2, 1.0, 1.0) scales the text file points plots alters the scale of the axis so my x-axis runs from 2 to 2 and my y-axis runs from to 1.01, resulting in the following graph: Figure Maple graph with theoretical light curve and Clanfield data 38

40 I also created a Maple light curve using the ETD data to compare this with using exactly the same method: Figure Maple graph with theoretical light curve and ETD data The theoretical light curve appears to be a relatively good fit for the actual data, however it can be seen in the graphs of figures and that the light curve has a completely flat bottom. A more accurate fit for the actual data would display a slightly curved bottom and this can be obtained by accounting for the effects of limb darkening which are explained in the following chapter. 39

41 Modelling With Limb Darkening CHAPTER 6 Up until this point, all calculations have been completed under the assumption that the star has uniform brightness. However, this is unlikely to be the case and the effects can be best seen when considering a picture of the Sun (Figure 5.1.1). Figure A NASA SDO image of the Sun with Venus in transit ( Astronomy Picture of the Day, 2012) Figure shows Venus transiting across the Sun and it can be clearly seen that rather than being a disk of uniform brightness, the Sun grows darker towards the edges. This effect is known as limb darkening and it is assumed that this is the case for most if not all stars. 6.1 Transit Geometry To fully consider this effect, one must first understand the geometry of the transit by representing the star and planet as intersecting disks. I constructed figure to show how this geometry works. Figure Transit geometry 40

42 The triangle in figure is split into smaller sections, the areas of which must be calculated to work out a value for the eclipsed area of the star (yellow circle) when the planet (green circle) crosses its limb. In figure 6.1.2, s(t) is a distance given by: Where: s(t) = ξr Equation (Haswell, 2010, p. 104) R p = pr and ξ = a (sin2 (ωt) + cos 2 (i)cos 2 (ωt)) R The parameter ξ is calculated using Pythagoras theorem on the diagram shown in figure showing the geometry from the perspective of the observer. Where cos (i) scales the term acos (ωt) according to the angle of incidence. Figure Diagram showing the geometry of the planet orbit The transit geometry in terms of ξ can now be represented the using figure in the following way: The area of the part of the triangle within the planet, a 1, is given algebraically by: (Haswell, 2010, p. 104) a 1 = p2 R 2 α 2 The area of the part of the triangle within the star, a 2, is given algebraically by: (Haswell, 2010, p. 105) a 2 = R 2 β 2 41

43 The area of the full triangle, using equation 6.1.1, is: (Haswell, 2010, p. 105) triangle area = R ξr sin (β) 2 Equation Equation can be used to calculate the value of the angles a and b in radians as follows: cos(α) = p2 + ξ 2 1 2ξp and cos(β) = 1 + ξ2 p 2 2ξ (Haswell, 2010, p. 106) It can then be shown with the use of Pythagoras theorem that: sin(β) = 4ξ2 (1 + ξ 2 p 2 ) 2 2ξ Finally, (Haswell, 2010, p. 106) A e = R 2 (p 2 α + β 4ξ2 (1 + ξ 2 p 2 ) 2 ) 2 Equation There are three cases that must be considered when calculating the area of intersection, either the planet disc is completely outside of the star disc, the planet disc is completely inside of the star disc, or the planet disc covers the limb (outside edge) of the star disc. Case 1 If the planet disc lies completely outside of the star disc, the eclipsed area (A e ) is equal to zero. This is the case if: R p + R < s Substituting s = ξr and R p = pr gives: pr + R = R (1 + p) < ξr Finally dividing through by R gives: 1 + p < ξ Therefore: A e = 0 if 1 + p < ξ Case 1 Case 2 If the planet is crossing the limb of the star, the eclipsed area is as calculated before in equation A e = R 2 (p 2 α + β 4ξ2 (1 + ξ 2 p 2 ) 2 ) 2 This is the case if: R + R p < s R R p 42

44 Substituting s = ξr and R p = pr gives: R (1 + p) < ξr R (1 p) Giving: 1 p < ξ 1 p Therefore: A e = R 2 (p 2 α + β 4ξ2 (1 + ξ 2 p 2 ) 2 ) if 1 p < ξ 1 p 2 Case 2 Case 3 If the planet disc is completely outside of the star disc, the eclipsed area (A e ) is equal to the area of the planet which is πr p 2. This is the case if: R R p s Substituting s = ξr and R p = pr gives: R pr = R (1 p) ξr Giving: The eclipsed area in terms of R is πp 2 R 2 1 p ξ Therefore: A e = πp 2 R 2 if 1 p ξ Case 3 Gathering all of this information gives eclipsed area of the star: A e = { R 2 p 2 π (Haswell, 2010, p. 107) 0 if 1 + p < ξ R 2 (p 2 α + β 4ξ2 (1 + ξ 2 p 2 ) 2 ) if 1 p < ξ 1 + p 2 if 1 p ξ Case 1 Case 2 Case Light Reduction Due to Transit The light reduction caused by a transiting exoplanet can be calculated using the intensity I, specifically it s integral over the total area of the star, as follows: (Haswell, 2010, p. 109) F = I(r )da star area 43

45 Where r = rr is a distance from the centre point of the star to a point inside of the star s radius and 0 r 1 shown in figure Figure By calculating the integral with respect to r is can be seen that: (Haswell, 2010, p. 109) R F = I(r )2πr dr 0 This would simply be the area of the circle if I(r ) were a constant (as in the case of uniform brightness across the star) since: R F = 2πI r dr 0 = 2πI [ r 2 R 2 ] = 2πI ( R 2 2 ) = πir 2 0 The change in flux ΔF is the amount that the flux is reduced by when the planet crosses the star, it can be obtained by calculating the integral of the area that the planet is blocking (the occulted area) as before. ΔF = I(r )da occulted area Taking I(r ) to be a constant again gives the following: (Haswell, 2010, p. 112) ΔF = I da = IA e occulted area 44

46 Where A e is the eclipsed area as shown in section 6.1. When limb darkening is factored in, I(r ) can no longer be represented by a constant since the stellar disc becomes darker towards the edges. Therefore a normalised axial coordinate must be introduced r with the following: r = r R This corresponds to an inner circle with radius rr as shown in figure Figure a diagram showing the intersection of the star and planet in terms of the normalised axial cooridiante The red area in figure is obtained by increasing the value of r by a very small amount dr. This means that the radius of the star in terms of r is R = r + dr. Combining this with the equations obtained in section 6.1 for the eclipsed area A e gives the area of the red section as: (Haswell, 2010, p. 111) da(r) = da e dr dr This can be written in terms of the integral by changing the variables p to p r and ξ to ξ r. Therefore the resulting integral is: (Haswell, 2010, p. 111) 0 r da e dr dr = A e (rr, p r, ξ r ) = r2 A e (R, p r, ξ r ) Finally, by substituting the expression for A e obtained above into the integral for the change in flux (ΔF), to get: r=1 1 ΔF = I(r)dA(r) = I(r) d dr A er 2 (R, p r, ξ r ) dr r=0 0 45

47 (Haswell, 2010, p. 112) This gives an integral for the limb darkened change in flux due to the transit on an axially symmetric star disk. 6.3 Limb Darkening Laws It has already been said that the sun displays the effects of limb darkening (figure 6.1.1), an estimate of the effects of limb darkening in relation to the intensity (I) can be given as: I(r) = I(0) [1 u (1 a2 r 2 a 2 )] Equation ( Limb Darkening, n.d.) Where u is the limb darkening coefficient, a is the radius of the Sun (equivalent to R in my case) and r is the radial distance. These parameters are illustrated in figure Equation can be written in terms of the angle θ (figure 6.3.1) with the use of Pythagoras Theorem: ( Limb Darkening, n.d.) Figure a diagram showing the radial distance I(θ) = I(0)[1 u(1 cos(θ))] This is the case because the star does not have a solid surface and the stellar atmosphere can absorb or scatter photons. Since the photon has to travel through more of the stellar atmosphere from the observer s point of view at the limb of the disk, the star appears darker at its edges than in the centre. Figure is an adaptation of the above diagram (figure 6.3.1) that illustrates the extra distance travelled by a photon at the limb of the star. 46

48 Viewing Point Figure a diagram showing the emission of photons with respect to the viewing point Figure shows that a photon emitted from the centre of the star travelling in a straight line towards the observer and leaving the disk at point A has a shorter path that one emitted at an angle and leaving at point B. Since the photon emitted at point B appears to travel further through the stellar atmosphere than the one emitted at point A, it is said to have higher optical depth. Photons with a higher optical depth are less likely to escape through the atmosphere and be seen by the observer, thus the centre of the disk appears brighter, while the edges appear dimmer (Haswell, 2010, pp ). In other words, the star is said to display the effect of limb darkening. There are four limb darkening laws the first of which was used in this project to construct a theoretical curve in Maple. The equation states: I(μ) = 1 u(1 μ) I(1) Linear Law (Haswell, 2010, p. 101) Equation is the linear limb darkening law, where 0 < u < 1 represents the limb darkening coefficient of the star and μ = cos (θ). The other laws are: I(μ) I(1) = 1 u l(1 μ) v l μln (μ) I(μ) I(1) = 1 u q(1 μ) v q (1 μ) 2 I(μ) I(1) = 1 u c(1 μ) v c (1 μ) 3 Logarithmic Law Quadratic Law Cubic Law (Haswell, 2010, p. 101) The above laws are equally relevant and could have been used, but the linear one has been selected for this project. This is because while the others have the potential to provide more accurate results 47

49 (and allow more freedom for model construction), they rely on more unknown variables than the linear law which only relies on one (u). Therefore, this is appropriate when compared with the complexity of the other laws. 6.4 Maple with Limb Darkening Gathering all the information, it is possible to use Maple to create a theoretical light curve that takes the limb darkening effect into account. It can be seen from the code in figure that the light curve now also depends of u the limb darkening coefficient. Figure Maple screenshot showing the limb darkening code The steps in figure are explained as follows: ds = d 2 + y 2 is as before (section 5.2.2) i = 1 u(1 1 ds 2 ) is the intensity using the linear limb darkening law F is the calculated flux, the integral of the intensity as explained in section 6.2 i1 is a midpoint contained in the eclipsed area of the star rmax, rmin, dmax as before (section 5.2.2) if y < rmin then dmin = rmin 2 y 2 else dmin = 0 as before (section 5.2.2) A is the eclipsed area of the star calculated based on the transit geometry of section 6.1 A1 substituted the parameters into the code df1 is an approximation of the change in flux which is the area of intersection multiplied by the intensity midpoint p1l plots the first half of the theoretical light curve with limb darkening p2l plots the second half of the light curve which is a reflection of the first half since the curve is symmetric p3, p4 as before (section 5.2.2) 48

50 DF5 is the difference in flux obtained by calculating the area of the planet over the intensity p5l plots the dip in the light curve by subtracting the flux ratio ( DF5 F ) from the initial flux value (1) Figure shows the process of inputting my data and constructing the final graph. It can be seen that several values have been used for the limb darkening coefficient (L) as this will construct multiple light curves making it possible to deduce a more accurate approximate value. Figure Maple screenshot showing the data input method with limb darkening The scaling is done in the same way as chapter 5, resulting in the following light curve when using the Clanfield data: Figure Maple graph with limb darkening theoretical light curve and Clanfield data I then computed a second limb darkening graph (Figure 6.4.4) that included more theoretical light curves to see if I could get a better idea of which value of u would be most appropriate. Figure Maple graph with extra values of u 49

51 Again, I created a Maple curve using the ETD data to compare with my own which can be seen below in figure Figure Maple graph with limb darkening light curves and the ETD data It can be seen from the above figures that a limb darkening model is appropriate for the transit. This is because the light curve is not flat bottomed. When looking at my own data (figures and 6.4.4), it appears that the limb darkening co-efficient is relatively high, because the data points look to fit in best with the more steep theoretical light curves. 50

52 Discussion of Results 7.1 Original Aims CHAPTER Colour Filters Before beginning this project, I was keen to research the results of viewing a transit through colour filters (as opposed to the clear filter that was used). It is often advantageous to view objects in our own solar system through colour filters, as shown in Figure 7.1.1, and I wanted to investigate if this could be the case for transit events as well. I considered that there could be subtle advantages to using colour filters in this type of astronomy and that the mathematics could be interesting as there is not currently an abundance of research in this particular area. However it is considered unlikely that colour filters would dramatically affect the images of a transit as the objects are so far away and only the parent star is being observed because the planet is not large or bright enough to be seen directly in the vast majority of cases. The camera used for my observations has several filters including red, yellow, blue and clear. It also has an illumination filter which combines the red, yellow and blue light. It takes several seconds to change between the filters and around 5 more seconds to Figure 7.1.1: Table showing the advantages of different colour filters for some objects within the solar system ( Choosing a Colour/Planetary Filter, 2008). download the new camera signal, the exposure of the image then needs to be around 40 seconds. Colour imaging was included in my project plan (Appendix 9.1) and discussed briefly with David Harris who concluded that we would obtain less than one image per three minutes using this technique. This means that there would be less images overall for analysis and it would not be possible to reject any anomalies. Therefore we concluded that due to the time restrictions and complexity of the task there would be little or no advantage in collecting such images. However I believe it would still be interesting to research colour imaging in the future. The ETD website contains some data that has been collected through colour filters and so this could be a good place to begin when considering future work. 51

53 7.1.2 Observations When I originally created my project plan (Appendix 9.1), my main aims were to use the Clanfield Observatory equipment to collect images of a transit to analyse, create a light curve and mathematically model the transit using Maple thus demonstrating the importance of mathematics to exoplanetary science. Unfortunately I was unable to collect my own data due to persistent bad weather. However, I don t feel that this hindered the overall project too much as I still managed to visit the observatory, learn the process behind image collection and use previous images collected by David Harris to fulfil my other aims Light Curve Fitting Obtaining a suitable light curve was a very important and difficult task. I used AIP4Win to analyse the images and then Excel to analyse the data and create a light curve. However, many of the first light curves I created did not show promising results (an example can be seen in Figure 7.3.1). My first step to try and overcome this was to create a V Ens light curve, which takes the average of several comparison stars and subtracts this from the target star (Figure 7.3.1). Unfortunately this was also unsuitable as there appears to be a peak in the middle of the transit. Figure One of my first graphs (left) and a V-Ens graph (right) The only way around this problem was to be persistent with choosing different comparison stars until eventually managing to create the one by selecting the best results and using a ten point average. As this curve was still not as conclusive as I d have liked, I created another light curve with data from the ETD to compare it with. 52

54 7.1.4 Modelling For the modelling section of my project, I put my data into Maple to compare with a theoretical light curve. This section was more successful than I thought it would be given that the data did not form the curve I was expecting in Excel. The curves that included a limb darkening coefficient fit particularly well with the data collected at Clanfield. This section went well with the only exception of being unable to make the transit animation work. 7.2 Conclusions HAT-P-25b is a larger planet than any of those in our Solar System orbiting a star that is very close in size and mass to the Sun. The orbit is much smaller than the planets in the Solar System giving an orbital period of just 3.65 days and duration as viewed from Earth of around 2.67 hours. A vast amount can be learned about a planetary system from its transit light curve. Values such as that planet radius, orbital period, semi major axis and impact parameter can all be obtained when mathematics is applied, making it possible to create a model of the system. It has also been shown in this project that one can begin to approximate previously unknown factors such as the limb darkening coefficient described in chapter 6 which cannot be calculated directly. It can be difficult to obtain data for an exoplanet transit due to poor weather conditions as well as atmospheric distortion on Earth. These problems are resolved by space based telescopes such as Kepler (chapter 8) which do not suffer in the same way and can consequently attain very accurate results. 7.3 Future Work For future work in this area, I would recommend the collection of more sets of data for the same transit on separate occasions wherever possible. The results from numerous transits could then be averaged to produce more accurate results. If time permits, transit data could be collected through colour filters. In addition to the benefits previously discussed, they could also be used to further understand the effects of limb darkening. This is because limb darkening is more pronounced for the shorter wavelengths of the violet end of the visible spectrum than the longer wavelengths of red light ( Limb Darkening, n.d.). Another interesting area of research is the habitability of planets based on their location with respect to the parent star. Future work could include a calculation of the habitable zone for the particular star that has been studied. 53

55 CHAPTER 8 Telescope Missions and Current Research In recent years, an abundance of scientific research has taken place in the field of extrasolar planets. This chapter outlines the work that has been completed so far in the detection of new planets and the study of those already known in the hope of gaining a better understanding of their properties. 8.1 COROT The COROT (Convection Rotation and planetary Transits) was a French mission in association with the European Space Agency that began in December It was a space based mission whose purpose was study both transiting exoplanets and stars using a 27cm telescope (Mason, 2008, p. 14). This mission lasted a total of 2136 days ( The Project Main Steps, 2014) and during this time it discovered 32 extra solar planets and in excess of 100 more candidates that have yet to be confirmed (Atkinson, 2013). Figure 8.1.1: An artist s impression of the COROT mission ( Corot Update, 2009) Although the numbers discovered may not seem like much in comparison to the total catalogue of discovered exoplanets, it must be remembered that the COROT mission was one of the first of its kind, paving the way for future research. 8.2 Kepler Like COROT, the Kepler mission was also space based. It was operated by NASA and designed to look for planets using the transit method by observing thousands of stars simultaneously (Haswell, 2010, p. 206). The main focus is to look for terrestrial planets (see Chapter 1) which are defined by NASA as one half to twice the size of Earth ( Kepler Overview, n.d.). 54