Observation and Characterisation of Exoplanets

Author: Lucy Stickland

Date: May 2014

School of Physics and Astronomy, Cardiff University

Supervisor: Dr E L Gomez

Assessor: Dr S Ladak

Abstract

This report explores the variety of detection techniques used today in the quest for extrasolar planets, the recent increase in Earth sized planets is investigated and relationships between stellar and planetary companion parameters analysed for currently known exoplanets. Three known transiting planets Hat-P-25b, Wasp-43b and Wasp-2b are selected for observation with the Sedgwick telescope and photometry carried out on all three, once tested on the exoplanet ‘Qatar 1b’. A transit light curve is produced for each planet and subsequently used to calculate the semi major axis, orbital period, radius and density of each. To increase the accuracy of the parameters calculated, a programme was created called ‘The Exoplanetary Pixelisation Model’ to fit a theoretical light curve to the data recorded computationally rather than by eye and this model was tested on Hat-P-25b.

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Contents

1. Introduction………………………………………………………………………………….page.3

2. Review of currently known exoplanets …..…………………………………………………page.6

3. Planet selection for observations…………………………………………….………………page.8

4. Photometry…………………………………………………………………………….……..page.9

4.1. Photometry Practise on Qatar 1b..……..……..……………………………………...….page.9

4.2. Errors………………………………………………………………………....…….........page.11

4.3. Results from Photometry on Qatar 1b……………………………………………….…..page.11

4.4. Photometry on Hat-P-25b, Wasp-43b and Wasp-2b………………..……….………......page.12

4.5. Results from Photometry on Hat-P-25b, Wasp-43b and Wasp-2b……………….......…page.13

4.6. Exoplanetary Pixelization Transit Model (EPTM).……………………………………..page.14

5. Conclusion..……………………………………….…………………………………….……page.16

6. Future Work…………………………………………………………………………….……page.16

7. References………………………………………….…………………………………….......page.16

8. Appendix 1 – EPTM Code for Hat-P-25b......……………………………….………………page.18

1. Introduction

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Since the first discovery of an exoplanet orbiting a solar type star was made in 1995 (Mayor & Queloz, 1995), a further 1779 have been detected to date using a variety of techniques that will be explored in this report. The number of such detections has increased rapidly in recent years suggesting that as technology and observing techniques continue to improve, there will be a continuing exponential growth in the number of exoplanets found.

In February 2014 NASA announced the discovery of a further 715 exoplanets confirmed by the Kepler mission, four of which have sizes within 2.5 times the size of Earth and reside in the habitable zone1. This increase in known exoplanets is represented in Figure 1 showing that exoplanet detection is a hugely growing area of research, predominantly growing in success at the smaller mass end. This is vital for improving our knowledge on our place in the galaxy and is forever taking us one step closer to the finding of extra-terrestrial life.

Figure 1. Graphical representation of the number of known planets announced by NASA in February 2014 shown in orange compared to the number previously known shown in blue. Each bar from left to right represents planets that are Earth size, Super Earth size, Neptune size and finally Jupiter size and larger respectively. Image taken from ‘NASA Ames/W Stenzel’2.

Exoplanet detection is possible through a variety of detection techniques - initially it was the technique of radial velocity that dominated the field but in recent years the transit technique for detecting exoplanets has overtaken. Planets are found through radial velocity by detecting a change in the stellar radial velocity (along the line of sight) by measuring the Doppler shift using high resolution spectrometers (Santos, 2008). The star and planet orbit the system’s centre of mass - as the star moves away from the observer its light is redshifted and then blue shifted as it moves towards the observer. The amplitude of these variations are greatest and hence easiest to detect for stars with massive planets and small orbital radii making giant gas planets of a few hundred Earth masses the easiest to detect. For example, the change in velocity “of a star induced by the presence of a Jupiter-like planet (with a mass of 318 Earth masses and an orbital period of 12 years) is 13 m/s, while for an Earth-like planet this value decreases to a mere 8 cm/s” (Santos, 2008).

A limitation of radial velocity occurs from the fact that measurements are made along the line of sight. Such measurements give a projected radial velocity that allows the ‘projected mass’ of the companion to be calculated – this

is the minimum possible mass of the exoplanet given by M min=M truesin(i) where i is the orbital inclination (Santos,

2008). A larger inclination angle means the orbit is tilted further from an observer’s line of sight; this gives the star a perpendicular velocity component that is not recorded and therefore results in a mass calculated that is less than the true value. The mass can be assumed to be the true mass if a transit can be observed using the photometric transit method because for this to be possible the transit must be in line with the observer’s line of sight and hence the

inclination angle is 90degrees giving M min=M true. Strictly speaking a transit is visible if it satisfies the equation

acos ( i )≤ RP+RS, where ais the semi major axis and RP and RSare the radius of the planet and star respectively

(Kane, 2007). It is evident from this that as in radial velocity, giant planets with small semi-major axes are most likely

1 www.nasa.gov/ames/kepler/nasas-kepler-mission-announces-a-planet-bonanza/#.U0Gk2O5ATjZ2 http://www.nasa.gov/sites/default/files/knownexoplanets.jpg

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to have a transits seen along the observer’s line of sight and are therefore the most likely to be detected. The physical explanation behind this is that the photometric transit method works by measuring the reduction in flux when a planet passes in front of its host star - the larger the planet, the greater this reduction would be and hence the more likely the transit is to be seen. The photometric signal of a transiting planet would look similar to Figure 2 where the base line at position 1 represents the total flux of the star when not blocked by the star, at position 2 the planet has started to move in front of the star and partially blocks the light so the flux measured reduces, the flux reaches a maximum dip when the whole planet is in front of the star and then rises again as the planet crosses the other side – using this curve the planet size can be found as will be explained in section 4.3. The curved shape of the transit is due to limb darkening which involves an increase in brightness from the edges to the centre of the star (see section 4.6) and due to the curved

edges of the star and planet overlapping. A problem arises through the fact that phenomena other than transiting planets can also produce similar signals. As an example, the OGLE (Optical Gravitational Lensing Experiment) announced approximately 200 transiting candidates (Udalski et al., 2002a,b) that upon radial velocity follow up were found to be mostly eclipsing binary stars (Santos, 2008).

Figure 2. Transit signal when a planet passes in front of its host star – image taken from3

Transit method results can be combined with radial velocity measurements to calculate the radius, mass and density of a planet and to investigate its internal structure (Udry & Santos, 2007). The density it estimated by combining the planet size found from the transit method and the planet mass found from radial velocity (by assuming a spherical planet with uniform density). The transit model is also useful since an absorption spectrum is created when a planet passes in front of its host star - by monitoring this the atmospheric composition of the planet can be deduced.

In this report transit light curves of four planets (Qatar 1b, Wasp 2b, Wasp 43b and Hat-P-25b) were obtained through photometry and used to calculate values for the semi major axis, transit duration, orbital period, ratio of planetary to stellar radii, planet radius and planet density for each exoplanet through use of Equations (1-5). By equating Kepler’s 3rd law shown in Equation (1) with Equation (2) (see Seager & Mallen-Ornelas, 2003), Equation (3) is obtained to calculate the semi-major axis where a is the semi-major axis, P is the period (seconds), G is the

gravitational constant ( m3 kg−1 s−2 ¿, M p is the planetary mass (kg), M s is the stellar mass (kg), dt is the transit

duration (seconds) and R s is the stellar radius (m). Equation (4) relates the ratio of the planetary and stellar radii to the

dip in flux of the transit light curve caused by the planet-star system and is used later to calculate the radius of the planet where F is the flux (counts) and ∆ F is the maximum reduction in flux during transit. Equation (5) uses the planetary radius calculated through Equation (4) and the published planetary mass value to calculate the density of the planet where ρ is the planet density.

P2=4 π2 a3

G ( M s+M p ) (1)

P=aπdtR s

(2)

3 www.iac.es/proyecto/tep/transitmet.html4

a=G(M p+ M s)dt 2

4 Rs2 (3)

∆ FF

=( Rp

R s)

2

(4)

ρ=M p

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π R p3 (5)

Direct imaging is another detection method and is one of the most straight forward, it is not however as prosperous as radial velocity or the transit method due to difficulty in separating the planetary signal from that of the much brighter host star. Unlike radial velocity and transit detections, direct imaging is most successful when the planet’s orbit is face on as this increases the visibility of the planet. “The emission from an exoplanet can generally be separated into two sources: stellar emission reflected by the planet surface and/or atmosphere, and thermal emission from the planet” (Wright & Gaudi, 2012). The separation of the planet signal from that of its parent star is easiest for hot, giant planets at large orbital radii. The images are taken in the infra-red to increase the brightness of the planet compared to at visible wavelengths.

Like direct imaging, astrometry also works best for face on orbits. It is a technique that looks for the shift of a star around its centre of mass against background stars caused by an unseen orbiting planet and is performed by precisely measuring the star’s position in the sky over time. The shift observed is greatest for small stars orbited by large planets at large orbital radii and appears greatest against the celestial sphere for stars that are close by. Data from ‘The Extrasolar Planet Encyclopaedia’4 shows that only one planet (HD 176051 b) has been detected to date by this technique. One of the main problems with this method is that since it works best for planets with large orbital periods, observations have to be made for long periods of time to detect such a periodic wobble of a star.

Gravitational microlensing has detected many exoplanets towards the galactic bulge in recent years. An advantage to this technique is that it is sensitive to planetary systems throughout the Galaxy and not just in the solar neighbourhood (Griest & Safizadeh, 1998). Microlensing looks for a characteristic magnification light curve and relies on the alignment of a host star and planetary companion with a background source star. When a star passes between the observer and source star it acts as a lens. Its gravitational field bends the source star’s light rays to produce two images with milliarcsecond separation. If the lens star has a planetary companion whose orbit coincides with the source star, further magnification is caused which lasts for typically a few hours to a day compared to the typical 1-2 month duration for lensing events due to stars (Bennet & Rhie, 2002). The main problem with this technique is that

microlensing events are rare - only about 3x106 galactic bulge stars are microlensed at any given time (Udalski et

al.1994; Alcock et al. 1997), whilst only ~2% of Earth mass planets orbiting the lens stars will be in the right position to be detected (Bennett & Rhie 1996) meaning the planet cannot be studied again after the event has passed.

The final detection method to briefly discuss is pulsar timing. A pulsar is a small, highly dense neutron star that has formed from a supernova after the accretion of mass from a binary star. Pulsars emit clock like pulses of radio waves and reveal an exoplanet through periodic variations in the precise timing of these pulses. Since the pulsar would have a small orbit around the system’s centre of mass, there is a slight time difference in the detection of consecutive pulses when it is moving away from Earth compared to towards us. The sensitivity of this method allows planets of very small mass to be found, the smallest being ‘PSR 1257 12 b’ at only 0.02 times the mass of the Earth (Curie & Hanson, 2007).

4 http://exoplanet.eu/catalog/?f=%22transit%22+IN+detection5

2. Review of Currently known Exoplanets

The bias towards the detection of gas giants explained above led to detections initially suggesting that giant planets are most common. Since then the development of specially designed high resolution spectrographs with high precision and stability have allowed smaller planets with larger orbital radii to be detected and have led to an ever increasing number of known Earth like planets around Solar type stars. These high resolution spectrometers are designed in order to reduce as much as possible the effect of stellar oscillations that hide the signal from a small planet, the effect of which is further reduced by taking observations that last at least 15 to 30 minutes on target in order to average the oscillations out (Udry & Santos, 2007). Figure 3 shows the improvement in measurement precision as a continuous decrease in the minimum mass detected over the last 30 years, now reaching precisions of just a fraction the mass of Earth. It also illustrates the transformation from radial velocity to transit domination in the last few years and the ever growing number of detections. The contribution to detections by imaging, microlensing, pulsar timing and astrometry is indicated by colour. Data used for Figures 3-5 was taken from ‘The Extrasolar Planet Encyclopaedia’.

Figure 3. Known exoplanets in Earth masses plotted against the year of discovery with the detection method used indicated by colour. Earth, Neptune and Jupiter masses are labelled as reference points.

The bias in transit and radial velocity detections towards planets with small semi major axes and with imaging towards larger semi major axes is illustrated in Figure 4. It is evident from this graph that the size of planetary orbits differs much more around a star of a certain temperature than is seen in our solar system and shows little correlation between the two factors. Figure 5 investigates the relationship between the mass of host stars and their planetary companions. Taking 1 solar mass as an example, it is clear that a specific mass star can host a vast range of planetary masses, not seen to such extent in our own solar system. The apparent bunching of planets around stars of ~5000K (the temperature of the Sun) and 1 solar mass in Figures 4 and 5 respectively is likely due to missions such as Kepler being predominantly aimed at finding planets around Solar type stars. Such large variation in the planetary parameters supported by a 1 solar mass star raises the question that if we had the same amount of research on all stars whether the graphs below would in fact be filled with planets – this is something we are yet to find out. The planets analysed later in this project are all companions of Solar type stars, the parameters calculated therefore fitted in the strip of planets seen in both images below.

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Figure 4. Graph illustrating the relationship between the effective temperature of the host star and the semi-major axis of its planetary companion with Earth, Neptune and Jupiter plotted as reference points.

Figure 5. Illustration of the relationship between stellar mass and planetary mass with Earth, Neptune and Jupiter plotted as reference points.

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3. Planet Selection for Observations

To obtain transit light curves of exoplanets, three known transiting planets were chosen from the Exoplanet Transit Database (ETD)5 (Poddany, Brat & Pejcha, 2010) that had transits expected during the duration of this project. These were to be observed using the 0.83m Sedgwick telescope, part of the Las Cumbres Observatory in California. It is located at a latitude and longitude of 34.6875 degrees and 240.038889 degrees respectively, an altitude of 500m and a UTC Time Offset of -8hours6. To find suitable transits a plot of altitude against time was created using Staralt 7 for a variety of planets similar to Figure 6 using the Sedgwick Telescope parameters and coordinates of the transit. The transit time, altitude and Moon positioning on each graph was then analysed to determine the transit’s visibility. To have good visibility it must occur no closer than 30minutes to twilight and the Moon must not be nearby during the transit - these factors prevent extra light from filling CDD pixels which would distort the images and make the faint transits difficult to observe. Having the Moon nearby would produce too much blue light in the image due to reflection of light from the Sun, particularly evident at a full Moon. Furthermore, the planet’s altitude must remain greater than 30degrees for the duration of the transit because the lower the altitude, the more of the Earth’s atmosphere the light has to travel through and therefore the more extincted and reddened the final image becomes.

Figure 6. A plot of altitude against time, in this case for transiting planet GJ3470b. The two vertical dashed lines represent twilight and the curved dashed line in the bottom right hand corner represents the moon. On this date, the transit occurs between 06:25am-08:01am (UT) – the altitude during this time is above 30 degrees and the moon is not nearby making it a suitable transit to observe.

5 http://var2.astro.cz/ETD/6 http://lcogt.net/site/sedgwick7 http://catserver.ing.iac.es/staralt/

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The three transits chosen for study in this project and their published planet and stellar properties are listed in Table 1. GJ3470b was chosen due to being one of the most inflated low-mass planets known with a very low planet density, bridging the boundary between “super-Earths” and Neptunian planets (Nascimbeni et al, 2013). Hat-P-24b is an inflated hot Jupiter transiting a hot metal poor star whilst Wasp-13b is an inflated sub Jupiter mass planet with low density. The observing parameters required for these three transits are listed in Table 2.

Planet Planetary Mass (M_Earth)

Planetary Radius (R_Earth)

Period (days)

Stellar Mass (M_Sun)

Stellar Radius (R_Sun)

Stellar Metallicity(Fe/H)

Effective Stellar Temp (K)

GJ3470b 13.9+1.5-1.4 4.83+0.22-0.21 3.3 0.539−0.043+0.047 0.568−0.031

+0.037 0.20 +/- 0.10 3600 +/- 100

HAT-P-24b 216+/-9.85 13.6+/-0.79 3.4 1.191 +/-0.042 1.317+/-0.068 -0.16 +/-0.08 6373 +/- 80

WASP-13b 146+/-19 15.25 +/- 0.55 4.35 1.03−0.09+0.11 1.56+/-0.04 0.00+/-0.2 5826+/-100

Table 1. Current published values for each planetary star system chosen for observation taken from Demory et al (2013), Kippling et al (2010), Barros et al (2011) and Skillen et al (2009).

Planet FOV RA(deg)

FOV Dec(deg)

Transit Start Date

Transit Start Time (UT)

Transit End Date

Tansit End (UT)

Filter V(mag)

GJ3470 b 119.74165

15.37945 31/12/2013 06:25 31/12/2013 08:01 R 12.27

HAT-P-24 b 108.825 14.2625 06/12/2013 07:32 06/12/2013 11:12 R 11.818

WASP-13 b 140.1042 33.8825 02/12/2013 08:21 02/12/2013 12:14 R 10.51

Table 2. Information on the transits chosen for observation from GJ3470b, HAT-P-24b and WASP-13b and the observing parameters required.

Unfortunately due to technical issues with the telescope our observations were unsuccessful and three archive data sets were instead used for analysis by photometry for the rest of the project.

4. Photometry

4.1. Photometry Practise on Qatar-1b

Photometry is a technique concerned with measuring the flux of an object. Prior to analysing three large archive data sets, the aperture photometry technique to be used with GAIA was first practiced on a smaller data set, focusing on the target star ‘Qatar-1’ with transiting planet ‘Qatar 1b’. This practise allowed the procedure to be tested and the parameters calculated to be compared to those published on Qatar-1b.

To measure the transit light curve of Qatar-1b, the target star was identified using a star finder chart on the Exoplanet Transit database then five other stars in the image were chosen as calibrator stars. These stars were close to the target star to ensure that the atmospheric effect is similar for each and were of brightness within a few magnitudes of the target star to ensure that a full point spread function is obtained for each. An aperture was then placed over the target star, each calibrator star and one on an area vacant of stars, the last used to measure the background count. An example of what this would look like in GAIA is shown in Figure 7. The background was calculated in this way rather than using annular sky regions as the latter could be contaminated by nearby stars. Results were chosen to be in data counts rather than magnitudes.

All apertures used must be the same size, the optimum aperture radius was found by increasing the radius pixel by pixel around the brightest star in the image and plotting the radius against the data counts recorded as shown in Figure 8. An aperture size of 16 pixels was chosen for this data set as this is where the graph began to plateau (see

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Figure 8) and hence is the point at which the entire stellar signal is being recorded. The importance of recording the

entire signal is due to the dip in flux being only on average a couple of percent of the total flux.Figure 7 (left). An example fits file as seen in Gaia with equal sized apertures positioned around the target star, 5 calibrator stars and a background aperture over an area vacant of stars.

Figure 8 (right). Plot of signal measured in an aperture against aperture size to find where the graph plateaus and hence the optimum aperture size at which all signal is recorded.

After repeating readings over all fits files five light curves were plotted for the calibrated flux of the target star using each calibrator star by inputting the data recorded into Equation (6) where F is the calibrated flux, ST is the target star signal, SB is the background count, SC is the calibrator star signal and α is the normalising constant - Equation (6) taken from Howell (2006). Equation (6) shows that each curve was plotted by dividing the target star signal by each calibrator star’s signal respectively, all with the background count removed. This allowed the calibrator stars that produced good light curves to be found, those that did not were probably variable stars and were used no further. Variable stars have fluctuating brightness either caused by intrinsic variables such as shrinking/swelling of the star or by extrinsic variables such as an orbiting companion. Calibrator stars 3 and 5 were deemed suitable as the light curves produced by them showed the characteristic dip in flux of a transiting planet. The light curves produced using these two calibrator stars were then divided by a normalising constant, α, taken as the value of the baseline either side of the dip which represents the maximum calibrated flux of the star. This normalisation allowed both curves to be plotted on the same graph with a normalised baseline flux of approximately 1, a light curve was then fitted by eye to the data as seen in Figure 9.

F = ST−SB

Sc−SB

.1α

(6)

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Figure 9. Normalised calibrated plot of flux versus time for Qatar-1b using two calibrator stars, the solid line shows the transit light curve fitted by eye and the error of each point is represented by the error bars (see section 4.2).

4.2. Errors

N2 = NS + NB + npixel (ND + NR2) (7)

E(F) = ( ∂ F∂ ST )

2

∆ ST +( ∂ F∂ SC )

2

∆ SC+( ∂ F∂ SB )

2

∆ SB+( ∂ F∂ α )

2

∆ α (8)

Equations (7-8) taken from Howell (2006). The noise attributed to the target star, each calibrator star and the background was found using Equation (7) where N is the noise found respectively for each, NS is the source star signal, NB is the background count, npixel is the number of pixels in aperture, ND is the dark current and NR is the read out noise, (NS=0 when calculating the background noise as no target star signal was being recorded in the background aperture). ND and NR values were not provided in the fits files so could not be used, they are however insignificant in comparison to npixel with ND at approximately 11 electrons per pixel (Glenn & Kondepudy, 1994) and NR of 13 electrons per pixel

(Donald, 1995). The most significant contribution to the error in this equation is the random noise of the stellar signal NS as it is much higher than the background noise (the background noise is the signal recorded in the aperture when placed over an area with no visible source). The read out and dark noise is intrinsic noise added by the electrics of the CCD that come about from a thermal issue where electrons jump from the valence band to the conduction band with no light source and from hot pixels where the pixel wrongly thinks that light is present due to thermal noise. As well as these over-sensitive pixels there are unresponsive pixels known as ‘dead pixels’ that do not respond to light, reducing the efficiency of the CCD. Data from the Sedgwick Telescope had been bias-corrected and flat-fielded to account for such errors. The error in the flux F (see Equation (6)) was then calculated using Equation (8), solved

through use of the quotient rule where ∆ ST , ∆ SCand ∆ SB are the errors on the target star, each calibrator star and the

background respectively just found by Equation (7). The errors produced are represented as the error bars seen in figure 9.

4.3. Results from Photometry on Qatar-1b

Firstly, the normalised light curve in Figure 9 allowed an estimate to be made of the transit duration which was found to be (6900+/-2000) seconds, above the upper limit of the known value likely due to an increased error from poisson noise. Next, using Equation (3), the transit duration time could be used along with known values for the planetary mass, stellar mass and stellar radius to calculate the semi major axis - this was found to be (0.022+/-0.008)AU. Having

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now found the semi major axis, Equation (2) was used find a period of (1.46+/-0.73)days. By measuring the maximum reduction in flux during the transit, Equation (4) was used to calculate the ratio of the planet to stellar radii and gave Rplanet=0.16 R star which upon substitution of the known stellar radius gave Rplanet = (1.28+/-0.68)R J, where R J is the

radius of Jupiter. Using this radius and the known planet mass with Equation (5) the density of Qatar 1b was

calculated to be (0.689+/-0.192)g/cm3 . The errors in these parameters were obtained by plotting a best and worst fit

line on Figure 9 – these represent the two most extreme light curves that could be realistically plotted through the data according to the error bars calculated in Section 4.2. All of the values calculated, excluding the transit duration measurement, are within the limits of the published values indicating the precision and reliability of the photometric technique used. The calculated and published values for the parameters of Qatar 1b are summarised in Table 3.

Transit Duration Time (seconds)

Semi-major axis (AU)

Orbital Period (days)

Planet Radius (R J¿

Density (g/cm3 ¿

Calculated Values 6900+/-2000 0.022+/-0.009 1.460+/-0.730 1.280+/-0.680 0.689+/-0.192

Known Values 5802+/-66.52 0.0234+/-0.0003 1.420+/-0.000 1.160+/-0.045 0.690+/-0.091

Table 3. Comparison table of the parameters calculated for Qatar-1b versus the published values - published values taken from (Alsubai et al, 2011).

4.4. Photometry on Hat-P-25b, Wasp-43b and Wasp-2b

The use of Sextractor was explored for use on these data sets however after analysing both its benefits and limitations it was decided to be beyond the scope of this project and the previous method (GAIA) tested on Qatar 1b was taken forward. Sextractor would have been useful in minimising the time taken to take data compared to doing so through Gaia however it proved too time consuming for this project to write the star tracking code required to follow the movement of the stars in each fits file.

After finding the optimum aperture size for each data set in the same way as explained with Figure 8, over 300 fits files were analysed for the three new transiting planets in question, taking data on 33 calibrator stars in total. Choosing more calibrator stars gave more choice of which ones to use to create light curves - two were deemed suitable for Wasp-2b and Hat-P-25b and three for Wasp-43b. These calibrator stars were chosen as the light curves produced by them showed the characteristic dip in flux of a transiting planet, those that did not were likely to be variable stars as explained in Section 4.1. Using larger data sets than the 14 files used for Qatar-1b enabled far smoother and more accurate transit light curves to be produced - the light curve produced using the two calibrator stars of Hat-P-25b is shown as an example in Figure 10 below.

Figure 10. Normalised calibrated light curve for Hat-P-25b using 2 calibrator stars represented by the blue and green data. Error bars were calculated as explained in section 4.2.

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To further improve the transit light curves produced compared to that of Qatar-1b in Section 4.1, an average of the curves produced by each calibrator star was taken and normalised to produce a ‘supercalibrator’ ie.one averaged normalised curve of data points for each target star. The final three supercalibrated light curves produced are shown in Figures 11-13.

Figure 11. Averaged normalised calibrated light curve of Hat-P-25b.

Figure 12 (left). Averaged normalised calibrated light curve of Wasp-43b.

Figure 13 (right). Averaged normalised calibrated light curve of Wasp-2b.

4.5. Results from Photometry on Hat-P-25b, Wasp-43b and Wasp-2b

Using Figures 11-13 values for the transit duration and dip in flux were estimated and used to calculate the semi major axis, orbital period, planetary radius and planetary density in the same way as for Qatar 1b (using Equations 2 to 5), the values found are summarised in Table 4 along with known published values for comparison. The errors in these parameters were obtained using a best and worst fit line on each graph (Figures 11-13) as explained in Section 4.3. Unfortunately as seen in Figure 12, the observation of Wasp-43b’s transit was cut short, likely due to bad weather causing the telescope to close down. After analysing the data and comparing to known values it was found not to have reached the half-way point of the transit so was unusable in calculations for the semi major axis or period; it did however produce a planetary radius and density close to the published values showing that the curve produced did reach the transit’s total dip in flux. The radius obtained for Wasp-43b was slightly larger than the true radius but within the error range – this is likely to be caused by an overestimate of the dip in flux due to only having half of the

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light curve to work with. Having too large of a radius then leads to a density value smaller and slightly outside the error range of the published value.

All values for Wasp-2b are within the limits of the true value except for the period which is below the lower limit due to an underestimation of the transit time from the graph. The values for Wasp-2b were expected to be less accurate than Hat-P-25b due to the data appearing more noisy (see Figure 13) however, the right hand half of the graph has little noise so a curve was fitted to this and then assumed to be symmetrical to minimise the uncertainty in drawing the curve through the noisy data. Higher noise could be due to a number of reasons such as the transit being at low altitude which would distort the images received as explained earlier or the start of the transit may have been too close to twilight or have the moon nearby allowing extra light to fill the CCD pixels - this could explain the higher noise seen through the first half than the second of the light curve. A graph like Figure 6 could establish whether any of these factors is the reason for the high noise levels however due to using archive data this was not possible to plot as it is not clear which telescope was used to take the data and therefore no telescope parameters could be input into Staralt.

Hat-P-25b shows a far less noisy graph with the light curve passing through all but two error bars resulting in a reasonably distinct start and end to the transit allowing the transit time to be read off with confidence leading to a semi major axis and period within the limits of the known published values. It was however less clear where the maximum dip in flux is and was overestimated leading to a radius higher and density lower than the limits of their respective published values.

Hat-P-25b Project

Hat-P-25b Published

Wasp-43b Project

Wasp-43bPublished

Wasp-2b Project

Wasp-2b Published

Semi major axis (AU)

0.050+-0.005 0.0467 N/A 0.015+-0.000 0.019+-0.011 0.031+-0.011

Period (days)

4.094+-0.445 3.650+-0.000 N/A 0.813+-0.000 1.060+-0.090 2.150+-0.000

Radius (Rj) 1.366+-0.044 1.190+-0.081 1.090+-0.061 1.036+-0.019 1.113+-0.093 1.040+-0.060

Density (gcm-3)

0.295+-0.035 0.420+-0.070 1.777+-0.313 2.410+-0.080 1.309+-0.336 0.998

Table 4. Results for Hat-P-25b, Wasp-43b and Wasp-2b derived from the transit light curves obtained through photometry in this project. The published values given for comparison were taken from Charbonneau et al (2007), Cameron et al (2007), Blecic at al (2013), Gillon et al (2012) and Quinn et al (2012)8.

4.6. Exoplanetary Pixelization Transit Model (EPTM)

The EPTM is a theoretical model created to fit a theoretical transit light curve to data recorded rather than estimating a curve by eye –it was tested on Hat-P-25b through plotting in Python. Detailed information used to write this programme can be found in Brett, Durrance & Schwieterman (2010) from which the following formulas were taken

where V orb is the orbital velocity, G is the gravitational constant, M s is the stellar mass, M P is the planet mass,Dcentre

is the distance from the centre of the star to the centre of the planet, x pos and y pos are the distances in the x and y

direction to a pixel from the centre of the transit, Fb is the total flux blocked, Ω pix is the solid angle of a pixel, I 0 is

the band intensity at centre of star, µ is the limb darkening coefficient, d pix is the distance from the centre of the star to

a pixel and R s is the stellar radius. A complete copy of the code written for this project can be found in Appendix 1.

V orb=√ G ( M s+M P )a

(9)

Dcentre=√x pos2 + y pos

2 (10)

8 No errors provided in the published values for the semi-major axis of Hat-P-25b or the density of Wasp-2b.14

Fb=∆ Ω pix Σ pixels I 0[1−µ(1−√1−( d pix

R s)

2)] (11)The model was created by treating the planet as an array of pixels, the distance of each pixel from the centre of the star was calculated using Equation (10) at a series of time points matching those of the transit light curve data in question, the velocity of the planet across the star was calculated by Equation (9). Each pixel that was within the radius of the star would be contributing to a dip in flux and was summed over in Equation (11) at each time step to calculate the total flux blocked throughout the planet’s passage in front of the star. Subtracting these values from the stellar intensity gave the total flux observed at each time step; this was plotted as a line over the data of Hat-P-25b to get a more accurately fitted light curve as seen in Figure 14. µ is the limb darkening coefficient and accounts for the increase in brightness of the star from its edges to its centre and therefore the change in flux blocked per pixel as the planet crosses the star. This happens because light rays can escape from approximately 1optical depth within the photosphere meaning the light observed from the centre of the star comes from deeper within the star than that seen at the edges and is therefore coming from an area of higher temperature, is of shorter wavelength and hence, the star appears brightest in the centre and more light is blocked per pixel than at the edge. A comparison of the parameters calculated for Hat-P-25b using this model compared to the parameters found in Section 4.5 by fitting the curve by eye is displayed in Table 5 along with the published values.

Figure 14. Theoretical transit light curve fitted using Python to Hat-P-25b data using the EPTM – see Appendix 1 for the

parameters used.

Hat-P-25b Project (by Eye) – Figure 11

Hat-P-25b Project (by Model) – Figure 13

Hat-P-25b Published Values

Semi major axis (AU) 0.050+-0.005 0.0578+-0.004 0.0467

Period (days) 4.094+-0.445 5.051+-0.352 3.650+-0.000

Radius (Rj) 1.366+-0.044 1.298+-0.030 1.190+-0.081

Density (gcm-3) 0.295+-0.035 0.356+-0.022 0.420+-0.070

Table 5. Hat-P-25b parameters calculated from light curve fitted by eye and by the EPTM compared to published values.

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In Table 5 the errors were once again obtained using a best and worst fit line by shifting the theoretical curve to its maximum and minimum position that still allows it to still fit within the error bars. The values calculated from the model that differ from the published values are highlighted in red. It can be seen by this that the model has improved the accuracy of the radius and density by providing a better estimate to the total dip in flux but worsened the accuracy of the semi major axis and period by overestimating the transit duration. To create the light curve the published semi major axis was input into the EPTM, the light curve produced does not return this value suggesting that one of the other published values input is slightly out - either the planet mass, stellar mass, stellar radius or planetary radius. The stellar radius and planet mass are used in the calculation of the planet radius and density respectively; since these parameters came out close to the published values the published parameters used to calculate them are likely to be accurate leaving either the stellar mass or planetary radius as the likely parameter to be inaccurate. If the mass of the star was too large the planet’s velocity would be too and therefore the dip in flux would start sooner in Figure 14 than in Figure 11 which it does, however, it would also end sooner, which it does not. If the published planet radius is too large then the model would predict that flux is blocked sooner at the start of the transit and longer at the end than it actually is which could explain why the transit duration here is too long and therefore highlights the possibility that the published radius may be slightly too high. To test this in future the lower limit of the published radius could be input into the EPTM to see if this improves the calculated results, possibly leading to a more accurately known radius of Hat-P-25b.

5. Conclusion

Gaia was used for photometry on over 300 fits files to plot the transit light curves of Hat-P-25b, Wasp-43b and Wasp-2b from which the semi major axis, orbital period, planetary radius and density were calculated for each exoplanet. These values are summarised in Table 4 and lead to the conclusion that the photometric method used throughout this study can be highly accurate having led to 60% of the parameters calculated being within the limits of the published values, however its accuracy quickly reduces with increasingly noisy data. To fit a theoretical model to the data rather than fitting a light curve by eye the EPTM was created using Python and was tested on Hat-P-25b. It was found to improve the accuracy of the light curve in terms of measuring the total dip in flux however overestimated the transit duration – this is likely due to its high dependence on the accuracy of the published values entered into the model. Through analysis of known planetary data little relationship was found between the stellar temperature and planetary semi-major axis or the stellar mass and planetary mass indicating the vast range of companion parameters possible for any specific star, not expected to such an extent from our own Solar System. Figure 3 showed that there has been a continuing exponential growth of the number of exoplanets known since the first was detected, forever improving our knowledge on where we are in the galaxy, the evolution of exoplanets and any relationships present between their parameters and their host stars. As technology has improved over the years it also showed a continual decrease in the mass possible to detect, suggesting that as the precision of our instruments improves, the abundance of known Earth sized planets will continue to increase, perhaps leading to the detection of extra-terrestrial life in the not so distant future.

6. Future WorkThis project could be continued in the future to calculate the parameters of more planets, potentially focusing on constraining the parameters of Earth sized planets or for planets around more extreme stars than the Solar type host stars used in this project. The methods developed throughout this project could also be used to analyse multiple transits of the same planet – if variations between the transits are observed then it would indicate there may be a multiple planetary system present. The EPTM could be further tested on Wasp-2b and Wasp-43b and could be improved by extending the model to fit using curve fit. Finally, it would be beneficial to combine Sextractor with Python to perform automated photometry which would greatly reduce the time needed to take data, particularly useful when analysing large data sets.

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8. Appendix 1 – EPTM Code for Hat-P-25b

#Define Hat-P-25b parameters

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G= 6.67E-11M_star= 1.01*(1.9891E30) #kgM_planet= 0.567*(1.89813E27) #kga= 0.0465*(1.5E11) #semi major axis (m)R_planet=1.19*(6.9911E7) #radius of planet =1.19RjR_star=0.959*(6.955E8) #radius of star =0.959Rsunt_start=-9120 #seconds Total time data's taken over is 18240secondst_stop=9120 #seconds#t_points=1824 #Time point every 10secondst_points=109 #To have time array same size as number of data points for the planetIo=1 #For Fb equationu= 1 #Limb darkening coefficient for Fb equation

#create array with 't_points' as the number of time points to calculate positions for latert=linspace(t_start, t_stop, t_points)

#Find orbital velocity of planetv= sqrt((G*(M_star+M_planet))/a)print 'The orbital velocity of the planet is', v, 'm/s'

#Create an array of the position of the planet at each time point with respect to the centre of the star being 0x_pos= v*tprint 'x_pos is', (x_pos)

#Creating 2D array of planetary pixelspixels1=array([[0,0], [0,1], [0,2], [0,3], [0,4], [0,-1], [0,-2], [0,-3], [0,-4], [1,0], [1,1], [1,2], [1,3], [1,-1], [1,-2], [1,-3], [2,0], [2,1], [2,2], [2,-1], [2,-2], [3,0], [3,1], [3,-1], [4,0], [-1,0], [-1,1], [-1,2], [-1,3], [-1,-1], [-1,-2], [-1,-3], [-2,0], [-2,1], [-2,2], [-2,-1], [-2,-2], [-3,0], [-3,1], [-3,-1], [-4,0]], dtype=float)y=ones(41)

#pixels_radius=zeros(41)x_pix=zeros(41)y_pix=zeros(41)for i in range(41): #pixels_radius[i]=(1/pixels1[i,1])*R_planet x_pix[i]=(pixels1[i,0]/4)*R_planet y_pix[i]=(pixels1[i,1]/4)*R_planetprint x_pixprint y_pix

#Calculate the distance of each pixel from the centre of the star at each time pointd_pix=zeros((41, 109))Fb_t = zeros(109)for x in range(len(x_pos)): #for each time point Fb = 0 for i in range(41): #for each x_pix and y_pix value d_pix_a=sqrt(((x_pix[i]+x_pos[x])**2)+(y_pix[i]**2)) #+x_pos to each x value to find position of each pixel as the planet moves across the star d_pix[i] = d_pix_a if d_pix_a <= R_star:

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Fb += ((1-u*(1-sqrt(1-(d_pix_a/R_star)**2)))/41)/54 #divide by the number of pixels (41), multiply by the solid angle (1/54) found by adjusting to get graph to fit observed dip in observed graph. Fb_t[x] = Fbprint 'd_pix is', d_pix

#Plot curve figure(2)print Fb_tF=1-Fb_t #Fb is th eflux blocked at any pointT=t+9120 #to start at t=0 to fit graph of dataplot (T, F, 'k') #plot outside of a loopxlabel ('Time (s)')ylabel('Relative Flux')title('Hat-P-25b Transit Light Curve using EPTM')

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