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Summer Research Fellowship Programme of India's Science Academies

Detection of exoplanets using transit photometry

Sky Debnath

Department of Physics, National Institute of Technology Agartala, Jirania, West Tripura, Tripura 799046

Avinash A Deshpande

Astronomy and Astrophysics, Raman Research Institute, C. V. Raman Avenue, Sadashivnagar, Bengaluru 560080

Abstract

"Where did we come from? Are we alone? How life came into existence? How planets formed in our own Solar System? Is the Sun only star with planets? If not, how many are there? Out of those, how many are Earth-like? Of those, how many are habitable? Of which, how many can we visit and probably colonize?"- All these questions baffled us since infancy of humanity. The knowledge about the existence and characteristics of exoplanets is indispensable for giving us an answer. An Exoplanet or Extra-solar planet is a planet outside the solar system. Among about 200 billion stars in the Milky Way Galaxy, it is estimated that there can be 11 billion habitable earth-sized planets in this galaxy. This number is as high as 40 billion if planets orbiting the numerous red dwarfs are included. But as of 1 July 2019, there are only 4,096 confirmed exoplanets have been detected by the astronomers and therefore, a majority of them await for their discovery in our own galaxy alone. It was only over the past few decades that we first discovered evidence of their existence. The reason why it took so long to detect these objects is the fact that planets are dark objects and are very close to extremely bright source of light, their host star. In light, a planet is billion times fainter than its host star. Various methods are being used to detect them, one of which is transit photometry. On 7 March 2009, Kepler Space Telescope was sent by NASA to survey our part of Milky Way in search of earth-sized exoplanets using transit photometry method. My project is an attempt to learn about detection of exoplanets using the data available from the Kepler mission.

Keywords: Milky Way Galaxy, Kepler mission, transit photometry

INTRODUCTION

"This space we declare to be infinite... In it are an infinity of worlds of the same kind as our own." - Giordano Bruno (1584)

"We live on a planet which we call our mother Earth. This planet of ours revolve around a star which we call The Sun. There are seven more planets revolving around the sun and there are some other celestial objects too doing the same. The whole system together we call it our own SOLAR SYSTEM." - That was the only known planetary system for a long time until the first exoplanet was successfully detected. For centuries scientists, philosophers, and science fiction writers suspected that extrasolar planets existed, but science needs proof and there was no way of detecting them or of knowing their frequency or how similar they might be to the planets of our own Solar System. In the sixteenth century, the Italian philosopher Giordano Bruno, an early supporter of the Copernican theory that Earth and other planets orbit the Sun (heliocentrism) put forward the view that the fixed stars are similar to the Sun and are likewise accompanied by planets. In the eighteenth century, the same possibility was mentioned by Isaac Newton in the "General Scholium" that concludes his Principia. Making a comparison to the Sun's planets, he wrote "and if the fixed stars are the centres of similar systems, they will all be constructed according to a similar design and subject to the dominion of One." Various detection claims made in the nineteenth century were rejected by astronomers. The first evidence of an exoplanet (Van Maanen 2) was noted as early as 1917, but was not recognized as such. The first suspected scientific detection of an exoplanet occurred in 1988. Shortly afterwards, the first confirmed detection came in 1992, with the discovery of several terrestrial-mass planets orbiting the pulsar PSR B1257+12. The first confirmation of an exoplanet orbiting a main sequence star was made in 1995, when a giant planet was found in a four-day orbit around the nearby star 51 Pegasi​​.

Difficulties in Direct Detection of Exoplanets and Indirect Methods used for their Detection

An Exoplanet or Extrasolar planet is a planet revolving around some other star. If we are able to directly observe a planet, we can directly have the spectrum of its atmosphere using a conventional spectrometer or an integral field spectrometer. Thus it is possible to gather information about the physical and chemical characteristics of a planetary atmosphere. But the direct imaging of exoplanets is very challenging. The reason why it is difficult to detect these objects is the fact that planets are dark (i.e. don't have light of their own) and are very close to extremely bright source of light, their host star. In visible light, a planet is billion times fainter than its host star. So, we use various indirect methods to detect them among which Radial Velocity Method and Transit Photometry Method are the most successful.

Radial velocity method

The radial velocity (RV) method has played a fundamental role in exoplanet science. It is one of the most successful detection methods with over 1000 exoplanet discoveries to its credit. If it were not for RV measurements, we would probably not be studying exoplanets today. The basic principle behind the RV method for the detection of exoplanets is quite simple. One measures the line-of-sight (radial) velocity component of a star as it moves around the center of mass of the star-planet system. This velocity is measured via the Doppler effect, the observed shift in the wavelength of spectral lines due to the motion of the star.

Radial_velocity_doppler_spectroscopy.gif
    Radial Velocity of star causing periodic doppler shift in the spectral line. (Image source: https://commons.wikimedia.org/wiki/File:Radial_velocity_doppler_spectroscopy.gif​)

    If the mass of host star MS is known, using RV method we can easily calculate the mass of the planet MP, its orbital radius a, hence other parameters like orbital velocity VP can also be measured.

    If P is period of revolution,

    a  =  (G(Ms  +  Mp)4π2P2)13      (using    modified    Keplers  3rd  Law)\displaystyle a\;=\;\left(\frac{G(M_s\;+\;M_p)}{4\mathrm\pi^2}P^2\right)^\frac13\;\;\;(using\;\;modified\;\;Kepler's\;3rd\;Law)

    and

    VP  =  G(Ms  +  Mp)a\displaystyle V_P\;=\;\sqrt{\frac{G(M_s\;+\;M_p)}a}

    Transit photometry

    This method detects distant planets by measuring the minute dimming of a star as an orbiting planet passes between it and the Earth. The passage of a planet between a star and the Earth is called a "transit." If such a dimming is detected at regular intervals and lasts a fixed length of time, then it is very probable that a planet is orbiting the star and passing in front of it once every orbital period.

    transit.gif
    • 1
    Transit by a planet causing dip in the brightness. (Image Source: http://www.pitt.edu/~stepup/DataAnalysis.html )

    The dimming of a star during transit directly reflects the size ratio between the star and the planet: A small planet transiting a large star will create only a slight dimming, while a large planet transiting a small star will have a more noticeable effect. The size of the host star can be known with considerable accuracy from its spectrum, and photometry therefore gives astronomers a good estimate of the orbiting planet's size, but not its mass. This makes photometry an excellent complement to the RV method, which provides an estimate of a planet's mass, but not its size. Using both methods, combining mass and size, scientists can calculate the planet's density, an important step towards assessing its composition.

    Present Objective

    This project is an attempt to learn about analysing the data available from Kepler Mission which used Transit Photometry Method for detection of exoplanets.

    Scope

    Planetary systems come from protoplanetary disks that form around stars during the process of star formation. Planets may form within a few to tens (or more) of millions of years of their star forming. The planets of the Solar System can only be observed in their current state, but observations of different planetary systems of varying ages allows us to observe planets at different stages of evolution. If we can overcome certain technological limitations in the near future, it may be possible that we can observe the origin of life in any of exoplanet which will answer the biggest question of human existence-"Origin of life on Earth".

    AN OVERVIEW OF KEPLER SPACE TELESCOPE

    "If we find lots of planets like ours........ We'll know its like that we aren't alone and that some day we might be able to join other intelligent life in the Universe" - William J. Borucki, Principle Investigtor for NASA's Kepler Mission.

    About the Mission

    Kepler space telescope is a retired space telescope launched by NASA to discover Earth-size planets orbiting other stars. Named after astronomer Johannes Kepler, the spacecraft was launched on March 7, 2009 into an Earth-trailing heliocentric orbit. After nine years of operation, the telescope's reaction control system fuel was depleted, and NASA announced its retirement on October 30, 2018.

    Kepler has a fixed field of view (FOV) against the sky. Kepler's FOV covers 115 square degrees, around 0.25 percent of the sky, or "about two scoops of the Big Dipper". Thus, it would require around 400 Kepler-like telescopes to cover the whole sky. The Kepler field contains portions of the constellations Cygnus and Lyra.

    The scientific objective of Kepler is to explore the structure and diversity of planetary systems. This spacecraft observes a large sample of stars to achieve several key goals:

    • To determine how many Earth-size and larger planets there are in or near the habitable zone (often called "Goldilocks planets") of a wide variety of spectral types of stars.
    • To determine the range of size and shape of the orbits of these planets.
    • To estimate how many planets there are in multiple-star systems.
    • To determine the range of orbit size, brightness, size, mass and density of short-period giant planets.
    • To identify additional members of each discovered planetary system using other techniques.
    • Determine the properties of those stars that harbor planetary systems.

    Components of Kepler Space Telescope

    The Kepler Telescope is a wide-field Schmidt Telescope composed of Schmidt Corrector Assembly (SCA) which forms the entrance aperture to the instrument, the spherical Primary Mirror Assembly (PMA), the sunshade, Upper and Lower Telescope Housings, focus mechanisms, heaters and thermometers.

    Screenshot (32).png
     Kepler Flight System showing integrated photometer and space craft. 
      Screenshot (33).png
       Kepler photometer architecture. 
        Kepler Space Telescope (Image source : KSCI-19033: Kepler Instrument Handbook)

        Collection of Data

        The principle instrument in the Kepler is the photometer which measures the brightness variations of the stars. The photometer consists of the telescope, the focal plane array, and the local detector electronics. Kepler is a 0.95-meter (37-inch) aperture Schmidt-type telescope with a 1.4-meter (55-inch) primary mirror. For an astronomical telescope, Kepler’s photometer has a very wide field of view: it’s about 15 degrees across. The photometer features a focal plane array with 95 million pixels. It stared continuously at a large field of view to observe more than 100,000 stars simultaneously. The starlight enters the telescope, reflects from the primary mirror to the focal plane array of 21 modules each with two 50x25 mm 2200x1024 pixel CCDs. The pixels collect the photons of light from the stars. Every 6 seconds, the array “reads out” the number of photons in each pixel to an onboard computer for storage and initial processing. For the selected stars, the data (photon counts) accumulates in an on board computer, and is transmitted to Earth once each month. This data is accessible in Mikulski Archive for Space Telescope (https://archive.stsci.edu/kepler/). The data used for this project have been taken from Mikulski Archive for Space Telescope.

        EXOPLANETARY TRANSIT

        An exoplanetary transit will be visible only for the orientation of orbital plane that is sufficiently close to the "edge-on" (i≈90°). Inclination i is the angle between the line of sight and the orbital plane of the planet. For inclination angle i=90°, we will observe the planet to transit along the diameter of the stellar disc otherwise the transit will occur along a chord at a distance "bRS" from the centre of the stellar disc. Here "b" is known as impact parameter which is just how far the planet transists from the center of the stellar disc and is given by

        b =  a cosi  RS

        If P is the orbital period of the planet which can be obtained from the time gap between two consecutive transits, we can easily calculate the orbital radius a using eqn (1).

        Normalising No-Transit flux as 1 and taking the maximum dip in flux during the transit to be "Δ", Since the dip corresponds to the ratio of the projected areas of the planet and the star, we can write

        Δ  =    (RPRS)2\displaystyle \Delta\;=\;\;\left(\frac{R_P}{R_S}\right)^2

        But in real condition the light curve for the transit is not flat-bottomed due to the limb darkening of the stellar photosphere instead it's round bottomed. Taking "u" as limb-darkening coefficient we get (​Andrew Collier Cameron, 2016​)

        Δ  =  3(  1    u  +  u1b2    )3    u(RPRS)2\displaystyle \Delta\;=\;\frac{3\left(\;1\;-\;u\;+\;u\sqrt{1-b^2\;}\;\right)}{3\;-\;u}\left(\frac{R_P}{R_S}\right)^2
        Screenshot (34).png
          Schematic view of the regions of the celestial sphere from which full and grazing transits are visible.(image source: Methods of Detecting Exoplanets, page 90)
          Screenshot (35).png
          The light curve for uniformly  bright stellar disc
            HAT-P-3b.gif
            The light curve is round bottomed due to the limb darkening
              Transit Phase

              WORKING WITH KEPLER DATA

              Understanding the Lightcurve of Kepler 7 (KOI - K00097)

              Kepler-7 is a sun-like star located in the constellation Lyra in the FOV of the Kepler Mission.

              General Information about Kepler-7 
               PropertiesValue Units
              Kepler ID  5780885
               Mass1.347  solar mass (MSun)
               Radius1.843  solar radius (RSun)
               RA19.238768hours
               Dec41.089810 degrees
               Distance  950 ± 20pc 
              target pixe 1.png
                Target Pixel File of the first quarter of observation of Kepler7.

                Target Pixel Files (TPFs) from Kepler mission, contain movies of the pixel data centered on Kepler 7. TPFs can be thought of as stacks of images, with one image for every timestamp the telescope took data. Each timestamp is referred to as a cadence. These images are cut out ‘postage stamps’ of the full observation to make them easier to work with.

                Light curves are data which encapsulate the brightness of a star over time. Light curve is created from the Target Pixel File using Simple Aperture Photometry. Aperture Photometry is the simple act of summing up the values of all the pixels in a pre-defined aperture, as a function of time.

                light curve ot transit 1.png
                   Graph 1- Light curve  created from TPF of the quarter 1 of observation of Kepler 7.

                  Graph 1 (Fig 8) is obtained by extracting the normalised light curve data from the quarter 1 of Kepler-7. Stellar flux is shown along Y-axis. X-axis represent the BKJD (Barycentric Kepler Julian Date is a time unit used by Kepler Mission). Seven dips are clearly visible between 131–165 BKJD.

                  flat1.png
                    Graph 2- After flattening the light curve, Periodic transits can be observed in the light curve. 

                    In Graph 2 (Fig 9), lightcurve is flattened so as to make the stellar flux to be 1. It is observed that all the dips in flux are of same magnitude with uncertainity of 0.0002. Time gap between two consecutive dips is found to be approximtely 4.88 days. Randomly two consecutive transits are choosen as shown in Graph 3. Observation of two consecutive transit with same dip in flux is a clear indication of an object orbiting the star.

                    Figure_1.png
                      Graph 3 - Two consecutive transit with same dip(σ = 0.0002)

                      From Graph 3 (Fig-10), Orbital Period is obtained to be 4.88 days.

                      Using Kepler's 3rd Law, semi major axis, a = 0.095 A.U.

                      Transit.png
                      A single transit 
                        edited.png
                        Parameters are pointed out
                          Graph 4- A single transit

                          From Graph 4b (Fig: 11) findings are:-

                          Maximum depth in flux, Δmax = 0.0074 ± 0.0002 (limb darkening coefficient = 0 )

                          Total transit duration, T = 0.225 ± 0.001days

                          Ingress duration = 0.040 ± 0.001days

                          Full transit duration, t = 0.143 ± 0.001days

                          Egress duration = 0.041 ± 0.001days

                          Using eqn(3), Radius of the planet, RP = 17.03 Earth Radius (or 1.55 Jovian Radius)

                          RESULTS AND DISCUSSION

                          On analysing the light curve of the star Kepler-7, similar and periodic dips in the star flux are observed. These dips gave me the following information:

                          • Some object of considerable size is orbiting the star.
                          • The uncertainity in the normalised flux data is 0.0002.
                          • Orbital period of the object is 4.88 days.
                          • Semi-major axis of its orbit is 0.095 A.U.
                          • Its radius is calculated to be 17.03 times that of Earth or we can say 1.55 times of Jupiter radius. It is really a very big object.
                          • The total transit duration of the object is 0.224 days.

                          These results are compared with that in the Data Validation (DV) Report for Kepler ID- 5780885.

                          The value of Semi-major axis differs by 0.005 A.U. and Orbital period by 0.02 days.

                          CONCLUSION

                          I would like to conclude this report by answering few questions which may come to the reader's mind,

                          1. What is this project about?

                          Ans: This project deals with the transit method of detection of exoplanets. This is an attempt to learn how we can use transit photometry to understand the characteristic properties of an exoplanet.

                          2. Was the project successful?

                          Ans: Yes, in this project, using transit photometry it has been successfully shown that a Sun-like star Kepler-7 is accompanied by an orbiting companion of size 1.55 MJ.

                          3. What does this project lack?

                          Ans: This project didn't take further steps towards verifying that the object we are talking about is actually a planet and not anything else like grazing stellar binaries, transiting red/brown dwarfs or blended stellar binaries.

                          4. What are the shortcomings of Transit Method?

                          Ans: Transits can tell us so much more about the systems than anything else, but they are rare because they require chance orbital alignment with us. So we need to survey tens of thousands of stars to have a chance of finding just one. The dependence on orbital alignment means that transits are most likely to happen in systems where the planet is close to its host star, so the technique preferentially discovers this type of planet system.

                          5. What is the benefit of detecting exoplanets which are too far for us to go and colonize?

                          Ans: Being a part of this Solar System, there is a limitation to our knowledge about planetary system. Ability to observe other planetary systems can enhance our knowledge about planetary evolution. May be someday with more technological advancements, we will be able to observe the origin of life in an exoplanet or may find another intelligent species. But for all these, first step is to find an exoplanet. After that only we can solve these mysteries and who knows, may be someday in the distant future Man will step foot on one of those exoplanets.

                          6. What have I learnt during this project?

                          Ans: In this vast ocean of Astronomy, for the last two months I was more like swimming along the shore only, but I got an idea about its stretches. It was really a wonderful journey in this ocean of knowledge. But speaking about this specific project, I got to know more about exoplanets, their observation, I have learnt how to analyse the light curves. I also learnt about functioning of Kepler Space Telescope.

                          7. So, What is next?

                          Ans: After working on Transit photometry, I am looking forward to learn using Radial Velocity for exoplanetary detection. In future, I am willing to work on how we can find signatures of life on those exoplanets.

                          Bibliography

                          https://archive.stsci.edu/kepler/preview.php?dsn=KPLR011446443-2009231120729&type=SC

                          https://www.universetoday.com/category/kepler-mission-2/

                          https://www.nationalgeographic.com/magazine/2018/03/one-strange-rock-interactive-earth-solar-system-milky-way-galaxy/#close

                          https://exoplanets.nasa.gov/the-search-for-life/life-signs/

                          http://docs.lightkurve.org/tutorials/01-what-are-lightcurves.html

                          https://commons.wikimedia.org/wiki/File:Radial_velocity_doppler_spectroscopy.gif

                          http://www.astro.gsu.edu/lab/Supplemental_labs/supplemental_labs_files/exoplanets/Exoplanet_writeup.pdf

                          http://www.planetary.org/explore/space-topics/exoplanets/transit-photometry.html

                          https://en.wikipedia.org/wiki/Kepler_space_telescope

                          https://www.nasa.gov/kepler/faq#c

                          https://lco.global/spacebook/exoplanets/transit-method/

                          Data Validation Report for Kepler ID- 5780885.

                          ACKNOWLEDGEMENTS

                          I take this opportunity to express my thanks of gratitude to my guide Prof. Avinash A Deshpande for his useful comments, remarks and engagement through the learning process of this internship. Furthermore special thanks to Indian Academy of Sciences for providing me the golden opportunity to work on this wonderful project on the Exoplanets, which helped me to know about so many new things. I am really thankful to them. I would also like to thank friends especially Yash and Shubham who helped me a lot in completing this project within the limited time frame and special thanks to Subhrajeet and Paramartha for their moral support. Thank you all.

                          References

                          • Andrew Collier Cameron, 2016, Extrasolar Planetary Transits, Methods of Detecting Exoplanets,Astrophysics and Space Science Library, pp. 89-131

                          Source

                          • Table 1: en.wikipedia.org/wiki/Kepler-7
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