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Refining the Orbits of the Planets in HD 207832

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Refining the Orbits of the Planets in HD 207832 Refining the orbits of the planets in HD 207832 Emil Zadera Lund Observatory Lund University 2017-EXA119 Degree project of 15 higher education credits June 2017 Supervisor: Alex Mustill Lund Observatory Box 43 SE-221 00 Lund Sweden Abstract The star, HD 207832, has two Jupiter-like planets on orbits with poorly constrained ec- +0:22 +0:18 centricities. The eccentricities are 0.27−0:10 and 0.13−0:05 respectively. Notably, the two sigma error allows eccentricities up to 0.71 for one of the planets. Due to the large error bars, one aim of this project is to refine them. This is done by simulating the system for different initial eccentricities within the two sigma error bars for both of the planets. If a simulated system is shown to be unstable, the initial eccentricities used in the simulation, can not describe the observed system, HD 207832. In this project, it has been shown that the outer planet, in HD 207832, can not exceed an initial eccentricity of 0.6 in order for the system to remain stable. Furthermore, The level of chaos of the two planets, in each simulated system, is inves- tigated with the use of Fourier analysis. A code is written which calculates the Fourier transform of the eccentricities. The code then counts the number of peaks in the spec- trum which determines the level of chaos in the system. In this project, the use of Fourier analysis, to determine the level of chaos, is shown to be useful when comparing the chaos between simulations that have similar integration times. It is also shown that the outcome in each simulation is very sensitive to the fixed timestep used. It is highlighted that small changes in the timestep can change the outcome of the simulation in the sense of making a stable system, unstable. HD 207832, further, has a habitable zone, where a planet can support liquid water on its surface, that is located between the two Jupiter-like planets. Radial velocity measurements have yet not been able to detect any planet within this zone. In this project, stable orbits for a small planet, within the habitable zone, are thus searched for. This is done for the nominal system of HD 207832, and for the case when one sigma has been subtracted from the eccentricities of the two Jupiter-like planets. In this project, by the use of test particles, a few orbits are shown to be stable over at least 250 Myr in both of the simulated systems. It is thus possible that HD 207832 has a habitable planet that has not yet been detected. Key words: HD 207832 { eccentricity space { chaos { habitable zone { stability Popul¨arvetenskaplig beskrivning Vad Klarar Dessa Exoplaneterna Av? V˚ardatabas av dokumenterade exoplaneter ¨aruppe i tusental och nya planeter uppt¨aks varje dag. Ofta ¨ardessa planeterna bundna till en stj¨arna,precis som planeterna i v˚arat solsystem ¨arbundna till solen. Vissa av de planetiska systemen ¨argoda kandidater till att inneh˚allaplaneter som liknar jorden, medan andra har f¨orstr¨angaf¨orh˚allanden.Oavsett m¨angdenav exoplaneter, ¨ardet viktig att f¨oljaupp och kontrollera hur v¨aldessa planet- systemen ¨arbeskrivna av parametrarna som ¨argivna i v˚arandatabas f¨oratt kunna analy- sera dem vidare. I detta projekt bidrar vi med information till databasen, av de or¨akneligt m˚angaexoplanet-systemen, genom att titta p˚aett av dem. Detta system best˚arav tv˚a planeter, stora som Jupiter, som circulerar omkring en stj¨arnavid namn HD 207832. Vi inspekterar systemets parametrar och ¨aven m¨ojlighetenav att en oidentifierad, beboelig planet skulle kunna befinna sig i systemet. Varf¨orHD 207832? M¨atningarnasom togs f¨oratt verifiera detta systemet, tyder p˚aatt de tv˚abefintliga plan- eterna har en mycket stor os¨akerhet i sina parametrar. Enligt m¨atningarnaskulle plan- eterna kunna befinna sig p˚amycket exotiska banor. Med exotisk i detta samanhanget menar vi att skillnaden mellan den h¨ogstaoch den l¨agstahastigheten f¨oren planet i sin om- loppsbana skulle kunna vara stor. Hastigheten hoss en planet ¨arkopplad till planet-banans eccentricitet, som ¨aren parameter som avg¨orplanetens n¨armasteposition till stj¨arnan,och det l¨angsta avst˚andetfr˚an stj¨arnan. H¨arunders¨oker vi vilka m¨ojligaeccentriciteter som planeterna klarar av f¨oratt fortfarande holla sig stabila i systemet! Vi simulera systemet med olika eccentriciteter p˚aplaneterna och ser hur banorna evolverar. Det visar sig att m˚angaav de eccentriciteterna som os¨akerheten i parametrarna i v˚ardatabas till˚ater,rent sagt leder till kaotiska eller helt ostabila banor f¨orplaneterna. Vi har d¨arf¨orlyckats minska felet i parameterarna som beskriver HD 207832. Kan en habitabel planet befinna sig i HD 207832? Stj¨arnan,HD 207832, delar m˚angaegenskaper med v˚aransol n¨ardet kommer till dess storlek. Detta inneb¨aratt en habitabel planet, som liknar Jorden, skulle befinna sig p˚a ungef¨arsamma avst˚andfr˚anstj¨arnansom Jorden ¨arfr˚ansolen. Detta omr˚adetr˚akar vara precis mellan de tv˚aplaneterna vi k¨annertill i HD 207832. Fr˚agan¨arom den Jord-lika planeten klarar av att h˚allasig kvar i systemet mellan de tv˚agiganterna, utan att kastas ut p˚agrundav gravitationen som drar planeten mellan de andra himlakropparna. I detta projekt har vi i simuleringar satt in en tredje planet i systemet. Planeten ¨ar satt p˚aolika avst˚andfr˚anstj¨arnaninnom den beboeliga zonen runt HD 207832. I de flesta fallen klara sig den tredje planeten inte mellan de tv˚aJupiter-liknande planeterna. Den brukar allts˚akollidera eller utl¨osasfr˚ansytemet. Men det finns n˚agrafall, d¨arvi under vissa f¨oruts¨attningarkunnat beh˚alladen tredje planeten. Detta inneb¨aratt vi har hittat banor d¨aren beboelig planet skulle kunna befinna sig runt HD 207832. Acknowledgements This project has been a great experience and I would like to thank all the people who helped me to get where I am. Firstly, I want to thank Dr. Alexander James Mustill, who gave me the opportunity to take the project under his great supervision, and who introduced me to the tools used in planetary science. It was always exciting to bring my results to our meetings. Thank you for the great explanations, and for your great feedback and guidance. I also want to give my thanks to friends I spent so much time discussing with. You have been a great inspiration for my work. Lastly, I want to thank my parents and siblings, who motivated me all the way up to this point. Thanks for keeping up the contact. Especially thanks to Veronika, for always being there. Contents 1 Introduction 5 1.1 HD 207832 . .6 1.1.1 The Habitable Zone . .7 1.1.2 The Semi-Amplitude of a Low-Mass Planet . .8 1.2 The Orbital Elements . .8 1.2.1 The Conserved Quantities and Two Coordinates systems . .9 1.3 Planetary Evolution . 10 1.3.1 The Chaos in Planetary systems . 11 1.3.2 The Stability of Planetary systems . 12 1.3.3 The Hill Stability . 13 2 Method 15 2.1 The Hybrid Symplectic Integrator .................. 15 2.1.1 Working with Mercury6 ....................... 15 2.2 The Stability Analysis . 16 2.2.1 The Nominal System with Variable Eccentricities . 16 2.2.2 Exploring the Eccentricity Space . 17 2.2.3 The Analysis . 18 2.3 The Search for a Habitable Planet . 19 2.3.1 Simulating with Test Particle . 19 3 Results 21 3.1 Exploring the Eccentricity Space . 21 3.1.1 100 Myr Simulations . 23 3.2 The Possibility of a Habitable Planet . 24 4 Conclusions 26 4.1 The Eccentricity Space of HD 207832 . 26 4.2 The Level of Chaos . 27 4.3 The Habitable Planet . 27 A The Principle used in Mercury6 31 1 List of Figures 1.1 The nominal planetary system in HD 207832 (Haghighipour et al., 2012) and its habitable zone defined by the standard model (Kasting et al., 1993), with a radius between 0.85 AU and 1.65 AU. The inner and the outer planet are HD 207832b and HD 207832c respectively. .7 1.2 Some of the basic parameters used to describe a planetary orbit. The shad- owed area corresponds to the part of the orbit below a reference plane. The planet is moving in the direction of the arrow, away from the ascending node.9 1.3 The plot shows a time interval from a simulation of HD 207832, where the inner and the outer planet have the initial eccentricities 0.20 and 0.57 respectively. Notably, the amplitudes in the oscillations are found to be large. 11 1.4 The Fourier transform of the eccentricity for the outer planet in HD 207832. The simulations are for the nominal system but with the initial eccentricities, 0.40, for the inner planet, and 0.55, 0.56, and 0.57 for the outer planet respectively. The three systems are classified as fairly non-chaotic with a number of peaks in their power spectrum within [1-50), see Section 2.2.3. 12 1.5 The eccentricity space of the planet HD 207832c versus HD 207832b. The Hill stability limit divides the region where collisions between planets are allowed, from where they are not (Gladman, 1993). The parameters used for this plot can be found in Table 1.1 & 1.2. The nominal system with the eccentricities 0.13 and 0.27 for the respective planet, and the ellipses spanned by its upper one sigma and two sigma error bars, have been plotted. 14 2.1 The level of chaos is determined by the number of peaks that intersect the limit at 1% of the strongest peak.
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  • The Photoeccentric Effect and Proto-Hot Jupiters. I
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