The Stability of Observed Multiple-Planet Systems Karl W Jansson Lund Observatory Lund University 2012-EXA60 Degree project of 60 higher education credits (for a degree of Master) May 2012 Lund Observatory Box 43 SE-221 00 Lund Sweden The Stability of Observed Multiple-Planet Systems Karl W Jansson June 4, 2012 1 Abstract According to theories of planet formation, jovian planets are formed on relatively circular, non-inclined, wide orbits. Despite this, many observed jovian planets are found on tight eccentric orbits. This problem could possibly be solved with multiple-planet systems. A planet in a multiple-planet system is not only affected by the host star but also by the other planets. The interactions express themselves in that the planets transfer angular momentum between each other and the eccentricities and inclination of their orbits will oscillate. I have investigated if these planet-planet interactions can perturb a `circular' system enough to eject one planet and leave the rest of the system more tightly bound with orbits that have higher eccentricities and hence explain the observed exoplanets. I have studied the evolution of both the observed multiple-planet systems and systems I have generated myself both analytically with perturbation theory and numerically with N-body simulations. I have investigated how the stability of those systems depend on different parameters of the systems such as semi-major axes, inclinations and masses. My conclusions are that the stability of a multiple-planet system is a very sensitive function of the initial separations between the planets but also that the time between the first scattering event and disruption of the system (collision or ejection) is independent of the initial separations. I also find that multiple-planet systems are very chaotic in the sense that small changes in initial conditions lead to huge differences in the outcome. 2 3 Popul¨arvetenskaplig sammanfattning p˚asvenska Solen ¨arinte den enda stj¨arnanmed planeter i universum. Sedan 1995, d˚aden f¨orsta extra-sol¨araplaneten uppt¨acktes, har 770 exoplaneter observerats och fler uppt¨acks hela tiden. Enligt teorier om hur planeter formas f¨odsgasj¨attarmed cirkul¨ara,vida banor l˚angtfr˚ansin stj¨arnaprecis som planeterna i solsystemet. Trots detta har m˚angaav de observerade exoplaneterna v¨aldigtelliptiska (excentriska) banor som dessutom ¨araldeles f¨orn¨arastj¨arnan.F¨oratt l¨osadet h¨arproblemet har det, i litteraturen, f¨oreslagitsflera teorier. En ¨aratt en andra stj¨arnaflyger f¨orbiplanetsystemet och ¨andrarplaneternas banor. En annan ¨aratt man har planetsystem med mer ¨anen planet. I ett s˚adant fler- planetsystem p˚averkas planeterna inte bara av stj¨arnan utan av varandra ocks˚a.De h¨ar planet-planetinteraktionerna skulle d˚akunna ¨andraplaneternas banor s˚aatt planeterna kommer n¨armare varandra. Om tv˚aplaneter kommer riktigt n¨aravarandra kommer gravitationen mellan dem att ¨andraderas banorna ¨annu mer och efter att detta har up- prepats ett antal g˚angerkommer en av dem kastas ut ur planetsystemet. Resultatet av detta skulle vara att de kvarvarande planeterna f˚arbanor som ¨armer excentriska och ligger n¨armarestj¨arnan. Jag har, i det h¨arprojektet, unders¨okthuruvida teorin med flerplanetsystem skulle kunna fungera. Jag har unders¨okthur ett flerplanetsystem utvecklas med tiden b˚ademed analytiska approximationer och numeriska simuleringar. Mina resultat visar att tiden det tar f¨ortv˚aplaneter att komma inom ett visst avst˚andfr˚anvarandra ¨arv¨aldigtberoende p˚adet ursprungliga avst˚andetmellan dem. Mer ¨overaskande ¨aratt tiden mellan f¨orsta g˚angenplaneterna kommer riktigt n¨aravarandra och det att en planet blir utkastad ¨ar konstant, oberoende av den ursprungliga separationen mellan planeterna. Slutsatsen man kan dra av detta ¨aratt efter det att tv˚aplaneter i ett planetsystem har st¨ottp˚avarandra en g˚angser utvecklingen v¨asentligen likadan ut f¨oralla planetsystem. 4 5 Acknowledgements I first of all want to thank my supervisor Melvyn B. Davies for providing me with such an interesting project and for all inspiration and help he has given me. When I started with the thesis I had the idea that it would be a project slightly larger than my Bachelor's project. Oh so wrong I was. At times my motivation dropped and I thought I got nowhere just in time to have a meeting with Melvyn, who removed some of my question marks, got me super inspired and I was able to dive right back into work again. I think I now have started to realize what science really is: find a mediocre solution to your question only to discover two new questions you could investigate. Melvyn, you and Ross are also the reason I went from studying theoretical physics to astronomy. Thank you so much! I also want to thank everyone at the Astronomy Department. To me, we are one giant family. I love that if you have a question for someone you can just go to that person's office and knock on the door. I especially want to thank the other Master's students and the PhD students for making it so easy to crawl out of bed every morning, put on my clothes and sleep walk up S¨olvegatan to Astro. Hiva, you have been a great office mate and I am sorry for all the times I have interrupted you in your work to discuss mine. Thanks also goes to my friends at Theoretical Physics without which I would never have come this far. I just say: Go Teoretisk Elit! Finally I also want to thank my family and then especially my grandmother for all the support they have given me during all my studies. 6 CONTENTS 7 Contents 1 Introduction to exoplanets 9 1.1 What is an exoplanet? . 9 1.2 Multiple-planet systems . 9 2 Planetary systems 11 2.1 Derivation of the two-body problem . 11 2.2 Orbital parameters . 11 2.3 Properties of multiple-planet systems . 13 2.3.1 Mutual inclination, ∆I ........................ 13 2.3.2 Mutual Hill radius, rmH ....................... 14 2.3.3 Mean motion resonance . 14 3 Analytic approach for two-planet systems 15 3.1 Derivation of the secular oscillation equations . 15 3.2 Secular oscillations in the Solar System . 17 3.3 Comparison with N-body simulations . 20 3.3.1 The true Jupiter-Saturn system . 20 3.3.2 Jupiter and Saturn in mean motion resonance . 22 3.3.3 Jupiter and Saturn far from resonance . 24 3.4 Investigation of the dependence on properties of the two-planet system . 25 3.4.1 Fixed mass . 26 3.4.2 Separation between the planets . 27 3.4.3 Stellar mass . 29 3.4.4 Equal-mass planets . 31 4 Simulation setup 34 4.1 MERCURY6 ................................... 34 4.2 How to decide initial conditions . 35 4.3 Fitting method . 35 5 Three-planet systems with equal separations, ∆12 = ∆23 37 5.1 Simulation setup . 37 5.2 Timescales . 37 5.3 Planets involved in close encounters and ejections . 41 5.4 Evolution of the structure of the systems . 43 5.5 Conclusions . 46 6 Instability mechanisms 48 6.1 Stages in the evolution of a multiple-planet system . 48 6.2 Planet-planet interactions . 49 6.3 Fly-by stars . 50 6.4 Kozai mechanism . 50 7 The observed multiple-planet systems 52 7.1 Observation methods . 52 7.2 The observed systems . 53 7.3 Simulations . 54 CONTENTS 8 7.3.1 The Solar System . 54 7.3.2 HD 181433 . 57 7.3.3 HD 125612 . 58 7.3.4 HD 202206 . 60 7.4 Conclusions . 62 8 Three-planet systems with unequal separations, ∆12 6= ∆23 64 8.1 Simulation setup . 64 8.2 Close encounter timescale . 64 8.2.1 Spine-and-Ribs model . 65 8.2.2 Minimum-∆ model . 67 9 Low-mass planets in a protoplanetary disk 70 9.1 Description of the problem . 70 9.2 Simulations . 71 9.2.1 Close encounter timescales . 71 9.2.2 After the first close encounter . 74 10 Possible future extensions 78 References 80 A Derivation of orbital parameters for one planet 81 A.1 The two-body problem in two dimensions . 81 A.2 The position as a function of time . 86 A.3 The two-body problem in three dimensions . 88 B Derivation of the secular oscillation equations 89 9 1 Introduction to exoplanets 1.1 What is an exoplanet? A planet is, by definition by the International Astronomical Union, a celestial body that is orbiting a star or a stellar remnant (white dwarf, neutron star or black hole). It has to be massive enough that its self-gravity has made it round. It is not massive enough to start nuclear fusion and it must have cleared its neighborhood from other planetesimals. An exoplanet is a planet which orbits a star that is not the Sun. The first exoplanet orbiting a main-sequence star was detected in 1995 (Mayor & Queloz 1995) on a four day orbit around the Sun-like star 51 Pegasi. To date over 750 exoplanets have been detected and confirmed in the exoplanet database (exoplanet.eu, Schneider et al. (2011)). According to theories of planet formation, jovian planets (gas giants) are formed on relatively circular, non-inclined, wide orbits. Despite this, many observed exoplanets are found on tight eccentric orbits. 1 Observed exoplanets Solar System 0.9 0.8 0.7 0.6 0.5 Eccentricity 0.4 0.3 0.2 0.1 0 0.01 0.1 1 10 100 Semi-major axis (AU) Figure 1.1: Plot of the eccentricity and semi-major axis of all exoplanets (red crosses) in the exoplanet database (exoplanet.eu, Schneider et al.