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postgraduate thesis: Numerical exploration of the probability of capture into the 3:1 mean motion resonance
Title  Numerical exploration of the probability of capture into the 3:1 mean motion resonance 

Authors  
Advisors  Advisor(s):Lee, MH 
Issue Date  2013 
Publisher  The University of Hong Kong (Pokfulam, Hong Kong) 
Citation  Chan, K. [陳嘉豪]. (2013). Numerical exploration of the probability of capture into the 3:1 mean motion resonance. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b4979971 
Abstract  Meanmotion resonances (MMR) are frequently observed in extrasolar planetary systems. It is generally believed that the resonances result from the convergent migration of planets. The much larger number of systems near the 2:1 resonance compared to 3:1 resonance in both the radial velocity and the Kepler data is probably due to the difference in the capture behaviors of 2:1 and 3:1 resonances.
To study the capture probability of the 3:1 resonance, numerical threebody integrations with forced migration have been used to examine the dependence of the capture probability on migration rate, planetary masses, and initial orbital eccentricities. First, the numerical results have been confirmed with analytic theory in the adiabatic limit (Borderies & Goldreich 1984) and numerical results of the Hamiltonian model beyond this limit (Mustill & Wyatt 2011) for both the interior and exterior resonances in the circular restricted threebody problem. Then, the numerical exploration of the restricted threebody problem (R3BP) has been extended to cases with nonzero planet eccentricity in the adiabatic limit. The capture probability decreases with increasing planet eccentricity at small test particle eccentricity but does not depend strongly on the planet eccentricity at large test particle eccentricity. Interestingly, the critical eccentricity of the planet, below which resonance capture is certain, is much larger than the critical eccentricity of test particle which was not expected.
Finally, the numerical exploration has been extended to situations with different planetary mass ratio m1/m2. In the cases where both planets are initially on circular orbits, the critical migration rate for certain capture agrees with that of Quillen (2006) in the R3BP. However, it does not change monotonically with m1/m2 and peaks at m1/m2 = 1. For m1/m2 = 1, the resonance capture is certain when the eccentricities of the inner and outer planets are small and decreases as the eccentricities increase. In contrast, the capture probability is low when the eccentricities are small and the capture probability peaks at certain values of the eccentricities in the nonadiabatic limits. The capture probability as a function of planet eccentricities for mass ratios m1/m2 = 0.5 and 2 in the adiabatic limit has also been studied. The capture probability at m1/m2 = 2 shows similar behaviors with m1/m2 = 1 but the capture behaviors at m1/m2 = 0.5 are significantly different from the capture behaviors at m1/m2 = 1.
This research has explored the probability of resonant capture in several new regimes, including the elliptical restricted threebody problem, comparable mass cases in the adiabatic limit and the equal mass case in the nonadiabatic limits. This work enhances our knowledge in the capture behaviors of 3:1 MMR in different limits and is useful in the future studies of the period ratio distribution in extrasolar planet systems. 
Degree  Master of Philosophy 
Subject  Extrasolar planets. Astrophysics. 
Dept/Program  Physics 
Persistent Identifier  http://hdl.handle.net/10722/181537 
HKU Library Item ID  b4979971 
DC Field  Value  Language 

dc.contributor.advisor  Lee, MH   
dc.contributor.author  Chan, Kaho.   
dc.contributor.author  陳嘉豪.   
dc.date.accessioned  20130303T03:21:05Z   
dc.date.available  20130303T03:21:05Z   
dc.date.issued  2013   
dc.identifier.citation  Chan, K. [陳嘉豪]. (2013). Numerical exploration of the probability of capture into the 3:1 mean motion resonance. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b4979971   
dc.identifier.uri  http://hdl.handle.net/10722/181537   
dc.description.abstract  Meanmotion resonances (MMR) are frequently observed in extrasolar planetary systems. It is generally believed that the resonances result from the convergent migration of planets. The much larger number of systems near the 2:1 resonance compared to 3:1 resonance in both the radial velocity and the Kepler data is probably due to the difference in the capture behaviors of 2:1 and 3:1 resonances. To study the capture probability of the 3:1 resonance, numerical threebody integrations with forced migration have been used to examine the dependence of the capture probability on migration rate, planetary masses, and initial orbital eccentricities. First, the numerical results have been confirmed with analytic theory in the adiabatic limit (Borderies & Goldreich 1984) and numerical results of the Hamiltonian model beyond this limit (Mustill & Wyatt 2011) for both the interior and exterior resonances in the circular restricted threebody problem. Then, the numerical exploration of the restricted threebody problem (R3BP) has been extended to cases with nonzero planet eccentricity in the adiabatic limit. The capture probability decreases with increasing planet eccentricity at small test particle eccentricity but does not depend strongly on the planet eccentricity at large test particle eccentricity. Interestingly, the critical eccentricity of the planet, below which resonance capture is certain, is much larger than the critical eccentricity of test particle which was not expected. Finally, the numerical exploration has been extended to situations with different planetary mass ratio m1/m2. In the cases where both planets are initially on circular orbits, the critical migration rate for certain capture agrees with that of Quillen (2006) in the R3BP. However, it does not change monotonically with m1/m2 and peaks at m1/m2 = 1. For m1/m2 = 1, the resonance capture is certain when the eccentricities of the inner and outer planets are small and decreases as the eccentricities increase. In contrast, the capture probability is low when the eccentricities are small and the capture probability peaks at certain values of the eccentricities in the nonadiabatic limits. The capture probability as a function of planet eccentricities for mass ratios m1/m2 = 0.5 and 2 in the adiabatic limit has also been studied. The capture probability at m1/m2 = 2 shows similar behaviors with m1/m2 = 1 but the capture behaviors at m1/m2 = 0.5 are significantly different from the capture behaviors at m1/m2 = 1. This research has explored the probability of resonant capture in several new regimes, including the elliptical restricted threebody problem, comparable mass cases in the adiabatic limit and the equal mass case in the nonadiabatic limits. This work enhances our knowledge in the capture behaviors of 3:1 MMR in different limits and is useful in the future studies of the period ratio distribution in extrasolar planet systems.   
dc.language  eng   
dc.publisher  The University of Hong Kong (Pokfulam, Hong Kong)   
dc.relation.ispartof  HKU Theses Online (HKUTO)   
dc.rights  The author retains all proprietary rights, (such as patent rights) and the right to use in future works.   
dc.rights  This work is licensed under a Creative Commons AttributionNonCommercialNoDerivatives 4.0 International License.   
dc.source.uri  http://hub.hku.hk/bib/B4979971X   
dc.subject.lcsh  Extrasolar planets.   
dc.subject.lcsh  Astrophysics.   
dc.title  Numerical exploration of the probability of capture into the 3:1 mean motion resonance   
dc.type  PG_Thesis   
dc.identifier.hkul  b4979971   
dc.description.thesisname  Master of Philosophy   
dc.description.thesislevel  Master   
dc.description.thesisdiscipline  Physics   
dc.description.nature  published_or_final_version   
dc.identifier.doi  10.5353/th_b4979971   
dc.date.hkucongregation  2013   