Gauss's method for secular dynamics, softened

Touma, Jihad; Tremaine, Scott; Kazandjian, M. V.

doi:10.1111/j.1365-2966.2009.14409.x

Cited by 85 publications

(108 citation statements)

References 23 publications

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“…In a closely related 20 Note that the timescale of eccentricity modulation due to planet-disk interaction is almost certainly much shorter than that corresponding to semi-major axis evolution (Goldreich & Tremaine 1980;. 21 Possible mechanisms that may be responsible for maintenance of global lopsided modes of disks include disk self-gravity (Dury et al 2008;Mittal & Chiang 2015; see also Touma, Tremaine, &Kazandjian 2009 andBatygin 2012 for a related discussion) and hydrodynamic forces (Larwood et al 1996;Xiang-Gruess & Papaloizou 2014). Moreover, excitation of disk eccentricities may arise from external perturbations by passing or bound stars in star formation environments that are well-known to exhibit enhanced stellar multiplicity (Duchêne & Kraus 2013).…”

Section: Discussionmentioning

confidence: 99%

Capture of planets into mean-motion resonances and the origins of extrasolar orbital architectures

Batygin

2015

Monthly Notices of the Royal Astronomical Society

113

142

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The early stages of dynamical evolution of planetary systems are often shaped by dissipative processes that drive orbital migration. In multi-planet systems, convergent amassing of orbits inevitably leads to encounters with rational period ratios, which may result in establishment of mean motion resonances. The success or failure of resonant capture yields exceedingly different subsequent evolutions, and thus plays a central role in determining the ensuing orbital architecture of planetary systems. In this work, we employ an integrable Hamiltonian formalism for first order planetary resonances that allows both secondary bodies to have finite masses and eccentricities, and construct a comprehensive theory for resonant capture. Particularly, we derive conditions under which orbital evolution lies within the adiabatic regime, and provide a generalized criterion for guaranteed resonant locking as well as a procedure for calculating capture probabilities when capture is not certain. Subsequently, we utilize the developed analytical model to examine the evolution of Jupiter and Saturn within the protosolar nebula, and investigate the origins of the dominantly non-resonant orbital distribution of sub-Jovian extrasolar planets. Our calculations show that the commonly observed extrasolar orbital structure can be understood if planet pairs encounter mean motion commensurabilities on slightly eccentric (e ∼ 0.02) orbits. Accordingly, we speculate that resonant capture among low-mass planets is typically rendered unsuccessful due to subtle axial asymmetries inherent to the global structure of protoplanetary disks.

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Section: Discussionmentioning

confidence: 99%

Capture of planets into mean-motion resonances and the origins of extrasolar orbital architectures

Batygin

2015

Monthly Notices of the Royal Astronomical Society

113

142

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“…Because quadrupole and octopole expansions are in terms of the semimajor axis ratio, their accuracy is diminished for weakly hierarchical systems. For noncoplanar systems, one can use seminumerical averaging (Michtchenko & Malhotra 2004;Michtchenko et al 2006b;Migaszewski & Goździewski 2009b) or adaptions of Gauss' method (Touma et al 2009), which are valid for high e and I. However, these studies neglect resonant and short-term perturbations.…”

Section: Discussionmentioning

confidence: 99%

Secular Orbital Dynamics of Hierarchical Two-Planet Systems

Veras

Ford

2010

ApJ

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The discovery of multi-planet extrasolar systems has kindled interest in using their orbital evolution as a probe of planet formation. Accurate descriptions of planetary orbits identify systems that could hide additional planets or be in a special dynamical state, and inform targeted follow-up observations. We combine published radial velocity data with Markov Chain Monte Carlo analyses in order to obtain an ensemble of masses, semimajor axes, eccentricities, and orbital angles for each of the five dynamically active multi-planet systems: HD 11964, HD 38529, HD 108874, HD 168443, and HD 190360. We dynamically evolve these systems using 52,000 long-term N-body integrations that sample the full range of possible line-of-sight and relative inclinations, and we report on the system stability, secular evolution, and the extent of the resonant interactions. We find that planetary orbits in hierarchical systems exhibit complex dynamics and can become highly eccentric and maybe significantly inclined. Additionally, we incorporate the effects of general relativity in the long-term simulations and demonstrate that it can qualitatively affect the dynamics of some systems with high relative inclinations. The simulations quantify the likelihood of different dynamical regimes for each system and highlight the dangers of restricting simulation phase space to a single set of initial conditions or coplanar orbits.

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“…Since most of the questions of interest in the modeling of processes near MBH involve integration over timescales much longer than an orbital period, one alternative to direct N -body simulations is "N -wire" simulations (Touma, Tremaine & Kazandjian 2009), which treats the dynamics directly in terms of the orbit-averaged mass wires (Sec. 2.2) with timesteps > P .…”

Section: Other Approaches To Modeling Dynamics Near a Mbhmentioning

confidence: 99%

Stellar Dynamics and Stellar Phenomena Near a Massive Black Hole

Alexander

2017

Annu. Rev. Astron. Astrophys.

133

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Most galactic nuclei harbor a massive black hole (MBH), whose birth and evolution are closely linked to those of its host galaxy. The unique conditions near the MBH: high velocity and density in the steep potential of a massive singular relativistic object, lead to unusual modes of stellar birth, evolution, dynamics and death. A complex network of dynamical mechanisms, operating on multiple timescales, deflect stars to orbits that intercept the MBH. Such close encounters lead to energetic interactions with observable signatures and consequences for the evolution of the MBH and its stellar environment. Galactic nuclei are astrophysical laboratories that test and challenge our understanding of MBH formation, strong gravity, stellar dynamics, and stellar physics. I review from a theoretical perspective the wide range of stellar phenomena that occur near MBHs, focusing on the role of stellar dynamics near an isolated MBH in a relaxed stellar cusp.

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Gauss's method for secular dynamics, softened

Cited by 85 publications

References 23 publications

Capture of planets into mean-motion resonances and the origins of extrasolar orbital architectures

Capture of planets into mean-motion resonances and the origins of extrasolar orbital architectures

Secular Orbital Dynamics of Hierarchical Two-Planet Systems

Stellar Dynamics and Stellar Phenomena Near a Massive Black Hole

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