Properties of von Zeipel-Lidov-Kozai oscillations in triple systems at the quadrupole order: relaxing the test particle approximation. (arXiv:2011.03294v2 [astro-ph.SR] UPDATED)

<a href="http://arxiv.org/find/astro-ph/1/au:+Hamers_A/0/1/0/all/0/1">Adrian S. Hamers</a>

Von Zeipel-Lidov-Kozai (ZLK) oscillations in hierarchical triple systems have

important astrophysical implications such as triggering strong interactions and

producing, e.g., Type Ia supernovae and gravitational wave sources. When

considering analytic properties of ZLK oscillations at the lowest (quadrupole)

expansion order, as well as complications due to higher-order terms, one

usually assumes the test particle limit, in which one of the bodies in the

inner binary is massless. Although this approximation holds well for, e.g.,

planetary systems, it is less accurate for systems with more comparable masses

such as stellar triples. Whereas non-test-particle effects are usually taken

into account in numerical simulations, a more analytic approach focusing on the

differences between the test particle and general case (at quadrupole order)

has, to our knowledge, not been presented. Here, we derive several analytic

properties of secular oscillations in triples at the quadruple expansion order.

The latter applies even to relatively compact triples, as long as the inner

bodies are similar in mass such that octupole-order effects are suppressed. We

consider general conditions for the character of the oscillations (circular

versus librating), minimum and maximum eccentricities, and timescales, all as a

function of $gamma equiv (1/2) L_1/G_2$, a ratio of inner-to-outer orbital

angular momenta variables ($gamma=0$ in the test particle limit). In

particular, eccentricity oscillations are more effective at retrograde

orientations for non-zero $gamma$; assuming zero initial inner eccentricity,

the maximum eccentricity peaks at $cos(i_mathrm{rel,0}) = -gamma$, where

$i_mathrm{rel,0}$ is the initial relative inclination. We provide a Python

script which can be used to quickly compute these properties.

Von Zeipel-Lidov-Kozai (ZLK) oscillations in hierarchical triple systems have

important astrophysical implications such as triggering strong interactions and

producing, e.g., Type Ia supernovae and gravitational wave sources. When

considering analytic properties of ZLK oscillations at the lowest (quadrupole)

expansion order, as well as complications due to higher-order terms, one

usually assumes the test particle limit, in which one of the bodies in the

inner binary is massless. Although this approximation holds well for, e.g.,

planetary systems, it is less accurate for systems with more comparable masses

such as stellar triples. Whereas non-test-particle effects are usually taken

into account in numerical simulations, a more analytic approach focusing on the

differences between the test particle and general case (at quadrupole order)

has, to our knowledge, not been presented. Here, we derive several analytic

properties of secular oscillations in triples at the quadruple expansion order.

The latter applies even to relatively compact triples, as long as the inner

bodies are similar in mass such that octupole-order effects are suppressed. We

consider general conditions for the character of the oscillations (circular

versus librating), minimum and maximum eccentricities, and timescales, all as a

function of $gamma equiv (1/2) L_1/G_2$, a ratio of inner-to-outer orbital

angular momenta variables ($gamma=0$ in the test particle limit). In

particular, eccentricity oscillations are more effective at retrograde

orientations for non-zero $gamma$; assuming zero initial inner eccentricity,

the maximum eccentricity peaks at $cos(i_mathrm{rel,0}) = -gamma$, where

$i_mathrm{rel,0}$ is the initial relative inclination. We provide a Python

script which can be used to quickly compute these properties.

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