A constant parameter in a three body system?

Question

In a two body system, there are a lot of orbital parameters not changing over time, like eccentricity or the orbital plane.

A constant parameter can be a combination of two or more, like even though kinetic and potential energy is always changing in an elliptical orbit, they always sum up to a constant energy.

Is there any equivalent parameter that does not change for the smaller satellite in a three body system? Can that parameter be used to confirm if you are observing the same asteroid after a fly-by of, let's say Jupiter?

fyi Three-body versus CR3BP tags; combine them? – uhoh Jul 31 '19 at 12:50 — uhoh, Jul 31 '19 at 12:50

TildalWave · Accepted Answer · 2015-12-17T07:33:56.223

4

That would be the Jacobi integral ($C_\text{J}$, or $C_\text{H}$ in Hill's problem):

In celestial mechanics, Jacobi's integral (also The Jacobi Integral or The Jacobi Constant; named after Carl Gustav Jacob Jacobi) is the only known conserved quantity for the circular restricted three-body problem. Unlike in the two-body problem, the energy and momentum of the system are not conserved separately and a general analytical solution is not possible. The integral has been used to derive numerous solutions in special cases.

For a worked example, as requested specifically about application of CRTBP conserved quantity for a Jupiter flyby (in this case a time-frequency analysis of a Sun-Jupiter-comet system), see e.g. Time-frequency analysis of the restricted three body problem: Transport and resonance transitions, Luz V. Vela-Arevalo and Jerrold E. Marsden, Georgia Institute of Technology (PDF).

edited Dec 17 '15 at 07:33

answered Dec 17 '15 at 07:21

TildalWave

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So that is what it is! Finally the universe is starting to make sense. – SE - stop firing the good guys Dec 17 '15 at 07:23
Is that the research which allowed the estimation of a comet impact into Jupiter latitude with about $\pm 180 ^{\circ}$ accuracy? :) – SF. Dec 17 '15 at 08:43
@SF. No. Latitudes only go to $\pm 90°$ :P – TildalWave Dec 17 '15 at 14:31

score 3 · Answer 2 · answered Dec 17 '15 at 19:29

There is another parameter known as Tisserand's parameter which stays constant for a given body in 3 body probelm. This is more useful in identifying any given asteroid or comet. Given by

$T = \dfrac{a_e}{a}+2\sqrt{\dfrac{a}{a_e}(1-e^2)}cos(i)$

where e subscript represents parameters of perturbing body and no subscript for small body. It stays constant for a body even after the close encounter and is very useful in identifying if a newly discovered asteroid is an old body that underwent close encounter.

http://farside.ph.utexas.edu/teaching/336k/Newtonhtml/node122.html

A constant parameter in a three body system?

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