# A constant parameter in a three body system?

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?

• – uhoh
Commented Jul 31, 2019 at 12:50

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).

• So that is what it is! Finally the universe is starting to make sense. Commented Dec 17, 2015 at 7:23
• Is that the research which allowed the estimation of a comet impact into Jupiter latitude with about $\pm 180 ^{\circ}$ accuracy? :)
– SF.
Commented Dec 17, 2015 at 8:43
• @SF. No. Latitudes only go to $\pm 90°$ :P Commented Dec 17, 2015 at 14:31

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