Tisserand's criterion

[1][2] While all the orbital parameters of an object orbiting the Sun during the close encounter with another massive body (e.g. Jupiter) can be changed dramatically, the value of a function of these parameters, called Tisserand's relation (due to Félix Tisserand) is approximately conserved, making it possible to recognize the orbit after the encounter.

In a circular restricted three-body system, one of the masses is assumed to be much smaller than the other two.

Two observed orbiting bodies are possibly the same if they satisfy or nearly satisfy Tisserand's criterion:[1][2][3] where a is the semimajor axis (in units of Jupiters semimajor axis), e is the eccentricity, and i is the inclination of the body's orbit.

The relation defines a function of orbital parameters, conserved approximately when the third body is far from the second (perturbing) mass.

[3] The relation is derived from the Jacobi constant selecting a suitable unit system and using some approximations.

It is assumed, that far from the mass μ2, the test particle (comet, spacecraft) is on an orbit around μ1 resulting from two-body solution.

First, the last term in the constant is the velocity, so it can be expressed, sufficiently far from the perturbing mass μ2, as a function of the distance and semi-major axis alone using vis-viva equation Second, observing that the

Substituting these into the Jacobi constant CJ, ignoring the term with μ2<<1 and replacing r1 with r (given very large μ1 the barycenter of the system μ1, μ3 is very close to the position of μ1) gives