Tisserand's parameter


Tisserand's parameter is a value calculated from several orbital elements of a relatively small object and a larger "perturbing body". It is used to distinguish different kinds of orbits. The term is named after French astronomer Félix Tisserand, and applies to restricted three-body problems in which the three objects all differ greatly in mass.

Definition

For a small body with semi-major axis, eccentricity, and inclination, relative to the orbit of a perturbing larger body with semimajor axis, the parameter is defined as follows:
The quasi-conservation of Tisserand's parameter is a consequence of Tisserand's relation.

Applications

The parameter is derived from one of the so-called Delaunay standard variables, used to study the perturbed Hamiltonian in a three-body system. Ignoring higher-order perturbation terms, the following value is conserved:
Consequently, perturbations may lead to the resonance between the orbital inclination and eccentricity, known as Kozai resonance. Near-circular, highly inclined orbits can thus become very eccentric in exchange for lower inclination. For example, such a mechanism can produce sungrazing comets, because a large eccentricity with a constant semimajor axis results in a small perihelion.