# Tisserand's parameter

Tisserand's parameter (or Tisserand's invariant) is a value calculated from several orbital elements (semi-major axis, orbital eccentricity, and inclination) of a relatively small object and a larger "perturbing body". It is used to distinguish different kinds of orbits. It is named after French astronomer Félix Tisserand, and applies to restricted three-body problems, in which the three objects all differ greatly in size.

## Definition

For a small body with semimajor axis ${\displaystyle a\,\!}$, eccentricity ${\displaystyle e\,\!}$, and inclination ${\displaystyle i\,\!}$, relative to the orbit of a perturbing larger body with semimajor axis ${\displaystyle a_{P}}$, the parameter is defined as follows:[1]

${\displaystyle T_{P}\ ={\frac {a_{P}}{a}}+2\cdot {\sqrt {{\frac {a}{a_{P}}}(1-e^{2})}}\cos i}$

The quasi-conservation of Tisserand's parameter is a consequence of Tisserand's relation.

## Related notions

The parameter is derived from one of the so-called Delaunay standard variables, used to study the perturbed Hamiltonian in a 3-body system. Ignoring higher-order perturbation terms, the following value is conserved:

${\displaystyle {\sqrt {a(1-e^{2})}}\cos i}$

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.