Tisserand's parameter

Tisserand's parameter (or Tisserand's invariant) is a combination of orbital elements used in a restricted three-body problem, named after French astronomer Félix Tisserand.

Definition

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

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

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

Applications

• T_J, Tisserand’s parameter with respect to Jupiter as perturbing body, is frequently used to distinguish asteroids (typically $T_{J}>3$) from Jupiter-family comets (typically $2<T_{J}<3$).
• The roughly constant value of the parameter before and after the interaction (encounter) is used to determine whether or not an observed orbiting body is the same as a previously observed in Tisserand's Criterion.
• The quasi-conservation of Tisserand's parameter constrains the orbits attainable using gravity assist for outer Solar system exploration.
• T_N, Tisserand's parameter with respect to Neptune, has been suggested to distinguish Near Scattered Objects (believed to be affected by Neptune) from Extended Scattered trans-Neptunian objects (e.g. 90377 Sedna).
• Tisserand's parameter could be used to infer the presence of an intermediate-mass black hole at the center of the Milky Way galaxy using the motions of orbiting stars.

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:

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

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.

See also

• Tisserand's relation for the derivation and the detailed assumptions

External links

• David Jewitt's page on Tisserand's parameter