Tisserand's parameter

Tisserand's parameter (or Tisserand's invariant) is a combination of orbital elements used in a restricted three-body problem.

1 Definition
2 Applications
3 Related notions
4 See also
5 External links
6 References

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:

$\frac{a_P}{a} %2B 2\cdot\sqrt{\frac{a}{a_P} (1-e^2)} \cos i$

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).

Related notions

The parameter is derived from one of so called Delaunay standard variables, used to study the perturbed Hamiltonian in 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 orbit inclination and eccentricity, known as Kozai resonance. Near circular, highly inclined orbits can thus become very eccentric (in exchange for lower inclination). As example, such mechanism can produce Sun-grazing comets.¹

¹Large eccentricity with constant semimajor axis means small perihelion.

External links

David Jewitt's page on Tisserand's parameter

References

Murray, Dermot Solar System Dynamics, Cambridge University Press, ISBN 0-521-57597-4
J. L. Elliot, S. D. Kern, K. B. Clancy, A. A. S. Gulbis, R. L. Millis, M. W. Buie, L. H. Wasserman, E. I. Chiang, A. B. Jordan, D. E. Trilling, and K. J. Meech The Deep Ecliptic Survey: A Search for Kuiper Belt Objects and Centaurs. II. Dynamical Classification, the Kuiper Belt Plane, and the Core Population. The Astronomical Journal, 129 (2006). preprint