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 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:[1]

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

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

External links

References

  1. Murray, C. D.; Dermot, S. F. (2000). Solar System Dynamics. Cambridge University Press. ISBN 0-521-57597-4.
  2. Merritt, David (2013). Dynamics and Evolution of Galactic Nuclei. Princeton, NJ: Princeton University Press. ISBN 9781400846122.
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