For a small body with semi-major axis , orbital eccentricity , and orbital inclination , relative to the orbit of a perturbing larger body with
semimajor axis , the parameter is defined as follows:[2][3]
Tisserand invariant conservation
In the three-body problem, the quasi-conservation of Tisserand's invariant is derived as the limit of the
Jacobi integral away from the main two bodies (usually the star and planet).[2] Numerical simulations show that the Tisserand invariant of orbit-crossing bodies is conserved in the three-body problem on Gigayear timescales.[4][5]
Applications
The Tisserand parameter's conservation was originally used by Tisserand to determine whether or not an observed orbiting body is the same as one previously observed. This is usually known as the
Tisserand's criterion.
Orbit classification
The value of the Tisserand parameter with respect to the planet that most perturbs a small body in the solar system can be used to delineate groups of objects that may have similar origins.
TJ, Tisserand's parameter with respect to
Jupiter as perturbing body, is frequently used to distinguish
asteroids (typically ) from
Jupiter-family comets (typically ).[6]
The minor planet group of
damocloids are defined by a Jupiter Tisserand's parameter of 2 or less (TJ ≤ 2).[7]
TN, Tisserand's parameter with respect to
Neptune, has been suggested to distinguish near-
scattered (affected by Neptune) from extended-scattered
trans-Neptunian objects (not affected by Neptune; e.g.
90377 Sedna).
TN, Tisserand's parameter with respect to
Neptune may also be used to distinguish Neptune-crossing
trans-neptunian objects that may be injected onto retrograde and polar Centaur orbits ( -1 ≤TN ≤ 2) and those that may be injected onto prograde Centaur orbits ( 2 ≤TN ≤ 2.82).[4][5]
Other uses
The quasi-conservation of Tisserand's parameter constrains the orbits attainable using
gravity assist for
outer Solar System exploration.
Tisserand's parameter could be used to infer the presence of an
intermediate-mass black hole at the center of the
Milky Way using the motions of orbiting stars.[8]
Related notions
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.