Tisserand's parameter (orTisserand's invariant) is a number calculated from severalorbital elements (semi-major axis,orbital eccentricity, andinclination) of a relatively small object and a larger "perturbing body". It is used to distinguish different kinds of orbits. The term is named after French astronomerFélix Tisserand who derived it^[1] and applies to restrictedthree-body problems in which the three objects all differ greatly in mass.

For a small body with semi-major axis${\displaystyle a}$, orbital eccentricity${\displaystyle e}$, and orbital inclination${\displaystyle i}$, relative to the orbit of a perturbing larger body withsemimajor axis${\displaystyle a_P}$, the parameter is defined as follows:^[2][3]

${\displaystyle T_P=\frac{a_P}{a}+2\cos i\sqrt{\frac{a}{a_P}(1-e^2)}}$

In the three-body problem, the quasi-conservation of Tisserand's invariant is derived as the limit of theJacobi 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]

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 theTisserand's criterion.

The value of the Tisserand parameter with respect to the planet that most perturbs a small body in theSolar System can be used to delineate groups of objects that may have similar origins.

• T_J, Tisserand's parameter with respect toJupiter as perturbing body, is frequently used to distinguishasteroids (typically${\displaystyle T_J>3}$) fromJupiter-family comets (typically).^[6]
• The minor planet group ofdamocloids are defined by a Jupiter Tisserand's parameter of 2 or less (T_J ≤ 2).^[7]
• T_N, Tisserand's parameter with respect toNeptune, has been suggested to distinguish near-scattered (affected by Neptune) from extended-scatteredtrans-Neptunian objects (not affected by Neptune; e.g.90377 Sedna).
• T_N, Tisserand's parameter with respect toNeptune may also be used to distinguish Neptune-crossing trans-Neptunian objects that may be injected onto retrograde and polar Centaur orbits ( –1 ≤T_N ≤ 2) and those that may be injected onto prograde Centaur orbits ( 2 ≤T_N ≤ 2.82).^[4][5]

• The quasi-conservation of Tisserand's parameter constrains the orbits attainable usinggravity assist forouter Solar System exploration.
• Tisserand's parameter could be used to infer the presence of anintermediate-mass black hole at the center of theMilky Way using the motions of orbiting stars.^[8]

The parameter is derived from one of the so-calledDelaunay standard variables, used to study the perturbedHamiltonian in athree-body system. Ignoring higher-order perturbation terms, the following value isconserved:

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

Consequently, perturbations may lead to theresonance between the orbital inclination and eccentricity, known asKozai resonance. Near-circular, highly inclined orbits can thus become very eccentric in exchange for lower inclination. For example, such a mechanism can producesungrazing comets, because a large eccentricity with a constant semimajor axis results in a smallperihelion.

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

• David Jewitt's page onTisserand's parameter
• Tisserand criterion

• Orbits
• Equations of astronomy

• Articles with short description
• Short description matches Wikidata