Tisserand's parameter
Encyclopedia
Tisserand's parameter is a combination of orbital elements
used in a restricted three-body problem.
, eccentricity 
, and inclination
, relative to the orbit of a perturbing larger body with semimajor axis 
, the parameter is defined as follows:
The quasi-conservation of Tisserand's parameter is a consequence of Tisserand's relation.
standard variables, used to study the perturbed Hamiltonian in 3-body system. Ignoring higher order perturbation terms, the following value is conserved
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.
Orbital elements
Orbital elements are the parameters required to uniquely identify a specific orbit. In celestial mechanics these elements are generally considered in classical two-body systems, where a Kepler orbit is used...
used in a restricted three-body problem.
Definition
For a small body with semimajor axis, eccentricity , and inclinationInclination
Inclination in general is the angle between a reference plane and another plane or axis of direction.-Orbits:The inclination is one of the six orbital parameters describing the shape and orientation of a celestial orbit...
, relative to the orbit of a perturbing larger body with semimajor axis , the parameter is defined as follows:The quasi-conservation of Tisserand's parameter is a consequence of Tisserand's relation.
Applications
- TJ, Tisserand’s parameter with respect to JupiterJupiterJupiter is the fifth planet from the Sun and the largest planet within the Solar System. It is a gas giant with mass one-thousandth that of the Sun but is two and a half times the mass of all the other planets in our Solar System combined. Jupiter is classified as a gas giant along with Saturn,...
as perturbing body, is frequently used to distinguish asteroidAsteroidAsteroids are a class of small Solar System bodies in orbit around the Sun. They have also been called planetoids, especially the larger ones...
s (typically
) from Jupiter-family comets (typically 
). - 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%27s Criterion.
- The quasi-conservation of Tisserand's parameter constrains the orbits attainable using gravity assist for outer Solar system exploration.
- TN, Tisserand's parameter with respect to NeptuneNeptuneNeptune is the eighth and farthest planet from the Sun in the Solar System. Named for the Roman god of the sea, it is the fourth-largest planet by diameter and the third largest by mass. Neptune is 17 times the mass of Earth and is slightly more massive than its near-twin Uranus, which is 15 times...
, has been suggested to distinguish Near Scattered ObjectsScattered discThe scattered disc is a distant region of the Solar System that is sparsely populated by icy minor planets, a subset of the broader family of trans-Neptunian objects. The scattered-disc objects have orbital eccentricities ranging as high as 0.8, inclinations as high as 40°, and perihelia greater...
(believed to be affected by Neptune) from Extended Scattered trans-Neptunian objects (e.g. 90377 Sedna90377 Sedna90377 Sedna is a trans-Neptunian object discovered in 2003, which was about three times as far from the Sun as Neptune. For most of its orbit it is even further from the Sun, with its aphelion estimated at 960 astronomical units , making it one of the most distant known objects in the Solar System...
).
Related notions
The parameter is derived from one of so called DelaunayCharles-Eugène Delaunay
Charles-Eugène Delaunay was a French astronomer and mathematician. His lunar motion studies were important in advancing both the theory of planetary motion and mathematics.-Life:...
standard variables, used to study the perturbed Hamiltonian in 3-body system. Ignoring higher order perturbation terms, the following value is conserved
Consequently, perturbations may lead to the resonance between the orbit inclination and eccentricity, known as Kozai resonance
Kozai mechanism
In celestial mechanics, the Kozai mechanism, or the Lidov-Kozai mechanism, causes a periodic exchange between the inclination and eccentricity of an orbit...
. 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