Bondi k-calculus

2013-06-10T07:10:39Z

131.252.247.116:

'''Tisserand's parameter''' (or '''Tisserand's invariant''') is a combination of [[orbital elements]]{{Vague|date=August 2011}} used in a restricted [[N-body_problem#Three-body problem|three-body problem]], named after French astronomer [[Félix Tisserand]].

==Definition==
For a small body with [[semimajor axis]] <math>a\,\!</math>, [[eccentricity (orbit)|eccentricity]] <math>e\,\!</math>, and [[inclination]] <math>i\,\!</math>, relative to the orbit of a perturbing larger body with semimajor axis <math>a_P</math>, the parameter is defined as follows:<ref>{{cite book |last1=Murray |first1= C. D.| last2=Dermot | first2=S. F. |year= 2000|title=Solar System Dynamics|publisher=Cambridge University Press |isbn=0-521-57597-4}}</ref>

:<math>T_P\ = \frac{a_P}{a} + 2\cdot\sqrt{\frac{a}{a_P} (1-e^2)} \cos i</math>

The quasi-conservation of Tisserand's parameter is a consequence of [[Tisserand's relation]].

== Applications ==
* T<sub>J</sub>, Tisserand’s parameter with respect to [[Jupiter]] as perturbing body, is frequently used to distinguish [[asteroid]]s (typically <math>T_J > 3</math>) from [[List of periodic comets|Jupiter-family comet]]s (typically <math>2< T_J < 3</math>).
* 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<sub>N</sub>, Tisserand's parameter with respect to [[Neptune]], has been suggested to distinguish Near [[scattered disc|Scattered Objects]] (believed to be affected by Neptune) from Extended Scattered [[trans-Neptunian objects]] (e.g. [[90377 Sedna]]).
* Tisserand's parameter could be used to infer the presence of an [[intermediate-mass black hole]] at the center of the [[Milky Way]] galaxy using the motions of orbiting stars.<ref name=DEGN>{{cite book|last=Merritt|first=David|title=Dynamics and Evolution of Galactic Nuclei|year=2013|publisher=Princeton University Press|location=Princeton, NJ|isbn=9781400846122|url=http://openlibrary.org/works/OL16802359W/Dynamics_and_Evolution_of_Galactic_Nuclei}}</ref>

==Related notions==

The parameter is derived from one of the so-called [[Charles-Eugène Delaunay|Delaunay]] standard variables, used to study the perturbed [[Energy#The Hamiltonian|Hamiltonian]] in a 3-body system. Ignoring higher-order perturbation terms, the following value is conserved:

:<math> \sqrt{a (1-e^2)} \cos i</math>

Consequently, perturbations may lead to the [[resonance]] between the orbital inclination and eccentricity, known as [[Kozai mechanism|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 comet]]s, because a large eccentricity with a constant semimajor axis results in a small perihelion.

==See also ==
*[[Tisserand's relation]] for the derivation and the detailed assumptions

== External links ==
* [[David Jewitt]]'s page on [http://www2.ess.ucla.edu/~jewitt/tisserand.html Tisserand's parameter]

==References==
{{reflist|colwidth=30em}}

[[Category:Orbits]]

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