View source for Astronomy:Tisserand's parameter

{{Short description|Named after Félix Tisserand}}
'''Tisserand's parameter''' (or '''Tisserand's invariant''') is a number calculated from several [[Astronomy:Orbital elements|orbital elements]] ([[Semi-major and semi-minor axes|semi-major axis]], [[Astronomy:Orbital eccentricity|orbital eccentricity]] and [[Astronomy:Orbital inclination|inclination]]) of a relatively small object and a larger "[[Astronomy:Perturbation|perturbing body]]". It is used to distinguish different kinds of orbits. The term is named after French astronomer Félix Tisserand who derived it,<ref name = "Tisserand1896">{{cite book |last1=Tisserand |first1=F. |year= 1896|title=Traité de Mécanique Céleste|publisher=Gauthier-Villards |volume= IV }}</ref> and applies to restricted [[Physics:N-body problem#Three-body problem|three-body problem]]s in which the three objects all differ greatly in mass.


== Definition ==

For a small body with semi-major axis <math>a\,\!</math>, orbital eccentricity <math>e\,\!</math>, and orbital 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 name = "MD2000">{{cite book |last1=Murray |first1=Carl D. | last2=Dermott | first2=Stanley F. |year= 2000|title=Solar System Dynamics|publisher={{wipe|Cambridge University Press}} |isbn=0-521-57597-4|author2-link=Dermott's law }}</ref><ref>{{Cite journal|last1=Bonsor|first1=A.|last2=Wyatt|first2=M. C.|date=2012-03-11|title=The scattering of small bodies in planetary systems: constraints on the possible orbits of cometary material: Scattering in planetary systems|journal=Monthly Notices of the Royal Astronomical Society|language=en|volume=420|issue=4|pages=2990–3002|doi=10.1111/j.1365-2966.2011.20156.x|doi-access=free|arxiv=1111.1858}}</ref>

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


== Tisserand invariant conservation ==
In the three-body problem, the quasi-conservation of Tisserand's invariant is derived as the limit of the [[Astronomy:Jacobi integral|Jacobi integral]] away from the main two bodies (usually the star and planet).<ref name = "MD2000" /> Numerical simulations show that the Tisserand invariant of orbit-crossing bodies is conserved in the three-body problem on Gigayear timescales.<ref name ="Namouni2022">{{Cite journal|last1=Namouni|first1=F.|date=2021-11-26|title=Inclination pathways of planet-crossing asteroids|journal=Monthly Notices of the Royal Astronomical Society|language=en|volume=510|issue=1|pages=276–291|doi=10.1093/mnras/stab3405|doi-access=free|arxiv=2111.10777}}</ref><ref name = "Namouni2024">{{Cite journal|last1=Namouni|first1=F.|date=2023-11-20|title=Orbit injection of planet-crossing asteroids|journal=Monthly Notices of the Royal Astronomical Society|language=en|volume=527|issue=3|pages=4889–4898|doi=10.1093/mnras/stad3570|doi-access=free|arxiv=2311.09946}}</ref>
== 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 [[Astronomy:Tisserand's criterion|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.
* T<sub>J</sub>, Tisserand's parameter with respect to [[Astronomy:Jupiter|Jupiter]] as perturbing body, is frequently used to distinguish [[Astronomy:Asteroid|asteroid]]s (typically <math>T_J > 3</math>) from [[Astronomy:Comet#Short period|Jupiter-family comets]] (typically <math>2< T_J < 3</math>).<ref>{{Cite web|url=http://www2.ess.ucla.edu/~jewitt/tisserand.html|title=Dave Jewitt: Tisserand Parameter|website=www2.ess.ucla.edu|access-date=2018-03-27}}</ref>
* The minor planet group of damocloids are defined by a Jupiter Tisserand's parameter of 2 or less ({{math|T<sub>J</sub> &le; 2}}).<ref name="Damocloids-Jewitt" />
* T<sub>N</sub>, Tisserand's parameter with respect to [[Astronomy:Neptune|Neptune]], has been suggested to distinguish near-[[Astronomy:Scattered disc|scattered]] (affected by Neptune) from extended-scattered [[Astronomy:Trans-Neptunian object|trans-Neptunian object]]s (not affected by Neptune; e.g. [[Astronomy:90377 Sedna|90377 Sedna]]).
* T<sub>N</sub>, Tisserand's parameter with respect to [[Astronomy:Neptune|Neptune]] may also be used to distinguish Neptune-crossing trans-neptunian objects that may be injected onto retrograde and polar Centaur orbits ({{math| -1 &le;T<sub>N</sub> &le; 2}}) and those that may be injected onto prograde Centaur orbits ({{math| 2 &le;T<sub>N</sub> &le; 2.82}}).<ref name ="Namouni2022"/><ref name ="Namouni2024" />
=== Other uses ===
*The quasi-conservation of Tisserand's parameter constrains the orbits attainable using [[Astronomy:Gravity assist|gravity assist]] for [[Astronomy:Solar System#Outer Solar System|outer Solar System]] exploration.
* Tisserand's parameter could be used to infer the presence of an [[Astronomy:Intermediate-mass black hole|intermediate-mass black hole]] at the center of the [[Astronomy:Milky Way|Milky Way]] 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=https://openlibrary.org/works/OL16802359W/Dynamics_and_Evolution_of_Galactic_Nuclei}}</ref>

== Related notions ==

The parameter is derived from one of the so-called Delaunay standard variables, used to study the perturbed [[Physics:Hamiltonian mechanics#Overview|Hamiltonian]] in a [[Physics:Three-body problem|three-body system]]. Ignoring higher-order perturbation terms, the following value is [[Conserved quantity|conserved]]:

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

Consequently, perturbations may lead to the [[Physics:Resonance|resonance]] between the orbital inclination and eccentricity, known as [[Astronomy: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 comets, because a large eccentricity with a constant semimajor axis results in a small [[Astronomy:Perihelion and aphelion|perihelion]].

== See also ==
* [[Astronomy:Tisserand's criterion#Tisserand's relation|Tisserand's relation]] for the derivation and the detailed assumptions

== References ==
{{Reflist
|refs=

<ref name="Damocloids-Jewitt">{{cite web
  |title      = The Damocloids
  |date       = August 2013
  |publisher  = UCLA – Department of Earth and Space Sciences
  |author     = Jewitt, David C.
  |url        = http://www2.ess.ucla.edu/~jewitt/damocloid.html
 }}</ref>

}} <!-- end of reflist -->

== External links ==
* David Jewitt's page on [http://www2.ess.ucla.edu/~jewitt/tisserand.html Tisserand's parameter]
* [https://farside.ph.utexas.edu/teaching/336k/Newtonhtml/node122.html Tisserand criterion]

[[Category:Orbits]]

{{Sourceattribution|Tisserand's parameter}}