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Tisserand's criterion is used to determine whether or not an observed orbiting body, such as a
comet A comet is an icy, small Solar System body that, when passing close to the Sun, warms and begins to release gases, a process that is called outgassing. This produces a visible atmosphere or Coma (cometary), coma, and sometimes also a Comet ta ...
or an
asteroid An asteroid is a minor planet of the Solar System#Inner solar system, inner Solar System. Sizes and shapes of asteroids vary significantly, ranging from 1-meter rocks to a dwarf planet almost 1000 km in diameter; they are rocky, metallic o ...
, is the same as a previously observed orbiting body. While all the orbital parameters of an object orbiting the Sun during the close encounter with another massive body (e.g. Jupiter) can be changed dramatically, the value of a function of these parameters, called Tisserand's relation (due to Félix Tisserand) is approximately conserved, making it possible to recognize the orbit after the encounter.


Definition

Tisserand's criterion is computed in a circular restricted three-body system. In a circular restricted three-body system, one of the masses is assumed to be much smaller than the other two. The other two masses are assumed to be in a circular orbit about the system's center of mass. In addition, Tisserand's criterion also relies on the assumptions that a) one of the two larger masses is much smaller than the other large mass and b) the comet or asteroid has not had a close approach to any other large mass. Two observed orbiting bodies are possibly the same if they satisfy or nearly satisfy Tisserand's criterion: :\frac + \sqrt \cos i_1 = \frac + \sqrt \cos i_2 where a is the
semimajor axis In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the lon ...
, e is the
eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off- center, in geometry * Eccentricity (graph theory) of a ...
, and i is the
inclination Orbital inclination measures the tilt of an object's orbit around a celestial body. It is expressed as the angle between a Plane of reference, reference plane and the orbital plane or Axis of rotation, axis of direction of the orbiting object ...
of the body's orbit. In other words, if a function of the
orbital elements Orbital elements are the parameters required to uniquely identify a specific orbit. In celestial mechanics these elements are considered in two-body systems using a Kepler orbit. There are many different ways to mathematically describe the same ...
(named
Tisserand's parameter Tisserand's parameter (or Tisserand's invariant) is a value calculated from several orbital elements (semi-major axis, orbital eccentricity and inclination) of a relatively small object and a larger " perturbing body". It is used to distinguish dif ...
) of the first observed body (nearly) equals the same function calculated with the orbital elements of the second observed body, the two bodies might be the same.


Tisserand's relation

The relation defines a function of orbital parameters, conserved approximately when the third body is far from the second (perturbing) mass. :\frac + \sqrt \cos i \approx The relation is derived from the
Jacobi constant In celestial mechanics, Jacobi's integral (also known as the Jacobi integral or Jacobi constant) is the only known conserved quantity for the circular restricted three-body problem.vis-viva equation In astrodynamics, the ''vis-viva'' equation, also referred to as orbital-energy-invariance law, is one of the equations that model the motion of orbiting bodies. It is the direct result of the principle of conservation of mechanical energy which ...
:(\dot \xi ^2+\dot \eta ^2+\dot \zeta^2) =v^2=\mu\left(\right) Second, observing that the \zeta component of the
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed sy ...
(per unit mass) \mathbf=\mathbf\times\mathbf is :\xi \dot \eta- \eta \dot \xi = h \cos I where I\,\! is the mutual inclination of the orbits of μ3 and μ2, and h=, \mathbf , =\sqrt. Substituting these into the Jacobi constant CJ, ignoring the term with μ2<<1 and replacing r1 with r (given very large μ1 the barycenter of the system μ1, μ3 is very close to the position of μ1) gives :\frac + \sqrt \cos i \approx {\rm const}


See also

*
Orbital elements Orbital elements are the parameters required to uniquely identify a specific orbit. In celestial mechanics these elements are considered in two-body systems using a Kepler orbit. There are many different ways to mathematically describe the same ...
*
Orbital mechanics Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws of ...
* ''n''-body problem


References

Orbits