Tisserand's Criterion
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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 warms and begins to release gases when passing close to the Sun, a process called outgassing. This produces an extended, gravitationally unbound atmosphere or Coma (cometary), coma surrounding ...
or an
asteroid An asteroid is a minor planet—an object larger than a meteoroid that is neither a planet nor an identified comet—that orbits within the Solar System#Inner Solar System, inner Solar System or is co-orbital with Jupiter (Trojan asteroids). As ...
, 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 long ...
(in units of Jupiters semimajor axis), 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-Centre (geometry), center, in geometry * Eccentricity (g ...
, 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 reference plane and the orbital plane or axis of direction of the orbiting object. For a satellite orbiting the Eart ...
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 o ...
(named Tisserand's parameter) 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 selecting a suitable unit system and using some approximations. Traditionally, the units are chosen in order to make μ1 and the (constant) distance from μ2 to μ1 a unity, resulting in mean motion n also being a unity in this system. In addition, given the very large mass of μ1 compared μ2 and μ3 :G (\mu_1+\mu_2) \approx 1 \approx G (\mu_1+\mu_3) These conditions are satisfied for example for the Sun–Jupiter system with a comet or a spacecraft being the third mass. The Jacobi constant, a function of coordinates ξ, η, ζ, (distances r1, r2 from the two masses) and the velocities remains the constant of motion through the encounter. :C_J=2 \cdot(\frac+\frac)+ 2n(\xi \dot \eta- \eta \dot \xi) - (\dot \xi ^2+\dot \eta ^2+\dot \zeta^2) The goal is to express the constant using orbital parameters. It is assumed, that far from the mass μ2, the test particle (comet, spacecraft) is on an orbit around μ1 resulting from two-body solution. First, the last term in the constant is the velocity, so it can be expressed, sufficiently far from the perturbing mass μ2, as a function of the distance and semi-major axis alone using
vis-viva equation In astrodynamics, the ''vis-viva'' equation 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 applies when the only force acting on an object i ...
:(\dot \xi ^2+\dot \eta ^2+\dot \zeta^2) =v^2=\mu\left(\right) Second, observing that the \zeta component of the
angular momentum Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...
(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 o ...
*
Orbital mechanics Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to rockets, satellites, and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and the law of universal ...
* ''n''-body problem


References

Orbits