Tisserand's criterion

Last updated

Tisserand's criterion is used to determine whether or not an observed orbiting body, such as a comet or an asteroid, is the same as a previously observed orbiting body. [1] [2]


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.


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: [1] [2] [3]

where a is the semimajor axis (in units of Jupiters semimajor axis), e is the eccentricity, and i is the inclination of the body's orbit.

In other words, if a function of the orbital elements (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. [3]

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

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.

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

Second, observing that the component of the angular momentum (per unit mass) is

where is the mutual inclination of the orbits of μ3 and μ2, and .

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

See also

Related Research Articles

In physics, a Langevin equation is a stochastic differential equation describing how a system evolves when subjected to a combination of deterministic and fluctuating ("random") forces. The dependent variables in a Langevin equation typically are collective (macroscopic) variables changing only slowly in comparison to the other (microscopic) variables of the system. The fast (microscopic) variables are responsible for the stochastic nature of the Langevin equation. One application is to Brownian motion, which models the fluctuating motion of a small particle in a fluid.

In mathematical physics, n-dimensional de Sitter space is a maximally symmetric Lorentzian manifold with constant positive scalar curvature. It is the Lorentzian analogue of an n-sphere.

In classical mechanics, the Laplace–Runge–Lenz (LRL) vector is a vector used chiefly to describe the shape and orientation of the orbit of one astronomical body around another, such as a binary star or a planet revolving around a star. For two bodies interacting by Newtonian gravity, the LRL vector is a constant of motion, meaning that it is the same no matter where it is calculated on the orbit; equivalently, the LRL vector is said to be conserved. More generally, the LRL vector is conserved in all problems in which two bodies interact by a central force that varies as the inverse square of the distance between them; such problems are called Kepler problems.

In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics.

In celestial mechanics, the specific relative angular momentum of a body is the angular momentum of that body divided by its mass. In the case of two orbiting bodies it is the vector product of their relative position and relative linear momentum, divided by the mass of the body in question.

In physics and astronomy, Euler's three-body problem is to solve for the motion of a particle that is acted upon by the gravitational field of two other point masses that are fixed in space. This problem is exactly solvable, and yields an approximate solution for particles moving in the gravitational fields of prolate and oblate spheroids. This problem is named after Leonhard Euler, who discussed it in memoirs published in 1760. Important extensions and analyses were contributed subsequently by Lagrange, Liouville, Laplace, Jacobi, Darboux, Le Verrier, Velde, Hamilton, Poincaré, Birkhoff and E. T. Whittaker, among others.

<span class="mw-page-title-main">Jacobi integral</span>

In celestial mechanics, Jacobi's integral is the only known conserved quantity for the circular restricted three-body problem. Unlike in the two-body problem, the energy and momentum of the system are not conserved separately and a general analytical solution is not possible. The integral has been used to derive numerous solutions in special cases.

In physics, precisely in the study of the theory of general relativity and many alternatives to it, the post-Newtonian formalism is a calculational tool that expresses Einstein's (nonlinear) equations of gravity in terms of the lowest-order deviations from Newton's law of universal gravitation. This allows approximations to Einstein's equations to be made in the case of weak fields. Higher-order terms can be added to increase accuracy, but for strong fields, it may be preferable to solve the complete equations numerically. Some of these post-Newtonian approximations are expansions in a small parameter, which is the ratio of the velocity of the matter forming the gravitational field to the speed of light, which in this case is better called the speed of gravity. In the limit, when the fundamental speed of gravity becomes infinite, the post-Newtonian expansion reduces to Newton's law of gravity.

In theoretical physics, massive gravity is a theory of gravity that modifies general relativity by endowing the graviton with a nonzero mass. In the classical theory, this means that gravitational waves obey a massive wave equation and hence travel at speeds below the speed of light.

A theoretical motivation for general relativity, including the motivation for the geodesic equation and the Einstein field equation, can be obtained from special relativity by examining the dynamics of particles in circular orbits about the Earth. A key advantage in examining circular orbits is that it is possible to know the solution of the Einstein Field Equation a priori. This provides a means to inform and verify the formalism.

In physics and fluid mechanics, a Blasius boundary layer describes the steady two-dimensional laminar boundary layer that forms on a semi-infinite plate which is held parallel to a constant unidirectional flow. Falkner and Skan later generalized Blasius' solution to wedge flow, i.e. flows in which the plate is not parallel to the flow.

<span class="mw-page-title-main">Oblate spheroidal coordinates</span> Three-dimensional orthogonal coordinate system

Oblate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the non-focal axis of the ellipse, i.e., the symmetry axis that separates the foci. Thus, the two foci are transformed into a ring of radius in the x-y plane. Oblate spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two largest semi-axes are equal in length.

In many-body theory, the term Green's function is sometimes used interchangeably with correlation function, but refers specifically to correlators of field operators or creation and annihilation operators.

In classical mechanics, a Liouville dynamical system is an exactly solvable dynamical system in which the kinetic energy T and potential energy V can be expressed in terms of the s generalized coordinates q as follows:

In polynomial interpolation of two variables, the Padua points are the first known example of a unisolvent point set with minimal growth of their Lebesgue constant, proven to be . Their name is due to the University of Padua, where they were originally discovered.

<span class="mw-page-title-main">Cnoidal wave</span> Nonlinear and exact periodic wave solution of the Korteweg–de Vries equation

In fluid dynamics, a cnoidal wave is a nonlinear and exact periodic wave solution of the Korteweg–de Vries equation. These solutions are in terms of the Jacobi elliptic function cn, which is why they are coined cnoidal waves. They are used to describe surface gravity waves of fairly long wavelength, as compared to the water depth.

In classical mechanics, the central-force problem is to determine the motion of a particle in a single central potential field. A central force is a force that points from the particle directly towards a fixed point in space, the center, and whose magnitude only depends on the distance of the object to the center. In a few important cases, the problem can be solved analytically, i.e., in terms of well-studied functions such as trigonometric functions.

Accelerations in special relativity (SR) follow, as in Newtonian Mechanics, by differentiation of velocity with respect to time. Because of the Lorentz transformation and time dilation, the concepts of time and distance become more complex, which also leads to more complex definitions of "acceleration". SR as the theory of flat Minkowski spacetime remains valid in the presence of accelerations, because general relativity (GR) is only required when there is curvature of spacetime caused by the energy–momentum tensor. However, since the amount of spacetime curvature is not particularly high on Earth or its vicinity, SR remains valid for most practical purposes, such as experiments in particle accelerators.

A proper reference frame in the theory of relativity is a particular form of accelerated reference frame, that is, a reference frame in which an accelerated observer can be considered as being at rest. It can describe phenomena in curved spacetime, as well as in "flat" Minkowski spacetime in which the spacetime curvature caused by the energy–momentum tensor can be disregarded. Since this article considers only flat spacetime—and uses the definition that special relativity is the theory of flat spacetime while general relativity is a theory of gravitation in terms of curved spacetime—it is consequently concerned with accelerated frames in special relativity.

In physics and mathematics, the Klein–Kramers equation or sometimes referred as Kramers–Chandrasekhar equation is a partial differential equation that describes the probability density function f of a Brownian particle in phase space (r, p). It is a special case of the Fokker–Planck equation.


  1. 1 2 Roy, John A.E. (Dec 31, 2004). Orbital Motion (4th ed.). CRC Press. p. 121. ISBN   9781420056884.
  2. 1 2 Gurzadyan, Grigor A. (Oct 21, 1996). Theory of Interplanetary Flights. CRC Press. p. 192. ISBN   9782919875153.
  3. 1 2 Danby, John M.A. (1992). Fundamentals of Celestial Mechanics (2nd ed.). Willman-Bell Inc. pp. 253–254. ISBN   9780943396200.