Jacobi integral

In celestial mechanics, Jacobi's integral represents a solution to the circular restricted three-body problem of celestial mechanics. [1]

The Jacobi integral is the only known integral for the 3-body restricted 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.

Definition

ynodic system

One of the suitable coordinates system used is so called "synodic" or co-rotating system, placed at the barycentre, with the line connecting the two masses μ₁, μ₂ chosen as X axis and the length unit equal to their distance. As the system co-rotates with the two masses, they remain stationary and positioned at (-μ₂,0) and (+μ₁,0)¹.

In the coordinate system $x,!,y,!$ , the Jacobi constant is expressed as follows:

:$C\_J=n^2(x^2+y^2)+2cdot\; (frac\{mu\_1\}\{r\_1\}+frac\{mu\_2\}\{r\_2\})-(dot\; x^2+dot\; y^2+dot\; z^2)$

where:

*$n=frac\{2pi\}\{T\}$ is the mean motion (orbital period T)
*$mu\_1=Gm\_1,!,mu\_2=Gm\_2,!$, for the two masses m₁, m₂ and the gravitational constant G
*$r\_1,!,r\_2,!$ are distances of the test particle from the two masses

Note that the Jacobi integral is minus twice the total energy per unit mass in the rotating frame of reference: the first term relates to centrifugal potential energy, the second represents gravitational potential and the third is the kinetic energy.

idereal system

In the inertial, sidereal co-ordinate system (ξ,η,ζ), the masses are orbiting the barycentre. In these co-ordinates the Jacobi constant is expressed by :

:$C\_J=2\; cdot(frac\{mu\_1\}\{r\_1\}+frac\{mu\_2\}\{r\_2\})+\; 2n(xi\; dot\; eta-\; eta\; dot\; xi)\; -\; (dot\; xi\; ^2+dot\; eta\; ^2+dot\; zeta^2)$

Derivation

In the co-rotating system, the accelerations can be expressed as derivatives of a single scalar function $U(x,y,z)=frac\{n^2\}\{2\}(x^2+y^2)+frac\{mu\_1\}\{r\_1\}+frac\{mu\_2\}\{r\_2\}$

[Eq.1] $ddot\; x\; -\; 2ndot\; y\; =\; frac\{delta\; U\}\{delta\; x\}$

[Eq.2] $ddot\; y\; +\; 2ndot\; x\; =\; frac\{delta\; U\}\{delta\; y\}$

[Eq.3] $ddot\; z\; =\; frac\{delta\; U\}\{delta\; z\}$

Multiplying [Eq.1] , [Eq.2] and [Eq.3] par $dot\; x,\; dot\; y$ and $dot\; z$ respectively and adding all three yields

$dot\; x\; ddot\; x+dot\; y\; ddot\; y\; +dot\; z\; ddot\; z\; =\; frac\{delta\; U\}\{delta\; x\}dot\; x\; +\; frac\{delta\; U\}\{delta\; y\}dot\; y\; +\; frac\{delta\; U\}\{delta\; z\}dot\; z\; =\; frac\{dU\}\{dt\}$

Integrating yields

$dot\; x^2+dot\; y^2+dot\; z^2=2U-C\_J$

where C_J is the constant of integration.

The left side represents the square of the velocity $v,!^2$ of the test particle in the co-rotating system.

¹This co-ordinates system is a non-inertial which explains the appearance of terms related to centrifugal and Coriolis accelarations.

See also

*Rotating reference frame

References

Carl D. Murray and Stanley F. Dermot "Solar System Dynamics" [Cambridge, England: Cambridge University Press, 1999] , pages 68-71. (ISBN 0-521-57597-4)

[1] Original research article:Jacobi, Carl Gustav Jacob (1836) "Sur le movement d'un point et sur un cas particulier du problème des trois corps," "Comptes Rendus de l'Académie des Sciences de Paris", vol. 3, pages 59-61. (Available on-line at: http://visualiseur.bnf.fr/StatutConsulter?N=VERESS3-1201640420309&B=1&E=PDF&O=NUMM-90217 .)