Jacobi integral

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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.

Contents

[edit] Definition

[edit] Synodic system

Co-rotating system.
Co-rotating system.

One of the suitable co-ordinates 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 co-ordinate system x\,\!,y\,\! , the Jacobi constant is expressed as follows:

C_J=n^2(x^2+y^2)+2\cdot (\frac{\mu_1}{r_1}+\frac{\mu_2}{r_2})-(\dot x^2+\dot y^2+\dot z^2)

where:

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.

[edit] Sidereal system

Inertial 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)

[edit] 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 - 2n\dot y = \frac{\delta U}{\delta x}

[Eq.2] \ddot y + 2n\dot 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 CJ 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.

[edit] See also

[edit] 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 .)