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Astron. Astrophys. 326, 433-441 (1997)

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4. A force equation for homogeneous, isotropic distributions in Boltzmann equilibrium

In this section, I derive the force equation and then introduce a Newtonian form for the interaction.

The only trick is to apply the operator [FORMULA] over expression ( 11 ) and to develop the expression

[EQUATION]

[EQUATION]

[EQUATION]

HYPOTHESIS 2: The interaction among particles is between pairs of particles, and is proportional to their masses, as a function of their distance (it is a central force). By this hypothesis, the total potential energy is the sum of the potential energies between pairs of particles ( [FORMULA] ). Hence,

[EQUATION]

[EQUATION]

[EQUATION]

This becomes

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

HYPOTHESIS 3: We assume homogeneity and isotropy. This means that [FORMULA] only depends on distance [FORMULA] and [FORMULA]. The operator [FORMULA] can be expanded and thus (I note [FORMULA] instead of [FORMULA] to avoid confusion; the same with [FORMULA] and [FORMULA] in the following equations)

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

I  rename [FORMULA], [FORMULA], [FORMULA] angle between [FORMULA] and [FORMULA] and assume that [FORMULA] (valid in the light of Hypothesis 2) and substitute some expressions using ( 12 ) and ( 14 ), taking into account the isotropy assumed in Hypothesis 3:

[EQUATION]

[EQUATION]

[EQUATION]

This is the force equation. It gives us a relationship between different correlation functions in a distribution that follows our hypothesis and the force that is represented by means of v.

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© European Southern Observatory (ESO) 1997

Online publication: October 15, 1997
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