The Potential Energy as a Function of Pressure

A useful expression of the potential energy Q as a function of the internal pressure P can be derived for a star in hydrostatic equilibrium. From (1.33), one may also write

G rm dM2

The equation of hydrostatic equilibrium (1.6) multiplied on the right side by dMr / (4 nr2gdr) = 1 yields dP G dMr2

dr Znr4 dr from which we may express dM2r in (1.45) and get fP(R) 3 r 3 iR fR 2

If P(R) is the pressure at the surface of the configuration, we get

In some cases, for example for a stellar core, one cannot take P(R) equal to zero, since the pressure of the surrounding layers is not negligible. However, in most cases, for example for a star as a whole, one may consider that the pressure P(R) at the surface is negligible and thus ignore the first term on the right-hand side of (1.48) and get rR

This is an expression frequently used for the potential energy of a configuration in hydrostatic equilibrium.

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