The Equations of Internal Structure

Four fundamental equations govern the internal structure of fluid spheres, and reveal the balance between internal pressure and gravity (see Sect. 6.1.2): the hydrostatic equation; the perfect gas equation; the conservation of mass; and the conservation of energy.

7.1.2.1 Low-Mass Objects

From the hydrostatic equation, we may obtain the magnitude of the pressure at the centre of a sphere Pc, assuming constant density. This approximation leads to a simple equation:

where G is the gravitational constant, M the mass of the object, and R its radius. For objects of low mass relative to the Earth, where the effects of compression are low, this equation provides a satisfactory approximation. The internal pressure may then be written as:

r being the distance from the centre, and p the mean density.

The temperature at the centre may be estimated by assuming convection, as a function of the temperature gradient dLnT/dLnP, empirically determined as close to 0.3, and by choosing conditions at the appropriate limits (see Sect. 4.4.2.2).

7.1.2.2 An Object That is a Compressible Sphere

In the case of massive objects, compression becomes important and the density becomes a function of pressure. The precise determination of the pressure, density and temperature profiles would require a full integration of the set of differential equations (hydrostatic equation, transfer equation, mass conservation, and energy conservation). An approximate analytical solution may be obtained using the poly-tropic relation between pressure and density:

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