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The balance between gravity and internal pressure is known as hydrostatic equilibrium. In this section we will derive an equation for this equilibrium condition.

Let us consider the forces acting on a small element of stellar material, as shown in figure 6. The lower face is at a distance r from the centre of the star and the upper face is at a distance r+r. Each of the faces is of area S. The volume of the element is therefore

S r

and, recalling that mass = density × volume, its mass is given by

r S r,

where r is the density of stellar material at radius r.

Figure 6:  A small element of mass inside a star.

There are two forces acting on the element in the radial direction: In hydrostatic equilibrium, the inward force is balanced by the outward force and so we can write:

Pr S = Pr+r S + (GMr / r2) r S r.

Rearranging gives:

Pr+r - Pr = - (GMr / r2) r r.

If we are considering an infinitesimal element, we can write,

Pr+r - Pr = (dPr / dr) r            (in the limit r -> 0, Pr / r = dPr / dr, where Pr = Pr+r - Pr ).

By combining these last two equations we obtain:

dPr / dr = - GMr r / r2.

This equation is known as the equation of hydrostatic support. For much of this course, we will omit the subscript r from P, M and , but it is important to remember that these quantities are all functions of r.

©Vik Dhillon, 27th September 2010