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
Sr
and, recalling that mass = density × volume, its mass is
given by
_{}_{r }Sr,
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:
An outward force due to the pressure exerted by the stellar material
on the lower face of the element. Note that the forces due to the pressures
on the side faces of the element exactly balance.
Recalling that pressure = force / area,
this outward force is equal to:
P_{r }S.
An inward force due to the pressure exerted by the stellar material
on the upper face of the element (P_{r+r })
and the gravitational attraction of all
the stellar material lying within a distance r of the centre of
the star. The gravitational force acts as if all of the mass interior to
the element were concentrated at the centre of the star and the
remainder of the star were neglected.
Recalling that we may write
Newton's second law in the form force = mass × acceleration,
we obtain the following expression for the inward force:
P_{r+r }S +
(GM_{r }/ r^{2})
_{}_{r }Sr,
where the mass is given by the
_{}_{r }Sr term and the acceleration is given
by GM_{r }/ r^{2}.
In hydrostatic equilibrium, the inward force is balanced by the outward force
and so we can write:
If we are considering an infinitesimal element, we can write,
P_{r+r } -
P_{r } = (dP_{r }/ dr)
r
(in the limit r -> 0,
P_{r }/
r =
dP_{r }/ dr, where
P_{r } =
P_{r+r } -
P_{r }).
By combining these last two equations we obtain:
dP_{r } / dr = - GM_{r }_{}_{r } / r^{2}.
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.