In deriving the equation of hydrostatic
support, it has been assumed that the gravity and pressure forces are
balanced in a star. How valid an assumption is this?
Let us consider again the element of mass in
figure 6.
We have seen that the outward force acting on the element
is given by
P_{r }S
and the inward force acting on the element is given by
P_{r+r }S +
(GM_{r }/ r^{2})
_{}_{r }Sr.
If the inward and outward forces are not equal, there will be a resultant
force acting on the element which will give rise to an
acceleration, a. This resultant force is given by
_{}_{r }Sr a, where we have simply multiplied
the acceleration of the element by its mass. Hence
we can write
Recalling that the acceleration due to gravity, g, is given
by GM_{r }/ r^{2}, and, for the
sake of brevity, dropping the subscript r, we may write:
dP / dr +
_{}g =
_{}a.
This is a generalised form of the
equation of hydrostatic support.
We are now in a position to determine what happens if there is
a resultant force acting on the element, i.e. if the sum of the two
terms on the left-hand side of the above equation is not zero.
Suppose that their sum is a small fraction _{}
of the gravitational term, i.e.
_{}g =
_{}a.
This means that there will be an inward acceleration given by
a = _{}g.
If the element starts from rest with this acceleration, its inward
displacement s after a time t will be given by:
s = ½at^{2} = ½_{}gt^{2}.
Rewriting the above equation in terms of t, we obtain:
t = (1 / _{}^{½}) ×
(2s / g)^{½}.
If we allow the element to fall all the way to the centre of the star,
we can replace s in the above equation by r and then
subsitute g = GM / r^{2}, giving:
This equation gives the time it would take a star to collapse if the
forces are out of balance by a factor _{}.
Fossil and geological records indicate that the properties of the Sun have
not changed significantly for at least 10^{9} years
(3 × 10^{16} s) and we know that the dynamical timescale
for the Sun is approximately 2000 s (calculated by substituting values
for the mass and radius of the Sun in the above expression for
t_{d}). Hence, in the case of the Sun, we find that
_{} can be no greater than 10^{-27}.
Not all stars are like the Sun, of course. During their lives,
all stars undergo periods of radial expansion and/or contraction
and at these times _{} will be much greater
than this value. In such cases, the generalised form
of the equation of hydrostatic support derived above must be used.
Nevertheless, most stars are like the Sun (i.e. on the
main sequence) and so we may conclude that:
the equation of hydrostatic
support must be true to a very high degree of accuracy.