vik dhillon: phy213 - the equations of stellar structure - minimum value for central pressure of a star
minimum value for central pressure of a star
Even though we have derived only two of the equations of stellar
structure and we have not yet looked at the chemical composition or
the physical state of the material of which stars are composed,
we can still determine a minimum value for the central pressure of a
star. Dividing the
equation of hydrostatic support
by the equation of mass conservation,
we obtain:
(dP / dr) / (dM / dr) =
(- GM /
r 2) /
4r 2.
On rearranging, we have:
dP / dM =
- GM / 4r 4.
This equation can be written in integral form as:
Pc - Ps =
- (GM / 4r 4) dM,
where the subscripts c and s refer to the centre and surface
of the star, respectively.
By swapping the integration limits and noting that the
mass of the star must increase outwards from zero at the centre of
the star, i.e. Mc = 0, we can write:
Pc - Ps =
(GM / 4r 4) dM.
We can obtain a lower limit to the value of the integral on the right-hand
side by noting that, at all points inside a star,
r < rs. Hence 1/r >
1/rs and
1/r 4 > 1/rs4.
So:
(GM / 4r 4) dM >
(GM / 4rs4) dM =
GMs2 / 8rs4.
Hence
Pc - Ps >
GMs2 / 8rs4.
If we approximate the pressure at the surface of the star to be zero,
i.e. Ps = 0, we obtain:
Pc >
GMs2 / 8rs4.
We know the values of Ms and rs
for the Sun and these can be inserted into the inequality to give:
Pc >
4.5 × 1013 Nm-2 = 4.5 × 108
atmospheres.
In light of this high value for the central
pressure, it may seem surprising that the solar material is gaseous. As
we shall see shortly, it is not an ordinary gas.
/
r 2) /
4
r 2
(GM / 4
(GM / 4
>
4.5 × 1013 Nm-2 = 4.5 × 108
atmospheres.