validity of spherical symmetry assumption
Stars are rotating, gaseous bodies, so we might expect them to be
flattened at the poles. But to what extent are they flattened?
If the departures from spherical symmetry are negligible, then
the variables in the equations of stellar structure
depend only on the distance from the centre of the star.
If the departures from spherical symmetry are significant, the
shape of the star must be taken into account.
Consider an element of mass m near the surface of
a star of mass M and radius r,
as shown in figure 8.
A small element of mass near the surface of a rotating star.
In the absence of rotation, the inward force of gravity on the element is
balanced by the outward force due to thermal pressure. With rotation,
however, an additional inwardly-acting force is required to keep the
element in circular motion. This centripetal force is also provided by
gravitation, which means that the element must move outwards to regions
of lower thermal pressure in order to maintain hydrostatic equilibrium.
The centripetal force is given by
where is the angular velocity of the
star. This force will not cause a serious
departure from spherical symmetry provided that it is much less
than the gravitational force acting on the element, i.e.
m2r << GMm / r2,
which on rearrangement gives
<< (GM / r3)½
It can be seen that the right-hand side of this equation is very similar to
the inverse of the expression for the dynamical timescale we obtained
(2r3 / GM)½.
In fact, given that =
2 / P, where P is
the rotation period of the star, we can say that rotation will have only
a slight influence on the structure of a star provided that the rotation
period is very long compared to the dynamical timescale of the star.
The Sun rotates once in just under a month. The dynamical timescale of the
Sun is around 2000 s. Hence, we may conclude that:
departures from spherical symmetry due to rotation can be neglected.
This statement is true for the vast majority of stars. There are some stars
which rotate much more rapidly than the Sun, however, and for these the
rotation-distorted shape of the star must be accounted for in the equations
of stellar structure.
©Vik Dhillon, 27th September 2010