dP/dr = - G M(r) x density / r**2
Let's make some simple approximations which will allow us to estimate the central pressure of the Sun. We have
(P(surface)-P(center))/(R(sun)-0) = - G 0.5 M(sun) x (M(sun)/vol of Sun) / (R(sun)/2)**2
The pressure at the surface of the Sun is tiny compared to the central pressure and so can be ignored. We then find
P(center) ~ 0.5 G M(sun)**2 / R(sun)**4 ~ 5 x 10**15 dynes per sq cm
This is roughly 5,000,000,000 atmospheres! The true value is about 70 times larger, but this estimate is not so bad, all things considered. Oh yes, you should remember that
P(center) ~ G M**2 / R**4
We know P(center) and so if we use some knowledge that we have about gas pressure, then we can infer the temperature at the center fo the Sun. For the Sun, the gas pressure is the normal gas pressure so that
Pressure = (number density) x constant x Temperature = N k T
In c.g.s units, the constant is 1.4 x 10**(-16) and is called Boltzmann'x constant.
Turning this expression around leads to
T = P / ( N x k ) ~ 5 x 10**15 / 1.7 x 10**24 / 1.4 x 10**(-16) Kelvins.
This works out to around 20,000,000 Kelvins! The true value is around 15,000,000 Kelvins and so this estimate is pretty good.
Well, when the pressure and gravitational forces are in balance, there is no acceleration and things just sit around. However, when the forces are out of balance, a net force is exerted and an acceleration must occur. Using Newton's Force Law (F = ma, 2nd Law of Motion) we have that
density x acceleration = - dP/dr - G M(r) density / r**2
The acceleration is the change in velocity over some time interval t. To get a rough handle on the effects of an imbalance, let us say that the system is slightly out of balance and that the amount that it is out of balance is given by
Imbalance ~ f x (Force of Gravity)
Here, f is a small number and represents the magnitude of the imbalance. If f = 1, then the imbalance is huge. If f is 0.001, then the imbalance is small. Given this definition, we have that
Force Imbalance = - f [ G M(r) density / r**2 ] ~ density x acceleration
Now, let's consider the case when the star is initially at rest (it is not moving) and then an imbalance arises. The imbalance causes the star to shrink with a speed of
[ V - 0 ] / t ~ f [GM/R**2]
The surface then moves inward at a speed of
V ~ t x f [GM/R**2
How long does it take the radius of the Sun to change by a significant amount? We consider motion which is on the order of amount f x R significant. In this event, How long does the change take? Well, because
time = distance / speed ~ t ~ f x R / (t x f GM/R**2)
we have that
t**2 ~ R**3/(GM)
Note that the time is roughly
time ~ sqrt(1/[ G x density ])
This result for the time is general and is known as the free-fall time! Remember this form. For Solar values and using G = 6.67 x 10**(-7) [c.g.s. units], we have that the
time to collapse is only ~ 1 hour !!!