00:01
Hello, in the question we have given that consider a star with mass of 15 solar mass and the radius of 3 solar radius.
00:09
So, use the equation of hydrostatic equilibrium to calculate the central pressure of the star in terms of the sun's central pressure.
00:18
So, we have given this mass and radius.
00:20
So, we have to use the hydrostatic equilibrium.
00:22
So, this is the equation for the hydrostatic equilibrium.
00:27
So, we will use this.
00:29
Now, we have to assume so, they have asked us to make some assumption and write them clearly.
00:35
So, we assume that this star has uniform density of rho naught.
00:54
So, then i can write this mass of star as equal to 4 by 3 pi r cube into rho naught because density is mass per unit volume.
01:10
So, mass will be density, sorry, volume times density.
01:13
So, this is what we get.
01:15
Now, if we plug this in the hydrostatic equilibrium, this formula, so, this will be dp by dr that is equal to minus g m is 4 by 3 pi capital r cube into rho naught square and then this r square is there over here.
01:34
So, rather than writing this capital r, we will write it as small r so that we can specify that it is at some particular point.
01:46
So, now from here, this r gets cancelled.
01:49
So, this is r square.
01:50
So, this just we get r from here.
01:53
And so, if we now differentiate, sorry, integrate this equation, so, integration of dp that will be equal to minus integration of this g into 4 by 3 pi rho naught square.
02:08
So, this will be over here and we get r dr from here.
02:12
So, now the integration limits will be from pc to 0 and from 0 to r.
02:17
So, this from here, we get a minus sign which will be cancelled over here.
02:21
So, pc will be equal to, so, we get g into 4 by 3 pi rho naught square into r square by 2.
02:34
So, after integrating this, so, integration of r dr is r square by 2.
02:38
So, this is what we get.
02:40
Now, we will do some manipulation over here.
02:43
So, pc will be equal to now this if i see what i do is so if i take this r square inside, so, if i do 4 by 3 pi r cube rho naught and this rho naught i take outside and if i have multiplied by r over here, so, i have to divide by r as well.
03:03
So, by doing that, i will get this thing.
03:07
So, that is g into rho naught divided by r and this 2 is there already.
03:17
So, now, what is this 4 by 3 pi r cube rho naught, this is the actual mass of that star...