Help Find Pressure Stratification

[Tsunoyukami]

Hey everyone, this is my first post here. I'm in my second year of university at the University of Toronto planning to do a major in physics and a minor in astrophysics and english (I know, the english is a twist).

I feel like my astrophysics class is moving too fast, in a sense. I understand the ideas behind each of the lectures and the conceptual ideas involved with each, but the I feel the math is beyond my understanding (at least at the speed that we move at).

I'll probably be in here more and more often over the next little while, trying to get comfortable with this stuff, but here's my current question:

1. Find the pressure stratification P(r) inside a star with mass M and radius R in which the density decreases linearly with r via the expression

D(r) = Dc(1-[r/R])

where Dc is the central density.

Alright, I've rethinked (rethunk, mayhaps) my approach and have come up with this:2. So, all I know is that I need to find an expression for the pressure which varies as a function of the radius. However, we know that the density also varies as a function of the radius, as does the mass (presumably, more mass is contained within the star if its volume increases).

So,

D(r) = m(r)/V(r)

Assuming that the star is a sphere, V = 4pi r^3 / 3, so,

D(r) = 3m(r) / 4pi r^3

Such that as r -> R, D -> 0, m -> MNow I need to find an expression for the pressure stratification, which I expect will be inversley proportional to r, (that is, the pressure at the centre of the star will be greater than that at its edges).

Pressure is defined as a force over a given area, so in the case of the star I will say this area is the entire surface area of the star at a given radius, such that:

SA = 4pi r^2

P(r) = F / 4 pi r^2

Now I simply need to find what the force IS. If the star is in hydrostatic equilibrium then the pressure force is equal to the gravitational force which can be written as:

Fg = -Gm(r)/r^2 where the mass is also a function of the radius, so that...

P(r) = -Gm(r) / 4 pi r^4Now, our term for D(r) = 3m(r) / 4 pi r^3 => 4 pi r^3 = 3m(r) / D(r)

P(r) = -Gm(r) / r [3m(r) / D(r)]
P(r) = -Gm(r) D(r) / 3m(r) r

But we know, from the question that D(r) changes with with respect to readius such that:

D(r) = Dc(1-[r/R])

P(r) = -Gm(r) Dc(1-[r/R]) / 3m(r) rDoes this make sense at all? I feel like I'm missing something since I am not including the M term (the entire mass of the star at radius R) which is provided in the question. All help will be greatly appreciated.

 

[SolsticeFire]

Hey, I just finished this assignment. I don't know if you'll check this before class but I sure hope so.

Using hydrostatic equilibrium, you have
dP(r)/dr = -GM(r)rho(r)/r^2

Since both mass and density are functions of r, you can make use of Mass conservation equation to solve for M(r) in terms of rho(o), r and R.

Then substitute the derived equation for M(r) and rho(r) in hydrostatic equation. Do some algebra before integrating both sides. Now you should have P(r) in terms of rho(o) and other r functions. Since there is still no M in the equation, you can go back to the derived M(r) equation and solve for M(R) and then solve in terms of rho(o). It should give you some M in the numerator and volume in the denominator (since you are solving for density). The problem is solved! Then just re-write the P(r) equation in terms of M, r and R. I hope this helps. Gah I hate staying up so late...