# Density of star from hydrostatic equilibrium and pressure

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1. Jul 31, 2016

### Dazed&Confused

1. The problem statement, all variables and given/known data
Assume that the pressure $p$ in a star with spherical symmetry is related to the density $\rho$ by the (distinctly unrealistic) equation of state $p= \tfrac12 k\rho^2$, where $k$ is a constant. Use the fluid equilibrium equation obtained in Problem 23 to find a relation between $\rho$ and $\Phi$. Hence show that Poisson's equation yields
$$\frac{d^2[r\rho(r)]}{dr^2} = -\frac{4 \pi G}{k}r \rho(r)$$
Solve this equation with the boundary conditions that $\rho$ is finite at $r=0$ and vanishes at the surface of the star. Hence show that the radius $a$ of the star is determinde solely by $k$ and is independent of its mass $M$. Show also that $M =(4/ \pi )a^4 \rho(0)$.

2. Relevant equations
$\nabla p = \rho \textbf{g}$ and $p + \rho \Phi = \textrm{constant}$. And $\nabla^2 \Phi = 4\pi G \rho$.

3. The attempt at a solution
Assuming the differential equation, I am able to do the rest of the question. The relation they are looking for is $\Phi = -k\rho$ and this can be easily shown by
$$\nabla p = k\rho'(r) \rho(r) \hat{\textrm{r}} = \rho \textrm{g}$$ and so
$$-\nabla \Phi = \textrm{g} = k\rho'(r) \hat{\textrm{r}}$$

from which you can get $\Phi$. However by the second relevant equation this must also mean that $-\tfrac12 k \rho^2 = \textrm{constant}$, or I'm missing something. I've also not been able to obtain the differential equation.

2. Aug 1, 2016

### Dazed&Confused

Not that it matters too much but it should be $M = (4/\pi)a^3\rho(0)$. Any ideas anyone?

Edit 2: the equation $p + \rho \Phi = \textrm{constant}$ is only for an incompressible fluid, so there is no contradiction.

Last edited: Aug 1, 2016
3. Aug 1, 2016

### TSny

Note that this is just $\nabla \Phi = -k \nabla \rho$.

To relate this to Poisson's equation, take the divergence of both sides.

4. Aug 1, 2016

### Dazed&Confused

Hi thanks for responding. I did realise this and found the relation they wanted. I took the Laplacian of this which did not work, so unsurprisingly taking the divergence of what you have said does not work either.

5. Aug 1, 2016

6. Aug 1, 2016

### Dazed&Confused

Yes I did use them. Could you try to see if you get the correct differential equation?

7. Aug 1, 2016

### TSny

Yes, I am getting the correct differential equation. Can you show your work?

8. Aug 2, 2016

### Dazed&Confused

Certainly. I used Poisson's equation on the relation, so $$\nabla^2 \Phi = 4 \pi G \rho = -k \nabla^2 \rho = -k\frac{1}{r^2} \frac{d}{dr} \left ( r^2 \frac{d\rho}{dr} \right )$$

from which I can't get the correct equation.

9. Aug 2, 2016

### TSny

OK, that looks good so far. Try re-expressing the right hand side.

10. Aug 2, 2016

### Dazed&Confused

Thanks I was able to get it.