Density of star from hydrostatic equilibrium and pressure

[Dazed&Confused]

Homework Statement

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 $M =(4/ \pi )a^4 \rho(0)$.

Homework Equations

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

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.

 

[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.

 

[TSny]

$$-\nabla \Phi = \textrm{g} = k\rho'(r) \hat{\textrm{r}}$$

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

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

 

[Dazed&Confused]

Hi thanks for responding. I did realize 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.

 

[TSny]

Make sure you are using the correct form of the Laplacian or the divergence in spherical coordinates.

https://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates#Del_formulae

 

[Dazed&Confused]

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

 

[TSny]

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

 

[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.

 

[TSny]

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

 

[Dazed&Confused]

Thanks I was able to get it.