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
<br /> \frac{d^2[r\rho(r)]}{dr^2} = -\frac{4 \pi G}{k}r \rho(r)<br />
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&#039;(r) \rho(r) \hat{\textrm{r}} = \rho \textrm{g} and so
-\nabla \Phi = \textrm{g} = k\rho&#039;(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&#039;(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 <br /> \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.