1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Density of star from hydrostatic equilibrium and pressure

  1. Jul 31, 2016 #1
    1. The problem statement, all variables and given/known data
    Assume that the pressure [itex]p[/itex] in a star with spherical symmetry is related to the density [itex]\rho[/itex] by the (distinctly unrealistic) equation of state [itex]p= \tfrac12 k\rho^2[/itex], where [itex]k[/itex] is a constant. Use the fluid equilibrium equation obtained in Problem 23 to find a relation between [itex]\rho[/itex] and [itex]\Phi[/itex]. Hence show that Poisson's equation yields
    [tex]
    \frac{d^2[r\rho(r)]}{dr^2} = -\frac{4 \pi G}{k}r \rho(r)
    [/tex]
    Solve this equation with the boundary conditions that [itex]\rho[/itex] is finite at [itex]r=0[/itex] and vanishes at the surface of the star. Hence show that the radius [itex]a[/itex] of the star is determinde solely by [itex]k[/itex] and is independent of its mass [itex]M[/itex]. Show also that [itex]M =(4/ \pi )a^4 \rho(0)[/itex].

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

    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 [itex] \Phi = -k\rho[/itex] and this can be easily shown by
    [tex] \nabla p = k\rho'(r) \rho(r) \hat{\textrm{r}} = \rho \textrm{g} [/tex] and so
    [tex] -\nabla \Phi = \textrm{g} = k\rho'(r) \hat{\textrm{r}} [/tex]

    from which you can get [itex] \Phi[/itex]. However by the second relevant equation this must also mean that [itex]-\tfrac12 k \rho^2 = \textrm{constant}[/itex], or I'm missing something. I've also not been able to obtain the differential equation.
     
  2. jcsd
  3. Aug 1, 2016 #2
    Not that it matters too much but it should be [itex]M = (4/\pi)a^3\rho(0)[/itex]. Any ideas anyone?

    Edit 2: the equation [itex]p + \rho \Phi = \textrm{constant}[/itex] is only for an incompressible fluid, so there is no contradiction.
     
    Last edited: Aug 1, 2016
  4. Aug 1, 2016 #3

    TSny

    User Avatar
    Homework Helper
    Gold Member

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

    To relate this to Poisson's equation, take the divergence of both sides.
     
  5. Aug 1, 2016 #4
    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.
     
  6. Aug 1, 2016 #5

    TSny

    User Avatar
    Homework Helper
    Gold Member

  7. Aug 1, 2016 #6
    Yes I did use them. Could you try to see if you get the correct differential equation?
     
  8. Aug 1, 2016 #7

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Yes, I am getting the correct differential equation. Can you show your work?
     
  9. Aug 2, 2016 #8
    Certainly. I used Poisson's equation on the relation, so [tex]
    \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 ) [/tex]

    from which I can't get the correct equation.
     
  10. Aug 2, 2016 #9

    TSny

    User Avatar
    Homework Helper
    Gold Member

    OK, that looks good so far. Try re-expressing the right hand side.
     
  11. Aug 2, 2016 #10
    Thanks I was able to get it.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Density of star from hydrostatic equilibrium and pressure
  1. Hydrostatic pressure (Replies: 2)

Loading...