I have some issue regarding the stellar structure. I know analytically the equation of state, and I have been asked to build the structure of the star from these two equations

$\frac{dP}{dr}=-\frac{G m \rho}{r^2}$

$\frac{dm}{dr}=4 \pi r^2 \rho$

The problem is that I have been told to integrate over density, so I don't really know how to transform these equations in order to do that. Any help is welcome!

  • $\begingroup$ Ae you sure you haven't been told to use mass as the integrating variable? $\endgroup$
    – ProfRob
    Apr 9, 2020 at 12:07
  • $\begingroup$ Yes, I was explicitly told to use the density as the integration variable. $\endgroup$ Apr 9, 2020 at 12:24
  • $\begingroup$ What do you know about the equation of state? Is it polytropic? $\endgroup$
    – ProfRob
    Apr 9, 2020 at 12:35
  • $\begingroup$ Not actually, it is an isothermal white dwarf with Sommerfeld corrections (arbitrary degeneracy) so it has to be integrated numerically. $\endgroup$ Apr 9, 2020 at 12:42
  • $\begingroup$ So if it is isothermal then you have P = f(rho). $\endgroup$
    – ProfRob
    Apr 9, 2020 at 12:52

1 Answer 1


If you have $ P = f(\rho)$, then $dP = f' d\rho$.

The equations of stellar structure become $$\frac{dP}{d\rho} = f'$$ $$ \frac{dm}{d\rho} = - \frac{4\pi r^4 f'}{Gm} $$

For a more general equation of state, $P = f(T, \rho)$, then you would also need an equation for $dT/d\rho$, which you would get from the equation for energy transfer, either radiative diffusion or convective transport. Note, that even in a white dwarf, this is required in the outer $\sim 1$% (by mass) of the star, where degeneracy and isothermality cannot be assumed, and it may significantly alter the derived radius at a given mass, depending on the mass (negligible at higher masses) and interior temperature (negligible if $T< \sim 5\times 10^6$ K).


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