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I'm currently studying the radius of a white dwarf and, in deriving some useful equations, I've seen my professor use the following relation: $\bar\rho=\frac{\rho_c}{6}$ when using a polytropic model $P(r)=\rho(r)^\gamma$ with $\gamma=\frac{5}{3}$, but I can't figure out why that holds...

I'm thinking maybe one can integrate $\frac{1}{V}\int_0^R \rho 4\pi r^2 dr = \bar\rho$ substituting the pressure where the density stands and using the initial conditions $P(0)=P_c=\rho_c^{\frac{5}{3}}$ and $P(R)=\rho(R)=0$, but I don't know what expression for $P$ I should be substituting there. Any help will be much appreciated!

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    $\begingroup$ It has nothing to do with the poly tropic model. It has to do with how p varies radially. $\endgroup$ May 27 '21 at 17:08
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    $\begingroup$ I haven't the slightest idea. $\endgroup$ May 27 '21 at 18:12
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    $\begingroup$ researchgate.net/figure/… $\endgroup$ May 27 '21 at 18:18
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    $\begingroup$ @ChetMiller it has everything to do with the adoption of a particular polytropic model. Given that you don't know where the result comes from, your initial comment is a bit puzzling. $\endgroup$
    – ProfRob
    May 27 '21 at 18:40
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    $\begingroup$ @ProfRob2 Very interesting. Thanks very much. $\endgroup$ May 27 '21 at 20:42
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This is a standard result obtained by numerically solving the Lane-Emden equation for a polytrope with $n=3/2$ (equivalent to an adiabatic index of $\gamma = 5/3$).

It can be found in any textbook or set of lecture notes that treat polytopes as a means of solving the stellar structure equations (e.g. these lecture notes).

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  • $\begingroup$ Clarifying, thanks!!! $\endgroup$ May 27 '21 at 18:42

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