# How to get the Chandrasekhar Limit from a plot?

at the moment I am trying to understand, how to obtain the Chandrasekhar mass limit from a plot like shown above.

Because for $n$ = 3, the mass is independent of the radius of the white dwarf. But in the green line, I see a dependence, which I do not understand. So do you have a formula to get a plot like in the picture?

And I am also interested, on how to find the Chandrasekhar mass numerically by solving the Lane-Emden equation (what I did already).

Best regards,

Tobias

• The English Wiki page on white dwarfs outlines how the limit comes about. See also Chandrasekhar's WD Wiki entry as well Commented Jul 20, 2018 at 11:24
• Yeah but I need the equation, where the plot comes from. Commented Jul 20, 2018 at 12:11
• Your link does not work for me. Please put the picture in your post. If you are talking about the mass-radius relationship for cold electrons, there is no (accurate) analytic formula for it. Commented Jul 20, 2018 at 21:20
• @RobJeffries ok, done. Yes, it is the mass-radius relationship. But if there is no analytic formula, how could they plot it? Commented Jul 21, 2018 at 7:07
• By plotting the results of numerical solutions to the equations of stellar structure. Commented Jul 22, 2018 at 15:37

To find the mass-radius relationship for a (non-rotating) star in general you solve the equations for hydrostatic equilibrium (Newtonian for white dwarfs, general relativistic for neutron stars) with an equation of state $\epsilon(P)$, which relates energy density to pressure. You can then find the mass through $$M(r) = \int_0^r dr \ 4\pi r^2\epsilon(r),$$ which is the mass contained within the radius $r$. The mass $M = M(R)$ of a given star (of radius $R$ can be parametrized by the energy density at the center $\epsilon_c = \epsilon(0)$. It can then be shown that stable stars are those that obey $$\frac{\partial M}{\partial\epsilon_c} \geq 0.$$ Thus, the limiting mass of a star is a local maximum of the mass as a function of central density (if it was a local minimum the star could get more massive and still be stable). The solution to the hydrostatic equilibrium equations also allows you to determine the mass-radius relationship, and with the knowledge of what the limiting mass is, this tells you what the radius of the largest mass star subject to that equation of state.