For finding the mass-radius curve of neutron stars, we can solve TOV Equations which are constraint equations got by solving Einsteins equations. The boundary conditions are $m(r=0)=0$ and $\rho(r=0)=\rho_c$. Then we put a physical condition that for $r\rightarrow r_*$(radius of star), $\rho\rightarrow 0$, and by this we can get the limiting radius $r_*$ and mass $m_*$ and plots likeMass Radius curve for TOV polytropic neutron star

and for density

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I understood and took it from MIT OCW

One can also do the same process for Modified Gravities and more involved $P(\rho)$ matter equations

  • I have seen many places where the mass-radius curve looks like

enter image description here

This curve is from the paper "Neutron stars in Einstein-Λ gravity: the cosmological constant effects G. H. Bordbar1, , S. H. Hendi1, and B. Eslam Panah"

  • Confusion: Why isn't the mass of neutron star cumulative as radius increases, i.e. it must be $0$ at $r=0$ and reach its maximum at $r=r_*$? Maybe I am not getting the right interpretation of these curves. So, I would be grateful for some explanation or references.

1 Answer 1


Your non-cumulative curves relate the total mass of the star to its surface radius. Different points represent different stars.

Your cumulative curves relate enclosed mass of a star versus radius of the shell enclosing the mass. Different points represent different radii within a given star.

  • $\begingroup$ Thanks for the answer!! So the perimeter is the central density to plot this curve?? $\endgroup$
    – user235005
    Mar 31, 2022 at 17:09
  • $\begingroup$ @user235005 I'm not sure. For the curve you showed, the star's radius is the x-axis, usually used for the independent variable. So, I'd guess they used a density of zero at that radius as the boundary condition. $\endgroup$
    – John Doty
    Mar 31, 2022 at 18:09
  • $\begingroup$ What can be the other perimeters for which we can draw the ($M_* - R_*$) curve? Bcoz $M_*, R_*$ is the result of solving TOV equations with initial conditions and a physical boundary constraint. So I guess the change in initial condition i.e. central density might be the perimeter! $\endgroup$
    – user235005
    Mar 31, 2022 at 20:59

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