White Dwarfs and Neutron Stars
Between 1930 and 1935, Subrahmanyan Chandrasekhar showed that white dwarf stars have a maximum mass. Stars beyond this mass limit become unstable to collapse. Chandrasekhar modeled the stars as degenerate Fermi gases. In the non-relativistic limit, the pressure is proportional to the mass density with an exponent of $5/3$
\[P = k \rho^{5/3}.\qquad {\rm (non-relativistic)}\]
This simple form for the equation of state is called a polytrope. As the pressure in the star increases, however, the thermal motion of the electrons reaches relativistic speeds. In the relativistic limit, the equation of state softens to
\[P = k \rho^{4/3}.\qquad {\rm (relativistic)}\]
This equation of state has an upper mass limit of $1.4~M_\odot$ for the degenerate electron gas, now called the Chandrasekhar limit. We now know that cold stars with masses that exceed this limit collapse to neutron stars.
The fully relativistic solution for spherical stars was found by Oppenheimer and Volkoff in 1939, building on the work of Tolman. This solution also has a limiting mass, and cold stars above this mass limit collapse to form black holes. The exact mass limit is difficult to determine, because of our incomplete understanding of the nuclear equation of state at high densities. A very general analysis places an upper mass limit at $3~M_\odot$. A combination of theoretical work and observational data suggests that the limit could be as low as $2.2~M_\odot.$ The mass limit for highly spinning stars could be about 20% larger.
The TOV Equations
The TOV equations are equations for a spherical star in hydrostatic equilibrium in general relativity. The pressure in the star is provided by the degeneracy pressure of an ideal Fermionic gas. This gas is characterized by the number density of particles $n$, the mass density $\rho$, the energy $\epsilon$, and the pressure $P.$ Finally, $m(r)$ gives the mass of the star contained within a radius $r$, so that $m(0) = 0,$ and $\Phi$ a gravitational potential that specifies the spacetime geometry. The TOV equations are then
\[\begin{aligned} \frac{dm}{dr} &= 4\pi r^2 \epsilon\\ \frac{dP}{dr} &= - (\epsilon + P)\frac{m + 4\pi r^3 P}{r(r-2m)}\\ \frac{d\Phi}{dr} &= - \frac{1}{\epsilon + P}\frac{dP}{dr} \end{aligned}\]
Numerical Solutions
Dimensionless Variables
Example
References
There are several good references on the TOV equations and the equations of state. For now, these can be consulted for more information.
- Aaron Smith, "Tolman–Oppenheimer–Volkoff (TOV) Stars," (2012) PDF.
- Richard R. Silbar and Sanjay Reddy, "Neutron stars for undergraduates," Am. J. Physics 72 892 (2004); DOI, arXiv.
- Irina Sagert, Matthias Hempel, Carsten Greiner, Juergen Schaffner-Bielich, "Compact Stars for Undergraduates," Eur. J. Phys. 27 577–610 (2006); DOI, arXiv.