Solution of the Tolman-Oppenheimer-Volkov equations

The class o2scl::tov_solve provides a solution to the Tolman-Oppenheimer-Volkov (TOV) equations given an equation of state, provided as an object of type o2scl::eos_tov. These classes are particularly useful for static neutron star structure: given any equation of state one can calculate the mass vs. radius curve and the properties of any star of a given mass.

Mathematical background:

In units where , the most general static and spherically symmetric metric is of the form

where is the polar angle and is the azimuthal angle. Often we will not write explicitly the radial dependence for many of the quantities defined below, i.e. .

This leads to the TOV equation (i.e. Einstein's equations for a static spherically symmetric object)

where is the radial coordinate, is the gravitational mass enclosed within a radius , and and are the energy density and pressure at , and is the gravitational constant. The mass enclosed is related to the energy density through

and these two differential equations can be solved simultaneously given an equation of state, . The total gravitational mass is given by

The boundary conditions are and the condition for some fixed radius . These boundary conditions give a one-dimensional family solutions to the TOV equations as a function of the central pressure. Each central pressure implies a gravitational mass, and radius and thus defines a mass-radius curve.

The metric function is

The other metric function, is sometimes referred to as the gravitational potential. In vacuum above the star, it is

and inside the star it is determined by

Note that this can be rewritten as

If the neutron star is at zero temperature and there is only one conserved charge, (i.e. baryon number), then

and this implies that is everywhere constant in the star. Alternatively,

where is the enthalpy density. Since , then if the entropy is everywhere a constant we also have

and is everywhere constant (even if there is more than one conserved charge).

The proper boundary condition for the gravitational potential is

which ensures that matches the metric above in vacuum. Since the expression for is independent of , the differential equation can be solved for an arbitrary value of and then shifted afterwards to obtain the correct boundary condition at .

The surface gravity is defined to be

which is computed in units of inverse kilometers, and the redshift is defined to be

which is unitless.

The baryonic mass is typically defined by

where is the baryon number density at radius and is the mass one baryon (taken to be the mass of the proton by default and stored in o2scl::tov_solve::baryon_mass). If the EOS specifies the baryon density (i.e. if o2scl::eos_tov::baryon_column is true), then this class will compute the associated baryonic mass for you.

In the case of slow rigid rotation with angular velocity , the moment of inertia is

where is the rotation rate of the inertial frame, is the angular velocity in the fluid frame, and is the angular velocity of a fluid element at infinity. The function is the solution of

where the function is defined by

Note that . The boundary conditions for are at and

One can use the TOV equation to rewrite the moment of inertia as

The star's angular momentum is just .

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