The class o2scl::tov_solve provides a solution to the Tolman-Oppenheimer-Volkov (TOV) equations given an equation of state (EOS), 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.
The EOS is typically specified using o2scl::eos_tov_interp which uses linear interpolation to interpolate a user-specified o2scl::table object. The Buchdahl EOS is given in o2scl::eos_tov_buchdahl, a single polytrope EOS is given in o2scl::eos_tov_polytrope, and a linear EOS is given in o2scl::eos_tov_linear.
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 and spherically symmetric star)
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
Alternatively, that this can be rewritten as
In this form, has no explicit dependence on
so it can be computed (up to a constant) directly from the EOS.
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. If one defines the "enthalpy" by
then
and thus or
. This is the enthalpy used by the o2scl::nstar_rot class.
Keep in mind that this enthalpy is determined by integrating the quantities in the stellar profile (which may be, for example, in beta-equilibrium). Thus, this is not equal the usual thermodynamic enthalpy which is
or in differential form
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 o2scl::tov_solve will compute the associated baryonic mass for you.
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