TOV (Tolman Oppenheimer Volkoff) Solutions
Last Updated: 1-14-2002
The TOV equations are those derived from Einstein's equations for spherically
symmetric, static perfect fluid systems. The equations can be found in Section 4.5.3 of
the report. Assuming a polytropic equation of state (EOS),
P = K \rho^\Gamma , one can integrate the equations. In the following, K = 1; this
makes the equations unit-less.
The solutions are found by integrating outward from the origin, r=0. They are
are parameterized by the central density, \rho(r=0), and the adiabatic constant
\Gamma in the EOS. A given TOV solution, or star, is characterized by its mass, M, and
its radius, R. Also, a given star will be associated with a given value of Max(2m/r),
where m = m(r) is the mass aspect function and is the mass contained
within a radius of r from the origin; the Max(2m/r) of a star is the maximum value
that 2m/r takes within the volume of the star. The maximum of Max(2m/r) over different
values of \rho(r=0) and \Gamma is approximately 6.1, when \rho(r=0) = 2.25 and \Gamma = 2.
Note, geometrized units (G = c = 1) are always used.
Below are plots of M, R, and Max(2m/r) over different values of \rho(r=0) and \Gamma.
Plots:
The following are plots of TOV solutions for various values of the central
denisity and \Gamma. In the plots, a curve's color is associated with its
value of \Gamma.
In the following, I plot various quantities versus \Gamma while only selecting
those stars with a Maximal Mass, Radius or value of Max(2m/r). The green line
identitifies where \Gamma = 4/3 .
Tests :
The following are plots of the relative change in the central density \rho(r=0) of a star
over time. The data was generated by my code, and demonstrates how the deviation due to
numerical errors of the computed solution to the TOV solutions leads to a stellar solution
that oscillates radially by a small amount. The numerical deviation acts as a perturbation
to the TOV solution, and thus the solution is no longer perfectly static.
Each plot is for K = 0.1 and \Gamma = 5/3. Two were generated using Huen's
Predictor-Corrector method to integrate the equations of motion, and the other two
use a 2nd-order Runge-Kutta (or Half-step) method. Two are near the maximum mass turnover
point (the "nearly unstable star" solution), while the other two are further away (the "stable
star" solution). No significant difference is seen between the two methods, either for
stable or nearly unstable star solutions. All critical searches were performed using
Huen's method.
to
- Half-step, Stable Evolution, \rho(r=0,t=0) = 0.1 ;
- Huen's, Stable Evolution, \rho(r=0,t=0) = 0.1 ;
- Half-step, Nearly Unstable Evolution,
\rho(r=0,t=0) = 1.0 ;
- Huen's, Nearly Unstable Evolution,
\rho(r=0,t=0) = 1.0 ;
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