Tolman-Oppenheimer-Volkoff Hydrostatic Equations Presentation

Radim Barta
Phys 210

Purpose
• Learn how the Runge-Kutta method works.
• Write computer code for the Runge-Kutta method in
Matlab (Octave) and Java
• Integrate the Tolman-Oppenheimer-Volkoff
hydrostatic equations of a star.
• Investigate how the equations are derived and look at
how the state equation affects the integral.
• Determine mass limits on different star types.

Tolman-Oppenheimer-Volkoff Equations

Tolman-Oppenheimer-Volkoff Equations
• Just some of the equations

Tolman-Oppenheimer-Volkoff
Equations
• After some equation manipulation we get the
following equations, which we then integrate
numerically.

Runge-Kutta Fourth Order Method
x n+1 = x n + h
• Most commonly used
method for numerical
integration
• Part of a predictorcorrector
group of
algorithms.
y n+1 = y n + (1/6)(k 1 + 2k 2 + 2k 3 + k 4 )
where
k 1 = h f(x n , y n )
k 2 = h f(x n + h/2, y n + k 1 /2)
k 3 = h f(x n + h/2, y n + k 2 /2)
k 4 = h f(x n + h, y n + k 3 )

Computer Code
• Matlab & Octave
• Script writing
• Matlab is a GUI
• It can make plots
• Has an extensive library
• Octave doesn’t require
access to hyper and runs
faster
while (p > 0)
i = i+1;
rho = rhonew;
dm = dr*funmass(r,rho)
m = m+dm
k1 = dr*funpress(r, p, m, rho)
k2 = dr*funpress(r+1/2*dr, p+1/2*k1, m, rho)
k3 = dr*funpress(r+1/2*dr, p+1/2*k2, m, rho)
k4 = dr*funpress(r+dr, p+k3, m, rho)
pnew = p+1/6*k1+1/3*k2+1/3*k3+1/6*k4
fprintf(fid,'1,,, %d,,, %f,,,
%e,,, %e,,, %e,,,\n',i,r,pnew,rho,m);
print(i,r,pnew,rho,m);
rhonew = (pnew/C)^(1/lamda);
r=r+dr;
p=pnew;
end;

Java
• Why rewrite the program in Java?
• I had Eclipse on my laptop
• If written correctly the classes can be run on Linux OS
and Windows
• Since it is a compiled language it has the potentiel to run
faster

Results

Results
• The Runge-Kutta method worked well on the simple
harmonic differential equations. Giving the sinusoidal
curves for position and velocity with time.

What I still want to do:
• Decipher what units the axis are in, and convert back
to STI for comparison with real stars
• Study more in depth how to choose proper state
equations
• And from this then be able to determine mass limits
on different types of stars
• Make my java program also create the plots