Physics 210 paper on the Tolman-Oppenheimer-Volkoff Hydrostatic ...

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star we need to set the boundary conditions. In this case
the solution at the surface of the star should match the
Schwartzshild external equations of a star (Equation (3)).
From this following the method shown on Wikipedia([5])
T µv = (µ + P )u µ u v + P g µv (1)
P = Cρ λ (2)
ds 2 = −e 2φ(r) dt 2 + (1 − (2Gm(r)) −1
(rc 2 dr 2 + r 2 d(ω) 2
)
Bounds:
P = 0
m = M
φ = 1 log(1 − (2GM
2 rc 2 )))
dm
dr = 4πr2 ρ
dP
dr = −(P + ρ)m + (4/pir3 P )
r(r − 2m)
(3)
The differential equations are then solved using numerical
integration. There are many methods of numerical
integration, but the Runge-Kutta method is widely used
and is more accurate than simpler than methods such as
Euler’s method. The method chosen is the fourth-order
Runge-Kutta method (Equation (4)) which is one of the
most powerful methods of numerical integration.
y′ = f(t, y), y(t 0 ) = y 0
y n+1 = y n + 1 6 h (k 1 + 2k 2 + 2k 3 + k 4 )
t n+1 = t n + h
for
k 1 = f(t n , y n )
k 2 = f(t n + 1 2 h, y n + 1 2 hk 1)
k 3 = f(t n + 1 2 h, y n + 1 2 hk 2)
k 4 = f(t n + h, y n + hk 3 ) (4)
slope = 1 6 (k 1 + 2k 2 + 2k 3 + k 4 ) (5)
IV.
RESULTS
Two distinct ways were used to numerically integrate
with the Runge-Kutta method. The first involved writing
a script for a program, Matlab and Octave, which was
than ran with some initial parameters within the confines
of the program. The second method was to create a program
in the object oriented language, Java. Both methods
allowed the problem of integration to be passed onto
the computer, however there were certain advantages and
disadvantages to each. The program Matlab was capable
of plotting the solutions within the same program, however
it is a paid program, thus available remotely, which
greatly reduced commutation speed. Octave which acts
like Matlab, running similar scripts, could run on any
terminal, but being a command line program provided
no plotting option. Java is a robust language used in
programming. It has the advantage of being a compiled
language. The program is compiled then ran, and as such
runs faster than the scripting languages. It can be used
to plot but for simplicity this was avoided. Java is also
a multi-platform language, allowing the program to be
used on almost any computer. (The scripts and source
code, should be available along with this publication.)
The outcome of the actual integration was mixed. The
solution to the simple harmonic motion differential equations
match the true sinusoidal solution found using calculus
(Figure 1). The use of many various programs offered
a valuable learning experience, but decreased the
number of solutions that were found and analyzed, due
to time constraints. As such the impact of various equations
of state was not adequately investigated. The data,
for various central pressures of stars, was saved to files
and plotted; but no actual limits on star radii or masses
were determined. The figures show the change in pressure
as radius increases; comparing the same central pressures
in stars made of ultra-relativistic gas (Figure 2) and nonrelativistic
gas (Figure 3). Similar graphs can be created
with different combinations of the data collected: mass,
energy density, and pressure as they vary with radius as
found by the Runge-Kutta Method. The graphs do not
show units or very physically important data as better
equations of state are necessary, but the program created
allows not only the initial pressure to be set but
also a setting of the state constants in Equation 2. The
plots shown use C = 1 and lambda as described in the
method.
V. CONCLUSION
Computers and a numerical integration method like
the fourth order Runge-Kutta method greatly enhance
sciences ability to investigate the properties of nature.
This has particular importance when dealing with astronomical
objects like stars. The Runge-Kutta method
showed accuracy and convergence to the correct solution
when used to integrate the differential equations for simple
harmonic motions. It further showed its elegance by
easily being scripted in two mathematical programs and
in the Java language.
The solutions to the Tolman Oppenheimer Volkoff
equations yielded relationships between radius, mass, energy
density, and pressure. The energy density and pres-