Mass radius relationship for SIRIUS B

[cooper607]

hi guys, I have been doing a research on white dwarf stars and chanrashekhar limit. I need to plot a graph for the mass-radius relationship of the dwarfs. from the equalization of the hydrostatic equilibrium pressure and the electron degeneracy pressure I found out the radius -mass relationship for non relativistic fermi gas.

my final result is

R=3.58*10^16 M^(-1/3)

but when I take relativistic fermi gas, pressure comes in order of 4/3 and the radius cancels out leaving the equation to be
M(limit)=1.44 M(sun)

Now for this maximum limit how can I plot a graph for radius vs mass. which values should i input for different masses? please help me

 

[Matterwave]

What do you mean plot a graph of radius vs mass for a limit? The limit is one value. It's a dot. What is the radius at this mass? If your final result is correct, you need only input this M(limit) into M.

 

[cooper607]

But the relativistic fermi gas plot is different than the non relativistic, i want to know how do they get this graph?

 

[Chronos]

The formula can be found here - http://burro.astr.cwru.edu/Academics/Astr221/LifeCycle/WDmassrad.html. It is interesting to note that the radius actually shrinks as mass increases.

 

[cooper607]

dear Chronos, I actually visited that page earlier, for relativistic fermi gas they derived the relation to be 1=M^(-1/3) , so radius actually cancels out from both sides. so how do they plot that graph where radius is zero when mass goes to 1.44? i didnt get any expression for the graph..can you tell me if there is any radius -mass relation for the relativistic curve ?

 

[Ken G]

The "relativistic curve" must really mean the "completely correct curve that bridges between the nonrelativistic and relativistic limits." So those two limits are the easy calculations, because the kinetic energy per particle is either p²/2m, or is pc. That gives the the blue and red curves respectively, but those are only true at the edges of the plot. The green curve is the correct curve, which uses the fully correct expressions for the momentum flux that goes into pressure, and the kinetic energy density. In units where momentum is measured in mc units, those expressions are, the angle-averaged momentum flux that goes into the isotropic pressure is 1/3 (from the angle averaging) times p²/root(1+p²), and the kinetic energy is root(1+p²)-1. If you use those two expressions, you obtain the green curve, which is correct in all limits. So the confusion is about whether "relativistic" means "including relativistic corrections" versus "taking the ultrarelativistic limit." They mean the former with that green curve.