- #1
Bishop556
- 37
- 4
I am confused on how to use the Runge Kutta method to solve for a relationship between the Chrandrasekhar Mass and radius on the following two equations of state:
dx/dr = (-5/3)*(M/r^2)*[sqrt(1+x^2) /x] where x(r=0) = x_c
dM/dr = +3*(r^2)*(x^3) where M(r=0) = 0
where M is the mass, r is the radius of the white dwarf star, and x is the Fermi momentum. Also, x_c is the central Fermi momentum which we can take as some arbitrary high value.
I understand that for both equations that I have two dependent variables on r, M(r) and x(r), but I do not know how I can relate them. I wish to find radius needed such that x(r) approaches zero as this would indicate the Chrandrasekhar Mass at the same given r. If both equations were not coupled, then this would be a simple problem, but they are. Note that both equations are dimensionless as they have been scaled appropriately.
How would one go about using the Runge Kutta Method to determine the mass/radius relation? I know I need to somehow find a relationship between x and M which is given in the second equation, but that seems difficult to code.
dx/dr = (-5/3)*(M/r^2)*[sqrt(1+x^2) /x] where x(r=0) = x_c
dM/dr = +3*(r^2)*(x^3) where M(r=0) = 0
where M is the mass, r is the radius of the white dwarf star, and x is the Fermi momentum. Also, x_c is the central Fermi momentum which we can take as some arbitrary high value.
I understand that for both equations that I have two dependent variables on r, M(r) and x(r), but I do not know how I can relate them. I wish to find radius needed such that x(r) approaches zero as this would indicate the Chrandrasekhar Mass at the same given r. If both equations were not coupled, then this would be a simple problem, but they are. Note that both equations are dimensionless as they have been scaled appropriately.
How would one go about using the Runge Kutta Method to determine the mass/radius relation? I know I need to somehow find a relationship between x and M which is given in the second equation, but that seems difficult to code.