dats13
Feb9-11, 07:08 PM
1. The problem statement, all variables and given/known data
Assume a star is in hydrostatic equilibrium and that the density of the star is follows
\rho \propto \frac{1}{r^{a}}
where \r is the distance from the centre of the star and \r is a constant.
Derive an relation for the pressure of the star as a function of \r.
2. Relevant equations
Hydrostatic Equilibrium:
\frac{dP}{dr} = - \rho(r) \frac{G m(r)}{r^2} (1)
Mass Equation:
\frac{\mathrm{d} M}{\mathrm{d} r}=\rho (r)4\pi r^{2} dr (2)
3. The attempt at a solution
I am assuming that the following boundary conditions must be satisfied.
\rho(0) = \rho_c \; \; \; \rho(R) = 0
m(0) = 0 \; \; \; m(R) = M_{tot}
The problem that I run into is satisfying these equations with the assumption
\rho \propto \frac{1}{r^{a}}
If I assume
\rho(r) = \rho_c(1-(r/R)^{a})
then the boundary conditions are satisfied, but the power law assumption is not.
Once I get the suitable function for \rho, then I can solve Eq. (2) for the Mass as a function of r and then (1) to get a function for the pressure.
But I am unsure of how to satisfy both the power law and boundary conditions for \rho.
Any suggestions would be greatly appreciated.
Assume a star is in hydrostatic equilibrium and that the density of the star is follows
\rho \propto \frac{1}{r^{a}}
where \r is the distance from the centre of the star and \r is a constant.
Derive an relation for the pressure of the star as a function of \r.
2. Relevant equations
Hydrostatic Equilibrium:
\frac{dP}{dr} = - \rho(r) \frac{G m(r)}{r^2} (1)
Mass Equation:
\frac{\mathrm{d} M}{\mathrm{d} r}=\rho (r)4\pi r^{2} dr (2)
3. The attempt at a solution
I am assuming that the following boundary conditions must be satisfied.
\rho(0) = \rho_c \; \; \; \rho(R) = 0
m(0) = 0 \; \; \; m(R) = M_{tot}
The problem that I run into is satisfying these equations with the assumption
\rho \propto \frac{1}{r^{a}}
If I assume
\rho(r) = \rho_c(1-(r/R)^{a})
then the boundary conditions are satisfied, but the power law assumption is not.
Once I get the suitable function for \rho, then I can solve Eq. (2) for the Mass as a function of r and then (1) to get a function for the pressure.
But I am unsure of how to satisfy both the power law and boundary conditions for \rho.
Any suggestions would be greatly appreciated.