Solving Star's Temperature: A Step-by-Step Guide

[adichy]

Homework Statement

Consider a star with a density distribution ⍴ = ⍴_0(R/r), where R is the star’s outer radius. The star’s
luminosity is L, and all of its energy is generated in a small region near r = 0. Outside that region the heat flow is constant.

a) Find the surface temperature of the star T_s assuming a black body.
b) Assuming the opacity is dominated by electron scattering at all radii (i.e., a constant κ_es), solve for the temperature as a function of radius inside the star, excluding the energy-generating region.
(Hint: the algebra will be easier if you rewrite the heat flow in terms of Ts.)

Homework Equations

L=AσT^4
L=M^3.5 (not too sure about this one)

The Attempt at a Solution

a)
Im given a density profile and so i find the mass

m(r)=∫4πR²ρ_0 (R/r)dr (since we're finding the surface temperature I figured the limits will be from 0→R
therefore M=2πR³ρ0

then I sub in the the 2 equations in the relevant equations part and M from above:
L=Aσ(T^4)=M^3.5=(2πR³ρ0)^3.5
and then rearrange to find T (I don't get anything simple/neat so that throws me off a little)

Im wondering if this method is wrong in tackling this problem.

b)
im lost on this part of the question, any help will be appreciated

thanks!

 

[DrClaude]

L=M^3.5 (not too sure about this one)

Do the units make sense?

L=Aσ(T^4)=M^3.5=(2πR³ρ0)^3.5

In that equation, what is A? What is σ?

 

[adichy]

Im missing some units, in which case L ∝ M^3.5
I was quoting the mass-luminosity relation M/M_solar =(L/L_solar)^a, what i wasnt sure about was using a=3.5 since there is not information regarding the type of star.
A is the surface area of the star=4piR^2 and σ is the stefan Boltzmann constant

 

[DrClaude]

A is the surface area of the star=4piR^2 and σ is the stefan Boltzmann constant

Then you got all you need to calculate $T$.

 

[fairymath]

For part b, use the temperature gradient and treat the opacity as a constant.

 

[DrClaude]

For part b, use the temperature gradient and treat the opacity as a constant.

Please do not revive dead threads. The OP hasn't been here in almost three years.

Thread closed.