# A star's temperature

1. Mar 26, 2014

1. The problem statement, all variables and given/known data

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.)

2. Relevant equations

L=AσT^4

3. 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 dont 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!

2. Mar 27, 2014

### Staff: Mentor

Do the units make sense?

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

3. Mar 27, 2014

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

4. Mar 27, 2014

### Staff: Mentor

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

5. Mar 27, 2017

### fairymath

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

6. Mar 27, 2017