Estimate the central temperature of the Sun

[Lengalicious]

Homework Statement

Homework Equations

The Attempt at a Solution

I'm only stuck on part b). I make a) to be 3.87E26 W, but it gives that anyway in part b). OK, so here's my attempt:

$$\int_{r=0}^{r=R}dT(r)T(r)^3 = -\frac{3L(r)}{16\pi acl_{mfp}}\int_{r=0}^{r=R}dr\frac{1}{r^{2}}$$

$$\frac{T(r)^4}{4}|^{r=R}_{r=0} \sim \frac{T(r)^4_{centre}}{4}$$

$$\frac{T(r)^4_{centre}}{4}\sim-\frac{3L(r)}{16\pi acl_{mfp}}[-\frac{1}{r}]^{r=R}_{r=0}$$

OK, so am I on the right lines? If so, this is where I get stuck. The last bit of the R.H.S ends up having a 1/0 once evaluated. Any help would be great :). Also I wasn't sure about the L(r). Isn't the luminosity constant at all r? I guess L being a function of r suggests the contrary. :S

 

[phyzguy]

The luminosity can't be constant at all r, because in the core energy is being produced. L(r) is the total power radiated from inside r, so it has to go to zero at r=0. You need to estimate the size of the core and make a guess at the functional form of L(r) inside the core, then you can assume L(r) is constant outside the core.

 

[Lengalicious]

Ok I was initially under the impression that was the solution, but then why would the question supply me with the solar luminosity if I integrate over the functional form?

 

[phyzguy]

You still need to know the total luminosity, since this normalizes the functional form of L(r). Try setting up the problem and you'll see.

 

[Lengalicious]

ok so after saying $$L(r)=4\pi r^2F_\odot$$

$$\frac{-T^4_{centre}}{4}=-\frac{3}{16\pi acl_{mfp}}\int_{r=0}^{r=R_\odot}dr\frac{4\pi r^2F_\odot}{r^2}$$

$$\frac{-T^4_{centre}}{4}=-\frac{3(4\pi R_\odot F_\odot)}{16\pi acl_{mfp}}$$

$$\frac{-T^4_{centre}}{4}=-\frac{3(\frac{L_\odot}{R_\odot})}{16\pi acl_{mfp}}$$

Is this right, or?

 

[phyzguy]

No. This assumes that L(r) increases steadily from r=0 to r=Rsun. In fact, only a small region in the core of the sun is generating energy, so you should try something like:

L = K * r^2 r < Rcore
L = Lsun Rcore < r < Rsun

You'll need to estimate Rcore, then calculate K, then integrate.