Calculating an effective temperature of a sphere

I've been working on this problem. I think I have the answer. But is it right?

Assume:

A1: a 3D sphere with radius R
A2: the sphere is filled uniformly with N light sources per cubic unit
A3: the luminosity of each light source is L
A4: for whatever reason, the luminosity of the source falls by (1-r/R)^2

Problem:

Using:

* R = 14 billion light years
* N = 1 light source per 1 cubic giga-light year
* L = solar luminosity * 200 billion

What is the effective temperature of the sphere?

My Answer:

0.188 K

The first thing I did is model a single light source, at the center of the sphere. Well, I actually started with a circle, and then grew it out from there. I could then move the sphere around inside the boundary of its starting position. This would tell how many watts are leaving the boundary, and more importantly, are hitting the boundary.

I did this by filling the sphere with boxes. Nothing too fancy.



Here's the 2D version if you want see, and view source:
https://mikehelland.github.io/hubble.../cmbenergy.htm

3D version, for just a single moveable sphere:
https://mikehelland.github.io/hubble...mbenergy3d.htm

Then, create a sphere of boxes around every box in the original sphere at the origin.

https://mikehelland.github.io/hubble...nergy3dall.htm

Now I have the watts reaching the boundary, which is 0.2 megawatts. That's assuming each source is 1 watt.

If I plug the rest into this:

[tex]\displaystyle T={\sqrt[{4}]{\frac {L}{4\pi R^{2}\sigma }}}[/tex]

by doing:

Code:

---

// in hundred million light years
var R = 140

// Radius in meters
const mly2mpc = 1 / 3.261564
const mpc2km = 3.08e19
var Rm = R * 100 * mly2mpc * mpc2km * 1000

var Lsol = 3.846e26
var suns = 2e11 // 200 billion suns in a galaxy


//Stefan-Boltzman constant
const sbc = 5.67e-8

T = (L * Lsol * suns / (4 * pi * sbc * Rm**2))**(1/4)

---

That gives me 0.188.

Is that right?

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