Protons fused per second to power the Sun's luminosity

[jayqubee]

Homework Statement

This is for my intro to General Relativity class, using Hartle's text Gravity: An intro to Einstein's GR.
12.1 "How many protons must combine to make one He nuclei every second to provide the luminosity of the Sun? Estimate how long the Sun could go on at this rate before all its protons were used up."

Homework Equations

L = 3.85×10²⁶J/s

The hint given by the professor, probably to save us from attempting to use the actual thermonuclear fusion chain:
4H → ⁴He

The Attempt at a Solution

At first I thought this may be a 4-vector and CM frame problem, but the reaction equation didn't include a γ term that would represent light carrying away energy.

I decided to take an overly simple approximation heavy route, and ended up with 6.8×10⁴⁵ protons fused per second and a corresponding Solar lifetime of 4160 years which is obviously ridiculous.

My method was:
75% of the sun is H therefore: 8.92×10⁵⁶ H atoms available for fusion

let n = #fusions/second

nγ = L

Approximating γ as the blackbody peak of 1.41eV (from Daniel V. Schroeder's Thermal Physics)

n ≈ 1.7×10⁴⁵ fusions/second = 6.8×10⁴⁵ protons fused/second

Then it follows that the lifetime of the sun t, is:

t ≈ 1.31×10¹¹ seconds ≈ 4160 yrs.

This does not seem like the approach I was supposed to take and is a ridiculous answer, but there was nothing on luminosity in the chapter as it is mostly on the Schwarzschild black hole and Kruskal-Szekeres coordinates. If I am supposed to use 4-vector methods, where do I get the released energy from given that the reaction is 4H → ⁴He?

 

[mfb]

Approximating γ as the blackbody peak of 1.41eV (from Daniel V. Schroeder's Thermal Physics)

The sun is not radiation gamma rays from fusion.

This problem is way easier than your approaches:
Just consider the rest masses of proton and helium to calculate the energy released per fusion reaction (you can neglect the energy carried away by neutrinos).