What is the Lifetime of a Proton in a Solar Mass Star's Core?

[nctweg]

Homework Statement

Estimate the lifetime of a proton against fusion to ⁴He in the center of a Zero-Age-Main-Sequence solar mass star. First calculate the energy generation, ε_pp in the center of the star from the p-p chain. Then convert this to the number of fusions (conversion of 4 protons to 1 ⁴He) per unit volume per second. Finally compare this rate to the central number density of protons to estimate the lifetime.

Constants: (C = stellar core values)

T_C = 13*10⁶ K
ρ_C = 78 g/cm³
X_H (Mass fraction Hydrogen) = 0.7

Homework Equations

ε_pp ~ 1.07*10^-6*(ρ_C)*(X_H)²*(T₆)⁴

The Attempt at a Solution

This was an exam problem - I know how to find the energy generation rate as well as the number density of protons. Neither are difficult. I do not, however, have any idea how to convert the energy generated into the number of fusions per volume per second.

If anyone has a first step, that's really all I'm looking for but I genuinely have no idea how to even begin. I was thinking it might involve going through the PP-I -> PP-IV branches individually but that seems far too complicated.

 

[phyzguy]

The energy generation rate (ergs/sec) is just the energy generated per fusion(ergs/fusion) times the number of fusion reactions per second (fusions/second). Do you know how much energy is generated per fusion?

 

[nctweg]

Wooh, that was actually a huge help. I have arrived at an answer (though I do not know how correct it is). Here is the work (sorry, don't know Tex but I'll do my best to make it neat):

To find the energy per fusion of 4 protons to 1 He, I just used Einstein's E=mc²:

E_He ~ 3.72*10⁹ ev

E_{4 Protons} ~ 3.752*10⁹ ev

ΔE = 32 MeV = 5.12*10^-5 Erg = energy released per fusion (4 Protons -> 1 He)

Then I found the number density using n = X_H*ρ/m_p

n = 3.26*10²⁵ protons or so

Then finally, I divided number of protons by number of fusions/second, but I am thinking that I need to multiply number of fusions/second by 4 to get the number of protons/second at which point I arrived at an answer of approximately 1.14*10^13 years.

This answer is quite high but we were asked on the exam if our estimate was reasonable and I know it shouldn't be all that close just taking the PP chain into account. Not sure how good my methods were but I'm at least thrilled to be able to arrive at an answer.

 

[phyzguy]

That sounds high. Is your εpp in ergs/sec for the whole star, ergs/sec/g, or ergs/sec/cm^3?

 

[nctweg]

Now that you mention it, I'm actually not entirely positive. My notes are not explicitly clear but I believe that it's ergs/s/g.

Oooooohhhh, hold on. Let me do some stuff.

 

[nctweg]

Ok, it was definitely in ergs/sec/g. I converted it over and got ε_pp = 90 ergs/sec/c³ or so. Ended up with 1.46*10¹¹ years which seems better.