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I'm looking for a very rough, order-of-magnitude approach to the main sequence.

What I mean is, I have a spherically-symmetric hydrogen distribution. I'm looking to get a rough approximation for density $\rho(r)$, temperature $T(r)$, and pressure $P(r)$ given the total mass $M$.

To do this, I need an equation of state. Then I need an approximation for the rate of fusion, which I can balance against the heat lost due to radiation (for which I can use Stefan-Boltzmann).

So, how can I throw this into a back-of-the-envelope calculation that tells me that, e.g. solar mass stars should live ~10^10 years, without crunching on some differential equations?

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To get an order of magnitude estimate you can just use the total mass $M$ and luminosity $L$ of the star and an assumption of your fusion process. Main sequence stars fuse Hydrogen in to Helium through the proton-proton chain, which converts 0.7% of mass into energy. So the estimated lifetime of the star would just be: $0.007\frac{Mc^2}{L}$ ($c$ is the speed of light). For example, the sun would have a lifetime ~$10^{11}$ years, but that's assuming that all Hydrogen is being converted to Helium via the proton-proton chain.

Given the scaling from observations of the Sun, you can use the mass-luminosity relation to estimate lifetimes for other main sequence stars.

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But that takes the luminosity as an assumption! – Mark Eichenlaub Jul 23 '14 at 19:30
You get the luminosity, $L$, from $T(r)$ and applying the Stefan-Boltzmann law ($L=4\pi R^2\sigma T_{eff}^4$). $T_{eff}$ would just be the temperature at the surface of your ball of Hydrogen. (That's the energy you're radiating away.) – Aaron Jul 23 '14 at 21:16

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