<p>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 <a href="http://en.wikipedia.org/wiki/Proton%E2%80%93proton_chain_reaction" rel="nofollow">proton-proton chain</a>, 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.</p>

<p>Given the scaling from observations of the Sun, you can use the <a href="http://en.wikipedia.org/wiki/Mass%E2%80%93luminosity_relation" rel="nofollow">mass-luminosity relation</a> to estimate lifetimes for other main sequence stars.</p>

CC BY-SA 3.0
2014-07-23T19:30:50.780Z
But that takes the luminosity as an assumption!
Mark Eichenlaub
CC BY-SA 3.0
2014-07-23T21:16:03.313Z
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
CC BY-SA 3.0
2014-07-22T07:27:06.743Z
2014-07-22T07:27:06.743Z
Aaron
<p>I'm looking for a very rough, order-of-magnitude approach to the main sequence.</p>

<p>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 $M$.</p>

<p>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).</p>

<p>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?</p>

<star><stellar-astrophysics><hydrostatic-equilibrium><main-sequence>
What's an order-of-magnitude main sequence star look like?
Mark Eichenlaub
# What's an order-of-magnitude main sequence star look like?

(Asked by: username_0 on 2014-07-21)

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?

**Tags:** star, stellar-astrophysics, hydrostatic-equilibrium, main-sequence

---

## 1 answers

### Answer by username_1 on 2014-07-22. Score: 2

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](http://en.wikipedia.org/wiki/Proton%E2%80%93proton_chain_reaction), 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](http://en.wikipedia.org/wiki/Mass%E2%80%93luminosity_relation) to estimate lifetimes for other main sequence stars.

> **Comments:**
> **username_0**: But that takes the luminosity as an assumption!  
> **username_1**: 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.)  

---

