The problem I am faced with is approximating the mass of the sun, and I thought that a fancy approach would be to approximate the sun as being made up entirely of hydrogen atoms, finding the resulting radius of the sun, and calculating from thereon. Take $N$ hydrogen atoms and place them in a vacuum, and the gravitational force that exists between them will pull them together into a (non?)homogeneous sphere, where the gravitational force balances out the pressure that wants to push the sphere outwards. I thought that I could equate these two forces and be done with it, but from here on out I'm not too sure how to proceed. Bernoulli's equation:
$$P + \rho g h + \frac{1}{2}\rho v^2 = C$$
does not seem to apply, as the gravitational acceleration $g$ varies over distance, and if I wanted to apply it I would have to integrate perhaps. Another approach I thought of was to use the ideal gas law:
$$PV = nRT$$
And take a spherical shell at radius $r$, find the difference in pressure acting to push the shell inwards and push the shell outwards, and equate this to the gravitational force, but even then I have my doubts. It also introduces a new element to consider, the temperature, which I am not too sure how to deal with. Is this the correct way forward?
P.S. This method is probably oversimplifying the problem and has no correlation to the sun's actual radius, at this point I am more interested in solving this theoretical problem.
Best Answer
Your overall method for determining the equilibrium for a star is correct. We use a combination of an assumed 'polytropic relation': $$P = K \rho^{1+\frac{1}{n}}$$ and the requirement that this balances the gravitational pressure at a radius r (for a spherical shell of thickness $dr$): $$4\pi r^2\frac{dP(r)}{dr} = -\frac{GM(r)}{r^2}4\pi r^2\rho(r)$$ where $M(r)$ is the mass enclosed by the shell. Using the polytropic equation together with the fact that $\frac{dM(r)}{dr} = 4\pi r^2\rho(r)$ gives a second order differential equation for $\rho(r)$, called the Lane-Emden equation. You can use the ideal gas law $$P = \rho\frac{k_BT}{m}$$ where $m$ is the atomic mass of hydrogen, to determine the temperature $T(r)$. Some issues are that firstly, you will need to know $n$, and secondly, solutions are known only for some values of $n$.
For some solutions, there is a finite radius $R$, the radius of the star, for which the density goes to zero. It turns out that, for stars like our sun, $n=3$ is a good model, and there is a finite radius. However, an exact (analytic) solution is unknown.
As a special case, if you take $n\rightarrow\infty$ in the polytropic equation, essentially giving the ideal gas equation itself, with the assumption of constant temperature throughout the star, the solution is called an "isothermal sphere" - but this does not model the sun very well.
For the general case, knowing this radius $R$ (assuming you want the experimental value), you can plug in the integration constants to get $\rho(r)$, and the mass is simply $$M = \int_{0}^{R}4\pi r^2\rho(r)dr$$
Also note that you will need one of the total mass or the radius of the star to find the other.