Suppose I know the Luminosity $L$, temperature $T$ and Mass $M$ of star. Assuming the star is very heavy so that we can treat it to be radiation-dominated star. This would imply that pressure inside the star goes (roughly) as,
$$
P(r) = \frac{\sigma T^4}{4 \pi c r^2}
$$
How can I calculate the radius of (run-of-the-mill) star $R$ by balancing the radiation pressure and gravitational pressure? For this purpose the equation of Hydrostatics may be used,
$$
\frac{dP}{dr} = – \frac{G\ m(r)}{r^2} \rho
$$
but since density $\rho$ is $r$ dependent I don't know how to deal with it. It would be nice if we can simply use the idea of balancing pressures since that's what defines the main-sequence star.
Best Answer
If you have $L$ and you have $T$, then nothing more complicated than Stefan's law is required. If $T$ is the effective temperature of the star then this gives an exact answer.
$$ R = \left(\frac{L}{4\pi \sigma_B T^4}\right)^{1/2}$$, where $\sigma_B = 5.67\times 10^{-8}$ in SI units.
If on the other hand you are trying to solve the structure from first principles then you need to learn about polytropes and the solutions of the Lane-Emden equation. A star supported solely by radiation pressure can be treated as a $n=3$ polytrope, which has no analytic solution.
On p.155-162 of Clayton's "Principles of stellar evolution and nucleosynthesis" you can find a treatment using polytropes and some tables with solutions for various values of $n$. The radius of a star is $$ R = \left[ \frac{(n+1)K}{4\pi G}\right]^{1/2} \rho_c^{(1-n)/2n} \alpha_1,$$ where $\rho_c$ is the (here unknown) central density, $n=3$ and $K$ is the constant in the polytropic equation of state (the exact value of $K$ depends on what proportion of the gas pressure is due to radiation pressure) and for a $n=3$ polytrope $\alpha_1=6.9$.
The mass is given by $$ M = -4\pi \left[ \frac{(n+1)K}{4\pi G}\right]^{3/2} \rho_c^{(3-n)/2n} \alpha_1^2 \left(\frac{d\phi}{d\alpha}\right)_{\alpha_1},$$ where $-\alpha_1^2 (d\phi/d\alpha)_{\alpha_1} = 2.02$ for a $n=3$ polytrope.
In the standard model, the ratio of normal gas pressure to total pressure is $\beta$, such that $\beta=0$ for a star solely supported by radiation pressure. It can be shown that the mass of such a star is given by $$ M = 18 \frac{(1 - \beta)^{1/2}}{\mu^2 \beta^2} M_{\odot},$$ where $\mu$ is the mean number of mass units per particle. Thus if you know $M$ and the composition, this gives you $\beta$.
The value of $K$ is then given by $$ K = \left[ \frac{9N_0^{4} k_B^{4}c}{4\mu^4 \sigma_B} \frac{(1-\beta)}{\beta^4}\right]^{1/3}$$ and $N_0$ is Avogadro's number.
This value of $K$ enables you to derive $\rho_c$ from the second polytropic relation and then substitute this into the first polytropic relation to get $R$. Good luck!