# How do stars change in size over their main sequence lifetime?

I know that main sequence stars increase in temperature and luminosity as they age. However, I do not know about how they change in size. Do they expand? If so, why do they expand? Do they grow very much? I’ve tried to search up the answer, but details were limited, so I would appreciate an answer here.

Very roughly.

The star is in hydrostatic equilibrium so $$\frac{dP}{dr} = -\rho g$$.

Replacing $$dP/dr$$ by $$P_c/R$$, where $$P_c$$ is the central pressure and $$R$$ the radius, and letting $$\rho \propto M/R^3$$ and $$g \propto M/R^2$$, then we get $$P_c\propto \frac{M^2}{R^4}$$ But the central pressure is proportional to $$\rho T/\mu$$, where $$T$$ is the interior temperature and $$\mu$$ is the mean molecular mass. So $$\frac{\rho T}{\mu} \propto \frac{M^2}{R^4}$$ $$\frac{MT}{R^3 \mu} \propto \frac{M^2}{R^4}$$ $$R \propto \frac{\mu M}{T}$$

Now during the main sequence lifetime, the mass stays (roughly) constant and $$T$$ stays (roughly) constant because the temperature sensitivity of hydrogen burning is high. However, $$\mu$$ increases because 4 hydrogens (4 protons + 4 electrons) are getting turned into 1 helium (1 nucleus + 2 electrons). So $$\mu$$ increases (in the central regions) from 0.5 to 4/3. Hence the radius increases.

A different way to think about it, more appropriate for solar mass stars and above, is in terms of the radiative energy transport. The increase in $$\mu$$ is also accompanied by a decrease in the number of free electrons per mass unit. This reduces the opacity in the central regions and decreases the temperature gradient. This means the star is bigger for roughly the same central temperature.
2. This changes the mean-molecular weight $$\mu$$ of the core with time.
3. The hydrostatic law demands for the stellar structure to fulfil $$\frac{{\rm d}P}{{\rm d}r} = - \rho g(r)$$ and $$P = \frac{\rho k_B T}{\mu}$$. As $$\mu$$ increases with time in the core, this decreases the pressure support there, and demand it to shrink. This again increases the luminosity and expands the outer stellar shells, increasing the stars radius.