# How to calculate change in absolute magnitude due to a change in stellar radius?

Suppose that the radius of a star increases by some factor, how does this affect the absolute magntiude of the star?

I know that $$M_1 - M_2 = \Delta M = 2.5 \log \frac{L_1}{L_2}$$, so if I knew the luminosities, I could make the calculation. I know that luminosity is given by the formula $$L = 4 \pi \sigma R^2 T^4.$$ However, this shows a dependence on not just on the radius but also on the temperature and I'm not sure how to account for a change of temperature on the luminosity.

If the luminosity changes purely because of a change in radius (at fixed temperature), then for a small perturbation you can argue that since $$L \propto R^2 T^4$$ then $$\frac{\Delta L}{L} \simeq 2\frac{\Delta R}{R}$$ and hence $$\Delta M \simeq -2.5 \log_{10} \frac{L + \Delta L}{L}\ .$$ If the temperature changes too, then $$\frac{\Delta L}{L} \simeq 2\frac{\Delta R}{R} + 4\frac{\Delta T}{T}$$ and you will have to take account of any temperature change on the stellar spectrum if $$M$$ is not the bolometric absolute magnitude.