# Luminosity of a white dwarf

Why in an H-R diagram the white dwarfs are at the bottom left? I understand that they would have a high temperature when they are formed, and so at that instant shouldn't they have a high luminosity as well because they have temperatures higher than those stars in the main sequence? And according to Stefan Boltzman law $$L$$ is directly proportional to $$T^4$$.

## 1 Answer

As you say, the bottom left branch that appears in a typical H-R diagram represents very hot yet low luminous stars. The thing you are missing here is that Stefan-Boltzmann law describes the power radiated per unit surface area by a black body at a certain temperature, while the luminosity of a star is defined as the total power radiated by it.

Therefore, the absolute luminosity is given in terms of the Stefan-Boltzmann law by $$L = 4 \pi R^2 \sigma T_{\mathrm{eff}}^4$$ with $$R$$ the radius of the star and $$T_{\mathrm{eff}}$$ the effective temperature of the star. Now, how can a star be very hot and dim at the same time? Well, from the equation we conclude that its radius must be very small, and thus the name white dwarfs for these stars.