# Is something wrong with my luminosity calculation?

A few million years after a white dwarf forms, its surface temperature reaches $$100000\text{K}$$, while its radius is $$0.01R_\odot$$. Would this mean that its luminosity is $$\Big(\dfrac{100000}{5778}\Big)^4 \cdot 0.01^2 = 8.972 L_\odot$$? If so, why is it that way (I do understand the Stefan-Boltzmann law), and does that mean that all neutron stars are less luminous than white dwarfs?

It is that way because the effective temperature is defined to be $$(L/4\pi \sigma R^2)^{0.25}$$.
The radius of a neutron star is about 10 km $$(1.4\times 10^{-5}R_\odot)$$. They are born with surface temperatures of around $$10^8$$ K. The coldest white dwarfs have effective temperatures of about 3000 K.
The luminosity ratio is $$\frac{L_{\rm NS}}{L_{\rm WD}} = \left(\frac{10^{8}}{3000}\right)^4 \left(\frac{1.4\times 10^{-5}}{0.01}\right)^2 = 2.4\times 10^{12}\ .$$ Thus the most luminous neutron stars are (briefly) much more luminous than the coldest white dwarfs. They would remain more luminous until they cooled to $$\sim 10^5$$K, which could take as long as $$10^7$$ years (Lattimer & Prakash 2004).
• Could you please clarify for $(L/4\pi \sigma R^2)^{0.25}$ as to where the division stops? – fasterthanlight Apr 27 at 1:15