The total luminosity of a 1M white dwarf is 0.03L. If the star radiates as a black body of surface temperature T = 27,000 K, determine its radius and mean density. Suppose a white dwarf has a thin non-degenerate atmosphere with negligi- ble mass compared to M. The opacity in this atmosphere is given by Kramer's law -112. Show that in this atmosphere M /2 T13/4 where L is the luminosity of the white dwarf (which is constant throughout the atmosphere). By considering the interface between the atmosphere and the degenerate interior (or otherwise), show that L x T7/2, where Te is the central temperature. Show that the central temperature and the luminosity vary with time t according to Te(t) = To(1+at)-2/5 and L(t) = Lo(1+at)-7/5, where Lo and To are the values of L and Te at t = 0 and a is a constant. If the star has a central temperature of 3 x 107 K, how long will it take to cool by a factor of 10?