2) [40pts] 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 -7/2. Show that in this atmosphere M (2) "*7134 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 LaT72, where To is the central temperature. Show that the central temperature and the luminosity vary with time t according to T.(t) = T.(1 + at)-2/5 and L(t) = L.(1 + at)-7/5, where Lo and To are the values of L and T. 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