The luminosity function of a population of objects is its observed distribution of lumi- nosities. White dwarfs have a luminosity function of approximately dN/dL ~ L-74, with a cutoff below a luminosity L ~ 3 × 10-5 Lo-Show that this follows from the properties of individual white dwarfs, as follows. a. To obtain the power-law dependence, assume that a constant rate of star formation leads to a constant rate of white-dwarf formation, dN/dt, from stars that complete their evolution. The white-dwarf luminosity depends primarily on temperature, as L α T. Finally, from Eq. 4.57, the white-dwarf cooling rate can be roughly approxi- mated as dT/dt α T . Use the chain rule to combine these three relations and obtain the functional form of dN/dL. b. As noted in section 4.2.3.3, after Eq. 4.59, the Universe has a finite age, and therefore the oldest white dwarfs (which have had the longest time to cool) are no cooler than -4000 K. The most common white dwarfs have a mass of M0.6Mo, for which