Thermodynamics, Statistical Physics, and Quantum Mechanics

196 SOLUTIONS

as the mass of the neutrinodecreases. Equating the kineticenergyfrom

(a) with(S.4.79.6), we see

c) Substituting(S.4.79.4) into (S.4.79.7), wedetermine that

This value is at least anorder ofmagnitude largerthan anyexperimental

results for neutrinomasses,implyingthat themodeldoes notexplain the

manner in which Draco is held together (see also D. W. Sciama, Modern

Cosmology and the Dark Matter Problem).

4.80 Einstein Coefficients (Stony Brook)

a) At equilibrium the ratesof excitation out of and back tostate 1should

be equal, so

Substituting(P.4.80.1),(P.4.80.2), and (P.4.80.3) into(S.4.80.1) gives

We may find theratio ofthe populationsfrom(S.4.80.2) to be

b) At thermal equilibrium the population of the upper stateshould be
smallerthan that of thelowerstate by the Boltzmann factor so
(S.4.80.3)gives