Thermodynamics, Statistical Physics, and Quantum Mechanics

THERMODYNAMICS AND STATISTICAL PHYSICS 195

For an inversesquare lawforce(gravitation), the averagekineticenergy

and potential energy arerelated as

For a veryrough estimate of thegravitationalpotential energy ofDraco,

consider theenergy of a sphere ofuniformdensity and radius

The averagekinetic energy may beapproximated by

Substituting(S.4.79.2) and(S.4.79.3) into(S.4.79.1), we find

b) If most of the mass ofDraco is inmassiveneutrinos, we may estimate
the energy byconsidering the energy of auniformdistribution offermions
in a box of volumeV.The energy of such afermionic gas has beenfound
in Problem 4.66:

Rewriting (S.4.79.5) in terms ofdensity andvolume gives

If the mass of theneutrino is too low, inorder tomaintain theobserved
density, thenumberdensity would increase, and the Pauliprinciple would
require the kinetic energy to increase. So, in (S.4.79.6), the energy increases