38 PROBLEMS
4.79 Dark Matter (Rutgers)
Fromvirial theorem arguments, thevelocity dispersions ofbrightstars in
dwarf elliptical galaxies imply that most of the mass in these systems is in
the form of “dark” matter - possibly massive neutrinos (see Figure P.4.79).
The central parts of the Draco dwarf galaxy may be modeled as an isother-
mal gas sphere, with a phase-space distribution of mass of the form
Here, is the local mass density in thegalaxy, is thevelocity disper-
sion, and is the mass of a typical“particle” in the galaxy. Measurements
on Draco yield and light years). is
the “core” radius, where the density has decreased by close to a factor of 2
from its value at
a)b)c)Using the virial theorem, write a very rough (order of magnitude)
relation between and
Assume that most of the mass in Draco resides in one species of
massive neutrino. Show how, if the Pauli exclusion principle is not to
be violated, the distribution function above sets a lower limit on the
mass of thisneutrino.
Using the observations and the result of part (a), estimatethis lower
limit (in units of and comment on whether current measure-
ments ofneutrinomasses allowDraco to be heldtogether in the man-
ner suggested.