Ay 124: Structure and Dynamics of Galaxies
Problem Set 4
Handed out: February 27th; Due back: March 11th
1
The Andromeda spiral (M31) has a stellar nucleus. Within a projected radius of 1
arcsec, stars in the nucleus have a line of sight velocity dispersion of 150 km s-1,
and also show a systemic rotation speed of 150 km s-1. If the total luminosity
within 1 arcsec is 3 106 L(sun), calculate the total mass and mass/light ratio for
the inner arcsec.
Assuming the nuclear density falls off as   r-2 and that mass is dominated by
solar mass stars (i.e. no black hole), estimate the radius at which:
(a) the relaxation time equals the likely age of the galaxy (1010 yrs),
(b) a constant density core of that radius would collapse on the same timescale,
(c) 10 solar mass black holes would sink into the center on the same timescale.
2
The giant elliptical galaxy M87 is orbited by a few thousand globular clusters.
There are roughly equal numbers of globulars per octave of projected radius out
to 400 arcsec and the velocity dispersion and rotational speed is independent of
radius at respectively 390 km s-1 and 110 km s-1.
If the surface brightness of M87 falls off as R-2 at R>60 arcsec, how does the
mass/light ratio scale with radius for 60 < R (arcsec) < 400?
What is the total mass within 400 arcsec. State any assumptions made. If the
V-band luminosity is 7.6 1010 L(sun), what is the mass/light ratio?
3. For a disk with a perfectly flat rotation curve, V=constant, if the co-rotation
radius is Rc, whhere are the inner and outer Linblad resonances located for a m=2
pattern? What happens for the cases m=1 and m=3? How would the results differ
for the case of a Keplerian rotation curve?
4
By considering the dispersion relation for a gaseous disk (BT Chapter 6), show
that the group velocity of a spiral density wave for a marginally stable disk (i.e.
Q=1), has a value equal to the sound speed.
5
If the space density of large galaxies of radius 30 h-1 kpc and circular velocity 350
kms sec-1 is 0.03 h3 Mpc-3, estimate the maximum likely redshift of formation, zF,
if the protogalaxies from which they formed collapsed by a factor f according to
the following assumptions: (i) the protogalactic clouds have a space density such
as to just “touch” one another prior to collapse, (ii) the internal density in each
protogalactic cloud is uniform at the onset of collapse but negligibly greater than
that of the background at that redshift.