# Set 3 ```Ay 124: Structure and Dynamics of Galaxies
Problem Set 3
Handed out: February 18th; Due back: March 1st
1
Osipkov-Merritt models describe spherical systems with anisotropic velocity
dispersions. In these models, the distribution function has the form f(r,v)=
f(Q(r,v)) where
Q(r,v) = (r) + &frac12; v2 + r  v2 / 2ra2
where ra is a constant called the anisotropy radius. Prove that at any given radius r
the distribution function is constant on nested ellipsoidal surfaces in velocity
space, and express the axis ratio of these ellipsoids in terms of r / ra.
2. Modify the “closed box” model for chemical evolution in a stellar system by
permitting an external supply or outflow of gas from the system. Assume a
time-independent heavy element yield p, complete mixing, instantaneous
recycling and Z &lt;&lt; 1. Show that if gas with metallicity f is being added
or removed at rate dMf / dt, the evolution of Z is governed by:
(dZ / dt ) Mg - p (dMs / dt ) = ( Zf – Z) (dMf /dt )
In the case of gas outflow at a constant fraction  of the star formation rate, i.e.
dMf / dt = -  (dMs / dt) , show that when the gas fraction is x
Z = p (1 +  )-1 ln [ 1 + ( 1 + ) (x-1 – 1)]
For a pristene (Z=O) mass inflow dMf / dt = +  dMs / dt, derive an analogous
expression. What is the fraction of stars with metallicity less than Z when the gas
metallicity is Z1? Taking Z1 = 0.03 (1.5 the solar value), what value of 
is required to resolve the G-dwarf problem whereby 2% of F,G dwarfs in the solar
neighborhood have Z &lt; (1/6) Z1 ? Comment on the result.
3. A galaxy redshift survey conducted to modest limits in the 1970’s is thought to be
contaminated by non-uniformities arising from the fact the Milky Way is centered
in a local supercluster whose galaxian space density falls off with distance as r-1.8 .
Using the V/Vmax test, indicate how an observer could verify this is a problem in a
magnitude limited sample. How many survey galaxies would be required to
demonstrate at the 3 confidence level that the survey volume is not uniformly
populated?
4. Field galaxies are thought to obey a Schechter luminosity function with faint end
slope . Calculate the total number and luminosity density in terms of *, L* and
, noting carefully the sensitivity to Hubble’s constant h, expressed in units of
100 km/s Mpc-1. Show that the number magnitude counts for such a population
obeys the expression N(&lt; m)  * L*3/2  ( + 5/2). Explain how this can be
used to normalize the luminosity function in the method adopted by Sandage
Tammann and Yahil.
Suppose that all galaxies have disks with radii R/R* = (L/L*)&frac12;. Calculate the
(Euclidean) probability of intercepting a galaxy disk along a path of length c/Ho
in terms of the Schechter parameters. Evaluate this probability for R* = 20h-1 kpc
* = 0.025 h3 and =-1.0. List two practical applications of such a calculation.
5. Utilising CCD photometry tabulated in Capaccioli et al Astron. J 96, 487 (1988),
or otherwise, demonstrate graphically the distinction between Hubble’s law,
Hubble’s modified law and de Vaucouleurs R&frac14; law for describing the projected
light distribution in an elliptical galaxy.
For the R&frac14; case use the expression for the luminosity interior to projected
radius R to show that the total luminosity L = 7.22Re2 Ie where Re is the
radius containing half the light and Ie is the surface brightness at R=Re.
Demonstrate that the mean surface brightness interior to Re is 3.61 Ie.
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