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) + ½ v2 + r v2 / 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 << 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 < (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(< 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*)½. 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 R¼ law for describing the projected light distribution in an elliptical galaxy. For the R¼ case use the expression for the luminosity interior to projected radius R to show that the total luminosity L = 7.22Re2 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.