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ATA 2010 Dynamics of our Galaxy: Assignment
Due September 30, 2010. Please email your assignment to me at kcf@mso.anu.edu.au, with your assignment as a single attached file in a common format like postscript or pdf. Scanned handwritten material is acceptable. Please hand this work in on time: otherwise, I may not be able to grade it. Remember: this assignment represents 30% of your assessment for this module. 1. Give laxation where number a simple order of magnitude argument to show why the retime for 2-bo dy encounters in a stellar system TR 3 /m2 n, is the velo city dispersion, m the typical stellar mass and n the density of stars. (10 marks)

2. (From B&T 99, question 2.3). The density distribution (r ) =
4 M rJ 3 4 r J r 2 (r + r J ) 2

(where M and rJ are constants) for a spherical galaxy is called a Jaffe mo del (after Walter Jaffe of Leiden University, who devised it). · Verify that the total mass is M. · Show that the potential is = GM r ln( ) rJ r + rJ

by (i) using Poisson's equation, and (ii) deriving the potential from the density distribution. · Show that the circular speed vc is approximately constant for r rJ and falls off like vc r -1/2 for r rJ . · Comment on Jaffe mo dels as mo dels for real galaxies. (10 marks for whole question) 3.a) In a spherical galaxy with potential (r ) = V 2 ln(r ) with V = 200 km s-1 , a star at r = 10 kpc has vt = 100 km s-1 , vr = 50 km s-1 , where vt and vr are the tangential and radial components of the stellar velo city. What are the maximum and minimum values of r for the star's orbit ? (Hint: energy and angular momentum are conserved). 3.b) Integrate this orbit numerically until it reaches its third apogalacticon (maximum r ).


· plot the orbit in its own plane · plot r (t) · what is the angular distance between successive apogalactica · check that the apo- and perigalactic distances agree with your estimates in 3.a). 3.c) In this same potential, sketch the following in the (E , J ) plane, where E = v 2 /2 + , J = r vt are the energy and angular momentum of a star: (i) lo ci of circular orbits, (ii) a few lo ci of constant apogalactica, (iii) a few lo ci of constant perigalactica. Say we allow no orbits with E > E , where E is some constant. What is the ratio of the apogalactic radii for circular and radial orbits with E = E ? Mark these two orbits on your (E , J ) plane. What do you conclude about the properties of orbits in the outer parts of stellar systems with a sharp energy cutoff ? 3.d) The infal l ti radius r to fall in infall time to the analytically). (40 me is the time taken for a particle initially at rest at to the center of the galaxy. What is the ratio of the rotation perio d at radius r ? (Hint: you can do this marks for whole question).

4. A spherical galaxy has a density distribution (r ) = o [1 + r 2 /a2 ]-1 . · Show that the enclosed mass M (r ) r 3 for r a (i.e. in the core) and M (r ) r for r a (i.e. in the outer halo). · Now take a population of massless test particles in the potential of this galaxy. Assume that this population is spherical, nonrotating, isothermal and isotropic, with velo city dispersion in each velo city component. What is the radial density distribution of this test particle population ? (Hint: use Jeans' equation) · At large r , this density distribution simplifies and its form depends on a dimensionless number. Give a physical interpretation of this dimensionless number. What is the condition that the density distributions of the test particle population and the galaxy itself have similar forms at large r ? (20 marks for whole question) 5) This is an essay question. Write a short essay (about 500 words) on the goals and metho ds of chemical tagging in galactic archaeology. The paper "Panoramic Spectroscopy" on the website, and the background slides at the end of the ATA2010.ppt file may be useful. Bonus points for including relevant material from elsewhere (20 marks)