Physics 701: Classical Mechanics
c
2015
Andisheh Mahdavi
Thornton Hall 425, San Francisco State University
Fall 2015, TuTh 5:10PM
Homework 9 due in class December 8
While I may have consulted with other students in the class regarding this homework, the solutions presented here
are my own work. I understand that to get full credit, I have to show all the steps necessary to arrive at the answer,
and unless it is obvious, explain my reasoning using diagrams and/or complete sentences.
Name
Signature:
1. Consider the following enclosed mass profile for a spherically symmetric galaxy. It provides a good description
of the total gravitating mass in large elliptical galaxies. The mass profile extends to infinity.
M (r) =
M0 r 2
(b + r)2
(a) What is the total mass of the galaxy?
(b) Find the density profile for this galaxy.
(c) Find the gravitational potential for this galaxy. State your chosen zero point for the potential. What is
the gravitational potential when b = 0?
(d) Find and sketch the circular velocity profile. What is the peak circular velocity?
(e) Find the total gravitational potential energy.
(f) Compare this model to the Plummer model. How are they similar? How do you explain the difference
between their total gravitational potential energies? What else do you see as being the biggest difference
between the two models? Why?
2. By analogy with our in-class derivation of the Jeans equations in cylindrical coordinates, derive the spherical
Jeans equation:
(a) Write down the Hamiltonian per unit mass in spherical coordinates for particles under a potential Φ. Use
Hamilton’s equations to solve for ṙ, θ̇, and φ̇ in terms of vr , vθ , and vφ , and for the velocity derivatives
in terms of the potential.
(b) Use these relations to expand out the collisionless Boltzmann equation
df
dt
= 0 in spherical coordinates.
(c) Simplify the above with the following assumptions: (a) the system is steady-state; (b) the system is
spherically symmetric (looks the same no matter which way you rotate it).
(d) Take the appropriate moments of this last equation to show that
2σ 2 νβ(r)
d
dΦ
νσr2 + r
= −ν
dr
r
dr
(spherical Jeans equation)
p
where ν is the number (or probability) density, σr ≡ hvr vr i is the radial velocity dispersion and σθ ≡
p
hvθ vθ i is the tangential velocity dispersion and where β(r) is the dimensionless anisotropy parameter
β(r) ≡ 1 − σθ2 /σr2
3. Calculate the radial velocity dispersion profile σr (r) for a spherical galaxy with constant anisotropy parameter
β, power-law light profile ν(r) ∝ r−α , and power law mass profile M (r) ∝ rγ , assuming that the galaxy ends
abruptly at radius r = rmax .
4. A spherical object of radius a has many millions of stars. You measure the number density of stars using a
telescope, and find that it is everywhere constant. Consulting a spectrograph, you decide that the radial and
tangential velocity dispersions are (unequal) constants.
1
(a) Show that this system cannot be self-gravitating (i.e., must contain dark matter) by finding M (r).
(b) If the total stellar mass of the system is M∗ , find the radius of the system a.
(c) Find the total density profile ρ(r). It should be familiar as we have seen it before in class. What is its
name, and do you have better insight now as to why it was given this name?
(d) Is this system radially or tangentially anisotropic?
(e) In the real world, we are rarely able to measure both σr2 and σθ2 independently. Suppose that in this
case we could only measure σr2 via the Doppler shift of spectral lines. What is the resulting fractional
uncertainty in our measurement of the amount of dark matter in this system? This is the so-called massanisotropy degeneracy of stellar dynamics. It can be resolved using more advanced techniques than the
Jeans equations.
2