Binney & Tremaine 10-4
[The Oort limit is the mass density in the solar neighborhood.]

Use your result to compute the mass of the dark matter within the solar system (out to 40 AU).
Should the dark matter have a noticeable impact on planetary dynamics?

Derive the Tully-Fisher relation from the Virial Theorem.
Be sure to state explicitly any necessary assumptions (about M/L, for example).

Does it make sense that galaxies of the same luminosity but very different scale length fall on the same Tully-Fisher relation?

A. Exponential Disks

In the first homework, you found an expression for the enclosed luminosity L(R) of an exponential disk. Assuming M/L is constant with radius and making the fudge of a "spherical" disk, use this expression to compute and plot V(R) for an exponential disk.
The actual rotation curve of a thin disk is given by BT eqn. 2-169. Plot this on the same diagram to see how close the spherical approximation comes.
You may like to have this tabulation, or better yet, this updated tabulation provided by our own Edmund Hodges-Kluck - Thanks Edmund!

B. Dark Matter Halos

Real disks are only approximately exponential, and their asymptotically flat rotation curves are usually interpreted to require extended halos of dark matter. Two forms of halo are generally considered: isothermal halos with constant density cores, and "NFW" halos motivated by numerical simulations of structure formation (BM eqns 8.57 & 8.58, respectively).

What do these density profiles imply for the total halo mass?

Derive the circular speed V(R) of a test particle orbiting in these mass distributions.
[Hint: dark matter halos can be presumed to be spherical.]
How do the shapes of these rotation curves compare?

C. Mass Models

Get the rotation curve data for NGC 2998.
[TIP: Delete the first line of this file (the one that is all zeros) or the fitting programs will choke.]

First, subtract off the rotation due to the luminous stars and gas. Do this for two choices of the stellar mass-to-light ratio: a) that given in the data file, and b) M/L = 0.5 for both bulge and disk components. Whatever is left we call dark matter.

[Hint: Velocities add and subtract in quadrature.]

Fit isothermal and NFW halos to the dark matter for this galaxy. You may write your own code, or use the fortran versions here: Pseduoisothermal halo code | NFW halo code. Note that getting fits with these codes to converge usually requires a reasonable initial guess for the parameters.

Plot the results for both halo types for both assumptions about M/L. Plot the original data together with lines for each of the mass components and the sum of all mass components. How good are these fits? Can you distinguish between

• halo types?
• M/L choices?

To get some feel for how the various parameters behave, check out Mihos's Rotation Curve JavaLab

Use the Galaxy Crash JavaLab of Chris Mihos to do this problem.
This JavaLab is pretty self-explanatory. To get it to run, click on "APPLET" (at upper left). This will bring up a window with 2 spiral galaxies and a bunch of controls. Play around with the applet and its controls to get a feel for what it does. This runs best on a PC. Some of the controls:

Under the LAB link, you will find several exercises. Do: