Binney & Tremaine 10-4
[The Oort limit is the mass density in the solar neighborhood.]
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?
In the first problem set, 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 "spherical" disks,
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.
[It'd be nice to have a version of this which is more finely sampled at small radii.]
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
[Hint: dark matter halos can be presumed to be spherical.]
How do the shapes of these rotation curves compare?
Acquire the rotation curve data for the galaxy
[TIP: Delete the first line of this file (the one that is all zeros) or the fitting programs will choke.]
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:
|Control||Theta||Phi||Peri||Red Galaxy Mass||Number of Stars||Friction||Big Halos|
|Function||Inclination Angle of each galaxy||Orientation Angle of each galaxy||Distance of Closest Approach||In terms of the Green Galaxy||Number of simulated points||Dynamical Drag between Particles||Extent of unseen dark matter|
Under the LAB link, you will find several exercises. Do: