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?
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!
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?
Get the rotation curve data for
[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: