Derive the density ρ(r) associated with the spherically symmetric Plummer potential Φ(r) = -GM/(r²+a²)^1/2.

Sparke & Gallagher problem 5.10

Hint: think about the role of surface brightness/surface density S = M/r² when you want to go from V² to V⁴.

A "spherical" exponential disk

Previously, 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.

A thin exponential disk

The actual rotation curve of a thin (non-spherical) exponential disk is given by
V²(r) = πGΣ₀R_dx²[I₀(x/2)K₀(x/2)-I₁(x/2)K₁(x/2)]
where x = r/R_d. Plot this on the same diagram to see how close the spherical approximation comes.
You may like to have this tabulation of "ikik," the term involving the Bessel functions.

Dark Mass

Observed rotation curves typically remain flat at a level comparable to the maximum velocity obtained by the disk alone. There are a number of cases in which the rotation curve is measured to remain flat beyond at least 10 scale lengths. Assuming V_flat = V_peak,disk, what is the ratio of total to luminous mass enclosed by 10 scale lengths?

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. Some of the controls:

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