If a disk galaxy has an exponential surface brightness profile like this: I=I_{0}exp(-r/h), where h is the scale length and I_{0} is the central luminosity density (in Lsun/pc^{2}), calculate:
If the galaxy has a constant stellar mass-to-light ratio (M/L)_{*}, derive an analytic expression for what the rotation curve of the galaxy should look like (ignore the bulge and halo of the Galaxy for this calculation). For your expression for the rotation curve, I want something that looks like V_{c}(r)=f(I_{0}, (M/L)_{*}, h, and r).
- the total luminosity of the galaxy (in terms of I_{0} and h)
- the half-light radius of the galaxy (in terms of h)
- the radius containing 98% of the total light (in terms of h).
Okay, now if the galaxy has a central surface brightness mu_{0}=19.2 mag/arcsec^{2}, (M/L)_{*}=1 Msun/Lsun,^{} and h=3.5 kpc, plot what the rotation curve of the galaxy should look like from r=0 to r=30 kpc. (Hint: if it looks flat, you screwed up.)
Now, the observed rotation curve is flat; at r=30 kpc, the circular velocity is still ~ 220 km/s. What is the mass needed to give this circular velocity? How much disk mass is there inside r=30 kpc? So how much dark matter do we need? So inside 30 kpc, what percentage of the Galaxy's mass is in dark matter?
Some studies suggest that the rotation curve of the Galaxy remains flat out to 150 kpc (or further!). If so, what fraction of the Galaxy's mass (inside 150 kpc) is in the form of dark matter?