Flat Rotation Curve Without Dark Matter?
A spiral galaxy at inclination i = 60 deg has an observed HI rotation curve that is flat at v_obs = 180 km/s from R = 5 kpc to R = 25 kpc. The stellar disk has a scale length of 3 kpc and a central surface brightness corresponding to a stellar mass of 5 x 10^10 M_sun. The HI mass is 8 x 10^9 M_sun, and the gas surface density is negligible beyond 15 kpc. A student claims that the flat rotation curve can be explained entirely by the visible mass if the stellar mass-to-light ratio is increased to 6 in solar units.
1. Compute the enclosed mass required at R = 25 kpc and compare it with the visible mass, correcting for inclination.
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The observed velocity must be corrected for inclination: v_c = v_obs / sin i = 180 / sin 60 = 180 / 0.866 = 208 km/s. The enclosed dynamical mass is M = v_c^2 R / G. With v_c = 208 km/s = 2.08 x 10^5 m/s and R = 25 kpc = 7.7 x 10^20 m, M = (2.08 x 10^5)^2 x 7.7 x 10^20 / 6.67 x 10^-11 = 4.3 x 10^41 / 6.67 x 10^-11 = 6.5 x 10^41 kg, which is 3.3 x 10^11 M_sun. The visible mass is 5 x 10^10 (stars) + 8 x 10^9 (gas) = 5.8 x 10^10 M_sun. Even at a mass-to-light ratio of 6, the stellar mass becomes 3 x 10^11 M_sun, which is close to the dynamical mass but leaves little room for gas and requires an implausibly high M/L for a spiral disk. At R = 15 kpc the gas is still present, so the shortfall is even harder to hide. The flat curve out to 25 kpc requires roughly 3 x 10^11 M_sun enclosed, about five times the visible mass at the stated M/L.
2. Evaluate the student's claim that raising the stellar mass-to-light ratio to 6 resolves the problem. Include the effect on the inner rotation curve.
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Raising M/L to 6 increases the stellar mass to 3 x 10^11 M_sun, which can nearly account for the enclosed mass at 25 kpc. But this creates two problems. First, the inner rotation curve would rise too steeply: at R = 5 kpc, the enclosed stellar mass for an exponential disk with scale length 3 kpc is already a large fraction of the total, and scaling it up by a factor of ~3 would overpredict the observed inner velocities, which are consistent with the lower M/L. Second, the required M/L of 6 is high for a star-forming spiral disk (typical values are 1-3 in optical bands), and it would imply a very bottom-heavy IMF that is inconsistent with the galaxy's colors. The flat curve from 5 to 25 kpc with a nearly constant velocity is naturally produced by a dark halo with an approximately isothermal profile, not by a truncated visible disk. The claim fails because it cannot simultaneously fit the inner and outer rotation curve.