If most of a spiral galaxy's visible mass is concentrated in its central bulge (as it appears to be), what does orbital mechanics predict the rotation curve should look like at large radii?
AFlat — velocity stays roughly constant because the disk provides a uniform mass sheet
BRising — stars at larger radii move faster because they are farther from the gravitational center
CDeclining — velocity should fall off at large radii, roughly as v ∝ 1/√r, like planets in the solar system
DOscillating — velocities alternate high and low depending on the density of spiral arms
From Kepler's laws: for a roughly point-mass or centrally concentrated mass M, orbital velocity v = √(GM/r), which decreases with radius as v ∝ 1/√r. This is exactly what we see in the solar system — Mercury orbits much faster than Neptune. If galaxies had most of their mass in the visible bulge, the same Keplerian decline would apply at large radii beyond the visible disk. The observed FLAT curve is therefore shocking and cannot be explained by the visible mass distribution.
Question 2 Multiple Choice
Galaxy rotation curves are observed to be flat at large radii — orbital velocity stays roughly constant far beyond the visible disk. What does this directly imply about the distribution of mass in the galaxy?
AThe galaxy has no mass beyond the visible disk, but the rotation is maintained by electromagnetic forces
BThe total enclosed mass must increase proportionally with radius, even where no visible matter is present
CGravity works differently at galactic scales, so the normal mass-velocity relationship does not apply
DThe flat curve reflects the average of many stars at different distances, masking the true Keplerian decline
For circular orbital velocity to be constant (v = constant), we need GM(r)/r = v² = constant, which means M(r) ∝ r — the enclosed mass grows linearly with radius. But at large radii, the visible stars and gas have already thinned out, so visible mass M(r) is roughly constant — not growing. Something invisible must be contributing mass that keeps increasing with radius. This is the dark matter halo: a roughly spherical distribution of non-luminous mass extending 5–10 times beyond the visible disk.
Question 3 True / False
The observation that galaxy rotation curves remain flat at large radii is consistent with the distribution of visible stars and gas in the galactic disk.
TTrue
FFalse
Answer: False
This is the core observational puzzle. The visible light in spiral galaxies is concentrated in the bright central bulge and thins out rapidly at large radii — the luminous disk essentially ends. If rotation curves were determined by visible mass alone, they should show a Keplerian decline (v ∝ 1/√r) at large radii. The observed flat curves require mass to keep increasing with radius even in regions where there is no visible matter, implying a dominant dark matter halo that the light distribution cannot account for.
Question 4 True / False
If orbital velocity v is constant at large galactic radii, the enclosed mass M(r) within radius r must increase proportionally with r.
TTrue
FFalse
Answer: True
This follows directly from the orbital mechanics: for a circular orbit, gravitational force equals centripetal force, giving v² = GM(r)/r. If v is constant, then GM(r)/r = constant, so M(r) = v²r/G ∝ r. The enclosed mass must grow linearly with radius. Since visible mass stops increasing beyond the luminous disk, the additional mass must be invisible — the dark matter halo. This is why flat rotation curves are the primary kinematic evidence for dark matter.
Question 5 Short Answer
Why do flat rotation curves require a dark matter halo rather than simply a redistribution of the galaxy's existing visible mass to larger radii?
Think about your answer, then reveal below.
Model answer: Because we can directly observe where the visible matter is — stars and gas traced by light emission and absorption — and it really does thin out at large radii. The 21-cm hydrogen line maps gas well beyond the stellar disk and also shows no sufficient mass increase. The required M(r) ∝ r growth at large radii exceeds what any redistribution of observed matter could provide. The dark matter halo must contain roughly 85% of the total galactic mass with no corresponding luminous component.
The key point is that astronomers have independent observational tracers of visible mass (optical starlight, hydrogen 21-cm, CO emission), and all of them confirm that luminous matter does not extend to large radii in sufficient density. The flat curves are measured at radii where the luminous mass is already roughly constant, yet velocity stays constant — requiring non-luminous mass that simply isn't there in visible form. Multiple independent lines of evidence (gravitational lensing, cluster dynamics, CMB) confirm this conclusion.