Five Simplified Cases:
One Component at a Time

Five geometries, five distinctive rotation signatures

For the same total mass ($10^{10}\,M_\odot$ for stellar components, $1.33 \times 10^{9}\,M_\odot$ for the gas case) and the same reference disk size $R_d = 2$ kpc:

Bulge alone peaks at $V \approx 127$ km/s near $R = 1$ kpc and declines steeply — the most centrally concentrated signature.

Thin disk alone reaches $V \approx 212$ km/s at $R = 8$–$10$ kpc and stays roughly flat afterwards.

Thick disk alone reaches similar $V \approx 208$ km/s but more slowly, with the maximum displaced to larger radii.

Gas ring alone, carrying only $\sim 13\%$ of the stellar mass scale, peaks at $V \approx 60$ km/s — modest but extended.

Spiral arms alone (10% mass excess with a narrower kernel) produce a curve very similar to the thin disk but slightly steeper at intermediate $R$ and declining faster at large $R$.

The bulge produces a sharp velocity rise: from $V_\text{tot} \approx 117$ km/s at $R = 0.5$ kpc to its maximum $V \approx 127$ km/s at $R = 1$ kpc, then declines steadily. The wave-field saturates by $R \approx 5$ kpc — beyond that, $M_\text{wave}$ stops growing. This is the signature of a 3D distribution with a very short coherence length ($ell_b = 0.41$ kpc): the field is intense at short distance and exponentially suppressed beyond. Pure bulges cannot maintain flat rotation curves; they need disk-scale companions.

The thin disk produces the most extended rotation curve: rising smoothly from $V \approx 100$ km/s at $R = 1$ kpc to $\sim 212$ km/s at $R = 8$ kpc, then staying flat to $R = 15$ kpc. The wave-field mass continues to grow steadily because $\ell_\text{thin} = 6.34$ kpc allows coherence over the full disk. This is the dominant component for most disk galaxies, producing the characteristic flat-rotation curve signature.

With the same total mass distributed over a $50\%$ larger scale, the thick disk produces a slower-rising curve that reaches a slightly lower peak ($V \approx 208$ km/s at $R = 10$ kpc). The longer coherence length $ell_text{thick} = 9.51$ kpc keeps the wave field active out to larger radii — the curve declines almost imperceptibly between $R = 10$ and $R = 15$ kpc. In a real galaxy, the thick disk carries only $\sim 25\%$ of the stellar mass, so its contribution is correspondingly modulated.

Despite carrying only $\sim 13\%$ of the stellar mass scale of cases 1–3, the gas ring produces a measurable rotation contribution: $V \approx 60$ km/s at large $R$. The curve rises gently (no central peak — the central hole suppresses the inner contribution) and continues to climb to the largest radii because of the long coherence $\ell_\text{gas} = 10.78$ kpc. The gas component is critical for shaping the outer rotation curve, particularly in gas-rich galaxies where it can account for a substantial fraction of the total wave field.

The spiral arm component shares the thin-disk geometry but with the narrower kernel $\ell_\text{arm} = 4.0$ kpc. The result is a rotation curve very similar to the thin disk at $R \lesssim 6$ kpc — slightly less efficient at low $R$, equally efficient at intermediate $R$ — but declining noticeably faster at $R > 10$ kpc. The shorter coherence length reflects the azimuthal concentration of the arms: they generate strong local wave fields but cannot maintain coherence over the full extent of the disk. In a real galaxy, the arms carry only $10\%$ of the thin-disk mass, so their contribution is small but distinctive.