---
title: Galaxy Rotation Curves and Dark Matter
draft: false
module: Galaxies and Dark Matter
moduleNumber: 10
lessonNumber: 3
order: 1003
summary: >
  The rotation curves of disk galaxies stay flat far beyond the light, demanding an
  extended halo whose density falls as the inverse square of radius. Decomposing the
  curve into disk, bulge, and halo, and fitting isothermal or NFW profiles, quantifies
  the missing mass. Gravitational lensing weighs the same mass without dynamics, the
  mass-to-light ratio climbs from stars to clusters, and the Bullet Cluster separates
  the collisionless dark matter from the colliding gas — evidence that MOND strains
  to match.
topics: [Galaxies and Dark Matter]
sources:
  - book: Carroll & Ostlie
    ref: "Ch. 24 §24.3; Ch. 25 §25.3 Dark Matter in Galaxies"
  - book: Ryden
    ref: "Ch. 7 — Dark Matter"
  - book: Maoz
    ref: "Ch. 6, Ch. 10"
---

The dynamics of galaxies do not match their light. A disk galaxy's rotation speed,
which should fall off beyond the luminous edge if the stars were all the mass, instead
stays constant to the largest measurable radii. The same excess appears in the
velocities of cluster galaxies, in the temperature of intracluster gas, and in the
deflection of light. This lesson turns the flat rotation curve into a halo density
profile, contrasts the isothermal and NFW models, adds the independent lensing and
mass-to-light constraints, presents the Bullet Cluster as direct evidence that the
dark mass is collisionless, and states where the MOND alternative succeeds and fails.

## Flat rotation curves and the missing mass

A star or gas cloud on a circular orbit at galactocentric radius $r$ obeys

$$
v_c^2(r) = \frac{G\,M(<r)}{r},
$$

for the mass $M(<r)$ interior to its orbit. Beyond the luminous disk, where the light
and hence (if light traced mass) the enclosed mass would be nearly constant, the speed
should decline as $v_c \propto r^{-1/2}$. Instead, the 21-cm line traces neutral
hydrogen far past the optical edge and shows $v_c(r)$ **flat** — constant to tens of
kpc, twice the optical radius or more. A flat curve inverts to

$$
M(<r) = \frac{v_c^2\, r}{G} \propto r,
$$

mass rising linearly with radius through a region containing negligible light. The
implied density profile follows from $\d M/\d r = 4\pi r^2 \rho$,

$$
\rho(r) = \frac{1}{4\pi r^2}\frac{\d M}{\d r} = \frac{v_c^2}{4\pi G\, r^2}
\propto r^{-2}.
$$

An extended halo with $\rho \propto r^{-2}$ produces exactly the flat curve. This is
the **dark halo**: mass that gravitates but does not shine, dominating outside the
stellar disk.

$$
% caption: Decomposition of a flat rotation curve into the falling contributions of
% the luminous bulge and disk plus a rising dark-halo term; only the halo keeps the
% total flat to large radius.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (9.0,0) node[right, black!70] {radius};
\draw[->, black] (0,0) -- (0,4.4) node[above, black!70] {circular speed};
% total flat curve
\draw[very thick] (0.2,0.3) .. controls (1.0,2.5) and (2.0,3.2) .. (3.2,3.3)
  -- (8.6,3.3);
\node[black, anchor=south] at (6.4,3.4) {total (constant)};
% bulge: peaks then falls Keplerian
\draw[black, densely dashed] (0.2,0.2) .. controls (0.7,2.0) .. (1.2,2.1)
  .. controls (2.0,1.6) and (3.4,0.9) .. (8.4,0.35);
\node[black, anchor=south west] at (1.3,1.9) {bulge};
% disk: peaks further out then declines
\draw[black, densely dotted] (0.2,0.15) .. controls (1.6,2.0) and (2.6,2.35) .. (3.2,2.3)
  .. controls (4.6,2.0) and (6.4,1.4) .. (8.4,1.05);
\node[black, anchor=south] at (3.2,2.35) {disk};
% halo: rises to plateau
\draw[acc, densely dash dot] (0.2,0.1) .. controls (2.0,0.9) and (4.0,2.3) .. (6.0,2.75)
  -- (8.4,2.9);
\node[acc, anchor=north east] at (8.3,2.7) {halo};
\end{tikzpicture}
$$

## Halo density profiles

Two profiles fit the data. The **pseudo-isothermal sphere** builds in a finite central
density with a core radius $r_c$,

$$
\rho_{\mathrm{iso}}(r) = \frac{\rho_0}{1 + (r/r_c)^2},
$$

which is constant for $r \ll r_c$ and falls as $r^{-2}$ for $r \gg r_c$, giving a
rotation curve that rises linearly through the core and flattens to
$v_\infty = \sqrt{4\pi G \rho_0 r_c^2}$ at large radius. It is the phenomenological
choice that reproduces observed curves with a central core.

Cosmological N-body simulations of collisionless cold dark matter instead produce the
**Navarro–Frenk–White (NFW)** profile,

$$
\rho_{\mathrm{NFW}}(r) = \frac{\rho_s}{(r/r_s)\,(1 + r/r_s)^2},
$$

with a **scale radius** $r_s$ and characteristic density $\rho_s$. It has a **cusp**,
$\rho \propto r^{-1}$, in the center and steepens to $\rho \propto r^{-3}$ in the
outskirts, passing through the $r^{-2}$ behavior near $r_s$. Integrating gives the
enclosed mass in closed form,

$$
M_{\mathrm{NFW}}(<r) = 4\pi \rho_s r_s^3
\left[\ln\!\left(1 + \frac{r}{r_s}\right) - \frac{r/r_s}{1 + r/r_s}\right].
$$

The two profiles differ most in the center: the isothermal core is flat while the NFW
cusp diverges. Whether real galaxies, especially dwarfs, have cores or cusps is the
**core–cusp problem**, one of the standing tensions of the cold-dark-matter model.

$$
% caption: The isothermal profile flattens to a constant-density core while the NFW
% profile cusps as the inverse of radius in the center and steepens to the inverse
% cube outside the scale radius; they agree near the scale radius where both fall as
% the inverse square.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (9.0,0) node[right, black!70] {log radius};
\draw[->, black] (0,0) -- (0,4.6) node[above, black!70] {log density};
% NFW: cusp rising to left, steep drop to right
\draw[acc, very thick] (0.3,4.2) .. controls (1.4,3.3) and (2.6,2.6) .. (4.0,2.0)
  .. controls (5.6,1.3) and (7.0,0.5) .. (8.4,0.2);
\node[acc, anchor=west] at (2.4,3.0) {NFW cusp};
% isothermal: flat core then r^-2 drop
\draw[black, densely dashed] (0.3,3.0) -- (2.6,3.0)
  .. controls (3.6,2.7) and (5.2,1.6) .. (8.4,0.5);
\node[black, anchor=south] at (1.4,3.05) {isothermal core};
% scale radius
\draw[black, densely dotted] (4.0,0) -- (4.0,2.0);
\node[black, anchor=north] at (4.0,0) {scale radius};
\end{tikzpicture}
$$

## Mass-to-light ratios

The dark mass shows up as a rising **mass-to-light ratio** $\Upsilon = M/L$, expressed
in solar units $M_\odot/L_\odot$. The stellar populations of galaxies have
$\Upsilon_\ast \sim 1$–$5$ in the optical. Dynamical masses give systematically higher
values as the aperture grows:

- **Inner disk** (starlight): $\Upsilon \sim 3$, consistent with stars alone.
- **Full galaxy** including the halo out to the last measured rotation point:
  $\Upsilon \sim 10$–$30$.
- **Groups and clusters** from the virial theorem: $\Upsilon \sim 200$–$400$.

The ratio climbs by two orders of magnitude from the stellar cores of galaxies to the
scale of clusters, tracking the increasing dark-matter fraction on larger scales. The
cluster value implies a matter density parameter $\Omega_m \approx 0.3$, far above the
$\Omega_b \approx 0.05$ in baryons, so most of the matter is non-baryonic.

## Gravitational lensing

Lensing weighs mass through its gravity alone, with no assumption of dynamical
equilibrium. A point mass $M$ deflects a light ray with impact parameter $b$ by the
general-relativistic angle $\alpha = 4GM/(c^2 b)$, twice the Newtonian value. A source
directly behind a lens is imaged into an **Einstein ring** of angular radius

$$
\theta_E = \sqrt{\frac{4GM}{c^2}\,\frac{D_{LS}}{D_L D_S}},
$$

where $D_L$, $D_S$, and $D_{LS}$ are the angular-diameter distances to the lens, to
the source, and between them. Measuring $\theta_E$ inverts directly to the mass inside
the ring,

$$
M(<\theta_E) = \frac{c^2}{4G}\,\theta_E^2\,\frac{D_L D_S}{D_{LS}}.
$$

**Strong lensing** — multiple images, arcs, and rings around cluster cores — probes
the mass in the dense center; **weak lensing** — the coherent few-percent tangential
distortion of thousands of faint background galaxies — maps the mass in the outskirts
statistically. Lensing masses agree with the dynamical and X-ray masses and exceed the
luminous mass by the same large factor, an independent confirmation that does not rely
on orbital motion.

$$
% caption: A foreground mass bends the light of a background galaxy into arcs and, for
% perfect alignment, a full Einstein ring whose angular radius fixes the projected mass
% inside it.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% observer
\fill[black] (0,0) circle (1.6pt);
\node[black, anchor=east] at (-0.1,0) {observer};
% lens mass
\fill[black] (4.2,0) circle (2.4pt);
\node[black, anchor=south] at (4.2,0.15) {lens mass};
% source behind
\node[black, anchor=west] at (8.3,0) {source};
\fill[black] (8.2,0) circle (1.4pt);
% deflected rays (bent around the lens)
\draw[black] (0,0) .. controls (2.2,0.9) and (3.6,0.95) .. (4.2,0.55)
  .. controls (5.2,0.2) and (7.0,0.1) .. (8.2,0);
\draw[black] (0,0) .. controls (2.2,-0.9) and (3.6,-0.95) .. (4.2,-0.55)
  .. controls (5.2,-0.2) and (7.0,-0.1) .. (8.2,0);
% Einstein ring at observer's sky (dashed image positions)
\node[black, anchor=south] at (1.9,0.75) {arc image};
\node[black, anchor=north] at (1.9,-0.75) {arc image};
\draw[acc, densely dotted] (4.2,0) circle (0.55);
\node[acc, anchor=west] at (4.85,-0.5) {Einstein radius};
\end{tikzpicture}
$$

## The Bullet Cluster

The **Bullet Cluster** (1E 0657-56) is a collision of two galaxy clusters caught in
the act, and it separates the mass from the gas. The hot X-ray-emitting intracluster
gas, which holds most of the **baryonic** mass, is collisional: as the clusters passed
through one another the two gas clouds rammed together, shocked, slowed, and lagged
behind at the center. The galaxies themselves, effectively collisionless points,
sailed through and now sit ahead of the gas on each side.

Weak-lensing reconstruction locates the gravitational mass. It coincides with the
galaxies — the collisionless component — and is **offset** from the gas, where most of
the baryons actually reside. If the excess gravity came from a modification of the
gravitational law sourced by the visible (mostly gaseous) matter, the lensing signal
would center on the gas. That it centers on the galaxies instead is direct evidence
for a dominant, collisionless, non-baryonic mass component that passed through the
collision unimpeded.

$$
% caption: In the Bullet Cluster the collisional X-ray gas is decelerated at the
% center of the collision while the lensing mass tracks the collisionless galaxies that
% flew ahead, placing the gravitating mass away from most of the baryons.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% left subcluster
\draw[acc, very thick] (1.4,0) ellipse (0.9 and 1.3);
\node[acc, anchor=south] at (1.4,1.35) {lensing mass};
\foreach \p in {(1.1,0.4),(1.6,-0.3),(1.3,-0.7),(1.7,0.6)} \fill[acc] \p circle (1.2pt);
% right subcluster
\draw[acc, very thick] (7.0,0) ellipse (0.9 and 1.3);
\node[acc, anchor=south] at (7.0,1.35) {lensing mass};
\foreach \p in {(6.7,0.4),(7.2,-0.3),(6.9,-0.7),(7.3,0.6)} \fill[acc] \p circle (1.2pt);
% central gas (lagged, between)
\draw[black, densely dashed] (3.3,0) ellipse (0.8 and 1.0);
\fill[black] (3.3,0) ellipse (0.8 and 1.0);
\draw[black, densely dashed] (5.1,0) ellipse (0.8 and 1.0);
\fill[black] (5.1,0) ellipse (0.8 and 1.0);
\node[black, anchor=north] at (4.2,-1.15) {X-ray gas (most baryons)};
\draw[->, black] (3.9,1.4) -- (2.0,0.8);
\draw[->, black] (4.5,1.4) -- (6.4,0.8);
\node[black, anchor=south] at (4.2,1.4) {mass displaced from gas};
\end{tikzpicture}
$$

## The MOND alternative

**Modified Newtonian Dynamics (MOND)** proposes that the discrepancy is not missing
mass but a departure from Newton's law at very low accelerations. Below a threshold
$a_0 \approx 1.2\times10^{-10}\ \mathrm{m\,s^{-2}}$ the effective acceleration
becomes $a = \sqrt{a_N a_0}$ rather than the Newtonian $a_N = GM/r^2$. For a star far
out in a galaxy,

$$
\frac{v_c^2}{r} = \sqrt{\frac{GM}{r^2}\,a_0}
\quad\Longrightarrow\quad
v_c^4 = GM\,a_0,
$$

a flat rotation curve with $v_c$ independent of $r$, and a built-in
$L \propto v_c^4$ Tully–Fisher relation with the correct normalization. MOND fits the
rotation curves of individual disk galaxies with a single universal parameter and no
per-galaxy halo. Its failures are on larger scales: it cannot fully account for the
mass of galaxy clusters without residual dark matter, it has no natural relativistic
extension that matches the CMB and structure formation, and — most directly — it does
not explain the Bullet Cluster, where the gravitating mass is displaced from the
baryons that a modified-gravity law would have to follow. The evidence favors dark
matter as a substance, with MOND capturing a still-unexplained regularity of galactic
dynamics.

## Summary

Flat rotation curves imply $M(<r) \propto r$ and a halo density $\rho \propto r^{-2}$
extending well beyond the light. The pseudo-isothermal profile has a constant-density
core, while the cosmologically motivated NFW profile cusps as $r^{-1}$ inside $r_s$ and
steepens to $r^{-3}$ outside, with the core–cusp question still open. The mass-to-light
ratio rises from $\Upsilon \sim 3$ in stellar cores to $\sim 300$ in clusters, giving
$\Omega_m \approx 0.3$ against $\Omega_b \approx 0.05$. Gravitational lensing weighs
the same mass without dynamics through the Einstein radius $\theta_E$, and the Bullet
Cluster displaces the lensing mass from the collisional X-ray gas, evidence for a
collisionless non-baryonic component. MOND reproduces individual galaxy rotation curves
and Tully–Fisher through a low-acceleration law $v_c^4 = GM a_0$ but fails on clusters,
in the CMB, and at the Bullet Cluster.[^co-dm][^ryden-dm][^maoz-dm]

[^co-dm]: Carroll & Ostlie, §24.3, §25.3 — the flat rotation curve of the Galaxy and other spirals, halo density profiles, and the mass-to-light ratio as evidence for dark matter.
[^ryden-dm]: Ryden, Ch. 7 — dark matter: rotation curves, cluster masses, gravitational lensing, and the Bullet Cluster.
[^maoz-dm]: Maoz, Ch. 6, Ch. 10 — dark-matter halos, the isothermal and NFW profiles, and the MOND alternative.