MOND and Alternatives to Galactic Dark Matter

A spiral galaxy rotates far too fast in its outskirts. Beyond the last bright stars, where the enclosed luminous mass has stopped growing, Newton's law predicts orbital speeds that fall away as $v\propto r^{-1/2}$ — and instead they stay flat, out to the edge of the neutral-hydrogen disk. The standard resolution invokes a vast halo of dark matter whose gravity holds the fast-moving gas in place. Modified Newtonian Dynamics, proposed by Mordehai Milgrom in 1983, refuses that halo. It keeps the baryons as the only matter and edits the law that turns a mass distribution into an acceleration, but only below a new universal constant $a_0\approx1.2\times10^{-10}\,\mathrm{m\,s^{-2}}$ — eight orders of magnitude weaker than the Sun's pull on Earth, a scale nothing in the Solar System ever probes. The audacity of the idea is that a single number reshapes an entire galaxy. Above $a_0$ , gravity is exactly Newtonian and the Solar System is untouched; below it, the effective pull is boosted to $g=\sqrt{g_N\,a_0}$ , which falls only as $1/r$ and makes rotation curves flat by necessity rather than by tuning a halo. From the baryons alone MOND predicts whole rotation curves — bumps, wiggles, and all — and forces the baryonic Tully–Fisher relation $M_b\propto v^4$ with almost no scatter. This article treats MOND as a sharp, falsifiable phenomenology beside dark matter: we derive its core law, work its numbers, and follow it to the galaxy clusters and cosmic microwave background where it strains, and where the gravitational-wave speed from GW170817 has already killed its first relativistic completion.

Key takeaways

Modified Newtonian Dynamics (MOND) leaves the baryons as the only matter and modifies gravity below a universal acceleration $a_0\approx1.2\times10^{-10}\,\mathrm{m\,s^{-2}}$ , boosting the effective pull to $g=\sqrt{g_N\,a_0}$ .
That single edit makes rotation curves asymptotically flat and forces the baryonic Tully–Fisher relation $v^4=G M_b\,a_0$ ; the modern radial-acceleration relation captures this with only $\sim$ 0.1 dex of scatter across 150+ galaxies.
Because the boost depends on total acceleration, MOND has a unique external-field effect that breaks the strong equivalence principle — a signature impossible for dark matter.
Galaxy clusters keep a factor $\sim2$ – $3$ mass excess even inside MOND, and the Bullet Cluster's offset lensing mass is its sharpest empirical wound.
Relativistic completions (TeVeS, aether-scalar-tensor) are needed for lensing and the CMB; GW170817 excluded the original TeVeS by pinning the gravitational-wave speed to $c$ .
Key idea: below a universal acceleration $a_0$ the effective gravity scales as $g\to\sqrt{g_N\,a_0}$ ; this one modification reproduces flat rotation curves and the Tully–Fisher relation from baryons alone, but does not straightforwardly extend to clusters and cosmology.

The core insight

Below a universal acceleration $a_0$ , the effective gravity scales as $g\to\sqrt{g_N\,a_0}$ ; this single modification reproduces flat rotation curves and the Tully–Fisher relation from baryons alone, but does not straightforwardly extend to clusters and cosmology.

Where this lives today

MOND survives as the most serious quantitative rival to dark matter at galaxy scales, and the modern battleground is precision. The SPARC database — 175 disk galaxies with Spitzer near-infrared photometry and resolved rotation curves — is where the radial-acceleration relation was pinned to $\sim0.1$ dex of scatter, the sharpest single test any theory of galaxies must pass. Gaia now probes the external-field effect directly in the outer Milky Way and in weakly bound "wide binary" stars, hunting for the sub-Newtonian boost of a broken equivalence principle. On the cosmological front, aether-scalar-tensor (AeST) gravity (Skordis and Złośnik, 2021) is engineered to reproduce the CMB acoustic peaks while keeping gravitational waves at $c$ , closing the loophole GW170817 opened under TeVeS. The Bullet Cluster still stands as the hardest evidence for genuinely collisionless mass, and no relativistic MOND has yet fit galaxies, clusters, lensing, and the CMB at once without some extra dark component — which is why the subject stays alive: a falsifiable programme with open verdicts, not a settled story.

The physical picture

Newtonian gravity is never in doubt where it has been tested. The Sun pulls on Earth at $\sim6\times10^{-3}\,\mathrm{m\,s^{-2}}$ , roughly eight orders of magnitude above $a_0$ , so every planet, binary star, and laboratory experiment lives deep in the regime where MOND and Newton agree exactly. Only in the thin, cold outskirts of galaxies — where the orbital acceleration $v^2/r$ has dropped below $a_0$ — does anything change. MOND's central claim is that this is not a coincidence but a law: rotation curves flatten at exactly the radius where the acceleration crosses $a_0$ . The first figure shows what that does to a galaxy's rotation curve.

Figure

Rotation curve of a disk galaxy: the Newtonian prediction from the visible baryons alone (blue) peaks and falls off like Keplerian r^−1/2, while the observed speed (amber points) stays flat near 180 km s⁻¹ — MOND boosts the baryonic curve to match once the acceleration drops below a₀

Where does the boost come from? Picture $a_0$ as a threshold in the relation between a source and the acceleration it produces. Above it, $g=g_N$ and everything is ordinary; below it the force law bends so that $g=\sqrt{g_N\,a_0}$ : a source that Newtonianly gives $g_N\propto1/r^2$ now yields $g\propto1/r$ , and the circular-orbit condition $v^2/r=g$ collapses to $v^2=\sqrt{G M_b\,a_0}$ , independent of radius. Flat curves are then inevitable, not fitted. And because the asymptotic speed depends only on the total baryonic mass and the universal $a_0$ , the baryonic Tully–Fisher relation $M_b\propto v^4$ drops out with no adjustable knobs — a relation dark matter must instead arrange through halo mass, concentration, and feedback. The cleanest modern statement of MOND's galactic success is the radial-acceleration relation (RAR): plot the observed centripetal acceleration $g_{\rm obs}=v^2/r$ against the acceleration $g_{\rm bar}$ the baryons alone would produce, point by point across thousands of measurements in hundreds of galaxies, and they collapse onto a single curve. At high acceleration it rides the Newtonian one-to-one line; below $a_0$ it peels upward exactly as $g\to\sqrt{g_{\rm bar}\,a_0}$ . The second figure is that relation — the empirical heart of the subject.

Figure

The radial-acceleration relation: observed acceleration g_obs versus the Newtonian baryonic acceleration g_bar, log–log. Above a₀ the data hug the one-to-one line g_obs = g_bar; below a₀ = 1.2×10⁻¹⁰ m s⁻² they bend up onto g_obs = √(g_bar a₀), with only about 0.1 dex of scatter across 150+ galaxies

The core equations

MOND is defined by one algebraic law and the constant it introduces; everything else is a consequence or a fitting form. The law states that the Newtonian acceleration computed from the baryons is not the acceleration a test particle feels — the two are related through an interpolating function of the ratio $g/a_0$ .

\mu\!\left(\frac{g}{a_0}\right) g = g_N = \frac{G\,M(<r)}{r^2} \qquad (1)

where: $g$ — the true acceleration a test particle experiences; $g_N$ — the Newtonian acceleration from the enclosed baryonic mass $M(<r)$ ; $a_0\approx1.2\times10^{-10}\,\mathrm{m\,s^{-2}}$ — the universal MOND acceleration scale; $\mu(x)$ — the interpolating function, with $\mu\to1$ for $x\gg1$ (Newtonian) and $\mu\to x$ for $x\ll1$ (deep-MOND). Reading it: when $g\gg a_0$ the function $\mu\to1$ and the equation is just Newton; when $g\ll a_0$ the left side becomes $g^2/a_0$ , so $g=\sqrt{g_N\,a_0}$ . The constant $a_0$ is astonishingly small, and sits suggestively close to cosmological scales — within a factor of order unity of $c H_0$ .

a_0 \approx 1.2\times10^{-10}\ \mathrm{m\,s^{-2}},\qquad 2\pi\,a_0 \approx c H_0 \sim c^2\sqrt{\Lambda/3} \qquad (2)

where: $c$ — the speed of light; $H_0\approx70\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ — the Hubble constant; $\Lambda$ — the cosmological constant. Numerically $c H_0\approx6.8\times10^{-10}\,\mathrm{m\,s^{-2}}$ , so $c H_0/2\pi\approx1.1\times10^{-10}$ , and $c^2\sqrt{\Lambda/3}\approx5\times10^{-10}$ — all within an order of magnitude of $a_0$ . MOND does not derive these coincidences; they merely hint that $a_0$ is set by the cosmic acceleration scale. The deep-MOND limit of equation (1), combined with the circular-orbit condition, yields the flagship prediction: a mass–velocity law with an exact slope of four.

v^4 = G\,M_b\,a_0 \qquad (3)

where: $v$ — the asymptotic flat rotation speed; $M_b$ — total baryonic mass (stars plus cold gas). This is the baryonic Tully–Fisher relation. Its slope is fixed at exactly 4 and its zero-point is fixed by $a_0$ alone, so MOND predicts a relation with no free parameters and essentially no intrinsic scatter. The transition between the two regimes is not derived from first principles; it is carried by a choice of $\mu(x)$ , of which two forms dominate the literature.

\mu(x)=\frac{x}{1+x}\ \text{(simple)},\qquad \mu(x)=\frac{x}{\sqrt{1+x^2}}\ \text{(standard)} \qquad (4)

where: $x=g/a_0$ — the dimensionless acceleration. Both satisfy the required limits $\mu\to1$ and $\mu\to x$ ; they differ only in how sharply gravity turns on near $a_0$ , and galaxy fits mildly prefer the "simple" form. Modern work often replaces $\mu$ entirely with a fitting function written directly in terms of the baryonic acceleration. That fitting function is the equation actually drawn through the radial-acceleration relation, and it interpolates smoothly from the Newtonian to the MOND branch with $a_0$ as its only scale.

g_{\rm obs} = \frac{g_N}{1-e^{-\sqrt{g_N/a_0}}} \qquad (5)

where: $g_{\rm obs}=v^2/r$ — the observed centripetal acceleration; $g_N$ — the baryonic acceleration from Poisson's equation. For $g_N\gg a_0$ the exponential vanishes and $g_{\rm obs}\to g_N$ ; for $g_N\ll a_0$ the denominator becomes $\sqrt{g_N/a_0}$ and $g_{\rm obs}\to\sqrt{g_N\,a_0}$ , recovering the deep-MOND law. The naive algebraic law (1) does not conserve momentum for a general mass distribution. AQUAL fixes this by deriving the force from a modified Poisson equation whose nonlinearity encodes the interpolating function.

\nabla\!\cdot\!\left[\mu\!\left(\frac{|\nabla\Phi|}{a_0}\right)\nabla\Phi\right] = 4\pi G\,\rho_b \qquad (6)

where: $\Phi$ — the MOND gravitational potential (with $g=-\nabla\Phi$ ); $\rho_b$ — the baryonic mass density. In spherical symmetry this reproduces the algebraic law exactly; away from it the two differ by a curl term. Because it follows from a Lagrangian, AQUAL conserves momentum, angular momentum, and energy. Finally, one number keeps MOND honest at the largest scales: applied to clusters, the theory does not fully close the mass budget.

\left.\frac{M_{\rm dyn}}{M_b}\right|_{\rm cluster}^{\rm MOND}\approx 2\text{–}3 \qquad (7)

where: $M_{\rm dyn}$ — the dynamical mass inferred from the hot X-ray gas or from lensing; $M_b$ — the observed baryonic mass. Even after the MOND boost, clusters demand a residual factor of two to three in unseen mass — MOND's central empirical difficulty.

Derivation AFlat rotation curves and the baryonic Tully–Fisher relation

The goal is to show that the deep-MOND limit of the force law makes a rotation curve asymptotically flat and simultaneously fixes the baryonic Tully–Fisher relation with a slope of exactly four and a zero-point set only by the universal constant $a_0$ — so both the flatness and the mass–velocity scaling are consequences of one modification, not per-galaxy fits. We take a test particle on a circular orbit far outside the baryons, where the enclosed mass has saturated to $M_b$ and the acceleration has fallen well below $a_0$ , and carry the algebra through to $v^4=G M_b\,a_0$ .

Step 1. Outside essentially all the baryons the mass distribution acts as a point of mass $M_b$ , so the Newtonian acceleration is $g_N=G M_b/r^2$ . Why: at large radius the detailed shape of the disk no longer matters — only the total enclosed baryonic mass does, which is exactly why the prediction depends on $M_b$ alone.

Step 2. At such radii $g\ll a_0$ , so the interpolating function sits in its low-acceleration branch, $\mu(x)\to x=g/a_0$ . Why: this is the defining assumption of MOND — the modification switches on precisely where the acceleration is sub- $a_0$ , and galaxy outskirts are the natural home of that regime.

Step 3. Substitute into the force law $\mu(g/a_0)\,g=g_N$ to get $(g/a_0)\,g=g_N$ , i.e. $g^2=g_N\,a_0$ , so $g=\sqrt{g_N\,a_0}$ . Why: squaring the acceleration on the left is what turns a $1/r^2$ source into a $1/r$ effective field — the whole flat-curve phenomenon lives in this one algebraic step.

Step 4. Insert $g_N=G M_b/r^2$ : $g=\sqrt{G M_b\,a_0}\,/\,r$ , which falls only as $1/r$ rather than $1/r^2$ . Why: an acceleration decaying as $1/r$ is exactly what a flat rotation curve requires, since $v^2/r$ must then be constant.

Step 5. Impose the circular-orbit condition $v^2/r=g$ : this gives $v^2/r=\sqrt{G M_b\,a_0}/r$ , and the radius cancels, leaving $v^2=\sqrt{G M_b\,a_0}$ . Why: the cancellation of $r$ is the mathematical statement that the rotation curve is asymptotically flat — no tuning, no halo, just the deep-MOND scaling.

Step 6. Square to obtain $v^4=G M_b\,a_0$ , the baryonic Tully–Fisher relation with slope exactly four. Why: because $a_0$ is universal the zero-point is fixed, so MOND predicts the relation and its tiny scatter, whereas dark matter must reproduce both by conspiracy. *As a numerical check: $v=180\,\mathrm{km\,s^{-1}}$ maps to $M_b\approx6.6\times10^{10}\,M_\odot$ for $a_0=1.2\times10^{-10}\,\mathrm{m\,s^{-2}}$ .*

Go deeper · Derivation B · 2 minThe external-field effect and the broken strong equivalence principle

Derivation BThe external-field effect and the broken strong equivalence principle

The aim here is to show that because MOND's boost depends on the total acceleration — not just the acceleration internal to a subsystem — a small self-gravitating system embedded in a larger galaxy's field behaves differently from an isolated one, so that a uniform external field bleeds into the internal dynamics and the strong equivalence principle fails. This external-field effect (EFE) has no analogue in Newtonian gravity or dark matter, where a uniform field only accelerates a subsystem bodily, making it one of MOND's sharpest fingerprints. We take a dwarf with weak internal field $g_{\rm int}$ deep inside a host's field $g_{\rm ext}$ , ordered as $g_{N,\rm int}\ll g_{\rm ext}\ll a_0$ .

Step 1. The interpolating function is evaluated at the total acceleration, and since the external field dominates, $\mu(|g_{\rm int}+g_{\rm ext}|/a_0)\approx\mu(g_{\rm ext}/a_0)$ . Why: MOND is nonlinear in acceleration, so the environment sets the operating point of $\mu$ — this is the algebraic root of the whole effect.

Step 2. Because $g_{\rm ext}\ll a_0$ , that value lies in the deep-MOND branch, $\mu(g_{\rm ext}/a_0)\approx g_{\rm ext}/a_0$ . Why: the host galaxy's outskirts are themselves sub- $a_0$ , so the dwarf is bathed in a low-acceleration field that pins $\mu$ to a small constant.

Step 3. The internal dynamics then obey $\mu\,g_{\rm int}^{\rm eff}=g_{N,\rm int}$ with $\mu$ frozen at $g_{\rm ext}/a_0$ , i.e. $(g_{\rm ext}/a_0)\,g_{\rm int}^{\rm eff}=g_{N,\rm int}$ . Why: with $\mu$ held constant by the environment, the internal force law is linear again — the dwarf is quasi-Newtonian even though it sits deep in the MOND regime.

Step 4. Solve for the effective internal field: $g_{\rm int}^{\rm eff}=(a_0/g_{\rm ext})\,g_{N,\rm int}$ . Why: the internal gravity is boosted over Newton by the constant factor $a_0/g_{\rm ext}$ — an effective gravitational constant $G_{\rm eff}=G\,a_0/g_{\rm ext}$ set entirely by the neighbourhood.

Step 5. The upshot: the dwarf shows no square-root MOND boost, only a rescaled $G$ , so a dwarf near a massive host displays weaker apparent dark matter than an identical isolated one — claimed as detected in gas-rich dwarfs and impossible in standard dark matter, where internal dynamics are shielded from a uniform external field. Why: this severs MOND from any dark-matter mimic, since internal kinematics come to depend on an object's gravitational environment, violating the strong equivalence principle. *The boost factor $a_0/g_{\rm ext}$ is exact in the stated ordering $g_{N,\rm int}\ll g_{\rm ext}\ll a_0$ .*

Go deeper · Derivation C · 1 minAQUAL — the Lagrangian that makes MOND conservative

Derivation CAQUAL — the Lagrangian that makes MOND conservative

The purpose of this derivation is to show why the algebraic law of equation (1) cannot be the whole story and how a modified, aquadratic Lagrangian repairs it: for a non-spherical mass distribution the naive law violates Newton's third law, so MOND must descend from a variational principle that guarantees conservation of momentum, angular momentum, and energy. We write the AQUAL Lagrangian, vary it, identify the interpolating function with its free function's derivative, and check that spherical symmetry hands back the algebraic law.

Step 1. Postulate the aquadratic Lagrangian density $\mathcal{L}=-\rho_b\Phi-\dfrac{a_0^2}{8\pi G}\,F\!\left(\dfrac{|\nabla\Phi|^2}{a_0^2}\right)$ , where $F$ is a free function replacing the Newtonian $\tfrac12|\nabla\Phi|^2$ . Why: making the kinetic term a general function of $|\nabla\Phi|^2$ rather than quadratic is the minimal way to build acceleration-dependence into the field theory — hence "aquadratic".

Step 2. Vary the action $\delta\!\int\mathcal{L}\,\mathrm{d}^3x=0$ with respect to $\Phi$ ; the Euler–Lagrange equation gives $\nabla\!\cdot\!\left[\mu(|\nabla\Phi|/a_0)\,\nabla\Phi\right]=4\pi G\,\rho_b$ with the identification $\mu(x)=F'(x^2)$ . Why: because the field equation follows from a variational principle, its solutions automatically conserve the Noether charges — this is precisely what the algebraic law lacked.

Step 3. In spherical symmetry the field equation integrates once over a sphere to give $\mu(g/a_0)\,g=G M(<r)/r^2$ , the algebraic law of equation (1); away from spherical symmetry the two differ only by a curl term. Why: this shows AQUAL is a genuine completion — it reproduces the simple law exactly where the simple law is trustworthy, and the external-field effect survives here as a rigorous consequence rather than a heuristic.

Limiting cases

The interpolating function $\mu(x)$ with $x=g/a_0$ organizes every regime, from the Solar System to the dwarf-galaxy environment.

Case	Result	Meaning
Newtonian regime	$g\gg a_0:\ \mu\to1,\ g=g_N$	High accelerations (Solar System, inner galaxies) recover Newton exactly; no deviation is observable.
Deep-MOND regime	$g\ll a_0:\ \mu\to x,\ g=\sqrt{g_N\,a_0}$	Low accelerations (galaxy outskirts, dwarf spheroidals) give the boosted $1/r$ field and flat rotation curves.
Asymptotic speed	$v_f=(G M_b\,a_0)^{1/4}$	Radius-independent; fixes the baryonic Tully–Fisher relation $M_b\propto v_f^4$ with zero predicted scatter.
External-field effect	$g_{N,\rm int}\ll g_{\rm ext}\ll a_0:\ g\approx g_N\,a_0/g_{\rm ext}$	A weak internal field inside a dominant external one is quasi-Newtonian with rescaled $G$ ; breaks the strong equivalence principle.
Cluster scale	$M_{\rm dyn}/M_b\approx2\text{–}3$ even in MOND	The boost is insufficient; residual unseen mass (neutrinos, missing baryons) is still required.

Worked examples

1 · From a rotation speed to a baryonic mass (confidence builder)

A gas-rich spiral has a flat rotation speed of $v=180\,\mathrm{km\,s^{-1}}$ . Using MOND alone, predict its total baryonic mass, and compare with what Newtonian dynamics would require if the speed held to $30\,\mathrm{kpc}$ .

Step 1. MOND fixes the mass through the baryonic Tully–Fisher relation $v^4=G M_b\,a_0$ , so $M_b=v^4/(G a_0)$ . With $v=1.8\times10^5\,\mathrm{m\,s^{-1}}$ , $v^4=(1.8\times10^5)^4=1.05\times10^{21}\,\mathrm{m^4\,s^{-4}}$ .

Step 2. Divide by $G a_0=(6.674\times10^{-11})(1.2\times10^{-10})=8.01\times10^{-21}$ : $M_b=1.05\times10^{21}/8.01\times10^{-21}=1.31\times10^{41}\,\mathrm{kg}$ . In solar masses, $M_b=1.31\times10^{41}/1.989\times10^{30}=\mathbf{6.6\times10^{10}\,M_\odot}$ — with no free parameter beyond the universal $a_0$ .

Step 3. For contrast, Newton with only this baryonic mass at $r=30\,\mathrm{kpc}=9.3\times10^{20}\,\mathrm{m}$ gives $v_N=\sqrt{G M_b/r}\approx\mathbf{97\,\mathrm{km\,s^{-1}}}$ , about half the observed speed — so Newton would demand a dark halo of roughly three to four times the baryonic mass to hold the gas at $180\,\mathrm{km\,s^{-1}}$ , while MOND supplies the same support from baryons alone.

2 · The MOND boost at a chosen radius

Take a point-mass galaxy of baryonic mass $M_b=5\times10^{10}\,M_\odot$ and ask what circular speed a test particle has at $r=20\,\mathrm{kpc}$ , using the radial-acceleration relation, equation (5). Compare with the Newtonian value.

Step 1. Compute the baryonic acceleration. With $M_b=5\times10^{10}\times1.989\times10^{30}=9.95\times10^{40}\,\mathrm{kg}$ and $r=20\times3.086\times10^{19}=6.17\times10^{20}\,\mathrm{m}$ , $g_N=G M_b/r^2=(6.674\times10^{-11})(9.95\times10^{40})/(6.17\times10^{20})^2=\mathbf{1.74\times10^{-11}\,m\,s^{-2}}$ . This is $g_N/a_0=0.145$ — well below one, so we are in the MOND regime.

Step 2. Apply the RAR fitting function. With $\sqrt{g_N/a_0}=0.381$ and $e^{-0.381}=0.683$ , the denominator is $1-0.683=0.317$ , so $g_{\rm obs}=1.74\times10^{-11}/0.317=\mathbf{5.50\times10^{-11}\,m\,s^{-2}}$ — a boost of about $3.2\times$ . (The pure deep-MOND estimate $\sqrt{g_N\,a_0}=4.57\times10^{-11}$ is a little lower, since $g_N$ is not yet far below $a_0$ .)

Step 3. Convert to a speed via $v=\sqrt{g_{\rm obs}\,r}=\sqrt{(5.50\times10^{-11})(6.17\times10^{20})}=\mathbf{184\,\mathrm{km\,s^{-1}}}$ . The Newtonian speed from the same baryons is $v_N=\sqrt{g_N\,r}=\mathbf{104\,\mathrm{km\,s^{-1}}}$ . MOND raises the rotation speed by $80\,\mathrm{km\,s^{-1}}$ at this radius — the discrepancy that dark matter would otherwise attribute to a halo. (As a coincidence check, $cH_0=6.8\times10^{-10}$ for $H_0=70\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ , so $cH_0/2\pi=1.08\times10^{-10}$ , within observational error of the fitted $a_0$ .)

Physics of the portrait

MOND is inseparable from the physicist who insisted, against the tide, that a single acceleration scale could stand in for an ocean of dark matter — and who then spent four decades sharpening that claim into testable law.

Origins · 1 minMordehai Milgrom and how this was discovered

Mordehai Milgrom

born 1946 · modified dynamics and the acceleration scale a₀

Mordehai (Moti) Milgrom, an Israeli physicist at the Weizmann Institute, published the founding idea of MOND in 1983 in three back-to-back papers in the Astrophysical Journal, when the dark-matter halo was already becoming orthodoxy. His insight was arithmetical before it was theoretical: the transition from Newtonian to anomalous behaviour in galaxies happens not at a characteristic length or mass but at a characteristic acceleration, always the same small number $a_0\approx1.2\times10^{-10}\,\mathrm{m\,s^{-2}}$ . From that one observation he built the modified force law, predicted flat rotation curves, and derived the baryonic Tully–Fisher relation and the external-field effect years before the data could test them. In 1984, with Jacob Bekenstein, he turned the heuristic law into the AQUAL field theory, giving MOND a conserving Lagrangian foundation. Milgrom has argued consistently that MOND should be judged as a predictive framework — one that anticipated the radial-acceleration relation — rather than dismissed for its troubles with clusters. Whether $a_0$ is a fundamental constant of a new dynamics or an emergent scale of dark-matter physics remains, in his framing, the question the data have not yet closed.

History

1983

— Milgrom proposes MOND in three papers in the Astrophysical Journal

1984

— Bekenstein and Milgrom build AQUAL, giving MOND a conserving Lagrangian

2004

— Bekenstein publishes TeVeS, the first workable relativistic MOND

2016

— McGaugh, Lelli and Schombert establish the radial-acceleration relation

2017

— GW170817 pins the gravitational-wave speed to c and excludes TeVeS

Going further

Texts. Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., §24 (dark matter and rotation curves) · Binney & Tremaine, Galactic Dynamics, 2nd ed., §2.6 and the modified-dynamics discussions · Mo, van den Bosch & White, Galaxy Formation and Evolution, Ch. 11 (disks and rotation curves) · Ryden, Introduction to Cosmology, 2nd ed., Ch. 8 · Dodelson, Modern Cosmology, 2nd ed. (structure growth and CMB constraints). Key numerical benchmarks: $cH_0\approx6.8\times10^{-10}\,\mathrm{m\,s^{-2}}$ and $cH_0/2\pi\approx1.1\times10^{-10}\,\mathrm{m\,s^{-2}}$ for $H_0=70\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ ; the deep-MOND relation $v^4=G M_b\,a_0$ gives $M_b=6.6\times10^{10}\,M_\odot$ at $v=180\,\mathrm{km\,s^{-1}}$ ; the RAR fitting function yields $g_{\rm obs}=5.5\times10^{-11}\,\mathrm{m\,s^{-2}}$ and $v=184\,\mathrm{km\,s^{-1}}$ at $r=20\,\mathrm{kpc}$ for $M_b=5\times10^{10}\,M_\odot$ ; the external-field boost factor is $a_0/g_{\rm ext}$ ; and $a_0=1.2\times10^{-10}\,\mathrm{m\,s^{-2}}$ is the standard fitted value.

Prerequisites. Galactic Rotation Curves · Evidence for Dark Matter in Galaxies · Galactic Dark Matter Halos and Density Profiles · Neutral Hydrogen 21-cm Line and Galactic Kinematics. Leads to. Galactic Dark Matter Halos and Density Profiles · Interstellar Medium of Galaxies · Hubble Sequence and Galaxy Morphology.

Ask the tutor: Ask me to walk through why the radius cancels in the deep-MOND limit, or to compute the baryonic mass your own galaxy's rotation speed implies.

Is MOND the same thing as "there is no dark matter"?

Not quite. MOND is the specific proposal that below the acceleration $a_0$ the relation between mass and gravity departs from Newton's, so baryons alone can explain flat rotation curves. It is more precise than a bare denial of dark matter: it makes numerical predictions — the Tully–Fisher slope, the radial-acceleration relation, the external-field effect — that dark matter does not automatically reproduce. It is also honest about its limits: at the cluster scale even MOND needs some extra unseen mass, and its cosmological versions may need a dark field. The cleaner statement is that MOND replaces dark matter at the galaxy scale with a modified law, while the largest scales remain contested.

How can MOND fit a rotation curve with essentially no free parameters?

Given a galaxy's near-infrared image and 21-cm gas map, you already know where the baryons are; multiplying the light by a stellar mass-to-light ratio (fixed within a narrow range by stellar-population models) gives the baryonic mass distribution. Feed that into the MOND force law and the entire rotation curve follows — including the small bumps where the light itself bumps, a pattern known as Renzo's rule. The only adjustable quantity is that mass-to-light ratio, constrained independently. Dark matter can fit the same curves, but needs a halo whose mass and concentration are extra parameters that must then be explained as correlating with the baryons.

What exactly is the external-field effect, and why is it a clean test?

In MOND the boost depends on the total acceleration a body feels, so a subsystem sitting inside a larger galaxy's field is influenced by that external field even though the field is uniform across it. A dwarf galaxy deep inside a host's gravity becomes quasi-Newtonian, with an effective $G$ rescaled by $a_0/g_{\rm ext}$ , and shows weaker apparent dark matter than an identical isolated dwarf. This is impossible in Newtonian gravity or dark matter, where a uniform external field only accelerates the whole subsystem and cannot reach its internal motions. Because it is a qualitative, yes-or-no difference — not a matter of fitting — the external-field effect is one of MOND's most falsifiable predictions, and it is being hunted in dwarfs and wide binaries.

Why do galaxy clusters break MOND?

Because the boost is not enough. Applying MOND to the hot X-ray gas and galaxies of a cluster still leaves the dynamical mass a factor of two to three above the observed baryonic mass, so some unseen matter — sterile or ordinary neutrinos, or missing baryons — is still required. The Bullet Cluster sharpens the problem: there the gravitational-lensing mass is offset from the X-ray gas and tracks the nearly collisionless galaxies, exactly as collisionless dark matter predicts and awkwardly for a theory in which gravity simply follows the baryons. Clusters are therefore MOND's central empirical difficulty, and no version of the theory has fully resolved them without conceding extra dark mass.

Why does MOND need a relativistic version at all?

The algebraic law and AQUAL are non-relativistic: they cannot describe how light bends, how the universe expands, or how the CMB acoustic peaks form, all of which need a covariant theory. Bekenstein's 2004 TeVeS supplied one and reproduced gravitational lensing, but it predicted gravitational waves travelling at a speed different from light — and GW170817's near-simultaneous arrival of gravitational waves and gamma rays constrained that difference to $\lesssim10^{-15}$ , killing the original TeVeS. Newer theories such as aether-scalar-tensor gravity keep $v_{\rm gw}=c$ and match the CMB, but whether any relativistic MOND can fit galaxies, clusters, lensing, and cosmology at once without a dark component is still open.

If MOND works so well for galaxies, why do most astronomers still favour dark matter?

Because dark matter succeeds across a far wider range of scales with one framework: the CMB acoustic peaks, large-scale structure growth, cluster lensing, and the Bullet Cluster all fall out of cold dark matter, while MOND handles galaxies beautifully but strains elsewhere and still lacks a fully successful relativistic form. The honest position is that MOND has uncovered real, precise regularities in galaxies — the tightness of the Tully–Fisher and radial-acceleration relations — that dark-matter simulations must work hard to reproduce. Many researchers therefore treat those regularities as clues any theory, including $\Lambda$ CDM, must explain, whether or not Milgrom's law is the final word.