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The inference of dark matter uses a gravitational law to turn observed motions and lensing into a mass distribution. modified gravity asks whether some discrepancies instead reveal a breakdown of that law. The strongest motivation at galactic scales is the regular relation between baryonic structure and galaxy rotation curves, including approximately flat outer rotation speeds and the baryonic Tully-Fisher relation.

In modified Newtonian dynamics, the low-acceleration spherical relation is $g\mu (g/a_0)=g_N$, with $\mu (x)\to 1$ at high acceleration and $\mu (x)\to x$ at low acceleration. For an isolated baryonic mass $M_b$ in the latter regime,Thus an asymptotically flat rotation curve and a fourth-power mass–speed relation arise from a common acceleration scale, without assigning an independent halo profile to every galaxy. This is a substantive phenomenological motivation. Distances, stellar mass-to-light ratios, gas distributions and the external gravitational environment must still be controlled.^[ref]

A nonrelativistic acceleration prescription does not itself specify light bending or cosmology. A relativistic completion must determine the metric followed by photons and matter. tensor-vector-scalar gravity illustrates the need for additional dynamical fields and a specified physical metric; lensing predictions then have to be calculated within that theory rather than imported unchanged from general relativity.^[ref]

Several observations challenge a simple replacement of dark matter by a single galactic acceleration law. galaxy clusters retain substantial mass discrepancies even after such a modification. In the bullet Cluster dark-matter separation, the dominant X-ray-emitting baryonic gas is displaced from the galaxies, whereas weak-lensing mass concentrations lie nearer the galaxies. Collisionless dark matter accompanying the galaxies provides a natural account of this separation. The inference uses a lensing model and does not constitute a theorem excluding every conceivable modified-gravity theory, but a successful alternative must reproduce the displaced lensing signal quantitatively.^[ref]

The cosmic microwave background, acoustic structure, expansion history and growth of cosmic structure provide additional simultaneous tests. A theory that fits galaxy rotation curves must also supply the appropriate cosmological perturbation evolution and lensing. Adding extra clustering fields can help, but may reintroduce an effective unseen gravitating component. Solar-system and binary-pulsar agreement with general relativity constrain alternatives in their tested regimes, although a modification designed to act only at very low acceleration is not automatically ruled out by a high-acceleration test.

A discriminating programme compares dynamics, which probes the potential accelerating nonrelativistic matter, with lensing, which probes the sum of the two scalar metric potentials. Scale-dependent discrepancies, gravitational slip and environmental screening can then be tested with controlled galaxy bias and selection effects. Galaxy-scale tensions in a dark-matter model must likewise be separated from uncertain baryonic astrophysics. Regular galactic scaling relations motivate modified gravity; clusters and the combined cosmological evidence strongly challenge its simplest dark-matter-free forms. Failure to detect a particular WIMP is not positive evidence for a particular modified-gravity theory.

Three tracers give complementary estimates of a cluster's gravitating mass. First, galaxy redshifts measure a line-of-sight velocity dispersion ${\sigma }_{\mathrm{los}}$. For a roughly stationary bound cluster, the virial theorem gives a cluster velocity-dispersion mass estimatewith a coefficient depending on the density profile and orbital anisotropy. The galaxy luminosity and plausible stellar mass-to-light ratio supply too little mass to bind typical rich clusters. Foreground interlopers, substructure, velocity anisotropy and a recent merger can bias the dynamical estimate; a large dispersion is not by itself proof of equilibrium.

Second, diffuse X-ray emission measures a hot intracluster plasma, not merely unresolved starlight. Spectral continuum and lines constrain its temperature, while surface-brightness profiles constrain its density after line-of-sight projection and emissivity modeling. If the gas has constant composition and is approximately spherical and in hydrostatic equilibrium,This cluster hydrostatic mass estimator uses $\mu m_p$ as the mean mass per pressure-supporting particle. The inferred potential requires substantially more mass than the stars plus measured gas, although the gas contains much of the visible baryonic mass. Turbulence, bulk motion, temperature structure and departures from hydrostatic equilibrium are important systematic errors.

Third, strong-lensing arcs and multiple images, together with coherent weak-lensing shear of background galaxies, measure projected gravitational deflection. In general relativity,This cluster gravitational-lensing mass estimate does not require the cluster galaxies or gas to be in dynamical equilibrium. It does require source-distance information, a lens model and control of point-spread-function-induced shapes, line-of-sight projection, unrelated line-of-sight structure and the mass-sheet degeneracy.

The convergence of dynamical, hydrostatic and lensing masses, exceeding the baryonic inventory, is the principal evidence. The Bullet Cluster adds a spatial test: collisional X-ray gas lags behind the galaxies during the merger, while the inferred lensing concentrations lie nearer the collisionless galaxy component. An additional weakly collisional gravitating mass accounts naturally for that separation.^[ref] The conclusion assumes a gravitational framework and does not identify the microscopic dark-matter particle; the different tracer systematics should be evaluated rather than treating three mass estimates as completely assumption free.

Treat the gravitational field as approximately spherical. The circular orbit condition givesIf every star had a representative mass $m$ and obeyed $L/L_{\odot }=(m/M_{\odot }{)}^3$, its mass-to-light ratio would be $(m/M_{\odot }{)}^{-2}$, giving $\boxed{m\simeq 0.47M_{\odot }}$. This is a one-mass estimate, not an actual average inferred uniquely from the data: a galactic stellar mass function, stellar remnants, dust and nonstellar mass alter the relation. The flattening of the galactic disk also changes the spherical conversion by an order-one factor.

Assume an isotropic, stationary spherical galaxy cluster. Its three-dimensional velocity dispersion obeys $⟨v^2⟩=3{\sigma }_{\mathrm{los}}^2$. For a uniform sphere, the virial theorem gives $3M{\sigma }_{\mathrm{los}}^2=3G{M}^2/(5R)$, henceA different density profile changes the coefficient, so $10^{15}{M}_{\odot }$ is the meaningful scale. For isothermal X-ray gas in hydrostatic equilibrium,Using $\mu =0.6$ and a representative slope $-d\ln {\rho }_g/d\ln r=2$–$3$ at $1\mathrm{Mpc}$ gives $M_X\simeq (1.3$–$1.9)\times 10^{15}{M}_{\odot }$. The temperature and extent alone cannot fix this coefficient without a density profile. The total luminosity is $5\times 10^{12}{L}_{\odot }$, so the previously inferred mass-to-light ratio gives

Three physically distinct contributions are unseen dark matter, hot diffuse intracluster gas omitted from the summed stellar mass, and faint stars or stellar remnants missed by transferring the Milky Way mass-to-light ratio to the cluster. The first two are actual additional components; the third changes the inferred stellar budget. Departures from virial equilibrium, velocity anisotropy or extra nonthermal pressure can also bias the dynamical or X-ray estimates, although they need not remove the discrepancy.

Assume virial equilibrium, structurally homologous early-type galaxies, a fixed relation between the measured central velocity dispersion ${\sigma }_c$ and the global random stellar motions, and a constant total mass-to-light ratio ${\rm Y}$. The virial theorem gives $M=k_v{\sigma }_c^2{R}_e/G$, where the dimensionless structure factor $k_v$ is common to the sample. With ${\mathcal{I}}_e$ denoting mean linear surface brightness inside the effective radius, $L=2\pi R_e^2{\mathcal{I}}_e$. ThusThe usual observed fundamental plane of elliptical galaxies has a tilt relative to these exponents because mass-to-light ratio varies with stellar population, chemical composition, and dark matter fraction; structural and dynamical nonhomology can also change $k_v$.

Printed convention. The source places the logarithm of a value expressed in $\mathrm{mag}\mathrm{arcse}{\mathrm{c}}^{-2}$ in the empirical fit. This differs from the usual logarithm of linear surface brightness. If ${\mu }_e$ is the quoted magnitude surface brightness,A coefficient $-0.8$ on $\log {\mathcal{I}}_e$ therefore becomes $+0.32$ on ${\mu }_e$, with an adjusted intercept; it does not become $-0.8\log {\mu }_e$. For a concrete distinction, changing ${\mu }_e$ from 20 to 25 decreases linear surface brightness by a factor 100, so the linear-intensity fit changes $R_e$ by $100^{0.8}\simeq 39.8$, whereas the literal printed logarithm changes it by $(25/20{)}^{-0.8}\simeq 0.837$.

The following part evaluates the printed numerical fit literally and labels that convention. If the fit was intended for linear surface brightness, its calibration must be specified; a conditional result for using the same coefficients with linear units is also given. No equality between these two conventions is assumed.

Evaluate the source's printed logarithm using the numerical magnitude value 21.0, with base-ten logarithms:This is the literal printed-fit convention identified above. Convert the mean visual surface brightness to a linear luminosity per projected area. At 10 parsecs, one square arcsecond subtends $100/(206264.806{)}^2\mathrm{p}{\mathrm{c}}^2$. The definition of absolute magnitude consequently givesThe constant $21.572=5{\log }_{10}(206264.806)-5$ is the conversion between the angular and physical projected area. Since the effective radius encloses half the total luminosity,For the stipulated stellar population of Solar-type stars, approximate the stellar mass-to-light ratio by $M_{\odot }/L_{V,\odot }$. Summing 100 such galaxies then givesThis assumes negligible interstellar extinction and that the supplied surface brightness is an intrinsic or already corrected value; an observed cosmological surface brightness requires additional dimming and passband corrections before this conversion.

For comparison only, if the empirical coefficients and intercept were instead meant for ${\log }_{10}({\mathcal{I}}_e/[L_{V,\odot }\mathrm{p}{\mathrm{c}}^{-2}])$, substituting $140.9$ in that fit would give $R_e=1.09\mathrm{kpc}$, $L_V=1.06\times 10^9{L}_{V,\odot }$, and $M_{\ast ,\mathrm{cluster}}=1.06\times 10^{11}{M}_{\odot }$. Those are conditional alternative results, not an unannounced correction of the printed units.

Treat the gravitational field as approximately spherical. The circular orbit condition givesIf every star had a representative mass $m$ and obeyed $L/L_{\odot }=(m/M_{\odot }{)}^3$, its mass-to-light ratio would be $(m/M_{\odot }{)}^{-2}$, giving $\boxed{m\simeq 0.47M_{\odot }}$. This is a one-mass estimate, not an actual average inferred uniquely from the data: a galactic stellar mass function, stellar remnants, dust and nonstellar mass alter the relation. The flattening of the galactic disk also changes the spherical conversion by an order-one factor.

Assume an isotropic, stationary spherical galaxy cluster. Its three-dimensional velocity dispersion obeys $⟨v^2⟩=3{\sigma }_{\mathrm{los}}^2$. For a uniform sphere, the virial theorem gives $3M{\sigma }_{\mathrm{los}}^2=3G{M}^2/(5R)$, henceA different density profile changes the coefficient, so $10^{15}{M}_{\odot }$ is the meaningful scale. For isothermal X-ray gas in hydrostatic equilibrium,Using $\mu =0.6$ and a representative slope $-d\ln {\rho }_g/d\ln r=2$–$3$ at $1\mathrm{Mpc}$ gives $M_X\simeq (1.3$–$1.9)\times 10^{15}{M}_{\odot }$. The temperature and extent alone cannot fix this coefficient without a density profile. The total luminosity is $5\times 10^{12}{L}_{\odot }$, so the previously inferred mass-to-light ratio gives

Three physically distinct contributions are unseen dark matter, hot diffuse intracluster gas omitted from the summed stellar mass, and faint stars or stellar remnants missed by transferring the Milky Way mass-to-light ratio to the cluster. The first two are actual additional components; the third changes the inferred stellar budget. Departures from virial equilibrium, velocity anisotropy or extra nonthermal pressure can also bias the dynamical or X-ray estimates, although they need not remove the discrepancy.

Use the low-redshift Hubble law, neglecting peculiar velocity and cosmological distance corrections:The angular radius is $15$ arcseconds, half the stated major-axis diameter. A spherical dynamical estimate givesIt estimates mass inside the observed disk radius, not the entire unobserved halo. Disk geometry changes the exact coefficient.

Using the same approximate distance for the distance modulus,For an illustrative stellar mass-to-light ratio of $1$--$5$ in solar $V$-band units, $M_{\ast }\sim 1.5\times 10^{10}$--$7.4\times 10^{10}{M}_{\odot }$, below the dynamical estimate. This suggests substantial nonluminous matter under the adopted assumptions. Gas, stellar-population uncertainty, inclination, extinction and geometry also affect the comparison; luminosity alone cannot uniquely fix a stellar mass. The single-distance approximation at $z\simeq 0.05$ is the intended introductory estimate.

Use a Tully-Fisher relation $L\propto v_{\max }^4$. A simple justification assumes comparable mean surface brightness, so $L\propto R^2$, and comparable total mass-to-light ratio, so $M\propto L$. The virial scaling $v^2\sim GM/R$ then gives $v^4\propto M^2/R^2\propto L$. Calibrating to the previous galaxy yieldsThis is a rough luminosity estimate: the empirical slope depends on passband and population, and the assumed constant surface brightness and mass-to-light ratio are approximations rather than exact properties of every spiral galaxy.

The low-redshift distance gives modulus about 36.28, so $M_V\simeq -20.28$ and $\boxed{L_V\simeq 10^{10}{L}_{\odot }}$. A stellar mass-to-light ratio of order one to a few would imply only $10^{10}$–several $10^{10}{M}_{\odot }$ in stars, below the dynamical estimate. Gas and a dark component can supply additional mass; the precise stellar estimate requires a population-dependent mass-to-light ratio and extinction correction.

An exponential luminous disk has most of its mass within a few scale lengths. Outside that concentration a mass-traces-light model tends toward a declining, approximately Keplerian speed, not an extended flat curve. Adding a spatially extended mass distribution explains the mismatch more naturally than merely rescaling the same concentrated light profile.

At this small redshift, $D\simeq cz/H_0\simeq 20.0$ Mpc. The outer angular radius gives $R_{\mathrm{out}}\simeq D\theta \simeq 21.3$ kpc. Approximating the dynamical mass by a spherical equivalent,The supplied Tully-Fisher relation gives $M_V\simeq -22.49$, and hence $L_V\simeq 9.9\times 10^{10}{L}_{V,\odot }$. Treating the population as solar-like gives the estimate ${{\rm Y}}_V\simeq 1M_{\odot }/L_{V,\odot }$, soThe dynamical estimate exceeds that by about a factor of five; a massive extended dark-matter component is the natural interpretation. Solar-like spectra do not exactly fix the population mass-to-light ratio, and a disk is not spherical, but order-unity corrections do not remove this large outer mass excess under the stated estimate.

The effective radius is one tenth of the optical outer radius: $R_e\simeq 2.13$ kpc. A flat rotation speed then gives $M_{\mathrm{dyn}}(<R_e)\simeq 4.9\times 10^{10}{M}_{\odot }$. Half the luminosity, with the same mass-to-light ratio, implies $M_{\mathrm{luminous}}(<R_e)\simeq 4.9\times 10^{10}{M}_{\odot }$.The concentration of light into a small fraction of the optical radius suggests a high-surface-brightness, centrally concentrated disk galaxy, plausibly an early-type spiral or lenticular with a substantial bulge. The kinematic and photometric data do not uniquely distinguish those morphologies. Disk geometry, an unknown inclination correction, or a varying stellar mass-to-light ratio would alter the exact numerical inner comparison.

Let $I$ denote a characteristic surface brightness and ${\rm Y}=M/L$ the relevant dynamical mass-to-light ratio. With comparable profiles, $L\propto I{R}^2$ and $M\propto V^{2}R/G$. Eliminating $R$ givesIf both $I$ and ${\rm Y}$ are independent of speed, converting luminosity to magnitude gives $M_V=\text{constant}-10{\log }_{10}V$. The empirical coefficient $-8$ instead means $L\propto V^{3.2}$, and thereforeFor constant surface brightness this requires ${\rm Y}\propto V^{0.4}$; for constant mass-to-light ratio it requires $I\propto V^{0.8}$. A mixture of these changes, including varying stellar populations or dark-matter fractions, can give the same observed slope. The simple derivation assumes comparable characteristic radii and structural coefficients.

The integrated spectrum combines stars moving toward and away from us within the galaxy, broadening their absorption by Doppler shifts. Assuming instrumental and individual-stellar widths have been removed or are negligible, the line-of-sight velocity dispersion isThe angular half-light radius at the low-redshift distance is $R_e\simeq 2.35$ kpc. For a pressure-supported, approximately isotropic equilibrium spheroid with a standard structural virial coefficient of order five, useThe coefficient represents assumed density profile, aperture and orbital structure; a different reasonable mass estimator changes the result by an order-unity factor. Treating $D$ as a luminosity distance and using $D_A=D_L/(1+z{)}^2$ would reduce the radius and mass by about four percent here.

The distance modulus is about $34.93$. The magnitude within $R_e$ therefore corresponds to $M_{V,\mathrm{half}}\simeq -19.23$ andDoubling the half-light luminosity givesThis ratio is compatible with an old stellar population plus possible dark matter. It is larger than a solar-like mass-to-light ratio, but does not by itself isolate a unique dark-matter fraction: stellar age, initial mass function, anisotropy and the virial coefficient are not specified. Counting the quoted half-light magnitude as the total would incorrectly double the inferred ratio.

For a spherical nonrotating tracer with luminosity density $\lambda$, radial velocity dispersion ${\sigma }_r$, enclosed luminosity $L(r)$, and constant mass-to-light ratio ${\rm Y}$, let $q=\lambda {\sigma }_r^2$ and $\gamma =G{\rm Y}$. DefineThe Spherical Jeans equation and the projected second stellar velocity moment give $p=2A+R{A}^{\prime }$. To differentiate without a singular lower-end contribution, write $A(R)={\int }_0^{\infty }q(\sqrt{R^2+z^2})dz$. If $R^{2}A$ vanishes at both endpoints, ${\int }_0^{\infty }Rp(R)dR=0$. A finite outer aperture instead retains its boundary stress term.

A stellar remnant is the object left after a star finishes its principal nuclear-burning evolution, such as a white dwarf, neutron star or black hole. Stellar remnants retain mass while often contributing little optical light, affecting an old stellar population's mass-to-light ratio.

An empirical relation between the luminosity of a disk galaxy and its rotation speed. A simple virial and surface-brightness argument gives $L\propto V^4/({{\rm Y}}^{2}I)$, so changes in mass-to-light ratio or surface brightness change the slope.