For an isolated approximately spherical mass distribution outside its luminous extent, retain the gravitational force $GMm/r^2$ and use the specified low-acceleration inertial law. circular motion then givesThe radius cancels when enclosed luminous mass is effectively constant. This is the asymptotic modified Newtonian dynamics result associated with the baryonic Tully-Fisher relation; an arbitrary extended disk cannot everywhere be replaced exactly by a point mass.

At the disk edge,Using $G=6.67\times 10^{-11}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$ givesThis is an order-of-magnitude isolated, low-acceleration estimate. The edge acceleration is below $a_0$, but not extremely small compared with it; a full interpolation law, disk geometry and an external gravitational field can affect a more precise mass fit.

For the spherical mass profile for a flat rotation curve, assume spherical symmetry where the halo dominates. A flat rotation curve gives $M(<r)=v^{2}r/G$, and differentiating $M(<r)=4\pi {\int }_0^r\rho (s)s^{2}ds$ yieldsHere $\rho$ is the density of the spherical gravitating model; it is not an exact deprojection of a thin luminous disk or a proof that the central density must diverge. At the outermost stated measured radius,This gives the requested galaxy-plus-halo mass estimate if the model is truncated at the outermost observed radius $2R$. The observation itself constrains enclosed mass within $2R$; any additional outer halo remains unconstrained. Extending $\rho \propto r^{-2}$ to infinity would give divergent mass. Subtracting the MOND-inferred luminous mass to obtain a Newtonian halo mass would mix independent model-dependent estimates without an independently measured luminous mass.

In the Bullet Cluster collision, most observed baryonic mass is hot gas. ram pressure slows and shocks the gas, while the galaxies pass through much more freely. weak gravitational lensing maps the projected gravitational mass without requiring that the gas be in hydrostatic equilibrium. The dominant lensing peaks lie closer to the galaxies than to the displaced X-ray gas. This supports a dominant weakly collisional dark matter component under the usual gravity model. It is stronger evidence than a mass-to-light discrepancy alone because distinct components separate during the same encounter. The inference needs background-source shape/redshift calibration, projection control and uncertainty in the lensing reconstruction; it does not identify a WIMP species or by itself exclude every modified-gravity theory.^[ref]

For dark-matter direct detection, consider XENON100 at Gran Sasso. A halo WIMP can undergo elastic scattering from a xenon atomic nucleus, leaving a recoil of only a few to tens of keV. The expected rate is very low, so a large target, low threshold and long exposure must be combined with background rejection. The nuclear-recoil kinematics of WIMP scattering gives $E_{R,\max }=2{\mu }_{\chi N}^2{v}^2/m_N$ in the nonrelativistic limit, where ${\mu }_{\chi N}$ is the reduced mass and masses and velocities use consistent units. This explains why low-energy response calibration matters.

The dual-phase xenon detector measures prompt particle scintillation S1 and delayed electroluminescence S2 from ionization electrons extracted into gas. Their time separation gives interaction depth; the S2 light pattern supplies transverse position. An inner fiducial volume uses xenon's self-shielding to suppress external backgrounds. The S2/S1 distribution statistically separates nuclear recoils from electron recoils; it is not perfect event-by-event identification. Underground siting reduces cosmic-ray backgrounds, while passive shielding, low-radioactivity components, xenon purification and a surrounding active detector veto reduce other events.^[ref]

Neutrons can also make nuclear recoils, so charge-to-light discrimination alone cannot eliminate them. Multiple-scatter rejection, shielding and background modelling remain necessary. Gamma-source and neutron-source calibrations map electron- and nuclear-recoil distributions, energy scale and selection acceptance. Position-dependent light collection and charge losses need correction; photon-counting fluctuations and uncertain low-energy yields limit threshold sensitivity. Blind event selection and independent sidebands guard against tuning cuts to apparent candidates. The 224.6-live-day, 34 kg search found two benchmark candidates against about one expected background event, consistent with background. Its interaction limit therefore depends on the assumed halo distribution, WIMP mass and nuclear response; it is not a detection or an exclusion of all dark matter.^[ref]

The original Feynman diagrams below distinguish elementary weak and electromagnetic vertices from an effective hadronic vertex. fermion arrows track fermion number; time runs left to right.