= Solution
A <galaxy cluster> deflects background <light> through its projected gravitational field. In the <thin-lens approximation for gravitational lensing>, $\boldsymbol\beta=\boldsymbol\theta-\boldsymbol\alpha(\boldsymbol\theta)$ maps image to source angle. Its <surface mass density> enters through
$$
\kappa=\frac{\Sigma}{\Sigma_{\rm crit}},\qquad
\Sigma_{\rm crit}=\frac{c^2D_s}{4\pi GD_lD_{ls}},\qquad
\mu=\frac1{(1-\kappa)^2-|\gamma_s|^2}.
$$
Here $\mu$ is signed <lensing magnification>; observed brightness uses $|\mu|$. The $D$'s are <angular diameter distances> to lens, source and between them; $\gamma_s$ denotes <lensing shear>, not a <Lorentz factor>. <strong gravitational lensing> creates arcs and multiple images near critical curves. For an axisymmetric lens, a tangential <Einstein ring> satisfies mean enclosed convergence one, giving
$$
\boxed{M_{\rm proj}(<D_l\theta_E)=\pi(D_l\theta_E)^2\Sigma_{\rm crit}}.
$$
This is projected enclosed <mass>, not a general three-dimensional <mass> profile inferred from a single arc.
<weak gravitational lensing> uses coherent distortions of many background galaxies. Their intrinsic orientations average approximately to zero, allowing an estimate of <reduced shear> $g=\gamma_s/(1-\kappa)$ and a projected <mass> map. The method traces all gravitating material without assuming <hydrostatic equilibrium>, which is valuable in merging clusters. Accurate work requires correction of the <point spread function>, calibrated shape measurement, source <redshifts>, foreground-member rejection and selection effects. Intrinsic shape noise, unrelated line-of-sight structures and lens-model degeneracies remain. The <mass-sheet degeneracy> leaves <reduced shear> unchanged under $\kappa\mapsto\lambda\kappa+(1-\lambda)$ and $\gamma_s\mapsto\lambda\gamma_s$, so absolute <mass> normalization needs additional information.
In the <Bullet Cluster>, <weak gravitational lensing> places the main projected <mass> peaks near the collisionless galaxies, displaced from the X-ray-emitting gas that contains most observed <baryon> <mass>. Comparing the lensing, optical and <X-ray> maps supports a dominant weakly collisional dark component under the usual gravity model; lensing alone does not identify its particle species. \b[Cluster lensing measures projected gravitational <mass> independently of gas <temperature> or dynamical equilibrium.]<A direct empirical proof of the existence of dark matter by Douglas Clowe and others>[Abstract and spatial comparison of weak-lensing mass, galaxies and X-ray gas]{ref}
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