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Gravitational lensing constrains a galaxy cluster's projected mass independently of its light output or dynamical equilibrium. The lensing convergence is projected surface mass density divided by ${\Sigma }_{\mathrm{crit}}=c^2{D}_s/(4\pi G{D}_d{D}_{ds})$, using the appropriate angular diameter distances. Strong gravitational lensing constrains the central mass through multiple-image geometry, and weak gravitational lensing constrains larger-scale projected mass through coherent image distortions. Converting these observations into a three-dimensional mass needs geometrical modelling and the relevant lensing degeneracies.

The three probes constrain the same deep gravitational potential by different observations. A large member-galaxy velocity dispersion requires a large binding mass. Applying the virial theorem, with a gravitational scale radius $R$ and approximately isotropic orbits, gives a cluster velocity-dispersion mass estimate of order $M\sim R{\sigma }_v^2/G$, with a profile-dependent numerical coefficient. The inferred mass-to-light ratio is much larger than the stellar value. Membership, orbital anisotropy and equilibrium assumptions affect the precision.

The hot intracluster medium radiates through thermal bremsstrahlung and line emission. Its X-ray spectrum measures temperature, while its surface brightness constrains gas density. Confining this gas requires a potential with characteristic $GM/R\sim k_{B}T/(\mu m_p)$. Combining resolved profiles through the cluster hydrostatic mass estimator gives the total mass, not just the gas mass. The detected gas contributes substantial baryonic mass but does not normally account for the full gravitating mass. Nonthermal pressure or a merging cluster can invalidate a purely thermal hydrostatic equilibrium estimate.

Finally, gravitational lensing measures mass through light deflection. Strong gravitational lensing supplies multiple images and arcs; weak gravitational lensing measures coherent background-image distortion over larger radii. The lensing convergence is projected surface mass density divided by the critical surface density, irrespective of whether the material emits light. Cluster gravitational-lensing mass estimates agree broadly with the high masses implied by galaxy motions and hot gas, while having different equilibrium assumptions. Together these observations support a dominant nonluminous component in galaxy clusters, after accounting for their stars and observed gas.

It is useful to rewrite the local lensing convergence asFor positive core radius, the requested limits areA singular isothermal sphere lens instead has $\kappa (\theta )={\theta }_E/(2|\theta |)$, which diverges at the centre. The finite core of the softened isothermal lensing potential regularizes this central lensing convergence; at large angular radius the two models have the same leading profile.

For an axisymmetric gravitational lens, the reduced deflection is $\alpha (\theta )=\bar{\kappa }(\theta )\theta$. The specified mean lensing convergence therefore givesThe derivative is smallest at the centre, where ${\beta }^{\prime }(0)=1-{\theta }_E/{\theta }_c$. If ${\theta }_c\ge {\theta }_E$, the mapping is monotone and an offset source has only one image. If $0<{\theta }_c<{\theta }_E$, the derivative is negative near zero and positive at large radius, producing two stationary points and a range of source offsets with three images. Thus the condition for a nonempty multiple-image region isThis condition permits multiple images; a particular source must additionally lie inside the multiple-image caustic of a softened isothermal lensing potential. At equality the critical radii collapse to the origin and there is no finite three-image region.

Put $r=|\mathbf{\theta }|$ and write the vector thin gravitational lens equation as $\mathbf{\beta }=(1-\bar{\kappa }(r))\mathbf{\theta }$. A tangential displacement at fixed radius has eigenvalue ${\lambda }_t=1-\bar{\kappa }$. A radial displacement has eigenvalue ${\lambda }_r=1-{\alpha }^{\prime }(r)$. From the definition of the mean lensing convergence,Since $\alpha =r\bar{\kappa }$, this gives ${\alpha }^{\prime }=2\kappa -\bar{\kappa }$. Consequently,The two requested gravitational-lensing critical curve conditions are thereforeThey correspond respectively to a tangential critical curve of an axisymmetric lens and a radial critical curve of an axisymmetric lens.

The tangential condition $\bar{\kappa }=1$ gives $\sqrt{{\theta }_t^2+{\theta }_c^2}={\theta }_E$, henceThis is a nonzero gravitational-lensing critical curve precisely when ${\theta }_E>{\theta }_c$. It is also the radius of the Einstein ring for an aligned source in this cored model, so the strength parameter ${\theta }_E$ itself is not the cored model's ring radius.

For completeness the radial condition is equally explicit. Subtracting the mean lensing convergence from twice the local one givesThus the radial critical curve of an axisymmetric lens has radiusagain nonzero exactly when ${\theta }_E>{\theta }_c$.

Choose a signed angular coordinate along the source-lens line. The singular isothermal sphere lens equation isFor $\beta >0$, the positive image is ${\theta }_{+}=\beta +{\theta }_E$. A negative image would require ${\theta }_{-}=\beta -{\theta }_E<0$, which is possible only for $\beta <{\theta }_E$. Here $\beta =4$ arcmin and ${\theta }_E=1$ arcmin, so there is only one image, at $\theta =5$ arcmin. Formally using the negative-image formula would produce a positive value, inconsistent with the branch from which it was obtained.

The paper defines its vector deflection as source minus image angle, ${\mathbf{\alpha }}_{\mathrm{paper}}=(\beta -\theta )\hat{\mathbf{\theta }}$. This is the negative of the usual reduced deflection in $\mathbf{\beta }=\mathbf{\theta }-{\mathbf{\alpha }}_{\mathrm{standard}}$. Therefore its relation to lensing convergence isFor the SIS, ${\mathbf{\alpha }}_{\mathrm{paper}}=-{\theta }_E\hat{\mathbf{\theta }}$. The two-dimensional polar-coordinate divergence givesUsing a plus sign with the printed inward deflection would incorrectly give negative convergence.

For a small intrinsically circular source, differentiate the thin gravitational lens equation. The radial and tangential source-image Jacobian matrix eigenvalues areThus the image is stretched tangentially around the lens centre, with no radial stretch in this idealized model. At $\theta =5$ arcmin, ${\lambda }_t=4/5$; the image is an ellipse with tangential-to-radial axis ratio $5/4$. It is not a complete ring. This local description assumes a source small enough that the Jacobian is nearly constant across it.

The signed lensing magnification isSince the lensing shear magnitude equals $\kappa$, it reduces to $\mu =(1-2\kappa {)}^{-1}$. At the image, $\kappa ={\theta }_E/(2\theta )=1/10$, henceThe image has positive parity and is $25\%$ brighter for the same intrinsic source. Gravitational lensing preserves surface brightness, so this flux amplification follows from its larger apparent solid angle. In evaluating the convergence, use the image angle $5$ arcmin, not the source angle $4$ arcmin.

Deflection changes one light ray's direction; focusing concerns relative directions of neighboring light rays and their bundle's cross section. To see the difference for a point-mass gravitational lens, send parallel incident light rays past the lens. A ray at impact parameter $b$ meets the optical axis approximately $f(b)=b/\hat{\alpha }=c^2{b}^2/(4GM)$ beyond the lens. Doubling $b$ quadruples this crossing distance. Thus the lens does not have a common focal length for all rays, although it can produce lensing magnification and an Einstein ring for a specified source-observer geometry.

The point-mass gravitational lens also illustrates why focusing must not be identified just with positive lensing convergence. Away from the central mass, differentiate the reduced deflection $\mathbf{\alpha }={\theta }_E^2\mathbf{\theta }/r^2$: its two-dimensional divergence is zero, so $\kappa =0$ there. Nevertheless its radial and tangential derivatives differ, giving lensing shear of magnitude ${\theta }_E^2/r^2$ and $\det A=1-{\theta }_E^4/r^4$. A vacuum ray bundle can be distorted and magnified. In the same ideal transparent geometrical optics model, surface brightness is preserved; the larger apparent solid angle supplies the increased received light.

The local and mean lensing convergence obey $2\kappa =2\bar{\kappa }+r{\bar{\kappa }}^{\prime }$. Hence the radial derivative of the lens mapping is $1-{\alpha }^{\prime }(r)=1-2\kappa +\bar{\kappa }$. Its zeros are radial gravitational-lensing critical curves, where radial stretching diverges and images merge on crossing the mapped gravitational-lensing caustic.

The combination of lensing shear and lensing convergence governing the local shape distortion of background sources. Galaxy ellipticities estimate it statistically under a specified ellipticity convention and intrinsic-orientation model. It is invariant under the mass-sheet degeneracy, so shape information alone does not generally fix absolute projected mass normalization.

The projected density $\Sigma (R)={\sigma }^2/(2GR)$ gives cylindrical mass $\pi {\sigma }^{2}R/G$, so the reduced deflection magnitude is the constant displayed angular Einstein radius. The signed thin gravitational lens equation is $\beta =\theta -{\theta }_E\mathrm{sgn}\theta$. A source with $0<\beta <{\theta }_E$ has two images; one with $\beta >{\theta }_E$ has one. The lensing convergence and lensing shear magnitude are both ${\theta }_E/(2|\theta |)$.

This circular lensing potential has reduced deflection $\alpha (r)={\theta }_{E}r/\sqrt{r^2+{\theta }_c^2}$, mean lensing convergence $\bar{\kappa }={\theta }_E/\sqrt{r^2+{\theta }_c^2}$ and local convergence $\kappa ={\theta }_E({\theta }_c^2+r^2/2)/(r^2+{\theta }_c^2{)}^{3/2}$. Its central convergence is finite for ${\theta }_c>0$. As the core shrinks to zero, the deflection tends to that of a singular isothermal sphere lens. This softening of the projected lensing potential must not be confused with replacing a three-dimensional density by a different cored profile.