1. Light paths and apparent positions

A foreground mass can alter a background source's light path, changing where its image appears. Treu's galaxy-lensing review describes β = θ − α in the thin-lens approximation. β is the source's unlensed angular position; θ is the image position.

Here α is the scaled deflection angle incorporating geometry, not simply the angle through which the ray bends near the lens. The review distinguishes α = (Dls/Ds)α̂. Dls and Ds are angular-diameter distances from lens to source and observer to source.

2. Rings versus arcs

An ideally centered source behind a circularly symmetric mass distribution produces an Einstein ring. θE is its angular radius, not diameter. Actual galaxies can have noncircular mass distributions, and sources have finite size; images can appear as arcs or multiple images. A missing complete circle does not mean missing lensing.

3. A ratio example within a simple model

For the review's singular isothermal sphere, θE = 4π(σ/c)²(Dls/Ds). σ is the model's velocity dispersion and c the speed of light, expressed in matching velocity units. The distance ratio is dimensionless and θE is in radians. This is a relation for a specified mass profile, not a universal galaxy-mass formula.

Hypothetically, with the same distance ratio, doubling σ makes θE four times larger. Seeing a ring four times larger does not by itself mean an actual galaxy has four times the mass. Check distances and the adopted mass distribution together.

4. What must be added to images?

Ruff and colleagues' SL2S paper combined lens models with stellar-dynamical analysis for 11 early-type galaxies. Redshifts and stellar velocity dispersions also entered the inference. Additional observations and assumptions lie between seeing a ring and separating stellar and dark-matter components.

Different mass models can reproduce images. The review's mass-sheet degeneracy and mass along the line of sight explain why matching image positions does not guarantee a unique mass solution. Read images, distance information, stellar kinematics and model assumptions separately.

5. Sources

Treu (2010), Strong Lensing by Galaxies: geometry and modeling. Ruff et al., arXiv:1008.3167: joint lensing/dynamics study.