  The two properties of the radial mass distribution of a gravitational lens
that are well-constrained by Einstein rings are the Einstein radius R_E and xi2
= R_E alpha''(R_E)/(1-kappa_E), where alpha''(R_E) and kappa_E are the second
derivative of the deflection profile and the convergence at R_E. However, if
there is a tight mathematical relationship between the radial mass profile and
the angular structure, as is true of ellipsoids, an Einstein ring can appear to
strongly distinguish radial mass distributions with the same xi2. This problem
is beautifully illustrated by the ellipsoidal models in Millon et al. (2019).
When using Einstein rings to constrain the radial mass distribution, the
angular structure of the models must contain all the degrees of freedom
expected in nature (e.g., external shear, different ellipticities for the stars
and the dark matter, modest deviations from elliptical structure, modest twists
of the axes, modest ellipticity gradients, etc.) that work to decouple the
radial and angular structure of the gravity. Models of Einstein rings with too
few angular degrees of freedom will lead to strongly biased likelihood
distinctions between radial mass distributions and very precise but inaccurate
estimates of H0 based on gravitational lens time delays.
