Degeneracies and scaling relations in general power-law models for gravitational lenses

The time-delay in gravitational lenses can be used to derive the Hubble constant in a relatively simple way. The results of this method are less dependent on astrophysical assumptions than in many other methods. For systems with accurately measured positions and time-delays, the most important uncertainty is related to the mass model used. Simple parametric models like isothermal ellipsoidal mass distributions seem to provide consistent results with a reasonably small scatter when applied to several lens systems. We discuss a family of models with a separable radial power law and an arbitrary angular dependence for the potential   ψ = r ^β F ( θ )  . Isothermal potentials are a special case of these models with   β =1  . An additional external shear is used to take into account perturbations from other galaxies. Using a simple linear formalism for quadruple lenses, we can derive H ₀ as a function of the observables and the shear. If the latter is fixed, the result depends on the assumed power-law exponent according to   H ₀∝(2- β )/ β   . The effect of external shear is quantified by introducing a 'critical shear' γ _c as a measure for the amount of shear that changes the result significantly. The analysis shows that in the general case H ₀ and γ _c do not depend on the position of the lens galaxy. Spherical lens models with images close to the Einstein radius with fitted external shear differ by a factor of   β /2  from shearless models, leading to   H ₀∝2- β   in this case. We discuss these results and compare them with numerical models for a number of real lens systems.