6 Modelling the lens

The time delay measured for SBS 1520+530 can be used to infer an estimate of H₀, via a model of the total gravitational potential responsible for the lensing effect. This model shall take into account the main lensing galaxy and a small galaxy cluster located to the North-West of the quasar.

Our models, already described in Faure et al. (2002), assume that different mass components along the line of sight are Pseudo-Isothermal-Elliptical-Mass-Distribution (PIEMD, Kassiola & Kovner 1993; Kneib et al. 1993; Kneib et al. 1996). In addition to the observational constraints already available in Faure et al. (2002), we now also try to pin down the flux ratio between the two quasar components, and to use it in the models. This ratio is difficult to estimate for SBS 1520+530: it varies with wavelength, it has different values in the continuum and in the emission lines, and even varies within the emission lines. Microlensing shall be blamed, but one should also keep in mind that our spectra were not taken at parallactic angle, with a relatively narrow slit, and at large airmass. The values we derive are therefore only tentative: $m_{\rm B} - m_{\rm A} = 1.40$ mag (i.e., $F_{\rm A}/F_{\rm B}=3.6$) in the emission lines, and $m_{\rm B} - m_{\rm A} = 0.83$ mag (i.e., $F_{\rm A}/F_{\rm B}=2.15$) in the continuum. The latter value was measured from K-band images (Faure et al. 2002).

Table 1 presents two lens models, computed for the two flux ratios, and matching a time-delay of $130\pm3$ days. Model (L) considers only the main lensing galaxy L (see 1) with a measured ellipticity $\epsilon =0.50 \pm 0.2$and with a position angle ${\rm PA}=-23 \deg \pm 7\deg$ (see Faure et al. 2002). Model (L+M+C) includes the main lensing galaxy (L), galaxy M, located at 2.6 $^{\prime\prime}$ North-East from the quasar image A (see Fig. 1), and a galaxy cluster, centered at $\sim$1$^\prime$ to the North-West of the quasar.

Modeling SBS 1520+530 is possible with a single lensing galaxy (i.e., galaxy L), only if $m_{\rm B} - m_{\rm A} > 1.15$. Given the errors on the image flux ratio, such a model is plausible. Taking the emission line flux ratio (Col. 3 in Table 1) as the true one, because it is less affected by microlensing, we find $H_0 = 63 \pm 9~\rm km~s^{-1}~Mpc^{-1}$. The error, once the image flux ratio is fixed, comes mainly from the position angle of the lens and its ellipticity.

If the magnitude difference between the images is less than 1.15, SBS 1520+530 can not be modeled by using one single galaxy. The model in Col. 2 of Table 1, with galaxy M and an intervening galaxy cluster gives $52 \pm~8~\rm km~s^{-1}~Mpc^{-1}$.

Finally, changing the flux ratio does not change much the value of H₀, for a given lens model. The flux ratio is important in the choice of model (i.e., multiple or single lens) but does not alter much the fit once the model is given.