4.1. Redshift of the galaxy
The redshift of the lensing galaxy can in principle be constrained by its color if we know its morphology. Besides the surface-profile fits, further evidence that we are observing an elliptical galaxy in front of HE 2149-2745 comes from the optical spectra of QSO A: no Mg ii absorption system is detected down to an (observed) equivalent width limit of Å (). Given the small impact parameter ( kpc at ), this result almost excludes a disk-like galaxy in the foreground at .
The lower limit derived for the galaxy V magnitude (Table 1) puts a lower limit of . This is the color an E galaxy would have at a redshift of or larger (using spectral energy distributions observed at from Coleman, Wu & Weedman 1980). A bound consistent with this redshift is obtained from the color (Bressan, Chiosi & Fagotto 1994). The implied absolute luminosity for is (considering K-corrections, km s-1 Mpc-1, and ), i.e., very close to . An upper limit for is difficult to establish, but for the observed the expected luminosity beyond becomes too large to be real. We thus arrive to .
4.2. Lens models
Because of the symmetry of the mass distribution expected for any regular galaxy, the deflection angle and the amplification should be very similar for the two images of HE 2149-2745. This is also true if we include an external shear. To explain the flux ratio of 4.3, the dependence of the amplification on the positions has to be very strong. This can be achieved if the images are located near a critical curve, implying high amplifications.
Given the required sensitivity of the models for small changes of the positions and the small number of constraints, a maximum likelihood model fitting is not appropriate for this system; instead, we use an analytical approach to find the possible model parameters considering the measurement uncertainties.
We use a singular isothermal elliptical mass distribution (SIEMD) as given by Kassiola & Kovner (1993). As can be seen from Fig. 2, the images are almost exactly located on the major axis of the galaxy. To simplify the calculations, we use the line as the major axis and project the center of the galaxy onto this line. We further include an external shear , whose source has to be located on the major or minor axis to be in agreement with the observed image positions. As observational parameters, we use the ratio of distances of the images from the center of the galaxy (nearly unity) and the amplification ratio . In addition, we force the two images to have different parity, which is a necessary condition to exclude the existence of more than two images. Even non-singular models (PIEMD) rule out the possible splitting of A in the radial direction.
On the main axis, the lens equation and the amplification for the SIEMD model with external shear read
Since a degeneracy in the models prevents the independent determination of and , we use the two above equations to define the parameter E,
With our data, we get (). The uncertainty in E is dominated by the errors in the galaxy position. If the mass distribution has the same ellipticity as the light (), an external shear of is needed to keep E inside the bounds. Because of the absence of very close galaxies or a rich cluster in the field, we do not expect such a large shear. For an ellipticity of , the minimal shear decreases to . Fig. 4 shows the possible parameters consistent with the measured positions and the flux ratio. Even for different parity of the images some very symmetrical models lead to more than two images (up to eight images are possible for SIEMD+shear models).
Only rough estimates for the absolute amplifications can be determined from the observations. For a best-fit model, we get . Considering the errors, a lower limit for of 27 (68% confidence) can be obtained.
To estimate the mass and velocity dispersion of the galaxy, we use a spherical model (). For lens redshifts of , the mass inside the Einstein radius is , and the velocity dispersion (, ). The implied mass-to-light ratio is in solar units.
The expected order of magnitude for the time delay is about weeks. A better estimate must await more stringent constraints on the galaxy position. Given the geometry of the system, off-center spectroscopy of the galaxy should be possible from the ground under excellent seeing conditions, or with STIS onboard the HST .
© European Southern Observatory (ESO) 1998
Online publication: September 30, 1998