Let us derive Eq. (3.4), first considering the case . We define and , so that , use Eq. (3.2) and begin with the -integration:
In the last step, we used that the integrand has one pole at where the corresponding residuum is ; the other pole at lies outside the unit circle and thus does not contribute. As a result we obtain for ,
Similarily, we derive for :
since the pole at lies again outside the unit circle; we thus find that for ,
Here we derive approximations for and for the case of non-critical clusters. This non-criticality implies that . We can expand the integral in (4.10), using
It is easily seen from (4.10) that for the following relation holds:
As it turns out, whereas the power series in (A2.2) converges, a Padé approximation behaves much better. From (A2.2) and (A2.3), we obtain for the lowest-order approximation (see e.g. Press et al. 1992, page 194 ff.)
and for the next higher order,
In Fig. 8, left panel, we compare with the approximations (A2.4) and (A2.5), and the expansion (A2.2) up to the term for a cluster with and the redshift distribution shown in Fig. 1 (i.e., for which ). We find that the approximations (A2.4) and (A2.5) are excellent for , good for , and clearly deteriorate for . As expected, approximation (A2.5) is always better than (A2.4). In Fig. 8, right panel, we compare with the approximations given above. We find qualitatively the same result, but the deviation of from the approximations (A2.4) and (A2.5) becomes important for smaller values of , , say. We note that these approximations become even better if the difference between mean redshift of the sources and the cluster redshift increases.
As a result, we obtain that the approximation (A2.4) for is sufficiently accurate over a wide range of (if approaches values close to 0.8, the cluster most likely is critical anyway). Therefore, one can use approximation (A2.4) for (most) non-critical clusters.
In the case of a non-critical cluster (or at positions of a critical cluster where ) the ratio
becomes according (4.10)
Therefore, for a fixed redshift distribution of the galaxies, the ratio R depends on only. If we identify the expectation values again with the local averages,
a comparison of (A3.2) with (A3.3) can in principle determine the local value of . In Fig. 9 we show R for a non-critical cluster as a function of (solid line) for the redshift distribution shown in Fig. 1 (). The following approximations for can be derived from (4.11) and the relations in Appendix 2:
good for , and
which is good for . We obtain Eq. (A3.4a) using the approximations (A2.4) for and . For Eq. (A3.4b) we use the approximations (A2.5) for and , and in the case of Eq. (A3.4c) we use the approximation (A2.5) for and the approximation (A2.4) for . We compare these approximations with in Fig. 9: for the chosen redshift distribution, (A3.4a) gives , approximation (A3.4b) is shown as the dotted line and is obviously a good approximation for small , whereas approximation (A3.4c) is shown as the dashed line and fits well up to . For higher surface mass densities the cluster is most probably critical and Eq. (A3.2) does not hold anyway, but has to be replaced according to Eq. (3.5); then, R becomes a function of and . Eq. (A3.4c) can be inverted to obtain as a function of the observable quantity R,
In Fig. 9 (right panel) we compare the surface mass density derived from Eq. (A3.5) with the true one and find indeed a good agreement for .
From Eq. (A3.5) one might conclude that can be directly obtained from observing without using any of the inversion techniques derived in the past. Unfortunately this is not the case because can not be determined precisely enough from local observations. To obtain a reliable estimate for a local surface mass density , one would need about 1000 galaxy images.
Thus, we try to derive the mean surface mass density by measuring : assuming a value for we perform the reconstruction according to (4.9) and derive from the mass and shear map the corresponding map of , which then depends on the assumed value for the mean mass density . Finally, we compare this with the measured , i.e., we search for which minimizes , averaged over the data-field . Again, we find that the mean surface mass density can not be determined in that way, because the local mean image ellipticities and can not be derived with sufficient accuracy given the observed number density of galaxy images even in very deep exposures.
We note that we tried to determine in various other ways from moments of image ellipticities: assuming a value for we performed the reconstruction and calculated from the mass and shear map the local expectation values of according to Eq. (3.5). Then, we compared that with the `measured' local means and minimized varying the value used for the reconstruction. All these attempts failed for resonable assumptions for the number density of galaxy images, showing that the image ellipticities provide not enough information to break the mass degeneracy in practice.
© European Southern Observatory (ESO) 1997
Online publication: July 3, 1998