Aperture Multipole Moments from Weak

The projected mass of a gravitational lens inside (circular) apertures can be derived from the measured shear inside an annulus which is caused by the tidal eld of the deeecting mass distribution. Here we show that also the multipoles of the two-dimensional mass distribution can be derived from the shear in annuli. We derive several expressions for these mass multipole moments in terms of the shear, which allow large exibility in the choice of a radial weight function. In contrast to determining multipole moments from weak-lensing mass reconstructions, this approach allows to quantify the signal-to-noise ratio of the multipole moments directly from the observed galaxy ellipticities, and thus to estimate the signiicance of the multipole detection. Radial weight functions can therefore be chosen such as to optimize the signiicance of the detection given an assumed radial mass proole. Application of our formulae to numerically simulated clusters demonstrates that the quadrupole moment of realistic cluster models can be detected with high signal-to-noise ratio S=N ; in ' 85 per cent of the simulated cluster elds S=N > 3. We also show that the shear inside a circular annulus determines multipole moments inside and outside the annulus. This is relevant for clusters whose central region is too bright to allow the observation of the shear of background galaxies, or which extend beyond the CCD. We also generalize the aperture mass equation to the case of`radial' weight functions which are constant on arbitrarily-shaped curves which are not necessarily self-similar.