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Adaptive moments are the second moments of the object intensity, measured using a particular scheme designed to have near-optimal signal-to-noise ratio. Moments are measured using a radial weight function interactively adapted to the shape (ellipticity) and size of the object. This elliptical weight function has a signal-to-noise advantage over axially symmetric weight functions. In principle there is an optimal (in terms of signal-to-noise) radial shape for the weight function, which is related to the light profile of the object itself. In practice a Gaussian with size matched to that of the object is used, and is nearly optimal. Details can be found in Bernstein & Jarvis (2002).

The outputs included in the SDSS data release are the following:

1. The sum of the second moments in the CCD row and column direction:
mrr_cc = <col2> + <row2>
and its error mrr_cc_err.
The second moments are defined in the following way:
<col2>= sum[I(col,row) w(col,row) col2]/sum[I*w]
where I is the intensity of the object and w is the weight function.
2. The object radius, called size, which is just the square root of mrr_cc
3. The ellipticity (polarization) components:
me1 = <col2> - <row2>)/mrr_cc
me2 = 2.*<col*row>/mrr_cc

and square root of the components of the covariance matrix:
me1e1err = sqrt( Var(e1) )
me1e2err = sign(Covar(e1,e2))*sqrt( abs( Covar(e1,e2) ) )
me2e2err = sqrt( Var(e2) )

4. A fourth-order moment
mcr4 = <r4>/sigma4
where r2 = col2 + row2, and sigma is the size of the gaussian weight. No error is quoted on this quantity.
5. These quantities are also measured for the PSF, reconstructed at the position of the object. The names are the same with an appended _psf. No errors are quoted for PSF quantities. These PSF moments can be used to correct the object shapes for smearing due to seeing and PSF anisotropy. See Bernstein & Jarvis (2002) and Hirata & Seljak (2003) for details.