The model fits yield an estimate of the axis ratio and position angle of each object, but it is useful to have model-independent measures of ellipticity. In the data released here, frames provides two further measures of ellipticity, one based on second moments, the other based on the ellipticity of a particular isophote. The model fits do correctly account for the effect of the seeing, while the methods presented here do not.
The first method measures flux-weighted second moments, defined as:
- Mxx = x^2/r^2
- Myy = y^2/r^2
- Mxy = x^y/r^2
In the case that the object's isophotes are self-similar ellipses, one can show:
- Q = Mxx - Myy = [(a-b)/(a+b)]cos2φ
- U = Mxy = [(a-b)/(a+b)]sin2φ
where a and b are the semi-major and semi-minor axes, and φ is the position angle. Q and U are Q and U in PhotoObj and are referred to as "Stokes parameters." They can be used to reconstruct the axis ratio and position angle, measured relative to row and column of the CCDs. This is equivalent to the normal definition of position angle (East of North), for the scans on the Equator. The performance of the Stokes parameters are not ideal at low S/N. For future data releases, frames will also output variants of the adaptive shape measures used in the weak lensing analysis of Fischer et al. (2000), which are closer to optimal measures of shape for small objects.Isophotal Quantities
A second measure of ellipticity is given by measuring the ellipticity of the 25 magnitudes per square arcsecond isophote (in all bands). In detail, frames measures the radius of a particular isophote as a function of angle and Fourier expands this function. It then extracts from the coefficients the centroid (isoRowC,isoColC), major and minor axis (isoA,isoB), position angle (isoPhi), and average radius of the isophote in question (Profile). Placeholders exist in the database for the errors on each of these quantities, but they are not currently calculated. It also reports the derivative of each of these quantities with respect to isophote level, necessary to recompute these quantities if the photometric calibration changes.Many more about those second order moments, and why they are called 'Stokes' parameters' can be found in references herein.