IP correction may already have been applied to the supplied Q and U map (for instance, within makemap or pol2map), in which case this routine will in effect determine a secondary IP correction that aims to remove any residual elevation-dependence left over by the earlier primary IP correction.

The IP model measured and applied by this routine can take one of two forms (selected using the MODELTYPE parameter). Firstly, the " single component" form:

p = A $+$ B$\ast$el $+$ C$\ast$el$\ast$el Qn_fp = p$\ast$cos( -2$\ast$( el - D ) ) Un_fp = p$\ast$sin( -2$\ast$( el - D ) )

where " el" is elevation (in radians), " p" is the fractional polarisation due to IP, (Qn_fp,Un_fp) are normalised Q and U corrections with respect to the focal plane Y axis, and (A,B,C,D) are the parameters of the model determined from the supplied input maps as described below.

Secondly, the " double component" form:

Qn_fp = A $+$ B$\ast$cos( -2$\ast$( el - C ) ) $+$ D$\ast$cos( -2$\ast$( el - E ) ) Un_fp = F $+$ B$\ast$sin( -2$\ast$( el - C ) ) $+$ D$\ast$sin( -2$\ast$( el - E ) )

This form has two extra free parameters (E and F) compared to the single component model.

For both model forms, the corrected output Q and U values are then given by:

Q_out = Q_in - Qn_tr$\ast$I_in U_out = U_in - Un_tr$\ast$I_in

where (I_in,Q_in,U_in) are the input I, Q and U values with respect to tracking north (e.g. Declination) and (Qn_tr,Un_tr) are the normalised Q and U corrections with respect to tracking north. These are determined by rotating the (Qn_fp,Un_fp) vector as follows:

Qn_tr = cos( 2$\ast$alpha )$\ast$Qn_fp - sin( 2$\ast$alpha )$\ast$Un_fp Un_tr = sin( 2$\ast$alpha )$\ast$Qn_fp $+$ cos( 2$\ast$alpha )$\ast$Un_fp

where alpha is the angle from tracking north to the focal plane Y axis, in sense of North through East, at the central epoch of the observation.

The (Q,U) values in the input maps are given with respect to tracking north. So if these maps had already been correct for IP with a pefect IP model, then in the absence of noise the (Q,U) of an astronomical source should be constant for all observations regardless of elevation (assuming the source is not variable). If the IP correction is not perfect, the (Q,U) measured in each observation will be offset away from the ’ true’ values by offsets that vary with elevation and azimuth. Thus if the (Q,U) values at a single point on the sky are plotted as a scatter plot, any imperfection in the IP model will be revealed by the points for different observations being distributed along an arc of some centro-symetric shape centred on the true (Q,U), with azimuth varying with distance along the arc. The above argument relies on the input (Q,U) values using tracking north as the reference direction. For instance, if they were instead to use the focal plane Y axis as the reference direction, then the true astronomical (Q,U) would not be constant (due to sky rotation) but would itself form some arc of a circle centred on the origin of the (Q,U) plane.

In practice, a separate circle is fitted to the input data at every point on the sky that is within both the AST and PCA masks. At a single point, each observation defines one position in the (Q,U) plane, and the best fitting circle passing through the (Q,U) positions for all observations is found. The centre of the circle defines the best estimate of the true astronomical (Q,U), and the offsets from the centre to each observation’ s (Q,U) position is a measure of the IP. Note, all reference to (Q,U) above refer to normalised (Q,U). The IP defined by each measured (Q,U) position is then rotated to use focal plane Y axis as the polarimetric reference direction instead of tracking north.

The above process, taken over all pixels in the AST and PCA masks, will typically produce many estimates of (Qn_fp,Un_fp) over a range of elevations. The parameters of the IP model (A,B,C,D) are then found by doing a weighted least squares fit to these (Qn_fp,Un_fp) values. The weighting function gives greater weight to pixels for which the residuals of the actual (Q,U) values from the fitted circle are smaller. It also favours circles in which the azimuth of each (Q,U) position is more closely correlated with its position around the circle.

IN = NDF (Read)

INLOGS = LITERAL (Read)

LOGFILE = LITERAL (Write)

MAPDIR = LITERAL (Write)

MODELFILE = LITERAL (Write)

MODELTYPE = LITERAL (Write)

OUT = NDF (Write)

• Single observation maps produced by skyloop can sometimes show IP that seems to vary with total intensity. In such cases, using a modified single component model in which:

p = A $+$ B$\ast$el $+$ C$\ast$el$\ast$el $+$ E$\ast$total_intensity q = p$\ast$cos( -2( el - D ) ) u = p$\ast$sin( -2( el - D ) )

results in corrected maps that show lower variation with elevation.