Keywords: mrtcal-prog-manual

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## Description of the radio projection

The unprojection-reprojection approach to add the pixel offsets is more accurate than simple offset addition to projected position, in particular far away from the equator of the coordinate system. If we look at the radio projection (its standard FITS name is Global Sinusoidal, abbreviated into GLS) of the full sky shown in Fig. , we can see that one can expect strong distortions near the pole. Note that the distortions depend on the declination of the object and NOT on the declination of the projection center3.4. In other words, the plane of projection is not tangent to the projection center, but to the equator instead. This is directly linked to the mathematical formula of the radio projection

|  |  | (3.4)
|  |  | (3.5)

where () is the projection center, () the object absolute coordinates, and () its offsets in the projected map. The key point in this projection is that is a function of and not . Hence, the deformation for wide fields of view will depend both on the distance to the projection center and the source declination! That's why the radio projection is deprecated and IAU recommends to replace it by the Sanson-Flamsteed projection (abbreviated in SFL).

Figure: Radio projection of the full sky (right ascension from -12 to 12 hours, declination from -90 to 90 degrees), for four projection centers (marked with a blue cross) at 0 right ascension and 0, 30, 60, 90 degrees declination. The parallels (resp. the meridians) are spaced by 30 degrees (resp. 2 hours).

The distortions near the pole may thus have a non-negligible impact on observations. Figures to show the effect in different conditions. The geometry of the multi-beam array shows no visible distortions, even one degree away from the source position, when the source position is located on the equator. However, the distorsions of the array geometry increase with the distance to the projection center when the source is located at high declination. Moreover, for the same source coodinates, the deformation at high source declination depends on the right ascension of the projection center (cf. Fig. ), but they are independent of the projection center declination (cf. Fig. )!

Figure: Effects of the radio projection on observations with HERA for source centers at 4h, 12h, 20h Right Ascension and 0, 44, 88 degrees declination (blue crosses). In this case, the projection center is aligned on the source center. The green polygon shows the shape of a square degree field in sky spherical coordinates. In addition, the red parallelograms show the resulting shape of a square multi-beam array as a function of the distance from the source center. Finally, the parallels (resp. meridians), shown as black dashed lines, are separated by 0.5 degree (resp. 1 degree).

Figure: Same as Fig. , except that the projection center is located 15 degrees south the source center. The deformations are independent of the declination of the projection center!

Figure: Same as Fig. , except that the projection center is located 1 hour angle west from the source center. The deformations depend on the right ascension of the projection center.

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