## In practice: Effect on Polaris observations

For reference, we first remind the observing strategy advised in the HERA manual. Two main points will help us explain the behavior on Polaris.

- The array is rotated by a given angle that ensures Nyquist sampling and optimal sky coverage in a single observation (i.e., each portion of the sky is observed only by one of the nine pixels during a scan). This requires two contiguous subscans.
- However, the observing strategy usually set up at Pico Veleta samples regularly the projected sky and not the sky itself as naive users tries to obtained rectangular projected maps. Moreover the HERA receiver array is a perfect square in focal plane and on the sky, but not in the projected map!

Under observer-friendly conditions, namely relatively small maps near the projection center and maps near the equator, these rules deliver the expected result. However, if those conditions are not met (large field of view and the source is located at high declination), the sky and the projected map will both be incorrectly sampled!

Maps near Polaris (declination 87:42:04.6) where observed during the project 219-07 (PI: P.Hily-Blant). The maps extend typically from (0,0) to (-1500,2000) arcsec (projected offsets). We focus here on the scans 26 and 40 covering the projection center, and scans 98 and 115 observed at the largest distance from the projection center.

Figure shows the scans observed near the projection center. They do not show any particular effect visible by eye. The array receiver is still square (rotated by 9.6 degrees3.6) on the projected map, and the coverage of the 2 scans is as expected. Figure shows the scan coverages in spherical coordinates (i.e., without projection). The spherical sky is correctly sampled.

Figure: Scans 26 (top) and 40 (bottom) relative coordinates (radio projection) of project 219-07 observing Polaris (declination 87:42:04.6). The observing strategy is the usual one, i.e. there were 2 OTF subscans (red: first subscan, black: second subscan), with a 9.6 degrees derotator angle, and a slight shift from one subscan to another to fill the gaps. These 2 scans cover the (0,0) reference position.

Figure: Same as Fig. (scans near the projection center) but showing absolute coordinates. In this case, the uniform mapping of the projected map results in uniform mapping of the sky.

On the other hand, the scans observed far away from the projection center are highly affected by the projection distortions. The projected receiver array is not square anymore. It is parallelogram-shaped in the projected map. As a consequence, the rows scanned in declination are shifted. Figure shows that the vertical scanning is not correctly sampled in the projected sky. This can be understood by looking at the scan in absolute coordinates (Fig. ): The scan does not follow a South-North direction suited for the 9.6 degrees derotator angle on the sky. To first order, the angle should have been different (the exact value is not computed here)3.7. On the other hand, the scan along the right ascension is correctly sampled, but the start and end points of the rows are actually shifted in projected coordinates compared to the scan 26 near the projection center. This leads to different edge effects in right ascension according to the declination of the source center.

Figure: Same as Fig. but showing scans 98 (top) and 115 (bottom) relative coordinates (radio projection). The same observing strategy is used. These 2 scans cover the (-1500,2000) offset position, far from the reference position. Note parallelogram shape of the receiver array in the projected map, and as a consequence the unexpected coverage of the vertical scan.

Figure: Same as Fig. (scans far from the projection center) but showing absolute coordinates. In this case, the attempt of uniform mapping of the projected map results in unexpected mapping of the sky. In particular, the 9.6 degrees derotator angle on the sky is obviously not suited here.