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Field shifts

A phase shift may be applied to the visibility data at the time a map is synthesized in order to translate the field centre. If the phase shift applied to the visibilities is

phase shift = q_uu + q_vv + q_ww

(16)

where (q_u, q_v, q_w) is constant then equation (2) becomes

phase = (p_u - q_u)u + (p_v - q_v)v + (p_w - q_w)w

(17)

whence equation (6) becomes

phase = [(p_u - q_u) - p₁(p_w - q_w)]u + [(p_v - q_v) - p₂(p_w - q_w)]v

(18)

Equations (7) become

x	=	- [cos $\displaystyle \theta$ sin $\displaystyle \phi$ + p₁(sin $\displaystyle \theta$ - 1)] - [q_u - p₁q_w]	(19)
y	=	- [cos $\displaystyle \theta$ cos $\displaystyle \phi$ + p₂(sin $\displaystyle \theta$ - 1)] - [q_v - p₂q_w]

From which we see that the field centre is shifted by

$\displaystyle \Delta$ x	=	q_u - p₁q_w	(20)
$\displaystyle \Delta$ y	=	q_v - p₂q_w

The shift is applied to the coordinate reference pixel. For the SIN projection (p₁, p₂) = (0, 0) and the shift is just

$\displaystyle \Delta$ x	=	q_u	(21)
$\displaystyle \Delta$ y	=	q_v

For the NCP projection the shift is

$\displaystyle \Delta$ x	=	q_u	(22)
$\displaystyle \Delta$ y	=	q_v - q_wcot $\displaystyle \delta_{0}^{}$

In the general case the correction for precession, although small, applies systematically to the whole field. For a shift of 1^o at $\alpha_{0}^{}$ = $\alpha_{p}^{}$ , and $\delta_{0}^{}$ = 30^o the whole map is shifted by about 0".1 in declination.

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