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Next: SYN projection equations in equatorial coordinates Up: No Title Previous: Introduction

Derivation of the SYN projection

From the basic synthesis equation, the phase term in the Fourier exponent is

phase = (e - e₀)^.B

(1)

where e and e₀ are the unit vectors pointing towards a point in the field and the field centre, B is a baseline vector, and we measure phase in rotations so that we don't need to carry factors of 2 $\pi$ . We can write

phase = p_uu + p_vv + p_ww

(2)

where (u, v, w) are components of the baseline vector in a coordinate system with the w-axis pointing from the geocentre towards the source and the u-axis lying in the J2000.0 equatorial plane, and

p_u	=	-cos $\displaystyle \theta$ sin $\displaystyle \phi$	(3)
p_v	=	-cos $\displaystyle \theta$ cos $\displaystyle \phi$
p_w	=	sin $\displaystyle \theta$ - 1

are the coordinates of (e - e₀), where ( $\phi$ , $\theta$ ) are the longitude and latitude of e in the (left-handed) native coordinate system of the projection with the pole towards e₀. Now, for a planar array we may write

n_uu + n_vv + n_ww = 0

(4)

where (n_u, n_v, n_w) are the direction cosines of the normal to the plane. Then

w = - $\displaystyle {\frac{n_u u + n_v v}{n_w}}$

(5)

Combining (2) and (5) we have

phase = [p_u - $\displaystyle {\frac{n_u}{n_w}}$ p_w]u + [p_v - $\displaystyle {\frac{n_v}{n_w}}$ p_w]v

(6)

From equations (3) and (6) the equations for the ``SYN'' projection for a planar synthesis array are thus

x	=	- [cos $\displaystyle \theta$ sin $\displaystyle \phi$ + p₁(sin $\displaystyle \theta$ - 1)]	(7)
y	=	- [cos $\displaystyle \theta$ cos $\displaystyle \phi$ + p₂(sin $\displaystyle \theta$ - 1)]

where

p₁	=	n_u/n_w	(8)
p₂	=	n_v/n_w

Next: SYN projection equations in equatorial coordinates Up: No Title Previous: Introduction