Jaw

(Visibility)

where b will usually vary with time as the earth rotates. The units of X, b and a are, respectively, m, m, and the dimensionless "rad". Thus, introducing k = 2n/X, the observational response (32) becomes m

Again, the exponential is the unit point source response for a point source at s. The visibility V (b) is seen to be the amplitude of R (t), the measured oscillatory response to I (a) at the projected fringe spacing b. It also contains the phase adjustment required to give the correct response R(t) for the actual sky distribution I (a). It is thus a complex number.

For a sufficiently brief observation, the fringe spacing and direction (or equiva-lently b) changes very little while the changing orientation 9 of B in inertial space (Fig. 7a) still leads to fringes translating across the source. The visibility function

V (b) thus is usually a slowly varying function. A brief measurement of R(t) might detect the passage of thousands of fringes at some (nearly) fixed b. An exception is an observation very, very close to the celestial pole, where the changing orientation of the fringes, and hence of b and V (b), can dominate the effect of fringe translation.

As stated, the exponential exp[ikB(t) • s] in (34) describes the frequency and phase of the fringe oscillations of a hypothetical point source of unit intensity at phase center in terms of the changing path-length difference B(t) • s (Fig. 7a), while V (b) is a complex multiplier that contains the phase offset and amplitude that derives from the source structure. Since the vectors B and s are known (in principle) at any given instant, a measurement of R(t), both phase and amplitude, determines V(b) for that particular time, or equivalently for that particular b. Knowledge of

V (b) for many different b allows one to construct an image of the source region. They are the Fourier components of the source structure.

In the Cartesian coordinates of the tangent plane (Fig. 7b), the components of a are x, y, and, in the associated Fourier plane, the components of b/X are u,v. Thus, one can expand the dot product in (33),

V(u,v) = I(x,y) exp[i2n(ux + vy)]dx dy (Visibility (7.35)

The components u and v are expressed as inverse radians while x and y are given in radians; the argument of the exponential is thus appropriately dimensionless. The function V (u,v) is another form of the visibility function.

Phase of visibility function Let us examine the phase information contained in the visibility function V (b). Consider the exponential in (33). Recall that the projected vector b is directed normal to the fringes and that b/X is the number of fringe cycles per radian (spatial frequency). Thus, b • a/X in (33) is the number of fringe cycles separating the phase center and the position a (see Fig. 7b). Multiply by 2n to obtain kb • a, the phase offset in radians between the two positions; this is the argument of the function)

exponential in (33). The real part of this exponential (the only part of concern to us) is cos(kb • a).

As an example, let the argument kb • a be a multiple of 2n radians at some instant of time for some element da at a. The cosine function is thus unity, and

V (b) simply contributes the amplitude I (a) da to the phase-center point response in (34) for this particular a; the visibility function contributes no phase shift. This is expected because the fringes cross the two positions (phase center and da) exactly in phase with each other (compare to Fig. 7b). The actual response R(t) due to this element, in this case, is the same (in phase) as that of the phase center. A unique case of this is when both positions are simultaneously viewed by the same fringe, namely when kb • a = 0, or b±a.

If, on the other hand, a source element is such that the phase center and source element give out of phase responses, we have kb • a = n; the cosine becomes -1. In this case, V(b) would shift the phase-center response by 180° to give the actual response R(t ) to this element.

For a (fixed) value of b, the integration (33) will yield a visibility of the form

V (b) = I0exp[ikb • aeff] where aeff is an effective coordinate produced by the integration over the extended source. Thus V (b) is seen to contain an amplitude I0 and a phase angle, kb • aeff, for that value of b. Multiplication by the phase-center response, exp [ikB(t) • s], yields, according to (34), the actual response R(t) for the extended source,

Here we see explicitly that the argument of the exponential in V (b) becomes a phase shift in the actual response; it is added to the phase angle at phase center in the argument of the exponential. In our examples above, the phase shifts kb • aeff were 0 radians and n radians respectively. This discussion illustrates how V (b) provides the phase and amplitude needed to correct the phase-centered response to the actual response for each value of b.

Sky brightness

The measured visibility V (b) is thus a Fourier amplitude/phase that allows us to reconstruct the sky brightness I (a). Those familiar with Fourier theory will recognize that (33) is the Fourier transform of the brightness distribution I (a). It can be shown (not here) that the expression can be inverted to obtain the sky brightness distribution, m

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