If there are multiple sources in the field of view, the observed response R'(t) would include the contributions of the several sources. (Here we choose the response function R' that is always positive, Fig. 2d). In the case of a rotating fringe pattern (Fig. 3), the responses of the different sources cancel or reinforce each other depending on the rotation angle of the earth. The modulation would go to zero if two sources (of equal intensities) were perchance modulating exactly out of phase with about the same frequency, and it could become very large a bit later when they are in phase. In general, the combined response for multiple sources would seem quite random with peaks of varying heights at irregular spacings.

### Bins on the sky

The shading method is a simple extension of the line-drawing method of Fig. 4. Instead of drawing lines at times of maximum response as in Fig. 4, the entire sinusoidal two-telescope point-source fringe pattern (i.e.,the two-telescope beam) is binned (shaded) onto a map of the sky as shown in Fig. 8a. This is done for each (small) time interval of the observation. The time intervals are of constant duration and are taken to be much shorter than the times between fringe maxima at a given bin.

Furthermore, the amplitude of each such sinusoidal shading is proportional to the measured antenna response in the time interval. This gives the most weight to the times when a source is at maximum visibility. The amplitude ("darkness") at a given sky bin is recorded as a number proportional to the value of this adjusted sinusoid at the bin. If the response is at peak, the sinusoidal shading would be pronounced, the analog of drawing one set of lines in Fig. 4. If the response is zero, no shading would occur.

Fringe pattern on sky

Fringe pattern on sky

Tangent plane

Shaded map (for one instant of time)

Sample sky bins

Tangent plane

Shaded map (for one instant of time)

Figure 7.8. Shading method of creating a sky map from interferometry data. For each time interval, each bin on the sky is assigned a value which is the product of the instantaneous value of the fringe pattern at the bin position and the instantaneous value of the actual response. (a) Sinusoidal shading pattern for one instant of time. Its amplitude will be large if a celestial source is on a visibility line. (b) Shading pattern for a single point source near the north celestial pole for a continuous 12-h observation as in Fig. 3. The central peak is located at the source position; cf. Fig. 4c. The random noise in a real observation would add irregularities to the map. The Bessel-function profile is sketched qualitatively.

### Sample sky bins

Figure 7.8. Shading method of creating a sky map from interferometry data. For each time interval, each bin on the sky is assigned a value which is the product of the instantaneous value of the fringe pattern at the bin position and the instantaneous value of the actual response. (a) Sinusoidal shading pattern for one instant of time. Its amplitude will be large if a celestial source is on a visibility line. (b) Shading pattern for a single point source near the north celestial pole for a continuous 12-h observation as in Fig. 3. The central peak is located at the source position; cf. Fig. 4c. The random noise in a real observation would add irregularities to the map. The Bessel-function profile is sketched qualitatively.

The contributions to each bin from all time intervals of the observation are then summed. The sinusoids from the many time intervals may have different orientations, amplitudes and phases. The resultant map of summed values is the desired representation of the sky.

In other words, at each instant of time, one paints onto the sky map the regions the telescopes can "see" proportionally to how much signal R'(t) is detected by the telescopes. When a source (or combination of sources) is most visible, the sinusoid peaks run across the source positions and they are binned with large amplitudes because R' is large. The sinusoids with maxima at the locations of the sources are thus given the most weight. In addition, stronger sources yield higher R' when they are visible, so their locations will consistently receive higher contributions. In this way, a sky map can be built up from the data. After superposition of the sinusoids for all the time intervals, the maximum values will lie at the source positions, and stronger and weaker sources will appear as such.

A 12-h observation of a single point source in the vicinity of the North Pole would, with such an analysis, yield circular shaded rings such as those of Fig. 8b. They are similar to a damped cosine function and are clearly a representation of the rings of our simple line-drawing method (Fig. 4c). This pattern is an example of a Bessel function. The rings are, as before, an artifact due to the imperfect filling of the Fourier plane.

For larger numbers of telescopes, each pair can be treated similarly. The shadings from all telescope pairs can be superimposed in a single set of bins. The final result of this process is a sky map that represents the data in an unbiased manner. It will nevertheless include signal, artifacts, and noise. Multiple bright sources in the sky would be evident, but each would appear with the artifacts characteristic of the observation, such as the rings in our two-telescope North-Pole case.

### Cross-correlation

Formally, in the shading method, the quantity that is summed for a given sky position a,8 at time t is the product of: (i) the fringe pattern on the sky (e.g., Fig. 8a), which is the interferometric response at time t to a unit point source at a,8, orfrom(14), RPS'(a,8,t) = 1 + cos 0 (a,8,t), and (ii) the detected power R'(t ).The latter could contain information about several different sources in the sky, including their strengths. This product is summed over all time intervals of the observation to obtain the final shading of the sky bin at a,8. The summation in integral form is

The integral (23) has the form of a cross-correlation function, seen before in (5.5). The product R'RPS' dt is the intensity one applies to the sky bin at a,8 at a given time, and the correlation C'(a,8) is the shading value at a,8 after summing all time intervals. One repeats the integral for each sky position a,8 and thus creates a sky map that shows the multiple sources in the beam with their respective strengths.

The function (23) can be understood to be a correlation as follows. The integral tests the extent to which the data function R'(t) correlates with, or matches, the expected response RPS'(a,8,t) for a source at position a,8. (The latter function is a "trial function". The correlation integral has a large value if the two functions R' and RPS' vary similarly in time and lesser values if the variations are uncorrelated.

If there is actually a source at the trial position a,8 and no other in the field, the data function R'(t) will vary exactly as the expected function RPS' for that position, and the correlation value will be larger than elsewhere in the field. Even if there are several sources in the field causing R' to be relatively complex, the correlation function will still recognize the similarities between the data and the trial functions at the appropriate positions a,8, and thus will produce large values at the several source positions.

(Cross-correlation function (7.23) for creating map)

One can see more clearly that similar functions in a correlation yield large values as follows. Subtract the time average values from the two functions to obtain new functions, R = R' — (R')av and RPS = RPS' — (RPS')av. These new functions will therefore have time averages of zero and the latter will have the form, from (14),

Substitute the above definitions into (23) and use the definition of a time average JR dt = Rav T, where T is the total integration interval, to obtain ft2

We thus find that

where C(a,8) is defined to be the unprimed correlation function based on the functions with zero averages, r t2

Expression (26) shows that maps based on the two correlation functions will be essentially identical except for a constant offset.

We can thus examine the behavior of (27) in lieu of (23). Suppose the two functions within the integral are identical; they will both be positive at the same times and negative at the same times because the average of each is zero. As noted regarding (5.5), their product will thus be positive at all times, and the integration (summation) over the entire observation will thus be large. If, instead, the functions are different in a random way, the products will vary randomly from positive to negative, and the integration will yield a value near zero.

The cross-correlation thus reveals the extent to which the observed function R' contains a signal RPS' indicative of a point source at sky bin a,8. The cross-correlation process thereby picks out the locations of the sources in the field of view, with amplitudes proportional to the source strength. This is the more conventional way to describe the equivalent shading approach. It has its own rationale, independent of the shading approach, even though the two are equivalent. If there are more than two telescopes, one would carry out a global cross-correlation operation that includes the contributions from each telescope pair.

### Equal weighting of time intervals

Finally, let us note a problem with this shading method as described here; it weights all time intervals equally. This can lead to uneven shading if some Fourier components are sampled more than others. Forexample, considerthe 12-hournorth-pole observation of Figs. 3 and 4c. Suppose that during half of this time, the earth turned only 1°. The fringes would continue to cross the source but the orientation (rotation angle) of the pattern would change negligibly. Then suppose that the earth rotated rapidly, so that it covered the remaining 179° in the next 6 hours. During the slow period, the shading method would pile up lots of wave-like patterns, all at nearly the same rotation angle, and during the fast period the waves would be spread out over a large range of angles.

Since one-half of the shadings derive from the slow period, the resultant map would show large-amplitude parallel sinusoidal ridges running through the concentric circles; the pattern of Fig. 8a would be superposed on that of Fig. 8b with equal weighting. This is indeed a fair representation of the data, but the large ridges would not represent the real sky very well. Rather, one might prefer to give equal weight to each wave orientation, even if one received more exposure than the others. The Fourier method described below does just this. One could also modify the cross-correlation method to take into account such an imbalance of exposures.

### Fourier analysis of sky brightness

The goal of our mapping has been to determine the intensity of the sky at every point (within, say, the 7' beam of our sample individual telescopes) where the brightness can vary arbitrarily from sky bin to sky bin. Several point sources in an otherwise empty sky are a limiting special case that is more easily solved. A continuous distribution can be described as a sum of Fourier sinusoidal (spatial) waveforms; the description of the brightness in these terms is known as Fourier analysis. We make use of complex numbers in this section. The reader may choose to read only the overview in the following subsection.

### Principle of aperture synthesis

Imagine that the brightness of the sky actually varies as a one-dimensional sinusoid that varies along the equator as a cosine wave, e.g., (1 + cos 0)/2, in the E-W direction, and with no variation in the N-S direction. It would resemble ocean waves frozen in time, like the waves of Fig. 8a. Let the spatial period of the brightness be 1.0''. In this case, the sky brightness would consist of only one Fourier spatial wavelength, &1 = 1.0''.

Let the fringe pattern of our antenna system have a similar spatial wavelength d& = 01 = 1.0''. This would be ideal for the detection of our artificial 1.0'' sinusoidal sky distribution. Consider the equatorial observation, Fig. 2. As the earth rotates, the sinusoidal fringe pattern scans eastward over the sinusoidal sky distribution, alternately yielding large and small responses as it comes into and out of phase with the sky distribution. The variation of the response R(t) would have large amplitude and would be sinusoidal, the same as for a point source.

Now let the brightness of the sky vary with a different period, say, 0.9". At any single instant of time, it would be out of phase with the 1.0" fringe pattern at some sky positions and in phase at others. As the earth rotates, the situation would remain the same in that, at any fixed time, some positions on the sky would still be in phase and others out of phase. Thus the response R(t) would not modulate as the earth rotates. Our 1.0" fringe pattern would not detect this 0.9" distribution.

Thus, we find that our 1.0" fringe pattern will preferentially detect a 1.0" sky distribution and not other spatial wavelengths. In fact, our pattern will select the 1.0" component of the sky distribution even if the sky brightness contains many other spatial wavelengths. Knowledge of the other spatial wavelengths of the sky brightness can be obtained only if other antenna spacings are used to yield the matching fringe periodicities. For example, a larger projected antenna spacing b could yield a 0.9" fringe pattern. This would detect a 0.9" spatial component of sky brightness. The amplitude of the response R (t) at each wavelength reveals the strength, or amplitude, of the corresponding component of sky brightness.

According to Fourier theory, any arbitrary function can be synthesized from a sum of sinusoidal waves with the appropriate amplitudes and phases. If these Fourier components are known, the arbitrary function can be constructed from the component sinusoids. In our case, a measurement with a given b yields one of the needed Fourier components. A complete set of b to some maximum resolution, with appropriate periodicities and orientations, can be used to construct a map of the surface brightness distribution of the sky.

The case of a single point-like source is an interesting special case. As long as the antenna fringe pattern has wavelength larger than the angular size of the source, a large response will be obtained, as in Fig. 2. This tells us that a true point source (a deltafunction of sky brightness) consists of all spatial Fourier wavelengths. A real source with (small) angular extent would be represented by all spatial wavelengths that are longer than the angular size of the source.

It follows that a single equatorial measurement (Fig. 2) would not distinguish between a point source and an extended wave of sky brightness. To be assured that the sky brightness truly consists only of a single small source rather than an extended sinusoid, one would have to measure all these spatial wavelengths with different baselines and would expect to detect fringes (modulation) at each of them. If modulation were found at only one of them, as in our equatorial case, one would be forced to conclude that the sky brightness distribution is a sinusoidal brightness wave as just described!

Similarly, a single polar (rotating) observation (Fig. 3) would not distinguish between a single point source and a bright spot surrounded by sinusoidal-like

(Bessel-function) rings of decreasing brightness (Fig. 8b). Measurements with different baselines would be required.

### Arbitrary sky brightness distribution

The response functions illustrated heretofore were for point sources only. Here we present the response to an extended source of arbitrary brightness distribution. Let the unit source vector s refer to the direction of the phase center (tangent point of the Fourier plane, Fig. 7b). The expression (21) gives the response RPS(t ) to apoint source at that position. Now define any other nearby sky position with the vector sum s + a. The vector a gives the offset from the phase center in radians. It is plotted on the tangent plane of Fig. 7b with position components x, y measured in angle (rad) in the E-W (right ascension) and N-S (declination) directions, respectively. We have also described the u ,v Fourier plane as a similar tangent plane, but it is a different mathematical space where the u,v coordinates of b/X are plotted in units of rad-1 (actually cycles/rad).

Let the specific intensity or brightness of the source as a function of position on the sky be designated I (a) (W m-2 sr-1). This quantity is power received onto 1 m2 per unit solid angle from element da by a telescope centered on position a of the source. (We introduce specific intensity in Section 8.4.) The response of the interferometer to a two-dimensional solid-angle element da of the source may be represented as the product of the unit point-source response, from (21), and I (a) da.

Integration of the product over the entire source region then yields the total response R to the extended source. One must carry out the integration over the entire beam of the individual telescopes, to > 7' in our earlier example. Thus, where B is the telescope separation (baseline) in meters and X is the wavelength of the radio-frequency radiation. (Radio astronomers often define B as the dimen-sionless B/X, but we choose not to do so.) This defines the magnitude of R(t) in terms of the actual sky brightness I. Note that da is the solid angle of the element (rad2 = sr) which we have indicated with d^ in other contexts (Sections 3.3 and 8.4).

Distortion due to the tangent-plane approximations is small for small beam sizes (usually < 1°), and a is nearly perpendicular to s in this approximation. Since b is by definition the component of B perpendicular to s and since a is nearly

(Response, extended source; W/m2)

perpendicular to s, we have the approximation

B • (s + a) = B • s + B • a « B • s + b • a Substitute into (28),

2n 2n

In inertial space, the baseline B and its projection b vary with time as the earth carries the two telescopes to different orientations. (In the earth frame of reference, it is the direction to the phase center s(t) that varies with time.) Expand the cosine function and suppress (but do not forget) the time dependence of B and b,

J be

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