B

(Angle between directions of constructive (7.1)

(a) Constructive (b) Destructive (c) Constructive , interference interference interference /

(a) Constructive (b) Destructive (c) Constructive , interference interference interference /

Figure 7.1. Two telescopes on the earth's equator, east-west of one another, receiving a plane wave from a distant source at different angles relative to the "baseline" connecting the telescopes. The signals are added (or multiplied) to yield a single output signal. (a) and (c) Constructive interference yielding a strong signal. (b) Destructive interference yielding, ideally, no signal. As the earth rotates, carrying the telescopes to different orientations, the output signal from a single source will alternately be strong and weak.

Figure 7.1. Two telescopes on the earth's equator, east-west of one another, receiving a plane wave from a distant source at different angles relative to the "baseline" connecting the telescopes. The signals are added (or multiplied) to yield a single output signal. (a) and (c) Constructive interference yielding a strong signal. (b) Destructive interference yielding, ideally, no signal. As the earth rotates, carrying the telescopes to different orientations, the output signal from a single source will alternately be strong and weak.

There are thus many directions of constructive interference from which the signal of a detected source could arrive and be detected. If the observer does not know, a priori, the location of the source, he or she can infer only that a source detected at maximum strength could lie in any one of these many directions. The intervening directions, e.g., at destructive interference, are excluded as possible source positions.

The rotation of the earth plays a key role here. It continuously reorients the telescopes and causes the directions of constructive/destructive interference to scan across a celestial radio source. This causes the amplitude of the summed radio-frequency oscillations to vary between a maximum value and zero as the angle 9 of the source increases uniformly; see "signal out" in Figs. 1a,b,c. This oscillation, or modulation, due to interference is at a much lower frequency than the radio frequency œ. The modulation allows one to identify the times of maximum signal when the source is in a direction of constructive interference.

The observation of Fig. 1 yields position information only in the left-right direction. The two telescopes can not locate a source in the other direction (in and out of the paper) because a source displaced a modest amount in this direction would exhibit maxima at the same times as the overhead source.

The determination of celestial source positions requires knowledge of the precise location of the telescopes relative to the stars at the times of the maxima. This is provided by knowledge of the geographic locations of the telescopes and the precise times of the occurrence of the maxima. The precise times specify the orientation of the earth.

This modulation of the signal between maxima and minima as the earth rotates also provides information about the angular size of the source. If the source is larger than the angle between the directions of strong reception, AO = X/ B, different parts of the source will simultaneously span the directions of constructive and destructive interference. Thus the signal will modulate very little (if at all) as the earth rotates. In other words, the presence of strong modulation of the signal indicates that the angular size AOs of the source is less than about AO, i.e., AOs < X/B.

Further positional and size information is obtained with telescope pairs of different spacings Bt and with different orientations, e.g., if the baseline runs in and out of the paper in Fig. 1. The baseline of a pair of telescopes is usually defined as a vector B with magnitude equal to the telescope spacing and with direction of the line joining the two telescopes. It is defined in inertial space, and it changes orientation as the earth rotation carries the telescopes to different relative positions in inertial space. Thus, B is continually changing. An array of telescopes provides many two-telescope vector baselines B simultaneously, and the rotation of the earth reorients them providing even more baselines during the course of a day. The information can yield high quality maps of the sky.

Equatorial observation

We now examine in detail a specific case, namely observations of an overhead source by two telescopes that lie relatively close together and east-west of one another on the earth's equator (Fig. 2a). We first consider that the two antennas are transmitting, rather than receiving in order to understand the telescope beam. Then we return to the (astronomical) receiving case and find the nature of the positional information provided by the observations.

Transmission of radiation Let the two equatorial telescopes in Fig. 2a each transmit purely sinusoidal waves with the same frequency in the direction approximately normal to their baseline. The two electromagnetic waves will interfere with one another as do the waves in a ripple tank experiment wherein two adjacent probes disturb the water surface with an oscillatory motion at a given frequency. An observer at the probes or telescopes finds that there are view directions of constructive interference (solid lines with arrows in Fig. 2a) alternating with directions of destructive interference. In the constructive directions, the phases of the two waves are always in phase. The

(a) Interference bands (b) Fan beams of 100% visibility

(a) Interference bands (b) Fan beams of 100% visibility

Figure 7.2. Radio interferometry with two telescopes on the earth's equator observing a region of the sky directly overhead on the celestial equator. (a) Directions of constructive (solid lines) and destructive (dashed lines) interference. At the time shown, the radiating point source A would be visible whereas point source A' would not. Extended source C would be only partially visible. (b) Fan beams of constructive interference. (c) Fringe pattern on sky showing (solid) lines of 100% visibility and dashed lines of invisibility. As the earth rotates the pattern translates toward the east (left). (d) Power received from sources A and C as the earth rotates, known as the response R'(t). The rapid (radio-frequency) electromagnetic oscillations of the radiation are illustrated (at greatly reduced frequencies) for the first peak. The fringe period of 67 ms is for a baseline of 10 km; see text. (e) Two-dimensional Fourier (u,v) plane showing (dot) the "spatial" frequency component measured in this experiment. If the observation were to be repeated 4 times during the next 6 hours, the points marked " x " at decreasing spatial frequencies would be sampled.

Figure 7.2. Radio interferometry with two telescopes on the earth's equator observing a region of the sky directly overhead on the celestial equator. (a) Directions of constructive (solid lines) and destructive (dashed lines) interference. At the time shown, the radiating point source A would be visible whereas point source A' would not. Extended source C would be only partially visible. (b) Fan beams of constructive interference. (c) Fringe pattern on sky showing (solid) lines of 100% visibility and dashed lines of invisibility. As the earth rotates the pattern translates toward the east (left). (d) Power received from sources A and C as the earth rotates, known as the response R'(t). The rapid (radio-frequency) electromagnetic oscillations of the radiation are illustrated (at greatly reduced frequencies) for the first peak. The fringe period of 67 ms is for a baseline of 10 km; see text. (e) Two-dimensional Fourier (u,v) plane showing (dot) the "spatial" frequency component measured in this experiment. If the observation were to be repeated 4 times during the next 6 hours, the points marked " x " at decreasing spatial frequencies would be sampled.

system thereby broadcasts a large power in these directions. No power is radiated in the directions midway between them; dashed lines in Fig. 2a.

In three dimensions the directions of constructive interference are fan beams (Fig. 2b). Each "fan" represents a plane of directions of constructive interference. These intercept the celestial sphere to give visibility lines, the solid vertical lines in Fig. 2c, also known as a fringe pattern. The visibility lines are circles on the celestial sphere that approximate great circles for directions nearly normal to the baseline. (A great circle on a sphere lies in the plane containing the center of the sphere, like the earth's equator or the meridian lines of longitude. The plane of a small circle does not intersect the center, e.g., the earth's latitude parallels.) As we learn below, the fan or fringe pattern actually is not a set of discrete lines, but rather is sinusoidal in the direction normal to the lines of maxima. This sinusoidal multiple-fan pattern is in effect the beam of the two-telescope system.

For directions approximately overhead, the angular spacing AO between the directions of constructive interference is, from (1), AO = X/B .If the telescopes are separated by B = 10 km and are broadcasting at frequency v = 6 GHz, the wavelength is X = c/v = 50 mm, and the angular spacing is AO = 5 x 10-6 rad = 1". If the telescopes are on different continents, B ~104 km, the spacing becomes AO = 0.001"

These spacings are very small compared to the beam size Obeam of one of the individual telescopes. If the dish diameter is d = 25 m,

The two telescopes would broadcast into this relatively wide 7' beam of directions, but this beam would contain ~ 420 cycles of the 1'' interference pattern (for B = 10 km) or 420 000 cycles of the 0.001'' pattern (for B = 104 km). The times for one fringe cycle to pass over a given celestial position due to the earth's rotation are 67 ms and 67 |s respectively. The period of the 6-GHz electromagnetic wave is much shorter; P = 1/v = 1.6 x 10-10 s = 0.16 ns. The 67 ms fringe period for B = 10 km is shown in Fig. 2d.

Now, consider the reception of radiation by the E-W equatorial telescopes. Let point-like source A (Fig. 2a) emit abroad band of radio frequencies, and let the two telescope receivers be tuned to the same single frequency. Since source A happens to lie in a zone of constructive interference in the figure, a wavefront from it will arrive at the two telescopes in phase. If the electric vectors of the electromagnetic waves from the two telescopes are summed, the result will oscillate at the radio frequency with a large amplitude. This is just the situation shown in Fig. 1a.

Obi ieam ieam

Reception

If, on the other hand, the point-like source of the radiation is in a direction of destructive interference (e.g. A'), the summed telescope signals will add to zero as in Fig. 1b. In intermediate regions, an intermediate signal would be obtained. An extended source C in general would have a greatly reduced, or zero, response depending upon the exact distribution of source brightness on the sky.

The lines of visibility (constructive interference) on the sky are again those shown in Fig. 2c; they run north-south (Fig. 2b). As the earth rotates, the pattern moves along the sky in the eastward direction (arrows), and source A passes from visibility to invisibility and back again with a sinusoidal response of period 67 ms or 67 |s for the cases given above. The corresponding frequencies are 15 Hz and 15 kHz. The 6 GHz radio frequency is greater by factors of more than 108 and 105 respectively.

Earth rotation

The response R'(t) in Fig. 2d is a plot of the power (amplitude squared) of the summed electric vectors of the electromagnetic signals received from source A at the two detectors as a function of time as the earth rotates. A low-pass filter averages out the very rapid radio-frequency oscillations shown in the first peak. (The prime in R' is used here to reserve R for the case where the average power is subtracted from the response curve.)

Here we have considered only a brief observation near the zenith. If another observation had occurred several hours later, the telescopes would have been carried by the rotating earth to a new orientation relative to the source. This results in a reduced effective baseline (baseline projected normal to the source direction). In turn, this yields an increased spacing of the visibility lines according to (1). As we demonstrate below, this additional set of lines helps refine our knowledge of the positions and angular sizes of the sources in the field.

The earth's rotation is thus doubly helpful: it causes the fringe pattern to pass over the source to yield the oscillatory detection that indicates the presence of a source and helps locate its position (Fig. 2d), and it reorients the telescopes so the spacing of the fringe pattern changes, which refines the results. (Figure 2e will be discussed later.)

Position of source

How is the response R'(t) used to provide information about the location of a point source of unknown celestial position? Knowledge of the physical locations of the telescopes in inertial space (derived from their location on the earth and the angle of the earth's rotation) allows one to locate precisely the positions of the visibility lines on the sky at any given time. One can plot these lines on the sky at an instant when the response of a point source is at one of its maxima.

Figure 2c is such a plot. The solid lines of position represent the positions upon which the source must lie plus or minus the uncertainty. If the signal is quite strong, the time of the visibility maximum could be quite precisely known, and the uncertainty in position quite small, say ±2% of the line spacing. In this case, the position of the source would be restricted to about 4% of the fringe period. Nevertheless, it could still lie on, or close to, any one of the ~420 visibility lines (for B = 10 km) within the overall 7' extent of our hypothetical beam.

If the observation of our point source were repeated after several hours, as suggested above, the new visibility lines with their greater spacing would be another set of lines of position upon one of which the source must lie. Clearly the source can lie only at positions that agree with both sets of lines, i.e., where the lines overlap within their uncertainties. This greatly restricts the possible locations of the source. Similarly, the telescopes could be relocated so that they lie, say, northeast/southwest of one another. The lines of position on the sky would then run normal to this direction (northwest/southeast). The intersections of these lines with the N-S lines would further restrict the possible positions of the source.

North-Pole observation

The fringe patterns on the sky can rotate as well as translate due to the earth's rotation. The rotation is best illustrated with two telescopes placed near the North Pole and viewing a region encompassing the north celestial pole (NCP; Fig. 3a). In our sketch, the baseline and telescope beam are not centered directly on the NCP. The fringes will rotate on the celestial sphere as the earth rotates. In the frame of reference of the earth, the source follows a circular path centered on the NCP and completes a cycle in 24 h (Fig. 3b). As the source passes through the fringe pattern, the response R '(t) will oscillate rapidly for part of the circle and slowly at others (Fig. 3c). In a typical arrangement, the fringe angular spacing would be <1'' so that a source more than 1° from the NCP would exhibit many thousands of fringe transits per day.

The time profile of these oscillations is unique to each possible source location: the number of oscillations per day gives the radius of the circular path, and the phase of the slow-fast modulation gives the azimuth. The rotation of the fringes on the sky thus yields two-dimensional positions whereas the equatorial observation of Fig. 2 yielded only one-dimensional lines of positions. If a visibility line (or an anti-visibility line) happens to lie directly on the NCP, sources at two opposing positions on the circle will yield identical responses. In these special cases, a given response pattern will be consistent with two opposing positions as candidates for the real source.

(a) Interference bands North celestial (b) Visibility bands; Earth frame

(a) Interference bands North celestial (b) Visibility bands; Earth frame

Figure 7.3. Radio interferometry with two telescopes at the North Pole observing point source A which is near, but not at, the celestial pole. (a) Interfering waves and directions of 100% visibility (solid lines) and invisibility (dashed lines). Point source A is on one of the former at the time shown. (b) Movement of point source A in frame of reference of the earth. The track is a circle about the north celestial pole (NCP). (c) Response R'(t) of the source A. It oscillates alternately slowly and rapidly; numbers indicate source positions given in (b). (d) Fourier plane showing circular track sampled by a continuous 24-h measurement with fixed projected baseline.

Figure 7.3. Radio interferometry with two telescopes at the North Pole observing point source A which is near, but not at, the celestial pole. (a) Interfering waves and directions of 100% visibility (solid lines) and invisibility (dashed lines). Point source A is on one of the former at the time shown. (b) Movement of point source A in frame of reference of the earth. The track is a circle about the north celestial pole (NCP). (c) Response R'(t) of the source A. It oscillates alternately slowly and rapidly; numbers indicate source positions given in (b). (d) Fourier plane showing circular track sampled by a continuous 24-h measurement with fixed projected baseline.

A pseudo image of the sky can be constructed graphically for a North-Pole observation such as that of Fig. 3. Again, we assume that a single point source is in the field of view. We further assume realistically that the visibility lines are closely spaced (e.g., 1") and that the NCP is sufficiently distant so that many fringes (say, ~700) cross the source during a 12-h period, or ~1 per minute on average. The procedure is the same as before, namely to make a short observation, say of 10 min, and to plot lines of visibility on the sky at a moment when the response is maximum. These are the lines of position; the source must lie on one of them.

(d)

y

• •

u

u

u

Telescopes Mastery

Telescopes Mastery

Through this ebook, you are going to learn what you will need to know all about the telescopes that can provide a fun and rewarding hobby for you and your family!

Get My Free Ebook


Post a comment