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Figure 7.4. Lines of 100% visibility on the sky for the situation of Fig. 3 with a single point source in the field of view. The lines are drawn for each orientation at a time of maximum response; they are lines of position upon which the source could lie. (a) Lines obtained from two brief observations separated by 2 h (earth rotation angle of 30°). (b) Lines obtained from four additional observations at 2-h intervals. The location of the point source (arrows) is becoming apparent. (c) Twelve sets of lines with 15° angular intervals (1 h) between observations. (d,e,f). Fourier u,v planes showing as dots the angles and spacings sampled in (a,b,c).

Figure 7.4. Lines of 100% visibility on the sky for the situation of Fig. 3 with a single point source in the field of view. The lines are drawn for each orientation at a time of maximum response; they are lines of position upon which the source could lie. (a) Lines obtained from two brief observations separated by 2 h (earth rotation angle of 30°). (b) Lines obtained from four additional observations at 2-h intervals. The location of the point source (arrows) is becoming apparent. (c) Twelve sets of lines with 15° angular intervals (1 h) between observations. (d,e,f). Fourier u,v planes showing as dots the angles and spacings sampled in (a,b,c).

Two such observations separated by ~2 h would yield two such sets of lines of position (Fig. 4a). The two sets are rotated from one another by 30°, the rotation angle of the earth in a 2-hour period. The source must lie somewhere on each set, so the true position must be at one of the many intersections. If the observations are repeated at 2-h intervals for 10 h, one can plot six sets of fringes at 30° intervals extending over 150° of rotation (Fig. 4b). In this case, the intersections begin to indicate the true source position (see arrows). If observations are made every 1 h for 11 h, the source position stands out markedly (Fig. 4c). This shows unambiguously that sufficient information lies in the data to locate uniquely a source position, except for a possible false image on the other side of the NCP.

### All-sky fringe pattern

Heretofore, we considered only sources nearly overhead, i.e., at 9 ~ 90°; this angle is defined in Fig. 1c. The location of the great and small circles of constructive

Figure 7.5. (a) Geometry which defines the directions of 100% visibility for two telescopes separated by distance B. The angle 0„ measured from the baseline is the angle of the nth line of 100% visibility where the n = 0 line is normal to the baseline. The electronic processing logic flow is shown. (b) Nineteen lines of 100% visibility on the celestial sphere spaced at equal intervals An of line number n. (c) Celestial sphere showing visibility lines and the earth's rotation axis for the case of the telescopes being placed such that their baseline intercepts the celestial sphere at declination 8 = ±50°. The earth's rotation causes the visibility pattern on the celestial sphere to rotate about the north celestial pole, i.e., about the earth's spin axis. The telescopes are continuously repointed toward the source region as the earth rotates. The track of a celestial source at 8 = +70° is shown in the earth frame of reference. Note that the view is from outside the celestial sphere.

Figure 7.5. (a) Geometry which defines the directions of 100% visibility for two telescopes separated by distance B. The angle 0„ measured from the baseline is the angle of the nth line of 100% visibility where the n = 0 line is normal to the baseline. The electronic processing logic flow is shown. (b) Nineteen lines of 100% visibility on the celestial sphere spaced at equal intervals An of line number n. (c) Celestial sphere showing visibility lines and the earth's rotation axis for the case of the telescopes being placed such that their baseline intercepts the celestial sphere at declination 8 = ±50°. The earth's rotation causes the visibility pattern on the celestial sphere to rotate about the north celestial pole, i.e., about the earth's spin axis. The telescopes are continuously repointed toward the source region as the earth rotates. The track of a celestial source at 8 = +70° is shown in the earth frame of reference. Note that the view is from outside the celestial sphere.

interference at all angles 0 < 6 < 180° follow from the geometry of Fig. 5a. Let 6n be the angle between the baseline and the direction of constructive interference such that the path to one telescope is exactly nk longer than that to the other. Here n is an integer and k is the wavelength of the electromagnetic wave. The geometry yields nX

cos 9n = — (Angle of constructive interference; n = integer; (7.3) B -(B/X) < n <+(B/X)

where B is the magnitude of the length of the baseline (in meters), n is an integer that ranges from + B/X to — B/X as On ranges from 0° to 180°. Thus the n = 0 line is at O0 = 90°. This gives the angles of all the visibility lines. On the celestial sphere (Fig. 5b), the lines form great circles (n = 0) or small circles (n = 0) about the baseline. Although only 19 lines are drawn in the figure, the actual number can be very large, 2B/X, or twice the baseline given in number of wavelengths. In the figure, the lines shown are drawn for equal intervals of An, at nX/B = +0.9, +0.8 . . . 0 . .. —0.8, —0.9, in order of increasing On and decreasing n.

The separation between adjacent lines follows from (3),

## Telescopes Mastery

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