0min ~ 1.22— (rad; circular aperture; X ^ d) (5.8)

d where X is the wavelength of the incoming radiation and d is the diameter of the mirror or antenna. A sketch of the appearance is presented in Fig. 7. This is called the Airy diffraction pattern, and the inner part, within 0min is called the Airy disk.

The diffraction phenomenon is equivalent to saying that the antenna can not distinguish the arrival directions that lie within angles 0min ~ X/d of the source directions; the two sources can not be resolved if they are closer together than this angle. (Even so, with sufficient signal, the response will be broadened or asymmetric, possibly revealing the presence of the second source.) As shown in Fig. 6d, the angle 0min corresponds to a shift of phase of only one wavelength across

Figure 5.7. Approximate appearance of the Fraunhofer diffraction pattern from a parallel beam imaged through a circular aperture. The radius of the first minimum (black) is 0m;n where sin 0min « 1.22 k/d for k ^ d. The inner bright circle is called the Airy disk.

the face of the telescope. This is an easy way to remember and rederive, approximately, the expression (6) or (8). One can think: "the telescope can not distinguish incident angles that yield less than 1 wavelength phase shift across the face of the telescope. The uncertainty in arrival direction is thus ~k/d.

Radio resolution

Diffraction is the primary factor that limits the resolution of radio telescope beams. For example, a 16-m dish antenna observing at v = 100 MHz (k = c/v = 3m) has a diffraction pattern, or half beam width, of k3

0min « 1.22— = 1.22— = 0.23 rad = 13° (radio) (5.9)

d 16

which is terrible! Observations at 5 GHz (k = 0.06 m) yield resolution improved by a factor of 50, or ~1/4 degree, comparable to the angular radius of the moon,

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