I don\'t understand how to do this problem: The Very Large Array (VLA) is a set

Question

I don't understand how to do this problem: The Very Large Array (VLA) is a set of 27 radio telescopedishes in Caton and Socorro counties, New Mexico. The antennas canbe moved apart on railroad tracks, and their combined signals givethe resolving power of a synthetic aperture 36.0 km indiameter. (a) If the detectors are tuned to a frequencyof 1.30 GHz, what is the angularresolution of the VLA? (µrad)

(b) Clouds of hydrogen radiate at this frequency. What must be theseparation distance of two clouds near the center of the galaxy,25000 light years away, if they are to beresolved? (ly)
(c) As the telescope looks up, a circling hawk looks down. Find theangular resolution of the hawk's eye. Assume that that the hawk ismost sensitive to green light having wavelength 500 nm and that ithas a pupil of diameter 12.0mm. (µrad)

(d) A mouse is on the ground 30.0 m below. By what distance mustthe mouse's whiskers be separated if the hawk can resolvethem? (mm)

I don't understand how to do this problem: The Very Large Array (VLA) is a set of 27 radio telescopedishes in Caton and Socorro counties, New Mexico. The antennas canbe moved apart on railroad tracks, and their combined signals givethe resolving power of a synthetic aperture 36.0 km indiameter. (a) If the detectors are tuned to a frequencyof 1.30 GHz, what is the angularresolution of the VLA? (µrad)

(b) Clouds of hydrogen radiate at this frequency. What must be theseparation distance of two clouds near the center of the galaxy,25000 light years away, if they are to beresolved? (ly)
(c) As the telescope looks up, a circling hawk looks down. Find theangular resolution of the hawk's eye. Assume that that the hawk ismost sensitive to green light having wavelength 500 nm and that ithas a pupil of diameter 12.0mm. (µrad)

(d) A mouse is on the ground 30.0 m below. By what distance mustthe mouse's whiskers be separated if the hawk can resolvethem? (mm)

(a) If the detectors are tuned to a frequencyof 1.30 GHz, what is the angularresolution of the VLA? (µrad)

(b) Clouds of hydrogen radiate at this frequency. What must be theseparation distance of two clouds near the center of the galaxy,25000 light years away, if they are to beresolved? (ly)
(c) As the telescope looks up, a circling hawk looks down. Find theangular resolution of the hawk's eye. Assume that that the hawk ismost sensitive to green light having wavelength 500 nm and that ithas a pupil of diameter 12.0mm. (µrad)

(d) A mouse is on the ground 30.0 m below. By what distance mustthe mouse's whiskers be separated if the hawk can resolvethem? (mm)

Explanation / Answer

(a)the diameter of the synthetic aperture is d = 36.0 km =36.0 * 103 m the detectors are tuned to a frequency of f = 1.30 GHz = 1.30 * 109 Hz we know that v2 - u2 = 2gS u = 0 and g = 9.8 m/s2 or S = (v2 - u2/2g) = (v2 -(0)2/2g) = (v2/2g) v = (2r/T) = 2r * f or S = ((2r * f)2/2g) = (22* r2 * f2/g) r = (d/2) = (36.0 * 103/2) = 18.0 * 103m S = r * or = (S/r) = ((22 * r2 *f2/g)/r) = (22 * r *f2/g) (b)the limiting angle of resolution of the circular apertureis min = 1.22 * (/D) we know that f = (c/) or = (c/f) where c is the velocity of light and has a value 3 *108 m/s f = 1.30 GHz = 1.30 * 109 Hz the minimum separation between the clouds be S1therefore we get min = (S1/r1) or S1 = min * r1 r1 = 25000 light years = 25000 * 9.46 *1015 m = 2365 * 1017 m (c)the angular resolution of the hawk's eye is min = 1.22 *(1/D1) 1 = 500 nm = 500 * 10-9 m D1 = 12.0 mm = 12.0 * 10-3 m (d)the distance the mouse's whispers be separated so that thehawk can resolve them be D2 therefore we get D2 = 1.22 *(1/2) we know that S2 = r2 * 2 or 2 = (S2/r2) S2 = 30.0 m and r2 = (D1/2) =(12.0 * 10-3/2) = 6.0 * 10-3 m or 2 = (S2/r2) S2 = 30.0 m and r2 = (D1/2) =(12.0 * 10-3/2) = 6.0 * 10-3 m

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