Large radio telescopes, like the one in Arecibo, Puerto Rico, can detect extreme

Question

Large radio telescopes, like the one in Arecibo, Puerto Rico, can detect extremely weak signals. Suppose one radio telescope is sensitive enough to detect a signal which lays down only 1.4 pW of power on an area of 4.6 x 1013 m2. (a) What would be the total power that would be received by the antenna, assuming that its diameter is 340 m? (b) What would be the power of a source at 24000 ly distance from Earth that could provide such a signal? A light-year is the distance light travels in one year.

Explanation / Answer

A) Here the intensity that whould be received by the Arecibo antennashould be equal to intensity of that signal on earth.         Then intensity received I_r = P_r/a         Pr =power received          a =area of cross section of antenna = D2/4          D = 340m Then a =(340m)2/4 Power on the earth Pearth = 1.4*10-12W Area on the earth A= 4.6 x 1013 m2 Then      I_r =I _e        Pr / a = P_earth / A     Power received by antenna P_r = (1.4*10-12W /4.6 x1013m2)*(340m)2/4                                                    = 2.7631*10-21W B) r =distance of the source to the antenna =2.4*104ly*9.46 *1015m/ly Powerof the source is Ps = 4r2I_e                                       = 4*( 2.4*104ly*9.46 *1015m/ly)2*(1.4*10-12W /4.6 x1013 m2)                                         =1.97*1016W Then      I_r =I _e        Pr / a = P_earth / A     Power received by antenna P_r = (1.4*10-12W /4.6 x1013m2)*(340m)2/4                                                    = 2.7631*10-21W B) r =distance of the source to the antenna =2.4*104ly*9.46 *1015m/ly Powerof the source is Ps = 4r2I_e                                       = 4*( 2.4*104ly*9.46 *1015m/ly)2*(1.4*10-12W /4.6 x1013 m2)                                         =1.97*1016W

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