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 1.3 pW of power on an area of 4.1 × 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 22000 ly distance from Earth that could provide such a signal? A light-year is the distance light travels in one year.

Answer 1

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(a)

Here the intensity that would be received by the Arecibo antenna should 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 = πD²/4

         D = 340m

Then a =π(340m)²/4

Power on the earth P_earth = 1.3*10^-12W

Area on the earth A= 4.1 x 10¹³ m²

Then      I_r =I _e

       Pr / a = P_earth / A

    Power received by antenna P_r = (1.3*10^-12W /4.1 x10¹³m²)*π(340m)²/4

                                                   = 2.87877*10^-21 W

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(b)

r =distance of the source to the antenna =2.2*10⁴ly*9.46 *10¹⁵m/ly

Power of the source is Ps = 4πr²I_e

                                      = 4π*( 2.2*10⁴ly*9.46 *10¹⁵m/ly)²*(1.3*10^-12W /4.1 x10¹³ m²)

                                        =1.7258*10¹⁶ W

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