Problems are listed in approximate order of difficulty. A single dot (•) indicates straigh...

Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.

••• Communications with interplanetary space probes. When an incoming digital signal has very low power, the message must be sent very , as this problem will show. (a) Consider a space probe, a great distance R from the earth, transmitting a radio signal at wavelength λ and power P0 from a radio dish of diameter d. On earth is a very large radio receiver with a dish of diameter D, collecting the radio-wavelength photons sent by the distant probe. Suppose the 1s and 0s of the digital message are represented by radio signals of high and low intensity, with a ratio of 4:1. Also suppose that in order to distinguish the signal from background noise, the receiver must collect at least 104 photons per “one bit” (and 2500 photons per “zero bit”). Derive an approximate formula for the highest bit rate (number of bits per second) at which the space probe can transmit a message which can be read by the receiver on earth. [Hint: Don’t forget diffraction effects: The outgoing radio signal diverges with half angle ∆θ ≈ λ/d.] (b) In 1995 NASA’s Galileo spacecraft entered orbit around the planet Jupiter, 800 million kilometers from earth Galileo has a main radio antenna 5 meters in diameter that was designed to transmit power P0 = 20 W at wavelength λ = 3 cm. NASA’s radio receiver on earth has a diameter of 70 meters (big!). Galileo’s main antenna was designed to transmit information at what bit rate? (c) Unfortunately, the main antenna on Galileo failed early in the mission. A small backup antenna operating at the same power and wavelength as the main antenna, but with an effective diameter of only 5 cm, was used to transmit data to earth. At what bit rate can this backup antenna send data? A typical image from Galileo’s camera contains about 106 bits. How long does it take for a picture to be sent from Galileo?