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At What Distance Does the Intensity of a One Million Watt Transmitter Equal That From the Cosmic Microwave Background?

-- | June 4, 2017

Question: If there was a 1,000,000 watt UHF television show broadcast into space, at what distance would it become the same intensity as the cosmic background radiation? Also, if the transmitter’s antenna is located at the equator how fast would the signal sweep across the closest exoplanet? — Ata

Answer: First, note that the intensity of electromagnetic radiation, which includes television transmissions, gets smaller as the inverse-square of the distance that the electromagnetic radiation travels. Also, since UHF frequencies are between 0.3 and 3 GHz, which happens to be within a transmission “window” through the Earth’s ionosphere, the transmission signal can get out into space. A 1 million Watt transmission has an intensity of 10^(11) erg/s, while the cosmic microwave background (CMB) has an intensity of about 10^(-18) erg/s/cm^2/ster/Hz at a frequency of 1 GHz (in the middle of the UHF band). If the receiver at the distant location is 1 cm^2 in size and is transmitted with a narrow frequency range of just one Hz, then to within an order-of-magnitude or so the distance at which the UHF transmission signal would equal that of the CMB is about (10^(11)/10^(-18))^(1/2) =~ 10^(14.5) cm =~ 3.3×10^(-4) light years =~ 21.2 Astronomical Units (AU). On astronomical distance scales this is really not very far. For example, Pluto’s closest distance from the Sun is 29.7 AU. As for the speed at which this transmission would sweep across the closest exoplanet, since UHF TV transmissions are not highly directional, those who might see this signal would likely measure a continuous transmission (assuming that it is not pulsed).

Jeff Mangum

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