A calculation from Cosmic Communications
Voicemail
for ET
There's one born every minute
New Scientist once profiled a service — racket — whereby you called a 900 number to leave voice messages for ET, to be beamed into space with a satellite dish. They contacted me, and since silence can imply complicity, I did the calculation.
The intention is to deliver voice messages to the stars. If you assume a voice message carries information at a data rate of C bits per second, working out how far the message can travel before it can no longer be received at that rate gives:
D2 ≤ 2π2P (RrRt/λ)2 / C N0
D is the distance; P the transmitted power; Rr and Rt the radii of the receiving and transmitting antennas; λ the radio wavelength; N0 the receiver noise level. This is a best-case scenario.
Cell phones code voice in the neighbourhood of C = 10,000 bits per second, so let us take that as the minimum data rate that can support voice communication. Cell phones are extremely stingy with spectrum and try to pack voice signals into as little space as possible, so the figure is a reasonable lower bound.
According to the website, Rt is approximately 2 metres and λ about 0.125 metres. Assume an Arecibo-sized listening antenna at the other end (Rr = 150 metres), chilled down to 3 degrees kelvin. Plugging all that in gives:
D ≤ (1.7 × 1013) √P metres
A light year is about 1016 metres.
Now let us assume they are transmitting at a whopping P = 1 megawatt. The signal will only be received properly about two light years away. However, it turns out that the actual FCC Part 15.247 rule under which they are operating allows only one watt of radiated power — so the actual distance is one thousand times less, roughly four times the distance from the sun to Pluto. And if I remember correctly, the nearest star is about four light years away.
One must remember that the above is a best-case scenario.