My question is this: How much power would a signal need to be detected for a given number of light years away.   I'm interested in the coefficient relationship between signal transmitting power times distance, and signal detection possibility.  The coefficient could be towards zero (a civilization pumps out a terawatt radio signal but the signal can only be barely detected 1 light year away), or the coefficient could be very high (send out a 1-milliwatt battery-watch-strength signal and it can eventually be easily detected by someone in the Andromeda galaxy).  It could be 1/x, 1/.01x, 1/sqrt(x), 1/x^2, 1/x^9, or whatever (where x is distance).  I know the coefficient lies SOMEWHERE between those two extremes (0 and infinity), and is related to the inverse square law with power, absorption characteristics of stellar dust, and the noise-to-signal ratio swamping by a sun.

While our sun (and many others stars) puts out it's peak power in the visible wavelength range (and by no coincidence-  most animal eyes on Earth have developed through evolution to pick up this strongest power wavelength range, in the so-called "visual" bandwidth), it also put out prodigious amounts out power (trillions of watts) in the radio frequency bandwidth (along with gamma ray, x-ray, infrared, ultraviolet, and the rest).  The radio frequency signal output strength of a sun is hardly inconsequential.  It can swamp out a signal.. like trying to hear a person's whispering at a rock concert.  You might be able to hear them if you were standing right next to them, but what if you were a mile away from the concert?  I know the answer is related to the LaPlace and Fourier transforms of the power of the signal in time (from communications theory).  I took some of the math in college (and found it fairly difficult to work with- convolving signal equations was a major pain).  In a given signal with a lot of random static, you can play with the math to get an information signal out of it (if there is one there to get), but there is a threshold limit in this process.

The detection ability plays a factor too, and I take our current ability (through Aricebo or the VLA) as baseline.  Anyone have a feel for the coefficient?  The coefficient means a lot, because it has a very big influence over whether SETI may be completely wasting their time, or if improving our detection ability might meaningfully improve our chances of detecting a civilization out there.  If we're not even close by orders of magnitude, then we shouldn't bother-  but otherwise, it may explain why we haven't heard any ET's yet, and maybe it's worth the effort to try and build bigger telescope arrays to increase our detection threshold.