Here’s some results. I placed a dipole antenna 1,000,000 meters away from a high gain 56 element yagi antenna. I could build that one. The dipole is transmitting 200 KW. A 70 ohm load on the 56 element input element was 20 uA, which comes to 20 uA * 70 ohms = 1.4 mV. Receiving voltage in an antenna is relative to distance. At a distance of 400 ly the voltage would be 3.7e-16 volts. So 56 elements is insufficient.

If we drop the temperature of the load to 190 K, dry ice, with 450 Hz BW (plenty BW to hear voices, at least on Earth) the noise voltage is

sqrt(4 * k * 190 K * 70 ohm * 450 Hz) = 18 nV

One can expect about 1 db gain per extra element in yagi antennas, which increases the voltage by a factor of 1.122. If we add 154 more elements to the 56 element yagi antenna

the voltage comes to

3.7e-16 V * 1.122^154 = 18 nV

That’s 154 el + 56 el = 210 elements. Sounds crazy.

According to parabolic dish antenna calculators the dish would need to be 4100000 meters in diameter, which is about 1/3rd the diameter of earth. So dish antennas won't work for me.

If we drop the load temperature to 1 K and use a special low temp amp and drop BW down to 100 Hz, then we only need 180 elements.

Anyway, 210 elements is a lot! I’ve seen photos a HAM radio guy with a 100 element antenna. You would have be good with antenna analysis software, make very accurate measurements, and know what you’re doing, and even then I’m not sure if you could get close to the required gain. Also it would have to be in a relatively quiet location on Earth due to the 18 nV input. That sounds like low input voltage, but it’s not. Trust me! I’ve spent lot of time working with metal detector designs that were considerably lower than that.

The bad thing about yagi antennas is it works at one frequency. Unacceptable to SETI, but maybe MM could tell me the magic frequency, if he knows it.

Dish antennas are wide BW, but it comes at a cost of being way to large. One third the diameter of Earth!

Does all my math work out?