Greetings to Altair: what is the probability of detecting a return message, assuming one was sent?

Question

Greetings to Altair was a 1983 transmission to Altair, 16.7 light years distant, sent on the 46m Stanford radio telescope, known as "The Dish". [Quast 2018]

Assuming an alien civilisation were to exist that detects the message and sends a single return message with comparable power to the original signal* (which might be a "million watt" transmitter giving the dish a 300-400 kW radar probe), the reply would reach the Earth some time after 2015.

How likely would we be to detect such a return message on Earth?

I'm anticipating answers might answer sub questions like: Would we passively detect such an event without a dedicated instrument being pointed in that direction? How big an instrument would be required to detect it? Have instruments of a class capable of detection been pointed in that direction, and how often?

(Feel free to make fairly swinging assumptions; I'm looking to ballpark this.)

Quast, Paul E. (2018). A profile of humanity: the cultural signature of Earth's inhabitants beyond the atmosphere. International Journal of Astrobiology, 1–21. doi:10.1017/S1473550418000290 

Answer 1

@ProfRob's link gives some very interesting numbers.

The Arecibo dish can detect [a million watt signal it emitted] at a distance of about 5000 light years. The dish has a diameter of 304m.

and confirms the inverse square law applies.

We assume the directionality of The Dish's (or the alien's) signal is equivalent to that of Arecibo

The Dish is 46m diameter, so the detection radius drops to $(46/304)^2$ or 2.3%, for a detection radius of 113ly. A 16.7 ly (Altair) signal is 300 times closer than Arecibo's limit, so a dish can be $\sqrt{300}$ = 17.4 times smaller or 17.4m.

Of Wikipedia's List of radio telescopes a great many are bigger than that; a more important discriminant might be the frequency of the signal.

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News has broken that we had a quick look for an hour on the 22nd August 2023, and that this was considered newsworthy. https://www.asahi.com/ajw/articles/14985246

At which point, I'd broadly estimate that a similar return message would almost certainly be lost. If this is the only time we've looked to see if there is a reply, assuming that the reply could have come at any time in the last eight years, there is a 1-in-70128 chance of detecting the signal.