PREMIUM: Brokenomics | Aliens

“The signal from the 1945 atomic bomb would take time to reach any potential aliens. Once they receive the signal, they would then start their journey to Earth. If they're traveling at 70% the speed of light, their starting point would need to be closer than the full 78 light-year radius our signal has reached.

To clarify, let's use "t" to represent the total amount of time from 1945 to 2023, which is 78 years. We can think of two parts to this time: the time the signal takes to reach the aliens, and then the time it takes for the aliens to travel to us. The speed of the signal is the speed of light (denoted as "c"), and the speed of the aliens is 70% the speed of light (or 0.7c).

If we let "d" represent the distance to the aliens in light-years, then the time for the signal to reach them is d/c (since time = distance / speed), and the time for the aliens to travel to us is d/(0.7c). Adding these two times together should equal the total time, t. In other words:

d/c + d/(0.7c) = t

d(1/c + 1/(0.7c)) = t

d = t / (1/c + 1/(0.7c))

If we substitute c = 1 (since we're measuring everything relative to the speed of light) and t = 78 years, then:

d = 78 / (1/1 + 1/0.7) = 78 / (1.7) ≈ 45.9 light-years.

This means that, if aliens detected our signal and then immediately started traveling towards us at 70% the speed of light, their star system would need to be less than about 45.9 light-years away in order for them to reach us by 2023.

According to the Hipparcos satellite data, there are approximately 64 known star systems within 45.9 light-years of Earth. This number could be somewhat higher if we included stars not cataloged by Hipparcos or if there were undiscovered stars or star systems within this distance.”