Question

Consider a civilization broadcasting a signal with a power of 2.0×104 watts. The Arecibo radio telescope, which is about 300 meters in diameter, could detect this signal if it is coming from as far away as 141 light-years. Suppose instead that the signal is being broadcast from the other side of the Milky Way Galaxy, about 70000 light-years away.

Answer 1

For light: Inverse Square Law 

The signal is weaken by the inverse square of the difference of distances this means that one over the square of the difference. 

Therefore, 70000LY / 141 LY =496.45 and that squared =246462.602 times less signal, or 1/246462.602.

 2 For a circular telescope like the Arecibo radio telescope that equals Pi times the radius squared. Sensitivity of the telescope depends on its area.
The new telescope needs 246462.60 times that area to have enough sensitivity 

(150 x 150 x Pi) = 70 685 sq m area for the Arecibo telescope and that times 246462.60
= 1.74212 x 10^10

What is the diameter of a circle with that area?

 
1. Divide the area by Pi = 5.548156 x 10^9
2. Take the square root of the result to find the radius = 74485.94293

3. Multiply by 2 to find the diameter = 148971.886