# National Aeronautics and Space Administration

## 1/r^2 falloff of reflected signal

### How strong is the reflected radar pulse?

How bright a source looks depends not only on how much energy it gives out but also upon how far away it is from the observer. You know that lights look dimmer the farther away you are from them - the same is true for any electromagnetic signal. This is because the energy from any light wave as it travels through space spreads out over a larger and larger surface area. The further away you are from a source, the less light (or signal) there is per unit area, and the less light you can collect with a "light bucket" of a given area. This concept is illustrated in the diagram below:

 In this schematic, light rays traveling outwards from a source are represented by arrows. You can see that the area (the squares) covered by the light rays increases as the light extends from the source. The same amount of light energy is spread out over more and more area. The total area is proportional to the distance from the source squared (think of the area of a sphere of radius R and how it increases the larger R is; A = 4 pi R2). The intensity is proportional to the distance squared.

This effect can be expressed mathematically as:

In this formula, d is the distance the observer is from the source. If you are 100 feet away from a 100 watt light bulb, it appears four times dimmer than if you were fifty feet away.

Your probes each release a pulse of about 1032 photons at a frequency of 1.5 MHz. These are emitted isotropically (in all directions), and spread out so that they cover a sphere of radius R, teh distance travelled. You plan to observe the return signals with the radio telescope at radio signals from Arecibo Observatory, the largest single dish radio telescope in the world. It has an effective surface area of 7854 square meters.

The distance to HTCas is 140 parsecs.
A parsec is equal to 3.0857 x 1016 m.

About how many photons do you expect to fall on Arecibo's collecting area (assuming no interference or absorption from the interstellar medium)? (Round to the nearest whole number)

 A service of the High Energy Astrophysics Science Archive Research Center (HEASARC), Dr. Alan Smale (Director), within the Astrophysics Science Division (ASD) at NASA/GSFC