## More About the Relationship Between Intensity and Distance

### More on Brightness as a Function of Distance

The intensity or brightness of light as a function of the distance
from the light source follows an inverse square relationship. Suppose
you were to use a light meter to measure an initial intensity
I_{i}, or brightness, a distance r from a light source. Suppose
that some time later the brightness of the light is either greater or
lesser; if the intensity diminished you would know that the source was
moving away from you and if it became brighter you would know that the
source was moving towards you (assuming the light source itself remained
the same).

This relationship can be illustrated by the diagram below, which
shows the apparent brightness of a source with luminosity L_{0}
at distances r, 2r, 3r, etc. Notice that as the distance increases, the
light must spread out over a larger surface and the surface brightness
decreases in accordance with a "one over r squared" relationship. The
decrease goes as r **squared** because the area over which
the light is spread is proportional to the distance squared.

An example of the "one over r-squared" relationship for light.

If M31 is moving with respect to the Earth, you should be able to see a change in its apparent brightness if you take two measurements at different times. Measuring this change would allow you to calculate its speed.