## Deriving the Relationship Between Intensity and Distance

### The Mathematics of the Relationship Between Intensity and Distance

If a galaxy is traveling away at a recession velocity V_{r},
then the increase in its distance from us during a time interval Δ
t would simply be V_{r} × Δ t. If its intensity at a
distance r is I_{0}, then the final intensity at the end of the
time interval would be given by:

**By convention, if the galaxy is moving away from us,
V _{r} and therefore Δ r, is positive, and the intensity
decreases, whereas if the galaxy is moving towards us Δ r is
negative and the intensity increases.** Substituting for Δ r
we get

Expanding this out yields the functional form of the relationship
between intensity and time for an object travelling at a speed
V_{r} in terms of its velocity and the time elapsed:

Looking at the expression in the denominator of this fraction, you can see that if the initial distance (r) is small compared to the distance the object travels in time Δ t, the expression reduces to:

If, on the other hand, the distance travelled in time Δ t is small compared to the initial distance of the object, the expression is approximately: