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The Mathematics of the Relationship Between Intensity and
Distance
If a galaxy is traveling away at a recession velocity Vr, then
the increase in its distance from us during a time interval delta t would
simply be Vr x D t. If its
intensity at a distance r is Io, 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, Vr
and therefore Dr, is positive, and the intensity decreases, whereas if
the galaxy is moving towards us Dr is negative
and the intensity increases. Substituting for D r we get
Expanding this out yields the functional form of the relationship between
intensity and time for an object travelling at a speed Vr 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 Dt, the
expression reduces to:
If, on the other hand, the distance travelled in time Dt is small compared to the initial distance of the
object, the expression is approximately:
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