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Deriving the Relationship Between Intensity and Distance

The Mathematics of the Relationship Between Intensity and Distance

diagram of intensity over distances

If a galaxy is traveling away at a recession velocity Vr, then the increase in its distance from us during a time interval Δ t would simply be Vr × Δ t. If its intensity at a distance r is I0, then the final intensity at the end of the time interval would be given by:


final intensity = initial intensity times initial 
	distance squared over final distance squared

By convention, if the galaxy is moving away from us, Vr 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

final intensity formula with final distance expanded out

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:

final intensity formula expanded again

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:

final intensity formula approximated for small change in distance

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:

final intensity approximation for small change in distance

 

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

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