Apparent magnitude?

Question

Star A, apparent magnitude 6.5, is at a distance of 10pc.

Star B, apparent magnitude 9.0, is at a distance of 1000pc.

which star is more luminous and by how many times?

Answer 1

Distance does not matter as magnitude is measured by how bright a star is seen from Earth. So star B would almost be 1.5x brighter than starA.

Answer 2

Geez, Richard!  Distance makes all the difference in the world where apparent magnitudes are concerned.

Star A has an absolute magnitude of 6.5.  (That is by the definition, set at 10 pc.)

Star B is 100 times farther away, meaning if its absolute magnitude was the same as Star A's, it would be 10,000 times fainter at that distance.  (Inverse square law.)  That is a difference of ten magnitudes.  In other words, its magnitude would be 16.5.  That's a difference in magnitude of 7.5 from 9.  Raise 2.51 to the 7.5 power and you get 1000.  Star B is 1000 times brighter than Star A.

Another way:

The difference in apparent magnitudes given is 2.5.  This is a nice number because 2.51^2.5 = 10.  Star A appears 10 times brighter than star B.  Star B is 100 times farther away, resulting in a difference of 10,000 times in magnitude.  Since Star A's apparent magnitude is 10 times Star B's, divide that into 10,000 and you get Star B being 1000 times brighter than Star A.

(You know where the 2.51 comes from, don't you?)