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I have a question: assume two identical stars, but one is 10 times farther than the other from the earth. If the nearest star has a relative magnitude of 5, what is the relative magnitude of the other?

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    $\begingroup$ Why can't you do this simple problem using the definition of the magnitude system and that received flux is proportional to the reciprocal of the square of the distance? $\endgroup$
    – ProfRob
    Feb 14, 2018 at 20:25
  • $\begingroup$ This is a good question, but it's also one that's very easily answered by looking at any primary or secondary source. $\endgroup$ Feb 15, 2018 at 14:29

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As @RobJeffries mentions, you can calculate the difference from the equations which define magnitude and flux. But this particular case is even covered in the Wikipedia article on apparent magnitude:

For objects within the Milky Way with a given absolute magnitude, 5 is added to the apparent magnitude for every tenfold increase in the distance to the object.

So if the nearest star has magnitude 5, the farthest one has magnitude 5 + 5 = 10.

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