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How can I compare the apparent magnitude of one object to another in percentages, e.g. the positive apmag of a dim star versus the negative apmag of the Sun?

EDIT: I'm an absolute amateur with mathematics. I know what a logarithm is, but that's about it!

EDIT: Currently I'm trying to get a comparison between the Sun and a red dwarf: measured from another star, hence a 'high' apmag of 3.12186514.

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If you want to compare the brightness of the objects, you could say that one is N times brighter or has a brightness that is only N percent of the other.

But if you want to compare the magnitudes of the objects, then the difference of the two encompasses the same information. Don't use percentages with magnitude values or differences.

If you had a pair of objects with magnitudes $-1.2$ and $2.5$, then the difference is $3.7$.

Since $\left( \sqrt[5]{100} \right) ^{3.7} = 30.2$, you can say that the brighter object is 30 times brighter, or the dimmer object is only 3% as bright as the partner.

Or you could say that one is 3.7 magnitudes brighter than the other.

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  • $\begingroup$ Thank-you! Would this work with any two objects? $\endgroup$
    – Kazon
    May 17 at 22:14
  • $\begingroup$ As long as they have defined magnitudes, yes. $\endgroup$
    – BowlOfRed
    May 17 at 23:17
  • $\begingroup$ I'm no maths expert, so I used this to compare the Sun with a red dwarf (apmag 0.993163478054013), and I ended up with the red dwarf being 13606035006% brighter. I assumed you got to 3% from 30.2 just by dividing by 10, but I now assume I'm missing something! $\endgroup$
    – Kazon
    May 18 at 0:19
  • $\begingroup$ Neither the sun nor any red dwarf has an apparent magnitude near 1. Show all the details (the apparent magnitude of both objects) in your question. $\endgroup$
    – BowlOfRed
    May 18 at 5:54
  • $\begingroup$ Will do! And this red dwarf's apmag is measured from another star, not from Earth. $\endgroup$
    – Kazon
    May 18 at 14:50
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How can I compare the apparent magnitude of one object to another in percentages, e.g. the positive apmag of a dim star versus the negative apmag of the Sun?

If the apparent magnitudes of the dim and bright stars are $M_D, M_B$ then assuming they are the same colors the ratio of their brightness can be calculated:

$$-2.5 \log_{10} \left( \frac{I_B}{I_D} \right) = (M_B-M_D)$$

or

$$ \frac{I_B}{I_D} = 10^{-(M_B-M_D)/2.5}$$

but we can get rid of that pesky leading minus sign by switching the order of subtraction

$$ \frac{I_B}{I_D} = 10^{(M_D-M_B)/2.5}$$

So for example if the apparent magnitudes of dim and bright are +3.7 and -1.2, the difference is almost 5, so we already know that the ratio will be almost 100 since every 5 is a factor of 100, every 2.5 is a factor of 10.

Numerically the ratio of brightnesses is about 91.2 or (91.2 - 1) × 100 = 9020% brighter.

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    $\begingroup$ What does the minus in superscript mean? Apologies; I keep forgetting to mention I'm not massively familiar with mathematics. $\endgroup$
    – Kazon
    May 17 at 22:27
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    $\begingroup$ @Kazon logarithms are confusing for sure. I've added a few more lines which might help a bit. That minus sign is the one that comes from the weird definition of magnitudes that makes negative values brighter and positive values dimmer. In all of science, I think that that negative sign has caused more people more headaches than any other, except perhaps the one in Chemistry that makes smaller pH more acidic than bigger pH. :-) $\endgroup$
    – uhoh
    May 18 at 1:23
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    $\begingroup$ I see! THank-you! I can see why they use magnitude and not percentages -- this results in some rather large numbers as the difference grows. $\endgroup$
    – Kazon
    May 18 at 20:38

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