## Apparent and Absolute Magnitudes

## Apparent Magnitude

*Apparent* magnitude m of a star is a number that tells
how bright that star appears at its great distance from Earth.
The scale is "backwards" and logarithmic. Larger magnitudes
correspond to fainter stars.
Note that brightness is another way to say the *flux of light*,
in Watts per square meter, coming towards us.

On this magnitude scale, a brightness ratio of 100 is set
to correspond exactly to a magnitude difference of 5. As magnitude
is a logarithmic scale, one can always transform a brightness ratio
B_{2}/B_{1} into the equivalent magnitude difference
m_{2}-m_{1} by the formula:

m_{2}-m_{1} = -2.50 log(B_{2}/B_{1}).

You can check that for brightness ratio B_{2}/B_{1}=100,
we have log(B_{2}/B_{1}) =log(100)=
log(10^{2}) = 2, and then m_{2}-m_{1}=-5, the basic
definition of this scale (brighter is more negative m).
One then has the following magnitudes and their corresponding relative
brightnesses:

magnitude m | 0 1 2 3 4 5 6 7 8 9 10
----------------------------------------------------------------
relative | 1 2.5 6.3 16 40 100 250 630 1600 4000 10,000
brightness |
ratios |

(Note that the lower row of numbers is just (2.512)^{m}.)

## Absolute Magnitude

*Absolute* magnitude M_{v} is the apparent magnitude the star
would have if it were placed at a distance of 10 parsecs from the Earth.
Doing this to a star (it is a little difficult), will either make it appear
brighter or fainter. From the *inverse square law for light*,
the ratio of its brightness at 10 pc to its brightness at
its known distance d (in parsecs) is
B_{10}/B_{d}=(d/10)^{2}.

Then, like the formula above, we say that its absolute magnitude is

M_{v} = m - 2.5 log[ (d/10)^{2} ].

Stars farther than 10 pc have M_{v} more negative than m, that is
why there is a minus sign in the formula. If you use this formula, make sure you
put the star's distance d in *parsecs* (1 pc = 3.26 ly = 206265 AU).

## Distance Determination

The above relation can also be used to determine the distance to a star
if you know both its apparent magnitude and absolute magnitude. This would
be the case, for example, when one uses Cepheid or other variable stars
for distance determination. Turning the formula inside out:

d = (10 pc) x 10^{(m-Mv)/5}

For example, for a Cepheid variable with M_{v} = -4, and m = 18,
the distance is

d = (10 pc) x 10^{[18-(-4)]/5} = 2.51 x 10^{5} pc.

Last update: April 10, 1998.

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