# Magnitude definition

A magnitude is really a flux ratio. It is defined as follows, where M's are magnitudes and F's are fluxes:

$\displaystyle M_1 - M_2 = 2.5 \times \log \left(\frac{F_2}{F_1}\right)$
(eqn 1)


The magnitude system (in the optical) was defined to be referenced to Vega. In other words, Vega is defined to be zero magnitude, and you would then define magnitudes of anything else as follows:

  $\displaystyle M = 2.5 \times \log \left(\frac{F_{\mathrm{Vega}}}{F}\right)$
(eqn 2)


When they looked at Vega with IRAS, they discovered that it did NOT look like they expected, and in fact it has a large infrared excess! Therefore, infrared magnitudes are defined with respect to what Vega would be, if it did not have an excess.

Generally, the zero points (e.g., the "Vega flux") are published for most of the bandpasses you might encounter. They are consolidated on the Central wavelengths and zero points page. Therefore, in order to convert the uJy that apex returns into magnitudes, use the equation 2 above, substituting these so-called "zero-point fluxes" in for "Fvega." Note that the zero-point fluxes are in Janskys and the fluxes returned by APEX are in microJanskys. You can find the zeropoints for many other bands on the web as well, such that you can freely convert between mags and flux densities.

Note that plain magnitudes get fainter (the number gets larger) as the distance of the object increases. This happens because Vega, your reference object, stays the same while such an example object moves farther away. BUT, colors (which are differences in magnitudes) are ratios of fluxes of the same object, and therefore independent of distance. This is powerful when you are studying objects whose distances you don't know, or comparing objects at a variety of distances.

To convert magnitudes back into fluxes (e.g., if you have optical or 2MASS magnitudes and need to get fluxes), you need to invert the equation above. Recall that to invert a logarithm (base 10), you have to raise both sides to the power of 10, e.g., if log x = y then x = 10^y.

# AB mags

BE CAREFUL to keep track of whether you are working with Vega-based magnitudes or AB mags. Vega magnitudes define things with respect to a Vega spectrum (as above), but in recent years, some folks (largely extragalactic folks) define things with respect to a flat spectrum source instead, and those are AB mags. Most Sloan folks (even those folks working with stars, and even those working with Sloan filters but not necessarily SDSS archival data) work in AB mags instead. For AB mags, you always use a flat reference spectrum, so the zero point is 3631 Jy for all bands.