The magnitude scale historically was simply a ranking system.
There was no real standard, other than that the brightest stars
were designated as first magnitude stars, slightly dimmer stars
were second magnitude stars, then even dimmer stars were third
magnitude, and so on.
To make things more rigorous, modern astronomers use a scale
where the difference in magnitude between two stars is determined
by the ratio of the flux of light from those stars. A magnitude
is defined such that an object which is 10 times brighter is
two and a half magnitudes lower.
m2-m1=-2.5 log10 (F2/F1)
When we see magnitudes in astronomy, the measurement is one of a star
being dimmer or brighter than something, but what are we comparing it to?
We need to have some object with which we measure everything else with respect
to. For visual light, this reference is Vega. In practice, a zero
magnitude flux is defined for each wavelength region of interest.
Common astronomical bands
Measuring the amount of light at every wavelength is a difficult
thing to do. If you have a finite amount of light coming in in different
wavelengths, then the more specific you are about the exact wavelength
you want, the more you limit yourself. In addition, filtering out light
of unwanted wavelengths can be both difficult and expensive. Typically,
astronomer have searched for filters that are easily made and replicated
that limit the range of wavelengths that can be detected to a small, but
not infinitesimally small, range. These regions of wavelength allowed by
a given filter are referred to as bands.
The following are the primary wavelengths and zero magnitude fluxes for
the UBV bands on the Johnson scale, RI on the Cousins scale, and JHK on the
Primary wavelength (nm)
Zero point flux (Jy)
If a star is giving off light of a given intensity I (often
assumed to be blackbody radiation,),
the flux of light coming off the star is
Notice that the flux as a function of wavelength is not the same as the
flux as a function of frequency. The reason for this is that wavelength
dependent flux is defined such that the total energy within some wavelength
range is conserved. Thus
The zero point magnitudes above are defined in terms of a frequency
dependent flux, not a wavelength dependent flux.