# AB magnitude

The AB magnitude system is an astronomical magnitude system. Unlike many other magnitude systems, it is based on flux measurements that are calibrated in absolute units, namely spectral flux densities.

## Definition

The monochromatic AB magnitude is defined as the logarithm of a spectral flux density with the usual scaling of astronomical magnitudes and a zero-point of 3631 Jansky,[1] where 1 Jansky = 1 Jy = 10−26 W Hz−1 m−2 = 10−23 erg s−1 Hz−1 cm−2. If the spectral flux density is denoted fν, the monochromatic AB magnitude is:

${\displaystyle m_{\text{AB}}=-{\frac {5}{2}}\log _{10}\left({\frac {f_{\nu }}{3631{\text{ Jy}}}}\right),}$

or

${\displaystyle m_{\text{AB}}=-{\frac {5}{2}}\log _{10}\left({\frac {f_{\nu }}{\text{Jy}}}\right)+8.90.}$

In cgs units of erg s−1 cm−2 Hz−1, it is:

${\displaystyle m_{\text{AB}}=-{\frac {5}{2}}\log _{10}f_{\nu }-48.600.}$

Actual measurements are always made across some continuous range of wavelengths. The bandpass AB magnitude is defined so that the zero point corresponds to a bandpass-averaged spectral flux density of 3631 Jansky:

${\displaystyle m_{\text{AB}}=-{\frac {5}{2}}\log _{10}\left({\frac {\int f_{\nu }(h\nu )^{-1}e(\nu )d\nu }{\int 3631{\text{ Jy}}(h\nu )^{-1}e(\nu )d\nu }}\right),}$

where e(ν) is the "equal-energy" filter response function and the (hν)−1 term assumes that the detector is a photon-counting device such as a CCD or photomultiplier.[2] (Filter responses are sometimes expressed as quantum efficiencies, that is, in terms of their response per photon, rather than per unit energy. In those cases the (hν)−1 term has been folded into the definition of e(ν) and should not be included.)

The STMAG system is similarly defined, but for constant flux per unit wavelength interval instead.

AB stands for "absolute" in the sense that no relative reference object is used (unlike using Vega as a baseline object).[3]

## Expression in terms of fλ

In some fields, spectral flux densities are expressed per unit wavelength, fλ, rather than per unit frequency, fν. At any specific wavelength,

${\displaystyle f_{\nu }={\frac {\lambda ^{2}}{c}}f_{\lambda },}$

where fν is measured per frequency (e.g., Hertz), and fλ is measured per wavelength (e.g., cm). If the wavelength unit is Ångstrom,

${\displaystyle {\frac {f_{\nu }}{\text{Jy}}}=3.34\times 10^{4}\left({\frac {\lambda }{\mathrm {\AA} }}\right)^{2}{\frac {f_{\lambda }}{{\text{erg}}{\text{ cm}}^{-2}{\text{ s}}^{-1}\mathrm {\AA} ^{-1}}}}$.

This can then be plugged into the equations above.

The “pivot wavelength” of a given bandpass is the value of λ that makes the above conversion exact for observations made in that bandpass. For an equal-energy response function as defined above, it is [4]

${\displaystyle \lambda _{\text{piv}}={\sqrt {\frac {\int e(\lambda )d\lambda }{\int e(\lambda )\lambda ^{-2}d\lambda }}}}$.

For a response function expressed in the quantum-efficiency convention, it is:

${\displaystyle \lambda _{\text{piv}}={\sqrt {\frac {\int e(\lambda )\lambda d\lambda }{\int e(\lambda )\lambda ^{-1}d\lambda }}}}$.

## Conversion from other magnitude systems

Magnitudes in the AB system can be converted to other systems. However, because all magnitude systems involve integration of some assumed source spectrum over some assumed passband, such conversions are not necessarily trivial to calculate, and precise conversions depend on the actual spectrum of the source in question. Various authors have computed conversions for standard situations.[5]