How does it work

Quantity of energy

The total number of photons N, per unit time per unit area, received through a filter F (of transmission \(T_\lambda\)) from a source of a flux density \(S_\lambda\) is:

\[\begin{split}N &=& \int_\lambda\,\frac{S_\lambda}{E_\nu}\,T_\lambda\,d\lambda\\ &=& \frac{1}{hc} \int_\lambda\,S_\lambda\,T_\lambda\,\lambda\,d\lambda\end{split}\]

Hence, if one consider \(T_\lambda\,\lambda\) as a distribution, the mean flux density \(\overline{f_\lambda}\) received through the filter F is:

\[\overline{f_\lambda} = \frac{\int_\lambda\,S_\lambda\,T_\lambda\,\lambda\,d\lambda} {\int_\lambda\,T_\lambda\,\lambda\,d\lambda}\]

and the magnitude (with the Pogson definition) is computed as:

\[m(F) = -2.5\log\left(\overline{f_\lambda}\right) + \textrm{ZP}\left(\overline{f_\lambda}\right)\]

where ZP is an offset, called the zero point of the photometric system. There are several photometric systems, differing by their definition of the zero point.

The Vega system

The historical system, most spread, is defined such as the \(\alpha\) Lyr star (Vega, which gives its name to the photometric system) has a magnitude 0 in all filters. In other words:

\[\textrm{ZP}\left(\overline{f_\lambda}\right) = -2.5\log\left(\overline{f_\lambda(Vega)}\right)\]

which can be directly written

\[m_{\textrm{Vega}}(F) = -2.5\log\left(\frac{\overline{f_\lambda}}{\overline{f_\lambda(\textrm{Vega})}}\right)\]

Vega was chosen because of its visibility from the northern hemisphere, its high flux, and the low amount of spectral lines in its visible spectrum.. This photometric systems does however not correspond to any remarkable spectral energy distribution. This led to the definition of two other systems: ST et AB.

The ST system

The ST system was defined such as a source with a constant flux density \(f_\lambda (\textrm{erg}\cdot\textrm{cm}^{-2}\cdot\textrm{s}^{-1}\cdot\overset{\lower.5em\circ}{\mathrm{A}}^{-1})\) against wavelength \(\lambda\) has a constant magnitude regardless of the filter. The zero point is chosen to provide a magnitude in V close to the magnitude in the Vega system (i.e., close to 0):

\[m_{\textrm{ST}}(F) = -2.5\log\left(\overline{f_\lambda}\right) - 21.1\]

The AB system

The AB system was defined such as a source with constant flux \(f_\nu (\textrm{erg}\cdot\textrm{cm}^{-2}\cdot\textrm{s}^{-1}\cdot\textrm{Hz}^{-1})\) against frequency \(\nu\) has a constant magnitude regardless of the filter. Here again, the ZP is chosen to provide a magnitude in V close to that of Vega system:

\[m_{\textrm{AB}}(F) = -2.5\log\left(\overline{f_\nu}\right) - 48.6\]

where \(\overline{f_\nu}\) is defined Koornneef et al. (1986) as

\[\overline{f_\nu} = \frac{\int_\nu\,S_\nu\,T_\nu\,d\nu\,/\,\nu} {\int_\nu\,T_\nu\,d\nu\,/\,\nu}\]

It is often useful to consider the pivot wavelength \(\lambda_p\) to easily convert \(\overline{f_\nu}\) into \(\overline{f_\lambda}\) (knowing \(\lambda\nu = c\) the speed of light):

\[\begin{split}\lambda_p^2 &=& \frac{\int_\lambda T_\lambda\,\lambda\,d\lambda} {\int_\lambda T_\lambda\,d\lambda /\lambda} \\ \overline{f_\nu} &=& \frac{\lambda_p^2}{c} \overline{f_\lambda}\end{split}\]

Hence, the magnitude in the AB system can also be written as:

\[m_{\textrm{AB}}(F) = -2.5\log\left(\overline{f_\lambda}\right) -2.5\log\left(\frac{\lambda_p^2}{c}\right) - 48.6\]

Units of spectral flux

The explanation here uses the non-SI unit \(\textrm{erg}\), which corresponds to \(10^{-7} \textrm{J}\). It is the unit of energy in the CGS system. The flux densities are often given in Janskys (\(\textrm{Jy}\)) in astronomy, which is defined as:

\begin{gather*} 1\ \textrm{Jy} = 10^{-23}\ \textrm{erg}\cdot\textrm{cm}^{-2}\cdot\textrm{s}^{-1}\cdot\textrm{Hz}^{-1}\\ = 10^{-26}\ \textrm{W}\cdot\textrm{m}^{-2}\cdot\textrm{Hz}^{-1} \end{gather*}