Stick number

Given a spectral flux density f, measured in ergs per second per square centimeter per hertz, the corresponding AB magnitude is defined as:

${\text{mag}}_{AB}=-2.5{\log }_{10}{f}_{\nu }-48.60$

Motivation for $f_{\nu }$ definition

The AB magnitude system is defined such that for any bandpass or filter being considered, the magnitude zero-point corresponds to a spectral flux density of 3631 Jy (1 Jy = 1 Jansky = 10⁻²⁶ W Hz⁻¹ m⁻² = 10⁻²³ erg s⁻¹ Hz⁻¹ cm⁻²). Thus a source with constant flux per unit frequency interval will have zero color (constant magnitude) in this system. (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 the Vega magnitude system).

Definition for $f_{\lambda }$ fluxes

Another common formulation for AB magnitudes is

${\text{mag}}_{AB}=-2.5{\log }_{10}\frac{S_{\nu }}{\mu Jy}+23.90$

This can be derived from the above definition using the definition of Jy, and the conversion between $F_{\nu }$ and $F_{\lambda }$ (see the Example below).

Fluxes from magnitudes from specific filters

To compute the spectral flux in ergs per second per square centimeter from AB-magnitudes of a filter, the filter response has to be characterised: The flux collected is dependent on the filter efficiency function $f(\lambda )$:

$F_{\nu }[\text{erg}{\text{ cm}}^{-2}{\text{ s}}^{-1}]=\int F_{\nu }[\text{erg}{\text{ cm}}^{-2}{\text{ s}}^{-1}{\AA }^{-1}]f(\lambda )d\lambda$.

In the simplest case of a top hat, with width $w$ and central position $\lambda$ (both in Angstrom):

$F_{\nu }[\text{erg}{\text{ cm}}^{-2}{\text{ s}}^{-1}]=F_{\nu }[\text{erg}{\text{ cm}}^{-2}{\text{ s}}^{-1}{\AA }^{-1}]\times w$

To get a relation between flux and magnitudes, we combine the equations

$S_{\nu }[Jy]=3.34\times 10^4{\lambda }^2{F}_{\lambda }[\text{erg}{\text{ cm}}^{-2}{\text{ s}}^{-1}{\AA }^{-1}]$

and

${\text{mag}}_{AB}=-2.5{\log }_{10}\frac{S_{\nu }[Jy]\times 10^6}{[\mu Jy]}+23.9$

(the latter can be derived from the definitions of Jy and AB magnitudes). This yields

${\log }_{10}{F}_{\nu }[\text{erg}{\text{ cm}}^{-2}{\text{ s}}^{-1}]=-m_R/2.5+23.9/2.5-6-{\log }_{10}(3.34\times 10^4\cdot {\lambda }^2/w)=-m_R/2.5-5\times {\log }_{10}(\lambda )-0.96$

Thus often the relation between AB magnitude in a certain band and fluxes in units of [erg s⁻¹ cm⁻² Å⁻¹] is written as

${\text{mag}}_{AB}=-2.5{\log }_{10}{f}_{\lambda }-C$(e.g.)

where $C$ depends on the filter definition.

As an example, for the Subaru R-band filter with $w\approx 1298\AA$ and central position $\lambda =6341\AA$ we get the relation:

${\log }_{10}{F}_{\nu }[\text{erg}{\text{ cm}}^{-2}{\text{ s}}^{-1}]=-m_R/2.5-5.45$

External links

• Conversion from AB magnitudes to Johnson magnitudes

References

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