Continuum fluxes

Converting between photon flux, energy flux and magnitudes is not readily obvious. Some useful formulas to remember are:

$$f_{\nu} = 6.626 \times 10^{-8} \, \lambda \, f'_{\lambda}$$ (1)

and;

$$m_{\rm AB} = -2.5 \log f_{\nu} + 8.9$$ (2)

where

• $f_{\nu}$ is the energy flux measured in ${\rm Jy}$ (Janskys, equivalent to 10^-26 W m^-2 Hz^-1),
• $f'_{\lambda}$ is the photon flux measured in ${\rm s}^{-1}\,{\rm m}^{-2}\,\mu{\rm m}^{-1}$,
• $\lambda$ is the wavelength measured in $\mu{\rm m}$,
• $m_{\rm AB}$ is the magnitude scale defined by Oke & Gunn (1983)¹.

The conversion to standard magnitudes depends on the calibration of the Vega flux in different wavelength bands. An example is shown in Table 1 for the Johnson-Cousins-Glass system from Bessell (1992)². Examples of monochromatic flux conversions are shown in Table 2, where the Vega magnitudes are approximated using a fit to the calibrations in Table 1 (see also Figure 1).

A further system used for the WFPC2 filters is based on the following formula

$$m_{\lambda} = -2.5 \log f_{\lambda} - 18.6$$ (3)

where $f_{\lambda}$ is the flux meaured in ${\rm W}\: {\rm m}^{-2}\, \mu{\rm m}^{-1}$. The conversion to AB magnitudes is given by ³

$$m_{\rm AB} = m_{\lambda} - 1.3 - 5 \log \lambda$$ (4)

where $\lambda$ is the wavelength measured in $\mu{\rm m}$. As with Vega magnitudes, the zero points of AB and WFPC2 magnitudes are the same around 0.55 $\mu{\rm m}$ (the V band). Figure 1 shows the relationship between these different magnitude systems from 0.2 to 5.0 $\mu{\rm m}$.

 

 

Table 1: Zero points for the Johnson-Cousins-Glass photometric system, normalized to Vega

band	B	V	R	I	J	H	K	L	M
effective wavelength ( $\mu{\rm m}$)	0.436	0.545	0.638	0.797	1.22	1.63	2.19	3.45	4.75
zeroth mag flux (Jy)	4000	3600	3060	2420	1570	1020	636	281	154
AB mag − Vega mag	−0.1	0.0	0.2	0.45	0.9	1.4	1.9	2.8	3.4

 

 

Table 2: Monochromatic flux conversions in the near infrared

band	wavelength	photon flux	energy flux	AB mag	Vega mag
	( $\mu{\rm m}$)	( ${\rm s}^{-1}\,{\rm m}^{-2}\,\mu{\rm m}^{-1}$)	( $\mu{\rm Jy}$)
z	1.00	1000	66.3	19.3	18.7
J	1.25	1000	82.8	19.1	18.1
H	1.65	1000	109.3	18.8	17.4
K	2.20	1000	145.8	18.5	16.6

  

Figure: The squares represent the zeroth-magnitude fluxes of the standard broad-band photometric system, effectively the low resolution ( ${\cal R} \sim 5$) spectrum of an A0 star (Vega). The black solid lines show the approximate wavelength range for each band; the dotted line is a fit to the B to M fluxes; the red solid line shows the AB magnitude zero point, and; the blue solid line shows the zero points for a system based on $\log f_{\lambda}$.