B.2 Formulae

B.2.1 Converting Between Fν and Fλ

B.2.2 Conversion Between Fluxes and Magnitudes

B.2.3 Conversion Between Surface Brightness Units

B.2.1

Converting Between Fν and Fλ

One Jansky (Jy) is defined as 10–26Wm-2Hz-1, so it is a unit of measurement of the spectral flux density, Fν.

To determine Fλ from Fν in Jy, use the following formula:

where λ is the wavelength in microns (μm), and β is a constant chosen from Table B.1 and depending on the units of Fλ. (This is simply derived, using the fact that dν/dλ = c/λ2.)

Table B.1: Constants for Converting Fλ and Fν.

Fλ measured in	β
Wm-2μm-1	3в10–12
Wcm-2μm-1	3в10–16
erg sec-1 cm-2 μm-1	3в10–9
erg sec-1 cm-2 е-1	3в10–13

Remember that 1W=107erg sec-1, and 1μm=104е.

B.2.2

Conversion Between Fluxes and Magnitudes

The spectral flux density Fν can be calculated from its magnitude as

where m is the magnitude and Fo the zero-point flux for the given photometric band. We list the central wavelengths and zero-point fluxes for the more commonly encountered photometric bands below in Table B.2. The CIT system was originally based on Beckwith et al. (1976, Ap.J., 208, 390); the UKIRT system is in fact based on the original CIT system, but with adjustments made largely owing to different filter bandpasses. It should be noted that for a given photometric band there will be small differences in the effective wavelength and zero-point flux from one observatory to another, and for astronomical objects with radically different colors, so these figures can only be treated as approximate.

Table B.2: Effective Wavelengths and Zero-points for Photometric Bands.

Band	λ[μm]	Fo[Jy] (CIT)	Fo[Jy] (UKIRT)
V	0.56	3540	3540
R	0.70	2870	—
I	0.90	2250	—
J	1.25	1670	1600
H	1.65	980	1020
K	2.2	620	657
L	3.4	280	290
L'	3.74	—	252
M	4.8	150	163
N	10.1	37	39.8
Q	20.0	10	10.4

B.2.3

Conversion Between Surface Brightness Units

Surface brightnesses are generally measured in Janskys arcsec-2, MJy steradian-1 or magnitudes arcsec-2. If you have a surface brightness Sν in MJy steradian-1, then you can use:

.

If you have Sν in magnitudes arcsec-2, you can simply use the formula and zero-points as given in the previous section for point sources.