Appendix B: Sky brightness units

Throughout this paper we have expressed the sky brightness in mag arcsec^-2. Since other authors have used different units, we derive here the conversions for the most used ones. In the cgs system, the sky brightness is expressed in erg s^-1 cm^-2 Å^-1 sr^-1. Now, if $m_{{\rm sky},\lambda}$ is the sky brightness in a given passband (in mag arcsec^-2) and $m_{0,\lambda}$ is the photometric system zeropoint of that passband, the conversion is obtained as follows:

$$% B_\lambda ({\rm cgs})= 10^{-0.4(m_{{\rm sky},\lambda}-m_{0,\lambda}-26.573)}$$ (B.1)

where the constant in the exponent accounts for the fact that 1 arcsec² corresponds to $2.35\times10^{-11}$ sr. For example, if we use the constant m₀ for the Johnson system V band given by Drilling & Landolt (2000), for a typical night sky brightness $m_{{\rm sky},V}=$ 21.6 mag arcsec^-2, this gives $B_V\simeq$ $3.7\times10^{-7}$ erg s^-1 cm^-2 Å^-1 sr^-1. For practical reasons, we have introduced the surface brightness unit sbu (see Sect. 4), which is defined as 1 sbu $\equiv$ 10^-9 erg s^-1 cm^-2 Å^-1 sr^-1. With this setting, the typical V sky brightness is 366 sbu.

Often $B_\lambda$ is expressed in SI units. The conversion from cgs units is simple, since 1 erg s^-1 cm^-2 Å^-1 sr^-1 $\equiv$10 W m^-2 sr^-1 $\mu$m^-1 and hence:

$$% B_\lambda (SI)= 10^{-0.4(m_{{\rm sky},\lambda}-m_{0,\lambda}-29.073)}$$ (B.2)

so that the typical V sky brightness turns out to be 3.7 $\times$ 10^-6 W m^-2 sr^-1 $\mu$m^-1.

Especially in the past, the sky brightness was expressed in S₁₀, which is defined as the number of 10th magnitude stars per square degree required to produce a global brightness equal to the one observed in the given passband. Since 1 sr corresponds to $3.282\times10^3$ square degrees, the conversion is given by the following equation:

$$% 1 \; S_{10}(\lambda) = 10^{-0.4(1.21-m_{0,\lambda})}\;\; \m... ...g s$^{-1}$\space cm$^{-2}$\space \AA$^{-1}$\space sr$^{-1}$ }.$$ (B.3)

Again, using the constant given by Drilling & Landolt (2000) for the V band, one obtains $1\;S_{10}(V)\equiv$ $1.24\times10^{-9}$ erg s^-1 cm^-2 Å^-1 sr^-1 $\equiv$ 1.24 sbu. For U, B, R and I the multiplicative constant is 1.38, 2.11, 0.57 and 0.28, respectively. Therefore, the sky brightness expressed in S₁₀ and in sbu are of the same order of magnitude in all optical broad bands. For example, the typical V sky brightness is 295 S₁₀(V).

Another unit used in the past is the nanoLambert (nL), which measures the perceived surface brightness and corresponds to 3.80 S₁₀(V). For the V passband this gives also $1\;{\rm nL} \equiv 4.72$ sbu.

Finally, to quantify the intensity of night sky emission lines, the Rayleigh (R) unit is commonly adopted. It is defined as 10⁶/4$\pi$ photons s^-1 cm^-2 sr^-1 and the conversion from erg s^-1 cm ^-2 sr^-1is given by the following expression:

$$% 1R \equiv 634.4 \; \lambda \; \mbox{erg s$^{-1}$\space cm$^{-2}$\space sr$^{-1}$ }$$ (B.4)

where $\lambda$ is expressed in Å. The intensity of non-monochromatic features, like the pseudo-continuum, is generally expressed in R Å^-1.

For a more thorough discussion of the sky brightness units the reader is referred to Leinert et al. (1998) and Benn & Ellison (1998).