It has been shown by Kocifaj et al. (2019) that night-sky luminance due to a ground-based light source (city or town) can be approximated for intermediate and/or long distances $D$ by the formula
(1)$$\begin{eqnarray*}
J \left( z \right) = {I_0}\ \frac{9}{{32\pi }}\frac{{\tilde{\tau }{\mathrm{ e}^{ - D\tilde{\tau }/\left( {3\tilde{H}} \right)}}}}{{\mu \tilde{H}D}}\left\{ {Ci\left( {aD} \right)\sin\left( {aD} \right) + \left[ {\frac{\pi }{2} - Si\left( {aD} \right)} \right]\cos\left( {aD} \right)} \right\},
\end{eqnarray*}$$where $z$ is the observational zenith angle, $\mu = \cos( z )$, ${I_0}$ the luminance of a city normalized to its area, i.e. the city interpreted as a point light source (${I_0}$ is measured in cd or lm sr−1). The functions $Si( x )$ and $Ci( x )$ introduced above are sine and cosine integrals, respectively (Press et al. 2007).The argument $x$ is the product of distance $\ D$ and parameter $a$, with
(2)$$\begin{eqnarray*}
a\ = \frac{{\tilde{\tau }}}{{\tilde{H}}}\ \left( {1 + \frac{{{{\tilde{\mu }}_0}}}{\mu }} \right),
\end{eqnarray*}$$where ${\mu _0}$ is the mean effective cosine of the emission angle (see section 2 of Kocifaj et al. 2019) and $\tilde{H}\ $ is the altitude up to which a homogeneous atmosphere would extend. Here we assume that $\tilde{H}\ $ is composed of the weighed contributions of molecular and aerosol constituents, i.e.
(3)$$\begin{eqnarray*}
\tilde{H} = \frac{{{H_\mathrm{ R}}{{\tilde{\tau }}_\mathrm{ R}} + {H_\mathrm{ a}}{{\tilde{\tau }}_\mathrm{ a}}}}{{{{\tilde{\tau }}_\mathrm{ R}} + {{\tilde{\tau }}_\mathrm{ a}}}},
\end{eqnarray*}$$where ${\tilde{\tau }_\mathrm{ R}}$ and ${\tilde{\tau }_\mathrm{ a}}$ are the Rayleigh and aerosol components of the total optical depth of a cloudless atmosphere except for ozone or water-vapour absorption bands, i.e. $\tilde{\tau } = {\tilde{\tau }_\mathrm{ R}} + {\tilde{\tau }_\mathrm{ a}}$ (Utrillas et al. 2000). Here, ${H_\mathrm{ R}}$ and ${H_\mathrm{ a}}$ are the Rayleigh and aerosol components of the atmospheric scaleheights. Aerosol scaleheight typically ranges from 1–3 km (Wu et al. 2011), while ${H_\mathrm{ R}}$ = 8 km is a commonly accepted value in many studies (see e.g. Waquet et al. 2009). The emission function for a city, i.e. the luminous intensity emitted by that city, is generally unknown, so an isotropic-like radiator appears to be a useful zero-order approximation, resulting in the following formula:
(4)$$\begin{eqnarray*}
J \left( z \right) = {W_0}\ \frac{9}{{64{\pi ^2}}}\frac{{\tilde{\tau }{\mathrm{ e}^{ - D\tilde{\tau }/\left( {3\tilde{H}} \right)}}}}{{\mu \tilde{H}D}}\left\{ {Ci\left( {aD} \right)\sin\left( {aD} \right) + \left[ {\frac{\pi }{2} - Si\left( {aD} \right)} \right]\cos\left( {aD} \right)} \right\},
\end{eqnarray*}$$where ${W_0}$ (lm) is the total amount of lumens directed above the horizontal; consequently, it is obtained as a superposition of both direct emission upwards and isotropic reflection from the ground. If the entirety of light approaching the sky is due only to diffuse reflection from the ground, then ${W_0} = G \cdot W_0^{\rm installed}$, where $G$ is the diffuse reflectance and $W_0^{\rm installed}$ is the total lumen output from all light sources in the city.
Citation: Kocifaj et al. (2019)
Citation function: Uses; Uses
Citation: Kocifaj et al. 2019
Citation function: Uses; Uses