1 Introduction

The goal of this section is to compute the integral, divided by $4\pi$, of the radiance received at a point inside the atmosphere, from all the other points of the atmosphere, while taking the atmospheric absorption into account.

2 Average volume radiance

Figure 1: Average radiance in a cylinder.

Consider a point $\bp$ inside the atmosphere, at a distance $r_p$ in meters from the axis. The total radiance $4\pi J_v(\bp)$ received at $\bp$, from all the points inside the atmosphere, is the integral over all the directions $\bw$ of the product of (see Fig. 1):

the integral over all the points $\bx$ along the path from $\bp$ to $\bq$ of the radiance $L(\bx)$ emitted per meter at $\bx$ (supposed here to be isotropic), times the atmospheric transmittance between $\bp$ and $\bx$, times the infinitesimal length element $\diff l$,
the infinitesimal solid angle $\diff\bw$.

Thanks to the translation invariance, the radiance emitted at $\bx$ is the same as the radiance emitted at $\bx^\bot$, and the transmittance $\mathfrak{t}(\bp,\bx)$ between $\bp$ and $\bx$ is related to the transmittance $\mathfrak{t}(\bp,\bx^\bot)$ by the equation $\mathfrak{t}(\bp,\bx)=\mathfrak{t}(\bp,\bx^\bot)^{1/\cos\phi}$, where $\varphi$ and $\phi$ are defined in Fig. 1. Finally, the infinitesimal length element $\diff l$ is equal to $\diff x/\cos\phi$, and the solid angle $\diff\bw$ is equal to $\cos\phi\diff\varphi\diff\phi$. The average radiance $J_v$ is thus: \begin{align} J_v(\bp)&=\frac{1}{4\pi}\int_{-\pi}^{\pi}\int_{-\pi/2}^{\pi/2}\int_{\bp}^{\bq^\bot}L(\bx^\bot)\mathfrak{t}(\bp,\bx^\bot)^{1/\cos\phi}\frac{\diff x}{\cos\phi}\cos\phi\diff\phi\diff\varphi\\ &=\frac{1}{4\pi}\int_{-\pi}^{\pi}\int_{\bp}^{\bq^\bot}L(\bx^\bot)\left(\int_{-\pi/2}^{\pi/2}\mathfrak{t}(\bp,\bx^\bot)^{1/\cos\phi}\diff\phi\right)\diff x\diff\varphi\\ &=\frac{1}{4}\int_{-\pi}^{\pi}\int_{\bp}^{\bq^\bot}L(\bx^\bot)f_0(\mathfrak{t}(\bp,\bx^\bot))\diff x\diff\varphi \end{align} where $f_0$ is defined by (see Fig. 2): \begin{equation} f_0(x)=\frac{2}{\pi}\int_0^{\pi/2}x^{1/\cos\phi}\diff\phi \end{equation}

Figure 2: Plot of the function $f_0$.

Instead of using the coordinates $(x,\varphi)$ to specify the point $\bx^\bot$, it is more convenient to use its cylindrical coordinates $(r,\theta)$ defined in Fig. 1. These coordinates are linked by the equation $[r\sin\theta,r\cos\theta]=[x\sin\varphi,x\cos\varphi-r_p]$, from which we get $x^2=r_p^2+r^2+2rr_p\cos\theta$. We can also express the infinitesimal surface element $\diff S$ using both coordinate systems, yielding $\diff S=x \diff x\diff\varphi=r \diff r\diff\theta$. Putting all this together we get: \begin{equation} \diff x\diff\varphi=\frac{\diff S}{x}=\frac{r}{\sqrt{r_p^2+r^2+2rr_p\cos\theta}}\diff r\diff\theta \end{equation}

We can thus rewrite the average radiance $J_v$ as \begin{equation} J_v(\bp)=\frac{1}{4}\int_{-\pi}^{\pi}\int_0^{r_s}L(\bx^\bot)f_0(\mathfrak{t}(\bp,\bx^\bot))\frac{r}{\sqrt{r_p^2+r^2+2rr_p\cos\theta}}\diff r\diff\theta \end{equation}

Finally, instead of using an angle $\theta$ measured from the opposite point to $\bp$, we can use an absolute coordinate $\theta$, represent $\bp$ by its coordinate $r_p,\theta_p$, and define $L$ and $T$ based on these absolute coordinates. Doing this gives the result presented in the conclusion.

3 Conclusion

The average volume radiance $J_v$ in an infinite cylinder can be computed from the radiance $L(r,\theta)$ emitted per meter, supposed isotropic, with the following formula: \begin{equation} J_v(r_p,\theta_p)=\frac{1}{4}\int_{2\pi}\int_0^{r_s}L(r,\theta)f_0(\mathfrak{t}(r_p,\theta_p,r,\theta))\frac{r}{\sqrt{r_p^2+r^2-2rr_p\cos(\theta-\theta_p)}}\diff r\diff\theta \end{equation} where $\mathfrak{t}(r_p,\theta_p,r,\theta)$ is the transmittance of the atmosphere between the points of coordinates $(r_p,\theta_p)$ and $(r,\theta)$, in a plane perpendicular to the axis, and where $f_0(x)=\frac{2}{\pi}\int_0^{\pi/2}x^{1/\cos\phi}\diff\phi$.