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 points of the surface, while taking the atmospheric absorption into account.

2 Average radiance inside the atmosphere

Figure 1: Average radiance in a cylinder.

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

the radiance $L(\bq)$ emitted from $\bq$,
the transmittance $\mathfrak{t}(\bp,\bq)$ of the atmosphere between $\bp$ and $\bq$,
the infinitesimal solid angle $\diff\bw$.

Thanks to the translation invariance, the radiance emitted at $\bq$ is the same as the radiance emitted at $\bq^\bot$, and the transmittance $\mathfrak{t}(\bp,\bq)$ between $\bp$ and $\bq$ is related to the transmittance $\mathfrak{t}(\bp,\bq^\bot)$ by the equation $\mathfrak{t}(\bp,\bq)=\mathfrak{t}(\bp,\bq^\bot)^{1/\cos\phi}$, where $\varphi$ and $\phi$ are defined in Fig. 1. Finally, the solid angle $\diff\bw$ is equal to $\cos\phi\diff\varphi\diff\phi$. The average radiance $J$ is thus: \begin{align} J(\bp)&=\frac{1}{4\pi}\int_{4\pi}L(\bq^\bot)\mathfrak{t}(\bp,\bq^\bot)\diff\bw=\int_{-\pi}^{\pi}L(\bq^\bot)\left(\int_{-\pi/2}^{\pi/2} \mathfrak{t}(\bp,\bq^\bot)^{1/\cos\phi}\cos\phi\diff\phi\right)\diff\varphi\\ &=\frac{1}{2\pi}\int_{-\pi}^{\pi}L(\bq^\bot)f_1(\mathfrak{t}(\bp,\bq^\bot))\diff\varphi \end{align} where $f_1$ is defined by (see Fig. 2): \begin{equation} f_1(x)=\int_0^{\pi/2}x^{1/\cos\phi}\cos\phi\,\diff\phi \end{equation}

Figure 2: Plot of the function $f_1$.

Instead of using the angle $\varphi$, it is more convenient to use the angle $\theta$, as defined in Fig. 1. These two angles are linked by the relation $\tan\varphi=r_s\sin\theta/(r+r_s\cos\theta)$, from which we deduce that $\diff\varphi=r_s(r_s+r\cos\theta)\diff\theta/(r_s^2+r^2+2rr_s\cos\theta)$. The average radiance can then be rewritten as: \begin{equation} J(\bp)=\frac{1}{2\pi}\int_{-\pi}^{\pi}L(\bq^\bot)f_1(\mathfrak{t}(\bp,\bq^\bot))\frac{r_s(r_s+r\cos\theta)}{r_s^2+r^2+2rr_s\cos\theta}\diff\theta \end{equation}

We can check the correctness of this result for the case where $L$ is constant, and the transmittance is 1, i.e. there is no atmosphere. In this case the average radiance should be $L$, and this is indeed what we get: $J=\frac{1}{2\pi}L\int_{-\pi}^{\pi}\frac{r_s(r_s+r\cos\theta)}{r_s^2+r^2+2rr_s\cos\theta}\diff\theta=L$.

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 coordinates $r_p,\theta_p$, and define $L$ and $T$ based on these absolute coordinates. Doing this gives the result presented in the conclusion.

3 Conclusion

The average radiance $J$ received in the atmosphere in an infinite cylinder can be computed from the radiance $L(\theta)$ emitted at the surface with the following formula: \begin{equation} J(r_p,\theta_p)=\frac{1}{2\pi}\int_{2\pi}L(\theta)f_1(\mathfrak{t}(r_p,\theta_p,r_s,\theta))\frac{r_s(r_s-r_p\cos(\theta-\theta_p))}{r_s^2+r_p^2-2r_pr_s\cos(\theta-\theta_p)}\diff\theta \end{equation} where $\mathfrak{t}(r_p,\theta_p,r_s,\theta)$ is the transmittance of the atmosphere between the points of coordinates $(r_p,\theta_p)$ and $(r_s,\theta)$, in a plane perpendicular to the axis, and where $f_1(x)=\int_0^{\pi/2}x^{1/\cos\phi}\cos\phi\,\diff\phi$.