1 Introduction

The goal of this section is to compute the irradiance at a point on the surface, due to the radiance of all the points inside the atmosphere, while taking the atmospheric absorption into account.

2 Irradiance from the atmosphere

Figure 1: Irradiance in a cylinder.

Consider a point $\bp$ on the surface. The irradiance $E_v(\bp)$ at this point, from all the points inside the atmosphere, is the integral over all the directions $\bw$ in the hemisphere above $\bp$ 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 cosine $\bw\cdot\bn$ of the angle between $\bw$ and the surface normal $\bn$ at $\bp$,
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 $\diff x/\cos\phi$, the scalar product $\bw\cdot\bn$ is equal to $\cos\varphi\cos\phi$ and the solid angle $\diff\bw$ is equal to $\cos\phi\diff\varphi\diff\phi$. The irradiance $E_v$ is thus: \begin{align} E_v(\bp)&=\int_{-\pi/2}^{\pi/2}\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^2\phi\cos\varphi\diff\phi\diff\varphi\\ &=\int_{-\pi/2}^{\pi/2}\int_{\bp}^{\bq^\bot}L(\bx^\bot)\left(\int_{-\pi/2}^{\pi/2}\mathfrak{t}(\bp,\bx^\bot)^{1/\cos\phi}\cos\phi\diff\phi\right)\cos\varphi\diff x\diff\varphi\\ &=2\int_{-\pi/2}^{\pi/2}\int_{\bp}^{\bq^\bot}L(\bx^\bot)f_1(\mathfrak{t}(\bp,\bx^\bot))\cos\varphi\diff x\diff\varphi \end{align} where $f_1$ is the function defined in the previous section.

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 $[r\sin\theta,r\cos\theta]=[x\sin\varphi,x\cos\varphi-r_s]$, from which we get $x\cos\varphi=r_s+r\cos\theta$ and $x^2=r_s^2+r^2+2rr_s\cos\theta$. 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} \cos\varphi\diff x\diff\varphi=\cos\varphi\frac{\diff S}{x}=\diff S\frac{x\cos\varphi}{x^2}=r\diff r\diff\theta\frac{r_s+r\cos\theta}{r_s^2+r^2+2rr_s\cos\theta} \end{equation}

We can thus rewrite the irradiance $E_v$ as \begin{equation} E_v(\bp)=2\int_{-\pi}^{\pi}\int_0^{r_s}L(\bx^\bot)f_1(\mathfrak{t}(\bp,\bx^\bot))\frac{r(r_s+r\cos\theta)}{r_s^2+r^2+2rr_s\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 $\theta_p$, and define $L$ and $T$ based on these absolute coordinates. Doing this gives the result presented in the conclusion.

3 Conclusion

The volume irradiance $E_v$ in an infinite cylinder can be computed from the radiance $L(r,\theta)$ emitted per meter, supposed isotropic, with the following formula: \begin{equation} E_v(\theta_p)=2\int_{2\pi}\int_0^{r_s}L(r,\theta)f_1(\mathfrak{t}(r_s,\theta_p,r,\theta))\frac{r(r_s-r\cos(\theta-\theta_p))}{r_s^2+r^2-2rr_s\cos(\theta-\theta_p)}\diff r\diff\theta \end{equation} where $\mathfrak{t}(r_s,\theta_p,r,\theta)$ is transmittance of the atmosphere between the points of coordinates $(r_s,\theta_p)$ and $(r,\theta)$, in a plane perpendicular to the axis.