1 Introduction

The goal of this section is to compute the transmittance between two points in the atmosphere.

2 Model

The transmittance $\mathfrak{t}(\bp,\bq)$ between two points $\bp$ and $\bq$ is the exponential of the opposite of the optical depth, itself equal to the integral of the sum of the absorption and scattering coefficients along the path between these two points. It depends on the wavelength, as the absorption and scattering coefficients do. However, if we look at averages over the short wave and long wave ranges, we can reduce this to 2 transmittance functions:

the transmittance $\mathfrak{t}_s$ in the short wave range, which depends on the scattering coefficient $k_s$ (there is no absorption in the short wave range),
the transmittance $\mathfrak{t}_e$ in the long wave range, which depends on the emissivity / absorptivity coefficient $k_e$ (there is no scattering in the long wave range).

The two cases are similar and in the following we thus simply drop the $s$ and $e$ indices. From the definition above we get: \begin{equation} \mathfrak{t}(\bp,\bq)=\exp\left(-\int_\bp^\bq k(\bx) \diff x\right)\label{eq:t} \end{equation}

In the Rama atmosphere the coefficients $k(\bx)$ only depend on $r$ and have the following form (see the Atmosphere Temperature and Pressure section): \begin{equation} k(\bx)=k(r)=k(0)\exp\left(\frac{\omega^2}{R_sT_a}\frac{r^2}{2}\right)\label{eq:k} \end{equation} where $r$ is the distance from $\bx$ to the rotation axis of Rama. Substituting this into Eq. \eqref{eq:t} yields an expression which can not be computed analytically. However, it can be approximated by replacing $k(\bx)$ with the first few terms of its Taylor series. This is justified because the maximum value of the exponent in Eq. \eqref{eq:k} is less than $1/2$, and because the approximation $e^x\approx 1+x+x^2/2$ is very good for $0<x<1/2$ — the relative error is less than 1.5%. This gives: \begin{equation} \mathfrak{t}(\bp,\bq)\approx\exp\left(-k(0)\int_\bp^\bq 1+Cr^2+\frac{C^2}{2}r^4\ \diff x\right)\ \mathrm{where}\ C=\frac{\omega^2}{2R_sT_a}\label{eq:tapprox} \end{equation}

Using the cylindrical coordinates $(r_p,\theta_p)$ and $(r_q,\theta_q)$ of $\bp$ and $\bq$, and by posing: \begin{align} \bx&=\bp+\mathfrak{t}(\bq-\bp)\\ l&=\Vert\bq-\bp\Vert\\ a&=C[r_p^2+r_q^2-2r_pr_q\cos(\theta_p-\theta_q)]\\ b&=2C[r_pr_q\cos(\theta_p-\theta_q)-r_p^2]\\ c&=Cr_p^2 \end{align} we get: \begin{align} \mathfrak{t}(\bp,\bq)&\approx\exp\left(-k(0)l\int_0^1 1+(at^2+bt+c)+(at^2+bt+c)^2/2\ \diff t\right)\\ &=\exp\left(-k(0)l\left[1+c+\frac{b+bc+c^2}{2}+\frac{a}{3}+\frac{ab}{4}+\frac{b^2+2ac}{6}+\frac{a^2}{10}\right]\right) \end{align} which is much cheaper to evaluate than an numerical integration of Eqs. \eqref{eq:t} and \eqref{eq:k}.

3 Conclusion

The transmittance between two points in Rama is given by \begin{align*} \mathfrak{t}(\bp,\bq)&=\exp\left(-k(0)\int_\bp^\bq \exp\left(\frac{\omega^2}{R_sT_a}\frac{r^2}{2}\right) \diff x\right)\\ &\approx\exp\left(-k(0)\Vert\bq-\bp\Vert\left[1+c+\frac{b+bc+c^2}{2}+\frac{a}{3}+\frac{ab}{4}+\frac{b^2+2ac}{6}+\frac{a^2}{10}\right]\right) \end{align*} where $k(0)$ is the scattering or emissivity coefficient on the rotation axis, depending on which wavelength range is considered, and where \begin{equation*} a=\frac{\omega^2}{2R_sT_a}[r_p^2+r_q^2-2r_pr_q\cos(\theta_p-\theta_q)]\quad b=\frac{\omega^2}{R_sT_a}[r_pr_q\cos(\theta_p-\theta_q)-r_p^2]\quad c=\frac{\omega^2}{2R_sT_a}r_p^2 \end{equation*}