Rama physics

We assume here that Rama is designed to provide an environment as close as possible to the one on Earth. This includes, in particular, the gravity, the temperature and the lighting. Our goal is to determine whether this is possible or not, at least in theory. We start with the gravity, and then study the lighting, temperature and winds, in several steps:

• We first compute the average lighting and temperature, and show that these average values, at the ground level, do not depend on the atmospheric properties.
• We thus continue by computing the lighting and temperature variations in the absence of atmosphere, and show that these variations are very small.
• This allows us to study an averaged, axisymetric case, in order to get approximate temperature, pressure and density profiles for the atmosphere. The result is that, under this hypothesis, the atmosphere temperature is uniform and the density profile is almost independent of the temperature variations.
• This "independence" of the density profile from the temperature allows us to suppose it given, in order to get an approximation of the temperature variations in the atmosphere, coupled with the temperature variations at the ground level, in the non axisymetric case.
• We finally study the wind velocity and the natural convection patterns in Rama, due to the buyoancy forces, themselves due to the temperature variations resulting from radiative transfer, obtained at the previous step.

Doing this study on a 3D model would be quite complex. In order to simplify computations, we use here an idealized model, namely an infinitely long cylinder, invariant under translation around its axis (but not invariant by rotation around this axis, unless stated otherwise, because of the 3 linear light sources 120° apart from each other). This allows us to use a 2D model in most cases, which is much simpler (for radiative transfer we still need a 3D model, but the translation symmetry is useful, even in this case, to reduce complexity).

The above steps are detailed in the following sections, using auxiliary results detailed in the Appendix, and using the notations and symbols defined below. The conclusions of this analysis are presented in the final section.

## Notations and symbols

We note scalars using a regular font (e.g. $r,\theta,z$), points and vectors using a bold font (e.g. $\bx,\bp,\bw$), operators using a caligraphic font (e.g. $\mathcal{E}, \mathcal{J}, \mathcal{S}$), and use the following constants and symbols:

 Constant Description Value Unit $r_s,r_e$ Rama's surface radius, external radius $8000, 9000$ $m$ $l_s$ Rama's average inner length $40000$ $m$ $p_s$ Rama's pressure at the surface $101325$ $Pa$ $\omega$ Rama's angular speed $2\pi/180$ $rad.s^{-1}$ $S$ Rama's average linear light source power $6.7\,10^6$ or $14.9\,10^6$ $W.m^{-1}$ $\Delta$ Rama's ratio between inner and outer long wave radiations $1.3$ or $2.9$ $\emptyset$ $\alpha$ Rama's average albedo $0.15$ $\emptyset$ $\bar{\alpha}$ Rama's "albedo" in the long wave range, $\Delta/(\Delta+1)$ $0.56$ or $0.74$ $\emptyset$ $\alpha_e$ Earth's average albedo $0.3$ $\emptyset$ $S_0$ Solar constant $1360$ $W.m^{-2}$ $R_s$ Specific gas constant of air $287.68$ $J.kg^{-1}.K^{-1}$ $c_v$ Heat capacity at constant volume of air $716.8$ $J.kg^{-1}.K^{-1}$ $\sigma$ Stefan-Bolztmann constant $5.67\,10^{-8}$ $W.m^{-2}.K^{-4}$ Symbol Description Unit $r,\theta$ Cylindrical coordinates $m,rad$ $\mathbf{e}_r,\mathbf{e}_{\theta},\mathbf{e}_z$ Basis vectors of the cylindrical coordinate system $\emptyset$ $\varphi,\phi$ Angles $rad$ $x,l$ Distances or lengths $m$ $t$ Time $s$ $\bp,\bq,\bx$ Points $m$ $\bn$ Normal vector $\emptyset$ $\bw$ Unit direction vector $\emptyset$ $\diff\bw$ Solid angle $sr^{-1}$ $Y$ Young's modulus $Pa$ $\nu$ Poisson ratio $\emptyset$ $\epsilon_r,\epsilon_{\theta}$ Radial and tangential strain $\emptyset$ $\sigma_r,\sigma_{\theta}$ Radial and tangential stress $Pa$ $p$ Pressure $Pa$ $\rho$ Volumetric mass density $kg.m^{-3}$ $e$ Internal energy $J$ $\lambda$ Wavelength $m$ $T,T_s,T_a$ Temperature (generic, at the surface, of the atmosphere) $K$ $I,L,\bar{L}$ Radiance in the short wave range, in the long wave range $W.m^{-2}.sr^{-1}$ $B$ Integrated black body radiance, $\sigma T^4/\pi$ $W.m^{-2}.sr^{-1}$ $J, \bar{J}$ Average of the radiance over all directions, in the short and long wave ranges $W.m^{-2}$ $E, \bar{E}$ Irradiance in the short wave range, in the long wave range $W.m^{-2}$ $P$ Phase function $\emptyset$ $\epsilon$ Emissivity $\emptyset$ $k_a,k_s,k_e$ Absorption, scattering and emissivity coefficients $m^{-1}$ $\mathfrak{t}, \tau$ Transmittance, optical depth $\emptyset$