tokens: the system cannot fulfil thermal and mechanical equilibrium conditions simultaneously unless some further assumptions are implemented. Since the initial state is in mechanical equilibrium and the upper and the lower layers have different densities, their temperatures cannot be the same. Thermal equilibrium, however, implies that both layers have the same temperature. In order to resolve this contradiction, JK proposed that one layer is optically thin and the second layer is optically thick. They also assumed that the system is adiabatic. In the optically thin regime, where opacity on one or both sides of the interface is negligible, continuity of E r holds on whereas T g is independent of E r . There are, however, further alternative ways to specify background state self-consistently. Jiang et al. ( 2013 ), for instance, argued that even when the background state is not strictly in thermal equilibrium, its evolution affects subsequent thermal evolution very slightly so long as the thermal time-scale is much longer than the instability time-scale. Under these circumstances, therefore, the initial configuration can be imagined to be in thermal equilibrium for exploring RT instability. Jiang et al. ( 2013 ) constructed an initial state with zero absorption opacity and effectively infinite thermal time-scale. As we mentioned earlier, JK assumed that one side of interface is in the adiabatic regime, and the other side is in the optically thin regime. Here, we follow JK assumptions for specifying the initial equilibrium state, which are applicable to some astrophysical systems. The momentum and the energy equations are written as where . Equation ( 34 ) is written based on the radiative equilibrium assumption. The parameter E represents the Eddington limit of the background state. We note that the introduced parameter E is then replaced by E 1 and E 2 for the lower and the upper layers, respectively. Also, we have Introducing the ratio of the radiation pressure to the gas pressure as r = p r / p g , we can then write and We note that the Eddington limit parameter E is a fixed input parameter; however, the parameter r is not a constant under our imposed conditions. 4.2 Linear perturbations After deriving the above equilibrium solutions, we can now linearly perturb RMHD equations. Following JK , we also apply adiabatic approximation by which we mean perturbed energy flux is neglected. It means that we require F ́ ≪ E t u ́. Using this approximation, we can assume that the opacity is a constant. Therefore, the parameter E becomes a constant, which is actually our implemented assumption. This parameter, however, may be different in the upper and lower layers. We thereby arrive at the following equations: As before, we assume that all the perturbed quantities are proportional to exp (ω t + i k x x ), and the above equations after lengthy but straightforward calculations reduce to the following equations: where parameters B and C depend on the input
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