Appendix A: Day-night temperature variations and chemical composition

Here we explore the implications of day-night temperature variations on the abundances of a condensing species. Suppose that on the day side, a given condensate (e.g. iron) condenses at a pressure level $P^\star_{\rm day}$ and a corresponding temperature $T_{\rm day}$. If the temperature on the night side $T_{\rm night}$ is smaller, at what pressure $P^\star_{\rm night}$will condensation take place?

Assuming an ideal gas, we write the Clausius-Clapeyron equation as

$${{\rm d}\ln p\over {\rm d}\ln T}=\beta$$ (A.1)

where p is the partial pressure of saturation of the condensing species and $\beta=L/{\cal R}T$ is the ratio of the latent heat of condensation L to the thermal energy ${\cal R}T$. For most condensing species of significance here, $\beta\approx 10$. We furthermore assume that $\beta$ is independent of T and P, which introduces only a slight error in our estimates.

By definition, on the day side, the saturation abundance of the condensing species, x=p/P is maximal and equal to $x^\star$ at $P=P^\star_{\rm day}$. On the other hand, the night side temperature is lower and the abundance becomes:

$$\ln x(P^\star_{\rm day})=\ln x^\star - \beta \ln (T_{\rm day}/ T_{\rm night}).$$ (A.2)

In order to reach condensation, i.e. $x=x^\star$, one has to penetrate deeper into the atmosphere. Equation (A.1) implies that

$${{\rm d}\ln x\over {\rm d}\ln P}=\beta\nabla_T -1,$$ (A.3)

and hence, on the night side,

$$\ln x(P)=\ln x(P^\star_{\rm day}) + (\beta\nabla_T-1) \ln(P/P^\star_{\rm day}),$$ (A.4)

assuming that $\nabla_T$ is constant. Using Eq. (A.2), one obtains the condensation pressure on the night side:

$${P^\star_{\rm night}\over P^\star_{\rm day}} = \left({T_{\rm day}\over T_{\rm night}}\right)^{\beta/(\beta\nabla_T -1)}.$$ (A.5)

Using $\beta\sim 10$, $\nabla_{\rm T}\sim 0.15$ and $T_{\rm day}/T_{\rm night}\sim 1.2$, one finds $P^\star_{\rm night}\sim 38 P^\star_{\rm day}$, a very significant variation of the condensation pressure. This implies that air flowing on constant pressure levels around the planet would lead to a rapid depletion of any condensing species on the day side, compared to what would be predicted from chemical equilibrium calculations. This can potentially also remove important absorbing gases from the day side, as in the case of TiO, which can be removed by CaTiO₃ condensation, or Na, removed by Na₂S condensation (Lodders 1999). Of course, most of the variation depends on the exponential factor $\beta/(\beta\nabla_{\rm T}-1)$, which is infinite in the limit when the atmospheric temperature profile and the condensation profile are parallel to each other.

In the discussion, we implicitly assumed $\beta\nabla_{\rm T}-1>0$; however, when the atmosphere is close to an isotherm, this factor can become negative. In this case the day/night effect is even more severe, as the condensing species is entirely removed from this quasi-isothermal region.