Ionization and transport in partially ionized multicomponent plasmas: Application to atmospheres of hot Jupiters

We study ionization and transport processes in partially ionized multicomponent plasmas. The plasma composition is calculated via a system of coupled mass-action laws. The electronic transport properties are determined by the electron-ion and electron-neutral transport cross sections. The influence of electron-electron scattering is considered via a correction factor to the electron-ion contribution. Based on these data, the electrical and thermal conductivities as well as the Lorenz number are calculated. For the thermal conductivity, we consider also the contributions of the translational motion of neutral particles and of the dissociation, ionization, and recombination reactions. We apply our approach to a partially ionized plasma composed of hydrogen, helium, and a small fraction of metals (Li, Na, Ca, Fe, K, Rb, and Cs) as typical for atmospheres of hot Jupiters. We present results for the plasma composition and the transport properties as a function of density and temperature and then along typical $P\text{−}T$ profiles for the outer part of the hot Jupiter HD 209458b. The electrical conductivity profile allows revising the Ohmic heating power related to the fierce winds in the planet's atmosphere. We show that the higher temperatures suggested by recent interior models could boost the conductivity and thus the Ohmic heating power to values large enough to explain the observed inflation of HD 209458b.

Figure 1

Composition of hydrogen plasma as a function of temperature for a mass density of $\rho ={10}^{-5}$ g/${\text{cm}}^3$. Solid lines show analytical results via Eqs. (17, 18, 19) and dashed lines numerical results. The normalization $n_0=n_{{\text{H}}_2}+n_{\text{H}}+n_{{\text{H}}^{+}}$ refers here to the total number density of hydrogenic species in the plasma.

Figure 2

Ionization degree of the PIP as a function of temperature for different mass densities.

Figure 3

Electrical conductivity of the PIP as a function of temperature for different mass densities.

Figure 4

Thermal conductivity of the PIP as a function of temperature for different mass densities.

Figure 5

Lorenz number of the PIP as a function of temperature for different mass densities.

Figure 6

Thermal conductivity of partially ionized hydrogen plasma as a function of temperature at a constant pressure of 1 bar. We compare our results with the arc-discharge experiment of Behringer and van Cung [31], which was evaluated using the local thermodynamic equilibrium assumption.

Figure 7

Total thermal conductivity $\lambda$ according to Eq. (29) as a function of temperature at ${10}^{-5}$ g/${\mathrm{cm}}^3$. The contributions of the translational motion of neutrals ${\lambda }_{tr}$ and of the heat of chemical reactions ${\lambda }_r$ and the electronic contribution are shown separately.

Figure 8

Pressure-temperature profiles of the atmosphere of HD 209458b. Shown are the four atmospheric models used in this work, three without an inversion, namely, G (yellow), L (red), and S (orange), and one with an inversion in the temperature, I (blue), located at about 0.03 bar. Further features of the models are displayed: the location of the radiative-convective boundary, the onset of the convective interior, and the characteristic temperatures $T_{\text{iso},1}$ and $T_{\text{iso},2}$.

Figure 9

Temperature $T$, ionization degree $\alpha$, thermal conductivity $\lambda$, and electrical conductivity ${\sigma }_e$ for different planetary interior models along the pressure axis of HD 209458b, specifically, for the four atmospheric models used in this work: G (yellow), L (red), S (orange), and one with an inversion, I (blue). Circles in the temperature profile represent the location of the radiative-convective boundary.

Figure 10

Electrical conductivity ${\sigma }_e$ for model I along the radius axis of HD 209458b, compared to the results of Batygin and Stevenson [11]. We show their original data (gray) as well as a shifted version (cyan) to ease the comparison (see the text).