ATMOSPHERIC ESCAPE BY MAGNETICALLY DRIVEN WIND FROM GASEOUS PLANETS. II. EFFECTS OF MAGNETIC DIFFUSION

In this work we use a numerical simulation code that was originally developed for the calculation of the acceleration of solar wind (Suzuki & Inutsuka 2005, 2006). At the surface of stars, various types of magnetic waves are excited (Matsumoto & Suzuki 2012, 2014). The excited waves propagate upward from the surface and then heat up the coronal region and drive the stellar wind by various dissipation processes of the waves (Goldstein 1978; Heyvaerts & Priest 1983; Terasawa et al. 1986; Kudoh & Shibata 1999; Matthaeus et al. 1999). It is thought that the Alfvén wave is especially important for the transfer of the energy. This mechanism of the energy transport from the surface to the upper atmosphere can be applied for objects that have their own magnetic field. Although details are uncertain, hot Jupiters are expected to have magnetic fields, so this mechanism is applicable to hot Jupiters. In our previous research, we extended the simulation code to calculate the atmospheric escape from gaseous planets, especially from hot Jupiters (Tanaka et al. 2014). Additionally, this mechanism is applied for the calculation of the structure of the atmosphere of brown dwarfs, and it is implied that the heating by the dissipation of MHD waves is also important for the brown dwarfs' atmosphere (Sorahana et al. 2014).

As for our MHD simulations, we basically follow Tanaka et al. (2014), except that we consider magnetic diffusion; we solve time-dependent equations and treat the propagation and dissipation of the MHD waves in a superradially open magnetic flux tube in the atmosphere. We adopt the same functional form for the superradial expansion factor, f(r), as in Tanaka et al. (2014), which was originally introduced by Kopp & Holzer (1976):

$$\begin{eqnarray}&&f(r)=\displaystyle \frac{{e}^{\frac{r-{r}_{0}-{h}_{1}}{{h}_{1}}}+{f}_{0}-(1-{f}_{0})/e}{{e}^{\frac{r-{r}_{0}-{h}_{1}}{{h}_{1}}}+1},\end{eqnarray} \tag{ 1 }$$

where h₁ is the typical height of closed magnetic loops, and the subscript 0 means the surface.

The effects of resistivity explicitly appear in the energy equation and induction equation:

$$\begin{eqnarray}&&\rho \displaystyle \frac{d}{{dt}}(e+\displaystyle \frac{{v}^{2}}{2}+\displaystyle \frac{{B}^{2}}{8\pi \rho }-\displaystyle \frac{{{GM}}_{\star }}{r})+\displaystyle \frac{1}{{r}^{2}f}\displaystyle \frac{\partial }{\partial r}\{{r}^{2}f[(p+\displaystyle \frac{{B}^{2}}{8\pi }){v}_{{\rm{r}}}\\ &&\quad -\displaystyle \frac{{B}_{{\rm{r}}}}{4\pi }({\boldsymbol{B}}\cdot {\boldsymbol{v}})-\displaystyle \frac{\eta }{4\pi }\displaystyle \frac{{B}_{\perp }}{r\sqrt{f}}\displaystyle \frac{\partial }{\partial r}(r\sqrt{f}{B}_{\perp })]\}\\ &&\quad +\displaystyle \frac{1}{{r}^{2}f}\displaystyle \frac{\partial }{\partial r}({r}^{2}{{fF}}_{{\rm{c}}})+{q}_{{\rm{r}}}=0,\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {B}_{\perp }}{\partial t}=\displaystyle \frac{1}{r\sqrt{f}}\displaystyle \frac{\partial }{\partial r}[r\sqrt{f}({v}_{\perp }{B}_{{\rm{r}}}-{v}_{{\rm{r}}}{B}_{\perp })+\eta \displaystyle \frac{\partial }{\partial r}(r\sqrt{f}{B}_{\perp })],\end{eqnarray} \tag{ 3 }$$

where ρ, ${\boldsymbol{v}}$, p, e, and ${\boldsymbol{B}}$ are the density, velocity, pressure, specific energy, and magnetic field strength, respectively. The subscripts r and ⊥ denote the radial and perpendicular components, respectively; $d/{dt}$ and $\partial /\partial t$ denote the Lagrangian and Eulerian derivatives, respectively. G and ${M}_{\star }$ denote the gravitational constant and the mass of a central object, respectively, and η is the magnetic resistivity. F_c and q_R are the thermal conductive flux and radiative cooling, respectively (Anderson & Athay 1989; Landini & Monsignori-Fossi 1990; Sutherland & Dopita 1993; see Tanaka et al. 2014 for details). In these equations, the effect of the superradial expansion of the magnetic flux tubes appears as the factor $r\sqrt{f}$. Note that the term with η in the energy Equation (2) indicates Joule (ohmic) heating and the term with η in the induction Equation (3) denotes diffusion of magnetic field lines.

The most dominant origin of the magnetic resistivity in the atmosphere of hot Jupiters is the collision between electrons and neutral particles. In such circumstances the magnetic resistivity η depends on gas temperature and an ionization degree as follows:

$$\begin{eqnarray}&&\eta \approx 234\displaystyle \frac{\sqrt{T(K)}}{{x}_{{\rm{e}}}}({\mathrm{cm}}^{2}\;{{\rm{s}}}^{-1}),\end{eqnarray} \tag{ 4 }$$

where ${x}_{{\rm{e}}}={n}_{{\rm{e}}}/n$ is the ionization degree (e.g., Blaes & Balbus 1994), n_e is the number density of electrons, and n is that of neutral and ionized hydrogen. If the second term on the right-hand side of Equation (3) is dominant over the first term, that is, the magnetic resistivity is large, the magnetic waves will diffuse out. Since the gas temperature and ionization degree will be low in the atmosphere of planets, the magnetic resistivity might not be negligible. Therefore, the resistive MHD calculation is needed to evaluate the non-ideal effects on the structure of the atmosphere and the mass loss.

Ionization degrees are determined by a method originally developed for Betelgeuse by Hartmann & Avrett (1984) (see also Harper et al. 2009). We adopt the solar abundance gas (Anders & Grevesse 1989)¹  and calculate the ionization and recombination of H, C, Na, Mg, Al, Si, S, Ca, Fe, and K in the gas phase. Since the typical surface temperature of hot Jupiters is ∼1000 K and it is too low to ionize hydrogen, heavy elements such as Na and K are the dominant ionizing sources in the lower atmosphere.

The inner boundary of the simulations is set at the photosphere in the solar and stellar wind calculations (Suzuki & Inutsuka 2005, 2006; Suzuki 2007; Suzuki et al. 2013). In this paper, we set the inner boundary at the position where ${p}_{0}={10}^{5}\;\mathrm{dyn}\;{\mathrm{cm}}^{-2}(=0.1\mathrm{bar})$. We fix the temperature at the inner boundary, and it is treated as a parameter. The density at the inner boundary is determined by the given temperature and p₀. The outer boundary radius is set to 360 planetary radii, and the number of grids is 6000. In the innermost region, dr corresponds to 0.0001 planetary radii, and dr increases gradually as the distance from the surface.

We set the planet mass to be the Jupiter mass throughout this paper. We inject velocity perturbations at the inner boundary, which excite upward-propagating Alfvén waves, and their value is treated as a parameter. In most cases, amplitude of the injected perturbations is 20% of the sound speed at the surface. In the case of the Sun, the velocity dispersion at the surface is caused by surface convection. On the other hand, several previous works have suggested that the convective-radiative boundary lies in a deep region of the atmosphere (e.g., Burrows et al. 2003). However, three-dimensional calculations of the atmospheric circulation of hot Jupiters suggest the existence of supersonic equatorial flow (e.g., Showman & Guillot 2002). Therefore, it is expected that turbulence can be created at the surface by the strong wind. We assume a broadband spectrum of the perturbations in proportion to $1/\nu$, where ν is the frequency of perturbations. To solve the MHD equations, we adopt a second-order Godunov scheme with the Method of Characteristics (Sano et al. 1999).

We here summarize several useful formulae to analyze and evaluate simulation results.

2.2.1. Scale Height

We treat the surface temperature T₀, the radius of a planet R_p, and the velocity dispersion $\delta v$ at the surface as parameters. We discuss the dependence of the atmospheric structure on these parameters. To evaluate them, we introduce the scale height of the atmosphere. The equation of hydrostatic equilibrium of the atmosphere is

$$\begin{eqnarray}&&\displaystyle \frac{1}{\rho }\displaystyle \frac{{dp}}{{dr}}+\displaystyle \frac{{{GM}}_{{\rm{p}}}}{{r}^{2}}=0.\end{eqnarray} \tag{ 5 }$$

M_p is the mass of a planet. If we assume an isothermal condition, the density profile can be expressed as follows:

$$\begin{eqnarray}\rho & = & {\rho }_{0}\mathrm{exp}[-\displaystyle \frac{r-{R}_{{\rm{p}}}}{{H}_{0}}\displaystyle \frac{{R}_{{\rm{p}}}}{r}]\\ & = & {\rho }_{0}\mathrm{exp}[-\displaystyle \frac{{mg}}{{k}_{{\rm{B}}}{T}_{0}}\displaystyle \frac{(r-{R}_{{\rm{p}}}){R}_{{\rm{p}}}}{r}],\end{eqnarray} \tag{ 6 }$$

where ${\rho }_{0}$ is the density at the surface, $r={R}_{{\rm{p}}}$, and H₀ is the pressure scale height of the atmosphere, denoted as

$$\begin{eqnarray}&&{H}_{0}=\displaystyle \frac{{k}_{{\rm{B}}}{T}_{0}}{{mg}},\end{eqnarray} \tag{ 7 }$$

where ${k}_{{\rm{B}}}$ is the Boltzmann constant, m is the mean molecular weight, and g is the gravitational acceleration at the planetary surface.

2.2.2. Poynting Flux

The behavior of the Alfvénic wave in the atmosphere, such as propagation, reflection, and dissipation, is a key to understanding the atmospheric structure. In the ideal MHD simulations the dissipation of the Alfvénic waves is done by the nonlinear mode conversion. The fluctuations of the magnetic pressure associated with the Alfvénic waves excite compressive waves. These waves eventually steepen to shocklets, which finally dissipate and heat the surrounding gas (Tanaka et al. 2014). This process has been examined in terms of the heating of the solar corona and the acceleration of the solar wind (Hollweg 1982; Kudoh & Shibata 1999; Suzuki & Inutsuka 2005). Recently, a radio observation by JAXA's spacecraft Akatsuki actually revealed radial distribution of compressive waves in the solar corona, which supports this scenario (Miyamoto et al. 2014).

In the resistive MHD simulations, the magnetic diffusion also contributes to the dissipation of MHD waves. When T₀ is lower, the dissipation of MHD waves in the low altitude where the magnetic resistivity is high affects the structure of the atmosphere. Therefore, an energetics argument of propagating MHD waves in the resistive MHD condition is important for understanding the properties of the atmospheric structure and mass loss.

To evaluate the propagation, reflection, and dissipation of the Alfvénic wave energy in the atmosphere, we summarize the Poynting flux carried by Alfvén waves below. The net Poynting flux arising from magnetic tension, which represents the energy flux of Alfvén waves measured from the comoving frame, can be written as follows:

$$\begin{eqnarray}&&{F}_{{\rm{P}}}={-B}_{{\rm{r}}}\displaystyle \frac{{v}_{\perp }{B}_{\perp }}{4\pi }.\end{eqnarray} \tag{ 8 }$$

The Poynting flux can be divided into an outward (parallel with B_r) part and an inward (antiparallel with B_r) part (Jacques 1977; Cranmer et al. 2007; Suzuki et al. 2013). In order to evaluate the inward and outward flux, we introduce Elsässer variables,

$$\begin{eqnarray}&&{z}_{\pm }={v}_{\perp }\mp \displaystyle \frac{{B}_{\perp }}{\sqrt{4\pi \rho }}.\end{eqnarray} \tag{ 9 }$$

By using the Elsässer variables, the net Poynting flux is

$$\begin{eqnarray}&&{F}_{{\rm{P}}}=\displaystyle \frac{1}{4}\rho ({z}_{+}^{2}-{z}_{-}^{2}){v}_{{\rm{A}}}.\end{eqnarray} \tag{ 10 }$$

In this equation, $\rho {z}_{+}^{2}{v}_{{\rm{A}}}/4$ and $\rho {z}_{-}^{2}{v}_{{\rm{A}}}/4$ correspond to the outward and inward components of the Poynting flux, respectively. Although in our simulations we inject the outward component of Alfvénic waves from the surface, the waves are eventually reflected inwardly.

In resistive MHD simulations, the propagation and dissipation of MHD waves are affected by magnetic diffusivity, and accordingly the atmospheric structure would be modified. We compare the ideal MHD and resistive MHD calculations and investigate how the magnetic diffusion affects the structure of planetary atmospheres and outflows.

Figure 1 compares the time-averaged atmospheric structure of resistive MHD cases with that of ideal MHD cases. Note that all atmospheric structures in this paper are time-averaged, and all properties have large time variability. The gas density in the upper atmosphere decreases more gradually in the ideal MHD cases because a larger amount of Alfvénic wave energy can reach the upper atmosphere and lifts up the gas by the magnetic pressure associated with the waves. On the other hand, in the resistive MHD cases Alfvénic waves are damped in the cool atmosphere at low altitudes because of the high resistivity, and then the gas density sharply drops without the support from the magnetic pressure. The Alfvénic waves that enter the upper atmosphere contribute to the acceleration of planetary winds. The wind speeds are quite slow in the resistive MHD cases, while the winds are accelerated to several tens of kilometers per second in the ideal MHD cases. As a result, the mass-loss rates in the resistive MHD cases are much smaller than those in the ideal MHD cases, because the density in the upper atmosphere is around two orders of magnitude less than that of the ideal MHD cases, and the velocity is also much smaller. If a planet is isolated or has a large distance from the central star, the planetary wind does not stream out in the resistive MHD condition except for intermittently driven outflows observed in the simulations because the average velocity is smaller than the escape velocity. However, if the planet is a hot Jupiter having small semimajor axis, the gas overflows easily from the small Roche lobe and is blown away by the stellar wind from the central star.

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In contrast to the large difference in the ρ and v_r profiles, the temperature structures of the resistive and ideal MHD cases are not so different from each other. Although the locations of the temperature inversion are slightly different, the upper atmospheres are heated up to $\sim {10}^{4}\;{\rm{K}}$ in both ideal and resistive MHD cases. Therefore, the heating by the dissipation of MHD waves is still important in the atmospheres of the resistive MHD cases.

We show the relation between the surface temperature, ${T}_{0}$, of gaseous planets and the atmospheric structure, and then we discuss the dependence of the mass-loss rate. In this subsection we fix the radius to the Jupiter radius and the amplitude of the velocity dispersion to 20% of the sound speed at the surface. Therefore, larger T₀ corresponds to larger MHD wave energy that is deposited to the magnetic flux tube. T₀ is thought to be mainly determined by the irradiation from the central star, more specifically, determined by the combination of the effective temperature of the central star and the semimajor axis of the planet. This implies that, in the framework of the wave-driven planetary winds, when we take planetary systems that have a similar central star, properties of the mass loss from hot Jupiters strongly depend on the distance from a central star.

3.2.1. Atmospheric Structure

The dependence of the atmospheric structure on T₀ is shown in Figure 2. In these simulations, the radius and mass of the planets are set to the radius and mass of Jupiter. In all cases, the gas temperatures in the lower atmosphere are almost isothermal, and they rise rapidly in the upper atmosphere because of the dissipation of the Poynting flux carried by Alfvénic waves. The dissipation of the energy in the upper atmosphere makes a high-temperature (∼10,000 K) corona-like region around the hot Jupiters. In all the cases, a low-temperature region forms just below the location of the temperature inversion because of the adiabatic expansion of the gas. As shown in Figure 2(a), the altitude of the temperature inversion is higher for higher T₀. This is mainly caused by the difference of the density in the upper atmosphere. In cases with higher T₀, the density in the upper atmosphere is higher because of the larger scale height; the density decreases more gradually with altitude (Equation (6)). Since larger heating is required to heat up the denser gas, the temperature inversion is located at a higher altitude in cases with larger T₀.

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While the temperature and density panels in Figure 2 show more or less similar profiles, there is a large difference in the radial velocity profiles. The cases with low surface temperature ${T}_{0}\leqslant 2000$ K yield quite slow wind velocities $\lt 3$ km s⁻¹. In contrast, in the case with high T₀ of 2500 K, gas is largely accelerated to attain a much faster velocity, $\approx 10$ km s⁻¹, than the other three cases. These results of the velocity and density profiles suggest that the wind structure can be categorized into two regimes: while slow and weak planetary winds stream out from cooler planets, fast and strong planetary winds are accelerated from hotter planets. We discuss this difference in Section 3.2.2.

Next, we examine the ionization degree that determines the magnetic resistivity in the atmosphere. To evaluate the non-ideal effects in the weakly ionized atmosphere, we introduce a magnetic Reynolds number,

$$\begin{eqnarray}&&{R}_{{\rm{m}}}=\displaystyle \frac{{vL}}{\eta }\end{eqnarray} \tag{ 11 }$$

where v and L are a typical velocity and length in the system. We use the scale height of the atmosphere (Equation (7)) and the amplitude of the velocity dispersion at the surface for L and v, respectively. Figure 3 shows the dependence of x_e and R_m on T₀. Since the typical value of T₀ is too low to ionize hydrogen atoms, the ionization degree is very low in the lower atmosphere. For example, ${x}_{{\rm{e}}}\sim {10}^{-10}$ in the cases with ${T}_{0}\;=$ 1000 and 1600 K, and x_e is still ∼10⁻⁸ even for ${T}_{0}\;=$ 2500 K. In this temperature region, the ionization degree is supported by the ionization of alkali metals, such as sodium and potassium, that have relatively low ionization potential. This low x_e leads to high magnetic resistivity and small magnetic Reynolds number. As a result, the magnetic Reynolds number in the lower atmosphere is very small, ${R}_{{\rm{m}}}\sim {10}^{-2}$ in the cases with ${T}_{0}\;=$ 1000 and 1600 K, and still ${R}_{{\rm{m}}}\sim 1$ for ${T}_{0}\;=$ 2500 K, although it is $\approx 100$ times larger owing to the higher x_e. The resultant low R_m means that the non-ideal effects are important in the lower atmosphere. In the upper atmosphere, however, high gas temperature and low gas density cause a sudden surge of x_e. It increases gradually in the region below the temperature inversion region as ρ decreases with altitude. Ionization degree is very high above there because the temperature rises high enough to ionize the hydrogen atoms.

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3.2.2. Two Regimes of the Poynting Flux

Figure 4 shows the net Poynting flux in the atmosphere for cases with different T₀. The net outgoing Poynting flux decreases rapidly with an increase of the altitude in the lower atmosphere, and it becomes almost constant in the upper atmosphere. The Poynting flux injected at the surface increases monotonically with T₀, but its behavior in the atmosphere is not simple; the net Poynting flux in the upper atmosphere is smaller for higher T₀ when ${T}_{0}\leqslant 1600\;{\rm{K}}$ (a), but it is larger for higher T₀ when ${T}_{0}\gt 1600\;{\rm{K}}$ (b). The Alfvénic waves are damped in the lower atmosphere, where the magnetic resistivity is high. The scale height of the atmosphere plays a key role in the resistive dissipation of the Alfvénic waves.

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The scale height is larger for higher T₀, which gives the large density in the upper atmosphere. On the one hand, the larger density in the upper atmosphere leads to a larger amount of mass loss, but on the other hand, it gives a negative effect on the wind through the enhancement of the resistive dissipation of the Alfvénic waves. The ionization degree is lower in the higher-density gas because of the efficient recombination, and consequently the magnetic resistivity is relatively higher. In Figure 3(b), the magnetic Reynolds number of the case with ${T}_{0}=1000$ K is low in the lower atmosphere, but it increases quickly in the upper atmosphere. In the case with ${T}_{0}=1600$ K, it is as low as the ${T}_{0}=1000$ K case in the lower atmosphere, but it increases more slowly than in the 1000 K case. Therefore, the magnetic Reynolds number remains relatively lower in almost the entire region than in the ${T}_{0}=1000$ K case. Because of the larger resistivity, a smaller fraction of the injected wave energy reaches the upper atmosphere in the ${T}_{0}=1600$ K case, compared to the ${T}_{0}=1000$ K case. This is the main reason why the transmitted Poynting flux shows the negative tendency on the surface temperature for ${T}_{0}\lt 1600$ K. However, for ${T}_{0}\gt 1600$ K, the Poynting flux exhibits a positive correlation with T₀ because the magnetic resistivity is lower for higher T₀.

Figure 5 compares the outward and inward energy flux by the Alfvénic waves (Equation (10)). Here we show three cases with T₀ of 1000, 1600, and 2000 K. Every case shows that the inward energy flux tracks the outward energy flux with a slightly smaller level in almost the entire region of the atmosphere. The net outgoing Poynting flux in Figure 4 is considerably smaller than the outward energy flux in Figure 5. These results indicate that the injected outgoing Alfvén waves are reflected back downward through the propagation in the atmosphere. A tiny fraction of the injected Alfvén waves can reach the upper atmosphere, which contributes to the heating and the acceleration of the gas. In general, a small fraction of the injected Poynting flux by Alfvén waves is transmitted to the upper atmosphere because they suffer dissipation and reflection, whereas the resistivity is one of the keys that control the dissipation of the waves. The fraction that enters the upper atmosphere determines the properties of the wind. For instance, if a large amount of the MHD wave energy reaches the upper atmosphere, it will increase the density there and drive outflows.

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The effect of the resistive dissipation of the Alfvénic waves is also seen in Figure 5. Both dissipation and reflection of the waves decrease the Poynting flux. The Poynting flux in the case ${T}_{0}=1000$ K (red lines) decreases much faster than that of the case ${T}_{0}=2000$ K (blue lines), which is caused by both resistive dissipation and reflection. The higher magnetic resistivity in the 1000 K case makes the Alfvénic waves dissipate faster, and the steeper density gradient results in strong reflection. In the 1600 K case (green lines), the Poynting flux quickly drops, which is much faster than in the other two cases. Since the scale height is larger than that of the ${T}_{0}=1000$ K case, the reflection of the Alfvénic waves is suppressed. In this case the resistive dissipation of the Alfvénic waves is the primary reason for the rapid drop of the Poynting flux, and the amount of energy that can reach the upper atmosphere is quite small.

3.2.3. Time Variability of Mass-loss Rate

Our simulations show large time variabilities of the atmospheric structure. Figure 6 shows the mass-loss rate for different T₀ cases with time. One can see that in the cases with lower ${T}_{0}\leqslant 1600$ K the outflow is not time steady, but the atmospheric material infalls during certain periods, which is seen as dashed lines. For example, in the case with ${T}_{0}\;=$ 1000 K, the outflow phases are relatively shorter and it is comparable to the total duration of the infall phases. In the case with ${T}_{0}\;=$ 1600 K, the integrated duration of the outflow phases is longer, and the infall phase disappears for higher ${T}_{0}\geqslant 2000$ K.

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These results suggest that in the cases with low ${T}_{0}\leqslant 1600$ K the atmosphere is almost hydrostatic because the damping of Alfvénic waves is too severe to drive steady winds, and the outflows occur only intermittently. For higher T₀ the intermittency of the planetary winds disappears, and more steady outflows are obtained.

In Figure 7 we compare the Poynting flux by the Alfvén waves in the active phase during 23.6–31.6 hr and that in the infall phase during 17.8–20.2 hr in Figure 6. In the outflow phase, a larger amount of the Poynting flux is carried by the Alfvén waves into the upper atmosphere, while in the infall phase, the Poynting flux in the upper atmosphere is strongly reduced by around four orders of magnitude. This result implies that the transmission of the Poynting flux into the upper layor controls the on–off nature of the atmospheric escape for the low-T₀ cases. However, we would like to note that the case with ${T}_{0}=1000$ K (not shown) shows more complicated behavior; Poynting flux during outflow phases does not always exceed Poynting flux during infall phases because of the rapid temporal variations.

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Here we describe the dependence of the atmospheric structure and the mass-loss rate on the radius of gaseous planets R_p. A number of theoretical works on the thermal evolution of the gaseous planets suggest that radii of gaseous planets, which tend to be around Jupiter radius, are affected by surrounding circumstances (e.g., Burrows et al. 2007). In addition, observations have shown that some hot Jupiters have enormously larger radii than are expected (Baraffe et al. 2010). Therefore, we treat R_p as a parameter and investigate the dependence on R_p. The mass-loss rate is very small for low T₀; therefore, we set ${T}_{0}=2000$ K in this subsection. We also fix the amplitude of the velocity dispersion at the surface, $\delta v=0.2{c}_{{\rm{s}}}$.

Figure 8 shows the dependence of the atmospheric structures on R_p. At low altitudes the temperature is almost constant (≈ surface temperature) in all the cases. It decreases initially but rises eventually by the heating owing to the wave dissipation. The temperature inversion is located at a higher altitude for larger R_p, because the scale height is larger; the slower decrease of the density leads to higher density in the upper region (Figure 8), which is not heated up to high temperatures. Accordingly, the bending location of the density gradient, which corresponds to the location of the temperature inversion, is systematically higher for larger R_p for the same reason.

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In all the cases, the temperature of the upper atmosphere is elevated to ∼10⁴ K, by the dissipation of MHD waves. However, the wind velocity is quite slow. Particularly in the cases with large R_p $\geqslant \;1.2{R}_{{\rm{J}}}$, the gas in the atmosphere is almost static or even infalls partly, whereas the cases with smaller radius show outflows with a few km s⁻¹.

Figure 9 shows the R_p dependence of the ionization degree and magnetic Reynolds number. The ionization degree, x_e, is quite low at low altitudes because of the low temperature there ($\approx 2000$ K) in all the cases. The dominant ion sources in the lower atmosphere are the alkali metals. In contrast, x_e increases to ${10}^{-4}-{10}^{-1}$ in the upper atmosphere in all cases. In the smaller R_p case, x_e starts to increase from a lower altitude and the final x_e is also higher in the upper atmosphere, because the recombination is slower there owing to the lower density.

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The mass-loss rate remains very small in these cases. The case with ${R}_{{\rm{p}}}={R}_{{\rm{J}}}$ gives the mass-loss rate of $2.6\times {10}^{-19}\;{M}_{\odot }\;{\mathrm{yr}}^{-1}$, and the case with 0.8R_J gives $2.3\times {10}^{-19}\;{M}_{\odot }\;{\mathrm{yr}}^{-1}$, which correspond to $1.7\times {10}^{7}\;{\rm{g}}\;{{\rm{s}}}^{-1}$ and $1.4\times {10}^{7}\;{\rm{g}}\;{{\rm{s}}}^{-1}$, respectively (here M_p = M_J and ${T}_{0}=2000$ K). In the ideal MHD calculations, the mass-loss rate in the case with ${R}_{{\rm{p}}}={R}_{{\rm{J}}}$ and ${T}_{0}=2000\;{\rm{K}}$ is $\sim 1.9\times {10}^{10}\;{\rm{g}}\;{{\rm{s}}}^{-1}$(Tanaka et al. 2014). Our results show that the large resistivity as a result of the low ionization degree in the low atmosphere significantly suppresses the mass loss from hot Jupiters. However, we would like to note that the wind shows large fluctuations. The possibility remains that the planetary wind streams out in an intermittent manner. Also, the Roche lobe of a hot Jupiter is small because of the small separation from the central star. Then, once slow gas material overflows the Roche lobe, it will be blown away by the stellar wind from the central stars.

The velocity amplitude at the surface, $\delta v$, is also an important parameter that controls the structure of the atmosphere and the wind of a planet. In this subsection, we fix ${R}_{{\rm{p}}}={R}_{{\rm{J}}}$, ${M}_{{\rm{p}}}={M}_{{\rm{J}}}$, and ${T}_{0}=2000\;{\rm{K}}$. In Figure 10, we show the atmospheric structure of the cases with different $\delta v$. Cases with different $\delta v$ give similar density and temperature profiles. On the other hand, larger $\delta v$ gives faster wind velocity. Because of the similarity of the density and the temperature shown in Figure 10, the ionization degree and the magnetic resistivity are also similar in all the cases. Therefore, it can be said that the difference of $\delta v$ at the surface has a small effect on the structure of the atmosphere except for the radial velocity.

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Although the difference of the density in the upper atmosphere is very small, the wind velocity is faster for larger $\delta v$, and it causes the increase of the mass-loss rate, as shown in Figure 11 for cases with ${T}_{0}=1000$ and 2000 K.

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We have investigated the effect of the magnetic diffusion on the dissipation of the Alfvénic waves, which suppresses the planetary winds, so far. On the other hand, the resistive dissipation also heats up the ambient gas via ohmic (Joule) heating, which plays a positive role in driving outflows. In this subsection, we examine the role of ohmic heating in a quantitative manner.

Current density is

$$\begin{eqnarray}&&{\boldsymbol{j}}=\displaystyle \frac{c}{4\pi }({\rm{\nabla }}\times {\boldsymbol{B}}),\end{eqnarray} \tag{ 12 }$$

and electric conductivity σ is related to the magnetic resistivity η by

$$\begin{eqnarray}&&\displaystyle \frac{1}{\sigma }=\displaystyle \frac{4\pi }{{c}^{2}}\eta .\end{eqnarray} \tag{ 13 }$$

Then, the ohmic heating rate per unit volume, ${Q}_{\mathrm{Ohm}}={{\boldsymbol{j}}}^{2}/\sigma$, is derived as

$$\begin{eqnarray}&&{Q}_{\mathrm{Ohm}}=\displaystyle \frac{\eta }{4\pi }{({\rm{\nabla }}\times {\boldsymbol{B}})}^{2}.\end{eqnarray} \tag{ 14 }$$

Figure 12 shows the ohmic heating rate in comparison with the ideal MHD heating (red solid) and cooling (green dashed) rates of the gas. The ideal MHD heating includes shock heating (see Section 2.2.2) in addition to adiabatic heating. In the lower atmosphere, the ohmic heating dominates over, or is comparable to, the ideal MHD heating because of the large resistivity. However, since the Alfvénic waves dissipate very rapidly, the ohmic heating rate drops drastically as the altitude increases.

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Resistivity depends sensitively on temperature. Figure 13 compares the ohmic heating rates for cases with different T₀. The higher-T₀ cases have smaller ohmic heating rates in the lower atmosphere, but this reverses in the upper atmosphere, because the dissipation of the Alfvénic waves is more gentle at low altitudes in these cases and a larger amount of the wave energy reaches the upper atmosphere. In addition, as shown in Figure 3, the magnetic Reynolds number is relatively lower in the upper atmosphere for higher T₀, which further increases the resistive heating.

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However, the ohmic heating rate is quite small in comparison to the shock heating except for the region close to the surface (Figure 12), so this effect is negligible in the upper atmosphere and gives little contribution to the atmospheric escape. Even in the low atmosphere where the ohmic heating is large, it does not play a primary role. The comparison between the ideal and non-ideal calculations in Figure 1(a) shows that the temperature profiles in the lower atmosphere are almost the same. In fact, the temperatures in the lower atmosphere in the resistive cases are barely higher than the ideal cases, and the difference is only several tens of kelvin. In summary, the ohmic heating does not play an important role in the entire structure of the atmosphere and the atmospheric escape.

In this work and the previous work, we have fixed the shape of the open magnetic flux tube that is described in Section 2.1. We presume that the overall structure of magnetic field lines of gaseous planets is similar to that of the Sun. The Sun has a multipole magnetic field, and the properties of open magnetic flux tubes are observed in detail (e.g., Tsuneta et al. 2008; Ito et al. 2010). However, the actual structures of magnetic fields in exoplanets are unknown. The overall structure of the planetary magnetic field affects the shape of the open magnetic flux tube. The most critically affected parameter in our modeling is the filling factor $f(r)$ (see Equations (1) and (2) of Tanaka et al. 2014). If the planetary magnetic field has a simple dipole structure, open magnetic flux tubes appear in the magnetic polar regions. In this case, heating in the upper atmosphere and the acceleration of gas flow by MHD wave energy occur only in the polar regions. Therefore, the actual appearance of flow is expected to be bipolar. In the simple dipole case of the magnetic field, the degree of superradial expansion of the open magnetic flux tube should be smaller, and thus the mass-loss rate is expected to be smaller because the reflection of the energy flux is expected to be enhanced. To investigate this more quantitatively, a three-dimensional calculation is required, which remains difficult at present. On the other hand, the observation of the bipolar outflow from an exoplanet is, in principle, possible in the detailed analysis of the differential spectroscopy for transiting hot Jupiters, unless the magnetic dipole axis is perpendicular to both the orbital plane of the exoplanet and the line of sight. This is because a blueshifted outflow before ingress and a redshifted outflow after the emersion, or vice versa, can be seen in this case. Once the mass flux of the bipolar outflow from the exoplanet is observed, we will possibly constrain the property of the magnetic field of the exoplanet by assuming the driving mechanism as described in this work.

As we mentioned in the Introduction, high-energy radiation such as X-ray and extreme-ultraviolet from a central star is important for the heating of the upper atmosphere and atmospheric escape from hot Jupiters. Here we compare the height where the heating by MHD waves is important and the penetration depth of stellar XUV. It is useful to introduce ${R}_{\mathrm{XUV}}$, the radius at which XUV is absorbed. ${R}_{\mathrm{XUV}}$ means that the optical depth for XUV reaches unity at that radius, and it corresponds to the radius where the column density of hydrogen becomes $\sim 5\times {10}^{17}\;{\mathrm{cm}}^{-2}$ (e.g., Murray-Clay et al. 2009; Rogers et al. 2011). We calculate ${R}_{\mathrm{XUV}}$ in each calculation and compare it with the region where the heating by the dissipation of the MHD waves becomes important.

Figure 14 shows the T₀ dependence of ${R}_{\mathrm{XUV}}$ and comparison with other considerable radii. As shown in Figure 2(a), the gas temperature slightly drops owing to adiabatic expansion and then is heated drastically to ∼10⁴ K. R_low is the radius at which the temperature becomes lowest and heating by MHD waves starts to become important, and ${R}_{8000\;{\rm{K}}}$ is the radius at which the temperature becomes higher than 8000 K. In this range of T₀, ${R}_{\mathrm{XUV}}$ is always located only slightly above R_low, and the value varies between 1.05R_p and 1.20R_p. This is consistent with previous works on the XUV heating in the atmosphere of hot gaseous planets (e.g., Murray-Clay et al. 2009). In addition, ${R}_{8000\;{\rm{K}}}$ always lies at a much higher altitude than ${R}_{\mathrm{XUV}}$.

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These result suggest that the region heated to ∼10⁴ K by the dissipation of the MHD waves is optically thin to XUV photons, and XUV can reach the altitude at which the temperature becomes lowest in our model. Photoionization, cooling from metal lines, and Lyα play important roles in the upper atmosphere of hot Jupiters. At this stage, our model does not include the effects of stellar XUV irradiation, the inclusion of which should improve our model. In the present modeling, the results cannot explain the observational results of the mass-loss rates adequately, since the mass-loss rates are largely reduced when we take into account the effects of the magnetic resistivity. This seems to indicate that the resulting mass-loss rate strongly depends on the details of the input physics. More realistic calculation is critically needed and will be part of our future work.