MAGNETOSPHERIC STRUCTURE AND ATMOSPHERIC JOULE HEATING OF HABITABLE PLANETS ORBITING M-DWARF STARS

In order to obtain a solution for the stellar wind, we use the SC version of BATS-R-US (Oran et al. 2013; Sokolov et al. 2013; van der Holst et al. 2014). The model is driven by the photospheric radial magnetic field (see Section 2), which is used to calculate the three-dimensional potential magnetic field above the stellar surface (Altschuler & Newkirk 1969). The potential field solution is in turn used as the initial condition for the magnetic field in the simulation domain. Once the initial potential field is determined, the model calculates self-consistently the coronal heating and the stellar wind acceleration due to Alfvén wave turbulence dissipation, taking into account radiative cooling and the electron heat conduction. Unlike most MHD models for the solar corona, the lower boundary of this model is set at the chromosphere, so that it does not initially assume a hot corona at its base. Instead, the heating is calculated self-consistently.

While the model works very well for the Sun, applying this model to M-dwarf stars is not so trivial. Our reference star, EV Lac, has stronger magnetic fields than the Sun, and there is a general lack of observations of the winds of other stars (see, e.g., Wood et al. 2004; Güdel 2007 and references therein). Our basic assumption is that the winds of M-dwarfs are accelerated in a similar manner to the Sun, with a combination of thermal acceleration (taken into account in the model) and dissipation of magnetic energy. An example of a different mechanism that the model cannot account for is stellar winds from highly evolved giants, which are likely driven by radiation pressure on dust grains (e.g., Lamers & Cassinelli 1999) or by radial pulsations (e.g., Willson 2000). Based on our assumption, several studies of Sun-like stars have been made using the SC model. Cohen et al. (2010) have applied the model to the active star AB Doradus, where they argued for the validity of the model for solar analogs. Similarly, Cohen et al. (2011b) performed simulation of HD 189733 driven by ZDI observations reproduced from Fares et al. (2010). The model has been used recently by Cohen & Drake (2014) to perform a parametric study on the stellar wind dependence on magnetic field strength, base density, and rotation period.

As stated above, we use the ZDI observation of EV Lac, as well as its stellar parameters of R_⋆ = 0.3 R_☉, M_⋆ = 0.35 M_☉, and rotation period, P_⋆ = 4.3 days (Morin et al. 2008), to drive the SC model. We also adopt the bolometric luminosity L_bol = 4.5 × 10²⁴ W from Morin et al. (2008), which corresponds to an effective temperature of 3400 K. We use a spherical grid that extends up to 100 R_⋆ so as to include the orbits of all three planets. Once a steady-state solution is obtained, we extract the stellar wind parameters of number density, n, velocity, u, magnetic field, B, and plasma temperature, T, at a given point along the orbit of one of the planets.

The SC solution is provided in the frame of reference rotating with the star (HelioGraphic Rotating coordinates or HGR). In this coordinate system, the $\hat{Z}$ axis is aligned with the rotation axis of the star, the $\hat{X}$ axis is aligned with the initial time of the ZDI observation (longitude "0"), and the $\hat{Y}$ axis completes the right-hand system. The GM model uses the Geocentric Solar Magnetospheric (GSM) coordinate system, which is identical to the Geocentric Solar Ecliptic (GSE) system for the case of a planetary dipole perpendicular to the ecliptic plane. In this special case, the planetary GSE/GSM coordinate system is defined with $\hat{X}$ pointing from the planet to the star (negative radial direction in the stellar frame of reference assuming a circular planetary orbit), $\hat{Z}$ is pointing to the north pole of the planet (perpendicular to the ecliptic plane and the plane of orbit), and $\hat{Y}$ completes the right-hand system. With a circular planetary orbit and an aligned planetary dipole, the conversion between the coordinate systems is

$$\begin{eqnarray} &X_{{\rm GSE}}=-r_{{\rm HGR}} &\\ &Y_{{\rm GSE}}=\phi _{{\rm HGR}} &\nonumber \\ &Z_{{\rm GSE}}=\theta _{{\rm HGR}}.& \nonumber \end{eqnarray} \tag{ 1 }$$

The orbital speed of the planet, U_orb, could be easily considered as a constant addition to $U^{{\rm GSE}}_y$. However, it is hard to estimate what would be the change in $B^{{\rm GSE}}_y$, which has a strong effect on the magnetospheric current system. After carefully confirming that the Alfvén Mach number with and without the addition of U_orb is essentially the same for all cases, we have decided to exclude this motion from our simulation.

The situation in which the planets cross into the sub-Alfvénic regime, and hence the closed magnetic loops, is similar to the sub-Alfvénic motion of Io in the magnetosphere of Jupiter, where the relative speed between Io and the ambient plasma defines the structure of the magnetosphere and the angle of the Alfvén "wings" (standing magnetic lobes as described in Neubauer 1980). In our simulation, we use the planetary frame of reference and assume the velocity field at the particular point in the wind solution represents the relative velocity between the planet and the coronal plasma.

The GM model solves the MHD equations on a Cartesian grid for the physical domain that includes the planet as the inner boundary, and the extent of the planetary magnetosphere. The model is driven from the outer boundary that is facing the star by the upstream stellar wind conditions, which can be fixed or time-dependent. This boundary is defined with inflow boundary conditions for all the MHD parameters for the case of super-Alfvénic stellar wind conditions. In the case of sub-Alfvénic stellar wind conditions, the boundary conditions for the pressure are changed to float in order to diminish numerical effects on the boundary from the inner domain (this does not happen for super-Alfvénic boundary conditions). For the same reason, we also set the upstream boundary very far from the planet. The boundary conditions for all the other boundaries are set to float for all the MHD parameters.

The inner boundary in GM is defined by the planetary parameters of radius, mass, magnetic field, and density (as described in Powell et al. 1999). In order to better constrain the velocities at the inner boundary, the GM model is coupled with a model for the IE, which is a completely separate model that solves for the electric potential in the ionosphere. The IE model provides the convection electric field, which is then used to calculate the velocities at the inner boundary of GM, along with the rotational velocity of the planet.

The coupling procedure, which is described in detail in Ridley et al. (2004) and in Tóth et al. (2005), begins by calculating the field-aligned currents, J_∥ = (∇ × B) · b, at 3R_p in the GM model. The currents are then mapped assuming a dipole planetary field to the ionospheric height of 120 km, using a scaling of B_I/B₃, where B_I and B₃ are the field strengths at the ionosphere and at the point of origin at 3R_p. Using the electric conductance tensor, Σ, an electric potential is solved with

$$\begin{equation} J_r(r_I)=\left[ \nabla _\perp \cdot (\Sigma \cdot \nabla \Psi) _\perp \right]_{r=r_I}, \end{equation} \tag{ 2 }$$

and this potential is mapped to the inner boundary of the GM at 2R_p (for numerical efficiency, the inner boundary is set higher than 1R_p to reduce the planetary dipole strength and increase the numerical time step). At the final stage, the electric field, E = −∇Ψ, and the bulk convection velocity, V = E × B/B² are calculated. The velocity field (along with the rotational velocity) is applied at the GM inner boundary.

The coupling of GM and IE models enables us to specify more realistic inner boundary conditions for the GM. This improved boundary specification allows us to estimate the energy input at the top of the planetary atmosphere due to its interaction with the stellar wind and the precipitating particles. Assuming the currents and the (scalar) conductance, σ, are known in the IE model, it is trivial to calculate the Joule Heating:

$$\begin{equation} Q=\mathbf {J}\cdot \mathbf {E}=J^2/\sigma, \end{equation} \tag{ 3 }$$

from the generalized Ohm's law with J = σE.

We note that, in a way, the electric conductance captures all the atmospheric parameters in it (chemistry, photoionization, etc.). For the case of the Earth, a more complex conductance can be used. The conductance depends significantly on the solar EUV flux (and correlates with the radio flux in the F10.7 centimeter wavelength, see, e.g., Moen & Brekke 1993), auroral particle precipitation (e.g., Robinson et al. 1987; Fuller-Rowell & Evans 1987), and other processes that are quite dependent on understanding the environment near the planet. Global models of the upper atmosphere, such as the one described by Ridley et al. (2002) can also be used, but they are driven by the observed solar luminosity and auroral precipitation, so it is difficult to drive them properly for other planets to determine the conductance patterns. In order to avoid further uncertainties, here we choose to use a constant Pedersen conductance, σ_p, of the order of the one used for the Earth (e.g., Ridley et al. 2004). The Pedersen conductance allows the magnetospheric currents to close through the ionosphere, and it depends on the collision frequency between electrons and ions, ν_{e, i}, and the electron plasma frequency, $\Omega ^2_e=n_e e^2/\varepsilon _0 m_e$:

$$\begin{equation} \sigma _p=\frac{\nu ^2_{e,i}}{\nu ^2_{e,i}+\Omega ^2_e} \sigma _0, \end{equation} \tag{ 4 }$$

with σ₀ = n_ee²/ν_{e, i}m_e, where n_e is the electron density, m_e is the electron mass, e is the electric charge, and ε₀ is the permittivity in free space. The extreme EUV irradiation of close-in planets around M-dwarf stars should reduce the altitude of their ionospheres to regions with higher electron density. It is not trivial to predict how the Pedersen conductance will change as a result of the increased EUV radiation, as it has a complicated dependence on the density variations of ions electrons and neutrals, ionization rates, atmospheric chemistry, and perhaps other factors. In the results section, we probe the sensitivity of our calculations to this by showing how a simple increase in the Pedersen conductance affects the Joule Heating for a given set of parameters.

The model wind solution for EV Lac shows an average speed of about 300 km s⁻¹ and the total mass loss rate is about 3 × 10⁻¹⁴ M_☉ yr⁻¹. While these values are close to those of the solar wind, the mass loss rate per unit stellar surface area from this diminutive M dwarf is an order of magnitude higher.

Figure 1 shows the steady-sate coronal and stellar wind solution for EV Lac. It shows the orbits of the three planets, selected coronal magnetic field lines, and color contours of the ratio between the dynamic pressure in the solution to that of a typical solar wind conditions at 1 AU (see background solar wind conditions in Table 3). The dynamic pressure of the ambient stellar wind at these close-in orbits is 10 to 1000 times larger than that near Earth. In addition, the magnetic field strength ranges between 500–2000 nT along the orbit of Planet A, between 200–800 nT along the orbit of Planet B, and between 100–200 nT along the orbit of Planet C. This is in contrast to a field strength of the order of 1–10 nT for typical solar wind conditions at 1 AU. Finally, the temperature of the ambient stellar wind along the orbits of the planets ranges from 300,000 K to over 2 MK. The typical solar wind temperature is about 10⁴ K.

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As seen in Figure 1, the stellar wind conditions change from sub- to super-Alfvénic along the orbits of all three planets. For Planet A and Planet B, we drive the GM simulation using both sub- and super-Alfvénic upstream conditions. For Planet C, we perform three GM simulations using sub-Alfvénic conditions, super-Alfvénic conditions with slow (more dense) stellar wind, and super-Alfvénic conditions with fast (less dense) stellar wind. Table 2 summarizes the upstream conditions used to drive the GM model for the different Planets.

Figure 2 shows the magnetospheric structures of Planet A and Planet B (the magnetospheric structure of Planet C is qualitatively similar). The most notable result is the dramatic change in the magnetospheric topology when the stellar wind upstream conditions change from sub- to super-Alfvénic. For sub-Alfvénic conditions the planetary field lines simply merge with the stellar wind field lines in an Alfvén-wings topology (Neubauer 1980, 1998). An Alfvén-wings configuration arises when a conducting obstacle moves in a plasma with a sub-Alfvénic speed and it is the result of the configuration of the current system that connects the external plasma with the currents flowing inside the body, along low flow cavities. It has been well observed and studied for the Jovian moons Io (Neubauer 1980; Combi et al. 1998; Linker et al. 1998; Jacobsen et al. 2007) and Ganymede (Ip & Kopp 2002; Kopp & Ip 2002; Jia et al. 2008). Some studies suggest that this configuration can be obtained at Earth during periods when the solar wind has a weak Alfvénic Mach number (Ridley 2007; Kivelson & Ridley 2008).

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For super-Alfvénic upstream conditions, an Earth-like magnetospheric configuration forms with the planetary field lines being draped over by the stellar wind, and a magnetopause bow shock being forming in front of the planet along with a magnetotail behind.

The transitioning between sub- to super-Alfvénic conditions occurs twice per orbit and within a short time. This has implications for the energy input into the upper atmosphere, which are discussed in Section 4.4.

Figure 3 shows the distribution of the height integrated ionospheric Joule Heating, Q_l, (in units of W m⁻²) in a format of polar plots. Each pair of panels shows the Joule Heating for one of the three planets extracted from a steady-state GM-IE simulation. The heating is clearly stronger for the closer orbit of Planet A than the more distant orbit of Planet C, for which the color scale of the plot has been extended to lower values of heating for clarity. The figure also shows that the heating is stronger for super-Alfvénic stellar wind conditions than for sub-Alfvénic conditions, and that the distribution is asymmetric, as expected, due to the asymmetric stellar wind magnetic field and magnetospheric field-aligned currents.

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For comparison with the case of the Earth, we ran the model using the planetary parameters of the Earth, and upstream conditions of a typical quiet solar wind, as well as with upstream conditions of a strong CME event. The upstream parameters of these reference runs are summarized in Table 3. Figure 4 shows the ionospheric Joule Heating for Planet A and for these reference Earth cases. It shows that the heating by the ambient stellar wind conditions at close-in orbits is about four orders of magnitude higher than the case of the ambient solar wind conditions at Earth, and is even higher than the heating during a strong space weather event on Earth.

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The top panel in Figure 5 shows the total area integrated power, P = ∫Q_lda, for all the solutions, where da is the surface element of the two-dimensional ionospheric sphere. It is consistent with the trend which is usually seen in Figure 3, with the power being the greatest for the closest planet. For the Planet A, the power reaches 10¹⁴–10¹⁵ W, which is 0.01%–0.1% of the total incident radiation of the M-dwarf star, assuming L_bol = 4.5 × 10²⁴ W, a = 0.06 AU (36 R_⋆), and P ≈ (1/4)L_bol(R_p/a)² ≈ 10¹⁸ W. Since the planetary albedo is likely not zero, much of this radiation will be reflected, and the significance of the Joule Heating can be even greater. For all cases, the heating in the super-Alfvénic regime is higher than the sub-Alfvénic one, and for the case of Planet C, the heating is greater for the super-Alfvénic regime with a slow stellar wind than for the super-Alfvénic upstream conditions with a fast stellar wind. This is due to the order of magnitude density variation of the stellar wind upstream conditions, which increases the wind's dynamic pressure (i.e., $\rho u^2_{sw}$). Moreover, the super-Alfvénic sectors are located near the helmet streamers, where the velocity component which is not parallel to the magnetic field (i.e., the non-radial component) is greater. As a result, and the upstream electric field, E = −u × B, which dictates the ionospheric electric potential and the coupling between the stellar wind and the planetary magnetosphere, is larger as well.

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In the middle panel of Figure 5, we show the Joule Heating power for different planetary magnetic field strengths for the sub- and super-Alfvénic upstream conditions in Planet A, along with the reference Earth cases. The power is greater for the weaker planetary fields due to the strong penetration by the stellar wind. As the field increases to 1G, the wind is pushed back and the energy input to the upper atmosphere is reduced. In addition, the total size of the heated polar cup is reduced too. Due to the uncertainties about the planetary magnetic field, here we assume a planetary dipole field aligned with the stellar rotation axis. We omit the effect of different dipole orientation, which could affect the energy input as a result of magnetic reconnection between the stellar wind and the planetary magnetosphere (e.g., a geomagnetic storm).

The bottom panel of Figure 5 shows the Joule Heating power for an ionospheric conductance of 0.25 (the value used for all simulations), 5, and 50 Siemens for the sub-Alfvénic case of Planet A, along with the reference Earth cases. As mentioned in Section 3.2, the actual ionospheric conductance depends on the atmospheric density, composition, the level of ionization, and the level of photoionization (the stellar EUV and X-ray flux). The power varies inversely with the value of the conductance. As the conductance represents the mobility of the ions, a lower value means that the ions are less mobile and collide more frequently with neutrals. As a result, the energy dissipation (or Joule Heating) increases. We note that for the case of close-in planets orbiting M-dwarf stars, the intense EUV radiation may boost the ionospheric conductance to much greater values and as a result, the Joule Heating may be lower than predicted here. However, due to the many uncertainties in defining the atmospheres of these planets, we leave the investigation of this particular issue to a future, separate study.

The results presented in Sections 4.2 and 4.3 are the steady-state solutions for the different planets, which are driven by the upstream stellar wind conditions extracted from the SC solution at particular locations along their orbit. As stated in Section 4.2, the magnetospheric structure undergoes significant change as the planets move from the sub- to super-Alfvénic plasma sectors along the orbit. In order to obtain the dynamic effect of such a transition, we performed a time-dependent simulation of Planet A starting at a sub-Alfvénic sector and ending in a super-Alfvénic sector. In this time-dependent simulation, the upstream conditions in GM are updated every 2:52 hr (10 degrees of the orbit of 4.3 days) so that the change in the driving stellar wind conditions is captured (in contrast to the single set of stellar wind conditions for the steady-state).

Figure 6 shows the magnetospheric structure, as well as the Joule Heating in the ionosphere at after 2:52 hr, 22:52 hr, 25:44 hr, and 26:36 hr. The planet moves to the super-Alfvénic sector around 25:00 hr, with the white line in the lower two panels representing the M_A = u_sw/v_a = 1 line. This line also helps to identify the magnetopause bow shock built in front of the planetary magnetosphere as the planet transitions to a super-Alfvénic sector. It can be clearly seen from the top four panels in Figure 6 that there is a significant heating of the upper atmosphere as the planet transitions between the sectors.

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Figure 7 shows the temporal change in the total power as a result of the planetary motion along the orbit. The total power increases by 50% as the planet moving from the sub- to super-Alfvénic sector. The heating in the sub-Alfvénic sector is 10 times higher in the dynamic simulation compared to the steady-state obtained at the same point, and the heating in the super-Alfvénic sector is 50% higher in the dynamic simulation compared to the steady-state obtained at the same point (in this case, the heating is almost 1% of the incident stellar radiative power). This is due to the fact that the dynamic simulation captures the temporal change in the magnetic field, which drives stronger field-aligned currents (the time derivative of the magnetic field is zero for the steady state, but can and does vary in the time-dependent case).

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Our results show in detail that the space environment of close-in exoplanets is much more extreme in terms of the stellar wind dynamic pressure, magnetic field, and temperature. Each of these parameters is about one to three orders of magnitude higher than the typical solar wind conditions near the Earth. The ultimate consequence of such an extreme environment, in the context of this paper, is the potential for stripping of the planetary atmosphere (in addition to atmospheric evaporation by the enhanced EUV/X-ray stellar radiation, Lammer et al. 2003). The process of atmospheric erosion due to the impact of an ionized wind is complex, and here we simply examine the degree to which the planetary magnetosphere is penetrated by the wind.

On this basis, a planetary magnetic field similar to that of the Earth seems to be strong enough to largely resist the stellar wind in the super-Alfvénic sectors. However, in the case of the sub-Alfvénic sectors there is no bow shock at all, and many field lines are connected directly from the stellar wind to the planet, resulting in poor atmospheric protection. As a matter of fact, Planet A is located for most of its orbit in a sub-Alfvénic plasma and the assumption of the existence of a bow shock (e.g., Grießmeier et al. 2004; Khodachenko et al. 2007; Lammer et al. 2007; Vidotto et al. 2011) may not be relevant for planets of this kind.

Grießmeier et al. (2004) have studied the atmospheric protection for hot-jupiter planets using scaling laws for the stellar mass loss and the planetary magnetic moment, and by estimating the atmospheric loss of neutral hydrogen. They concluded that hot-jupiters may have been significantly eroded during the early stages of the stellar system when stellar magnetic activity was high, and assuming that the mass-loss rate of young stars is comparatively high (about 100 times the current solar value, Wood et al. 2002). Other estimates of the mass loss rates of young stars suggest that the winds might not be so strong (only about 10 time the current Sun, Holzwarth & Jardine 2007; Sterenborg et al. 2011). In the case of the planets studied here, we do not include any atmospheric outflow. However, our simulations demonstrate that in order to estimate such an outflow and the degree of planetary protection afforded by a magnetic field, one should consider not only the magnitude of the stellar wind pressure, but also whether the surrounding space plasma is sub- or super-Alfvénic and the orientation of the stellar magnetic field, as it may allow stellar wind particles to flow directly into the planetary atmosphere.

The time-dependent results show that the magnetospheres of the planets under study experience significant changes in their topology and nature on timescales of a few hours, due only to their transition through different plasma conditions along their orbit. While the magnetospheres have an Earth-like structure during times when the planet passes through super-Alfvénic sectors, they have an Io-like Alfvén wing shape during times when the planet passes through sub-Alfvénic sectors. Planet A and Planet B reside in the sub-Alfvénic sectors for most of their orbits. Surprisingly, the super-Alfvénic sectors are associated with slower wind speed. These sectors are super-Alfvénic due to the order of magnitude increase in the plasma density, which reduces the Alfvén speed in the wind.

On the Earth, the Cross Polar Cap Potential (CPCP) is the difference between the maximum and minimum ionospheric electric potential. It is associated with the solar wind driver and it becomes saturated during major CME events (e.g., Reiff et al. 1981; Siscoe et al. 2002). In the case of the planets studied here, the strong driving stellar wind is reflected in a saturated CPCP, where the magnitude of the CPCP is 10²–10⁴ kV—much greater than a potential of about 50 kV for the ambient solar wind conditions and 1000 kV for CME conditions. Ridley (2007) and Kivelson & Ridley (2008) have discussed the dependence of the CPCP on the solar wind conditions. They showed that for larger v_A (and likely sub-Alfvénic flow), the solar/stellar wind conductance, Σ_A = (μ₀v_A)⁻¹ (μ₀ is the permeability of free space), is smaller than the ionospheric conductance, Σ_p, and as a result, the reflected fraction of the stellar wind electric field becomes larger than the incident one and the CPCP becomes saturated. In the case of our hypothetical planets, the stellar wind is always extremely strong. Therefore, it seems likely the CPCP in these planets is always saturated, unless Σ_p is reduced even further due to the space environment conditions so it is again smaller than Σ_A. This can prevent from the stellar wind electric field to reflect and the CPCP is again in a non-saturated state. As mentioned in Section 3.2, we leave the detailed calculation of the ionospheric conductance in close-in exoplanets to future study.

Our results show that there is a significant Joule Heating at the upper atmospheres of the planets as a result of interaction with the extreme stellar wind. Overall, the heating is two to five orders of magnitude higher than that of the Earth during quiet solar wind conditions. The quiet-time heating for close-in exoplanets is even higher than those obtained on Earth during a strong CME event. It is most likely that the heating is even greater during CME events on these close-in planets as the energy deposited in such events can be three orders of magnitude larger than a typical CME on Earth (Cohen et al. 2011a). The time-dependent simulation of Planet A shows that additional heating is available to the sharp and quick changes in the magnetospheric topology as the planet passes between the sub- to the super-Alfvénic sectors. In a way, these changes can be viewed as if a CME hits the planet twice in an orbit, leading to sharp and fast changes in the conditions of the driving stellar wind.

As expected, the heating decreases with the increase of the planetary field strength as the planetary magnetic pressure reduces the stellar wind forcing. Of course, this can be modified by the particular orientation of the planetary and stellar magnetic fields as further heating can be driven by magnetic reconnection, which accelerates the precipitating electrons to higher energies (Kivelson & Russell 1995; Gombosi 1999). Our simulations also show that the heating increases with the decrease of the Pedersen conductance in the ionosphere. This is important because the conductance captures the role of the upper atmosphere in the energy input from the stellar wind. In reality, the conductance is defined by the atmospheric parameters and conditions. By estimating these parameters using a more detailed modeling (e.g., Ridley et al. 2002, 2004), one can obtain a good estimation of the conductance and the overall heating of the top-side atmosphere by the interaction with the stellar wind.

While the Joule Heating of the upper atmosphere is far smaller than the planetary core Joule Heating necessary to explain the inflation of hot jupiter planets (Batygin & Stevenson 2010; Perna et al. 2010; Huang & Cumming 2012; Menou 2012; Rauscher & Menou 2013; Spiegel & Burrows 2013; Rogers & Showman 2014), it is still significant in terms of the energy balance of the atmosphere. For the ambient stellar wind conditions, the heating of 10–50 W m⁻² can reach 0.1%–3% of the total stellar irradiating input shown in Table 1. As stated above, it can be even greater for periods of CMEs, as we expect the CME rate in M-dwarf stars to be high (Güdel 2007). The additional heating at the top of the atmosphere is important for modeling the atmospheres of habitable planets (e.g., Kasting et al. 1993; Spiegel et al. 2008; Heng & Vogt 2011; Tian et al. 2014) as they affect the atmospheric temperature. Therefore, these models should take in to account the atmospheric Joule Heating in the context of their pressure boundary conditions. In addition, Joule Heating at the top of the upper atmosphere transfers the energy to the thermosphere below, driving changes in the temperature, density, and pressure in the form of acoustic and gravity waves. As the changes in the magnetosphere and Joule Heating repeat along the orbit, it would be interesting to study the timescale of the change in the driving force (i.e., the stellar wind conditions along the orbit), to the propagation timescale of the planetary perturbations. However, this study is beyond the scope of this paper.