Heating of the Atmospheres of Short-orbit Exoplanets by Their Rapid Orbital Motion through an Extreme Space Environment

We formalize our theoretical model for the possible heating of the planetary upper atmosphere of short-orbit exoplanets under the following underlying assumptions. First, we assume that due to the short orbit and the fast orbital motion of the planet, the variations in the background stellar wind magnetic field are also rather fast. Second, we neglect the possible existence of a planetary internal field, B_p , since the change in magnetic flux through the upper atmosphere should not be affected by the internal planetary field, only the external, stellar wind field (dB_p /dt = 0). We also neglect any field variations created by induced currents in the ionosphere. Finally, we assume a constant conductivity across the ionosphere. The last two assumptions are of course an approximation that needs further study. Nevertheless, our study here aims to provide some upper/lower limits for the effect.

We start with the integral form of Faraday's law, which states that a time-varying magnetic flux, Φ, generates an electromotive force, ${ \mathcal E }$ (EMF; a voltage generator):

$$\begin{eqnarray}&&{ \mathcal E }=-\displaystyle \frac{d{\rm{\Phi }}}{{dt}},\end{eqnarray} \tag{ 1 }$$

where the total magnetic flux, Φ, is defined as

$$\begin{eqnarray}&&{\rm{\Phi }}=\int {\boldsymbol{B}}\cdot {\boldsymbol{d}}{\boldsymbol{s}},\end{eqnarray} \tag{ 2 }$$

with B being the magnetic field and d s being a surface element. For a constant area, A, the magnetic flux is simply Φ = AB, where B here is the magnetic field which crosses the area A. For a constant area, Equation (1) becomes

$$\begin{eqnarray}&&-\displaystyle \frac{d{\rm{\Phi }}}{{dt}}=-A\displaystyle \frac{d{\boldsymbol{B}}}{{dt}}={ \mathcal E }={\boldsymbol{I}}R,\end{eqnarray} \tag{ 3 }$$

where I is the induced electric current driven by the EMF, and R is the electrical resistance. The electrical resistance can be related to the electrical conductivity, σ:

$$\begin{eqnarray}&&R=\displaystyle \frac{l}{\sigma a}.\end{eqnarray} \tag{ 4 }$$

We note that this conductivity has SI units of [S m⁻¹], and it is different from the height-integrated conductivity, with units of [S]. a is the cross section of the conductor/resistor through which the current I is flowing and l is the length of the current loop. With these definitions, we can express the total current, I , in terms of the current density, j , and the cross section, a:

$$\begin{eqnarray}&&{\boldsymbol{I}}=a{\boldsymbol{j}},\end{eqnarray} \tag{ 5 }$$

so Equation (3) becomes

$$\begin{eqnarray}&&-A\displaystyle \frac{d{\boldsymbol{B}}}{{dt}}={\boldsymbol{I}}R=a{\boldsymbol{j}}\displaystyle \frac{l}{a\sigma }=\displaystyle \frac{{\boldsymbol{j}}l}{\sigma },\end{eqnarray} \tag{ 6 }$$

and we can express the current density in terms of the rate of change of the ambient magnetic field:

$$\begin{eqnarray}&&{\boldsymbol{j}}=-\displaystyle \frac{A\sigma }{l}\displaystyle \frac{d{\boldsymbol{B}}}{{dt}}.\end{eqnarray} \tag{ 7 }$$

Figure 1 shows the geometry of the problem we formulate here. We assume a short-orbit exoplanet that sweeps through the interplanetary medium and the ambient stellar wind and that the ambient stellar wind magnetic field is radial in the frame of reference of the star due to the stretching by the stellar wind plasma. We neglect the azimuthal component of the ambient field since both the radial and the azimuthal field should contribute to the magnetic flux perpendicular to the ionosphere cross section. If a strong azimuthal field exists, it would only matter for the radial-azimuthal partition but not for the time-variations of the field magnitude. The star–planet line is defined as the X-direction (the planetary day side is facing the star) so that the ambient magnetic field is perpendicular to the planetary day side cross section, which is in the Y–Z plane. We focus our problem on the narrow conducting spherical layer at the top of the planetary atmosphere—the ionosphere, which is bounded by the ionosphere bottom, r_b , and the ionosphere top, r_t . The average altitude of the ionosphere's center is r_i = (r_b + r_t )/2, and the thickness of the ionosphere is d_i = r_t − r_b .

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In the limit where the ionosphere thickness is much smaller than the average height (r_i ≫ d_i ), and assuming that the ambient magnetic flux of the stellar wind is going through a spherical cross section bounded between r_b and r_t , the cross section can be approximated as

$$\begin{eqnarray}&&A={\int }_{{r}_{b}}^{{r}_{t}}2\pi {rdr}\approx 2\pi {r}_{i}{d}_{i}.\end{eqnarray} \tag{ 8 }$$

The change in the magnetic flux will drive a current in the ionosphere that will act to reduce that change (hence the minus sign in Equation (1)). Ideally, the induced current flows in a closed loop, which encloses the area through which the changing magnetic flux flows. In the more realistic scenario of the ionosphere, the current will flow through the path of highest conductivity, which will be defined by the local, nonuniform distribution of the ionospheric conductivity. As a lower limit for our problem, we consider the case where the current path, l, encloses half of the ionosphere cross section. This path is shown as dashed line in Figure 1(a). Since j is inversely proportional to l, this path provides the lower limit value for j, as well for the total JH. Using this path, we have

$$\begin{eqnarray}\begin{array}{rcl}l & = & \displaystyle \frac{1}{2}2\pi {r}_{b}+\displaystyle \frac{1}{2}2\pi {r}_{t}+2{d}_{i}=2\pi \displaystyle \frac{{r}_{b}+{r}_{t}}{2}+2{d}_{i}\\ & = & \,2\pi {r}_{i}+2{d}_{i}\approx 2\pi {r}_{i}.\end{array}\end{eqnarray} \tag{ 9 }$$

With these assumptions, and taking half of the cross section, A, Equation (7) becomes

$$\begin{eqnarray}&&{\boldsymbol{j}}=-\displaystyle \frac{1}{2}\displaystyle \frac{2\pi {r}_{i}{d}_{i}\sigma }{2\pi {r}_{i}}\displaystyle \frac{d{\boldsymbol{B}}}{{dt}}=-\displaystyle \frac{{d}_{i}\sigma }{2}\displaystyle \frac{d{\boldsymbol{B}}}{{dt}}.\end{eqnarray} \tag{ 10 }$$

Equation (10) can be written in the form of Ohm's law:

$$\begin{eqnarray}&&{\boldsymbol{j}}=\sigma {\boldsymbol{E}},\end{eqnarray} \tag{ 11 }$$

with ${\boldsymbol{E}}=-\displaystyle \frac{{d}_{i}}{2}\tfrac{d{\boldsymbol{B}}}{{dt}}$. The ohmic dissipation (JH), q, is given by q = j · E = j²/σ, or

$$\begin{eqnarray}&&q=\displaystyle \frac{{j}^{2}}{\sigma }=\displaystyle \frac{{d}_{i}^{2}{\sigma }^{2}{\left(\tfrac{{dB}}{{dt}}\right)}^{2}}{4\sigma }=\displaystyle \frac{{d}_{i}^{2}\sigma }{4}{\left(\displaystyle \frac{{dB}}{{dt}}\right)}^{2}.\end{eqnarray} \tag{ 12 }$$

Thus, the volumetric power of the JH as a function of the change in the ambient magnetic field is

$$\begin{eqnarray}&&q=\displaystyle \frac{{d}_{i}^{2}\sigma }{4}{\left(\displaystyle \frac{{dB}}{{dt}}\right)}^{2}\,[{\rm{W}}\,{{\rm{m}}}^{-3}].\end{eqnarray} \tag{ 13 }$$

The energy flux, Q, associated with the JH is given by multiplying Equation (13) by the width of the ionosphere, d_i :

$$\begin{eqnarray}&&Q={{qd}}_{i}=\displaystyle \frac{{d}_{i}^{3}\sigma }{4}{\left(\displaystyle \frac{{dB}}{{dt}}\right)}^{2}\,[{\rm{W}}\,{{\rm{m}}}^{-2}],\end{eqnarray} \tag{ 14 }$$

which is the total energy flux that is transferred to the ionosphere due to the change of the ambient stellar wind magnetic field.

In the limit of a very extended atmosphere, d_i ≈ r_t . In this case, the ionosphere cross section may be approximated as

$$\begin{eqnarray}&&A={\int }_{{r}_{b}}^{{r}_{t}}2\pi {rdr}=\pi ({r}_{t}^{2}-{r}_{b}^{2})\approx \pi {r}_{t}^{2}\approx \pi {d}_{i}^{2},\end{eqnarray} \tag{ 15 }$$

and the length of the current loop can be approximated as

$$\begin{eqnarray}&&l=\displaystyle \frac{1}{2}2\pi {d}_{i}+2{d}_{i}\approx 5{d}_{i}.\end{eqnarray} \tag{ 16 }$$

With these approximations, the current density becomes

$$\begin{eqnarray}&&{\boldsymbol{j}}=-\displaystyle \frac{\pi {d}_{i}^{2}}{2}\displaystyle \frac{\sigma }{5{d}_{i}}\displaystyle \frac{d{\boldsymbol{B}}}{{dt}}\approx -0.3{d}_{i}\sigma \displaystyle \frac{d{\boldsymbol{B}}}{{dt}}.\end{eqnarray} \tag{ 17 }$$

In this limit, the energy flux is

$$\begin{eqnarray}&&Q\approx {0.3}^{2}{d}_{i}^{3}\sigma {\left(\displaystyle \frac{{dB}}{{dt}}\right)}^{2}=0.1{d}_{i}^{3}\sigma {\left(\displaystyle \frac{{dB}}{{dt}}\right)}^{2}\,[{\rm{W}}\,{{\rm{m}}}^{-2}],\end{eqnarray} \tag{ 18 }$$

which is about half of the heating in the case where d_i ≪ r_i .

In a real ionosphere, the current does not flow in a wirelike, one-dimensional manner, but it flows in an extended region. The current, as well as its dissipated JH drop from a central region since the current acts to cancel the change in the magnetic flux. The size of the drop can be characterized by a "skin depth," δ, which can be estimated using Equation (5.165) from Jackson (1998):

$$\begin{eqnarray}&&\delta =\sqrt{\displaystyle \frac{2}{\sigma \mu \omega }},\end{eqnarray} \tag{ 19 }$$

with ω being the characteristic frequency of the time-varying magnetic field, and μ being the magnetic permeability of the material. Here we assume that μ = μ₀, the permeability of free space. If we assume that the dominant period, T, is of the order of 1 day (based on values shown in Section 3), then for ω = 2π/T and σ values of 0.1 and 10 S m⁻¹, we get a skin depth in the range of 100–1000 km. This means that the JH is not extremely local, but it is expanding from the skin around the ionospheric cross section, where the current flows. A more detailed estimate about the overall transfer of the VDJH requires a more detailed modeling effort.

VDJH depends on the variations of the interplanetary magnetic field (IMF) strength along the planetary orbit. Such detailed IMF data are not available for exoplanets (some observations were made for the stellar wind interaction with the interstellar medium; e.g., Wood et al. 2021), nor it is available for short orbits around the Sun (limited data at specific locations are available from the Parker Solar Probe; Raouafi et al. 2023). Due to the lack of observational constrains, we must rely on models to estimate the relevant stellar wind conditions. Physics-based models for the solar corona and the solar wind have been vastly validated against observations. Thus, they can provide the most reliable tool to study stellar winds of solar analogs, which are not observationally constrained. Such a model is the Alfvén wave solar model (AWSoM; van der Holst et al. 2014), which is based on the BATSRUS magnetohydrodynamic (MHD) model (Powell et al. 1999; Tóth et al. 2012). The model has been vastly used to study the stellar winds of Sun-like stars in the context of the space environment around exoplanets (e.g., Cohen et al. 2014, 2020; Garraffo et al. 2016, 2017; Dong et al. 2018; Vidotto et al. 2018; Alvarado-Gómez et al. 2019, 2022).

The model provides a three-dimensional numerical solutions for the solar/stellar wind from the stellar coronal base to a few tenths of stellar radii (depending on the specific star). The model is driven by the distribution of the photospheric radial magnetic field of the star, which is imposed as the inner boundary condition. These synoptic maps, or magnetograms, are obtained for the Sun using, e.g., the Michelson Doppler Imager (MDI; Scherrer et al. 1991) and the Helioseismic Magnetic Imager (Kosovichev & HMI Science Team 2007). For stars, low-resolution magnetic maps are obtained using the Zeemann–Doppler imaging (ZDI) technique (Donati et al. 1989). Using the observed boundary conditions, an analytic three-dimensional solution for the magnetic field is calculated using the potential field method (Altschuler & Newkirk 1969). This three-dimensional magnetic field is served as the initial condition in the model.

Once the initial, static (potential) conditions are defined, the model solves the set of nonideal MHD equations, which includes the conservation of mass, momentum, magnetic induction, and energy. The momentum equation includes an Alfvén wave pressure gradient term, which accelerates the solar wind, while the energy equation includes heating and cooling terms that depends on the Alfvén wave heating, electron heat conduction, and the radiative cooling. The model is run until an MHD, forced steady state is obtained in the three-dimensional simulation domain. We refer the reader to van der Holst et al. (2014) and Sokolov et al. (2021) for a complete description of the model. From this three-dimensional solution, simulated data can be extracted along a selected trajectory to provide the stellar wind conditions and magnetic field variations along a planetary orbit.

For our investigation here, we simulate the solar wind conditions during Carrington Rotation 1916, which represents solar minimum conditions (the model was driven by an MDI magnetogram). In the context of our investigation, the solar wind conditions should not vary much from solar minimum to solar maximum as only the wind's spatial distribution would change due to the change and reversal of the solar magnetic field. We extract this solar wind solution along two circular orbits close to the Sun—one at 0.05 au, and one at 0.1 au. These orbits represent the orbits of many known short-orbit exoplanets (e.g., the NASA exoplanetary encyclopedia ⁷ ). Thus, these orbital solutions represent typical variations of the IMF along the orbits of many exoplanets orbiting G-type stars in close-in orbits. For a case of an M dwarf exoplanet, we use the stellar wind conditions along the orbit of the planet Trappist-1e (Gillon et al. 2016) as calculated by Garraffo et al. (2017). While ZDI data are not available for Trappist-1, Garraffo et al. (2017) have used ZDI data of the proxy star GJ 3622. It has been demonstrated that stars with available ZDI data could be used as proxies for stars with a similar Rossby number and without available ZDI data (Garraffo et al. 2016, 2017, 2022).

Our goal in this paper is to demonstrate that VDJH could be significant in some exoplanets and it should be taken into account. The three synthetic data sets for B, as well as dB/dt, allow us to calculate the VDJH in a general manner, one based on the estimated conditions around an M dwarf star, and two that are based on the Sun itself (a G-type star). This range of conditions captures most of the possible intermediate exoplanet cases orbiting Sun-like stars.

Figure 2 shows the AWSoM solutions for the Sun and Trappist-1. The figure shows the distribution of the magnetic field strength and the number density over the equatorial plane in the simulation domain, along with the two hypothetical solar orbits and the orbit of the exoplanet Trappist-1e. It can be seen that the magnetic field of Trappist-1 is much stronger than that of the Sun (M dwarf stellar fields are much stronger than those of G-type stares), leading to a much stronger variations of the field strength along the orbit of Trappist-1e. The strongest variations are seen when the planet crosses the more dense regions near the helmet streamers of the stellar corona (the largest closed magnetic loops).

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