Transition from Super-Alfvénic to Sub-Alfvénic Stellar Wind Flow Passing by an Exoplanet, Using the Example of HD 209458b

The flow of super-Alfvénic or sub-Alfvénic plasma around a celestial body leads to the emergence of different types of magnetospheres: comet-like with a bow shock, or in the form of Alfvén wings resembling two tubes extended along the magnetic field lines of the flowing plasma. The interaction of magnetic exoplanets with stellar wind in these two different regimes has been considered in many studies. For example, Vidotto et al. in [1] investigated this issue with respect to exoplanets in the habitable zone around M dwarfs. The study [2] analyzed the reaction of magnetospheric current systems to changes in the Alfvénic Mach number (\({{M}_{{\text{A}}}}\)),which is the ratio of the flow speed around the planet and the Alfvén speed, and to variations in the planet’s proper magnetic field. Vernisse et al. [2] considered variations in \({{M}_{{\text{A}}}}\) due to changes in the flow velocity. Fischer and Saur [3] described the nonlinear interaction of Alfvén wings from two exoplanets. Two nearby exoplanets in the TRAPPIST-1 system (TRAPPIST-1b and 1c) were selected for analysis. The calculations were performed in MHD models, with neutral gas clouds simulating the planets. The authors concluded that detailed knowledge of star–planet interactions is necessary for exoplanet studies.

In this study, we consider one aspect of this interaction—the transition of the Alfvén Mach number in the stellar wind through unity and the corresponding changes in the structure of the exoplanet’s magnetosphere, using the example of the hot Jupiter HD 209458b. It is worth noting that similar problems have already been addressed in [4–6] regarding this object (HD 209458b) using 3D MHD modeling and taking into account the magnetic field of the hot Jupiter and its surrounding stellar wind. Zhilkin et al. [4–6] noted that the discussed issue is relevant for almost all hot Jupiters due to their proximity to the parent star. In addition, authors of [6] further examined the influence of a coronal mass ejection (CME) passing through the magnetosphere of HD 209458b on the transition of the planet’s flow regimes. The studies [4–6] present results of MHD modeling mainly in the equatorial plane or the planet’s orbit plane for the radial and azimuthal components of the stellar wind magnetic field, while in our study, we only model the magnetic field of the magnetosphere of HD 209458b in a paraboloid model with consideration for the vertical component of the stellar wind magnetic field. The modeling results are presented in the meridional noon-midnight plane of the magnetosphere. The gas envelope of the planet, calculated in the MHD modeling, is not defined in the paraboloid model. This also applies to the calculations of density, velocity, temperature, and Roche lobe. Furthermore, the papers [4–6] consider only the magnetic field of the exoplanet as the source of the magnetospheric field, while the paraboloid model includes the magnetic fields of the magnetospheric current systems in addition to the internal field of the planet, namely, the tail current system and the magnetopause current field, for which model parameters are determined based on conditions in the stellar wind.

The magnetic Mach number \({{M}_{{\text{A}}}} = {{V}_{{{\text{sw}}}}}{\text{/}}{{V}_{{\text{A}}}}\) is the ratio of the stellar wind speed \({{V}_{{{\text{sw}}}}}\) to the Alfvén speed \({{V}_{{\text{A}}}} = {{B}_{{{\text{sw}}}}}{\text{/}}{{({{\mu }_{0}}{{n}_{{{\text{sw}}}}}{{m}_{{{\text{sw}}}}})}^{{1/2}}}\), where \({{B}_{{{\text{sw}}}}}\) is the magnetic field of the stellar wind, or interplanetary magnetic field (IMF), \({{\mu }_{0}} = 4\pi \times {{10}^{{ - 7}}}\) H/m is the magnetic permeability of vacuum, \({{n}_{{{\text{sw}}}}}\) is the plasma density in the stellar wind, and \({{m}_{{{\text{sw}}}}}\) is the average mass of a particle in the stellar wind. The product \({{n}_{{{\text{sw}}}}}{\kern 1pt} {{m}_{{{\text{sw}}}}}\) is the mass density of the stellar wind.

The study [2] investigated how the structure resulting from the flow around a celestial body like the Moon changed with the variation in the velocity of the incoming plasma flow. The decrease in the velocity caused a transition from super-Alfvénic flow to sub-Alfvénic flow. Calculations were performed in a three-dimensional hybrid model. The authors noted that during this regime change, there was also a transition of currents from concentrated mainly in the equatorial plane to vertical, distributed along magnetic field lines.

Chané et al. [7] examined an extreme case where the Earth’s usual magnetosphere and the bow shock disappeared, and Alfvén wings formed with a length of \(600\,{\kern 1pt} {{R}_{{\text{E}}}}\), where \({{R}_{{\text{E}}}}\) is the Eath’s radius. In this case, \({{M}_{{\text{A}}}}\) and the magnetosonic Mach number were less than 1. The authors claimed that this was the first observation of Alfvén wings generated by Earth. The transition from super-Alfvénic to sub-Alfvénic flow was caused by very low solar wind plasma density. During this time, there was almost no auroral activity on Earth. The IMF was close to horizontal with an average value of 9.8 nT.

Ridley [8] numerically demonstrated that with an increase in the southward IMF, the Earth’s magnetosphere can transform into Alfvén wings. As an example, a CME was considered, which had a very strong southward magnetic field and very low plasma density. This led to the solar wind becoming sub-Alfvénic (\({{M}_{{\text{A}}}} < 1\)). Ridley [8] performed calculations in the MHD model of the University of Michigan. The IMF varied from –5 to –60 nT, while the density and speed of the solar wind remained constant: 5 cm^–3 and 400 km/s at a temperature of 250 000 K. \({{M}_{{\text{A}}}}\) varied from 8.2 to 0.7. Ridley [8] showed that the transition from super-Alfvénic to sub-Alfvénic regimes occurs continuously without a sharp change in the state of the magnetosphere of the body.

\({{M}_{{\text{A}}}}\) depends on three parameters: the speed of the plasma flow \({{V}_{{{\text{s}}w}}}\), the density of charged particles in it \({{n}_{{{\text{sw}}}}}\), and its magnetic field \({{B}_{{{\text{sw}}}}}\). We assume that the mass of charged particles in the flow \({{m}_{{{\text{sw}}}}}\) remains constant. In the problem considered in this study, we fix the experimentally found [9] distance from the center of the planet to the substellar point of the magnetopause (\({{R}_{1}}\)). This distance is found by the balance of the dynamic pressure of the stellar wind \({{P}_{{{\text{dynsw}}}}}\) and the magnetic pressure of the magnetosphere. This means that \({{P}_{{{\text{dynsw}}}}}\) should be const corresponding to the estimated scale of the magnetosphere \({{R}_{1}}\) (P_dynsw = \({{n}_{{{\text{sw}}}}}{{m}_{{{\text{sw}}}}}V_{{{\text{sw}}}}^{2}\) = const). In this case, either \({{n}_{{{\text{sw}}}}}\) and \({{V}_{{{\text{sw}}}}}\) do not change, or \({{n}_{{{\text{sw}}}}}V_{{{\text{sw}}}}^{2} = {\text{const}}\). Assuming that the first condition is satisfied (from which the second follows), we investigate how the magnetosphere changes with variations in \({{M}_{{\text{A}}}}\) due to changes in the component of the IMF parallel to the magnetic moment of the exoplanet. The orientation of the planet’s dipole is chosen to be southward, as with Earth.

The exoplanet HD 209458b was chosen because it is one of the few exoplanets for which the magnetic moment has been estimated (\( \sim {\kern 1pt} 1.6 \times {{10}^{{26}}}\) A m² [9]). It was discovered using the radial velocity method. Since the radius of the planet \({{R}_{{{\text{pl}}}}} = 1.38{\kern 1pt} \,{{R}_{{{\text{Jup}}}}} = \) \(9.54 \times {{10}^{7}}\) m [9], where \({{R}_{{{\text{Jup}}}}} = 69{\kern 1pt} \,911\) km is Jupiter’s radius, the magnetic field at the equator of this gas giant corresponding to the estimated magnetic moment is \({{B}_{{{\text{pl}}}}} = 18\,{\kern 1pt} 400\) nT.

HD 209458b is a hot Jupiter, located very close to the central star HD 209458, at a distance \(d = 0.047 {\text{AU}}\) = 7.1 × 10⁹ m. Orbital period \(T = \) 3.52 days. The central star is yellow, similar to the Sun, with a spectral class of G0V, an age of 4 Gyr, and a distance from the Sun of 47 pc.

Parameters of the stellar wind were estimated in [9]: the stellar wind speed \({{V}_{{{\text{sw}}}}} \sim 400 \times {{10}^{3}}\) m/s, and the stellar wind particle concentration \({{n}_{{{\text{sw}}}}} \sim 5 \times \) 10⁹ m^–3. According to the results of [10], the magnetic field on the surface of a Sun-like star is \( \sim {\kern 1pt} 1{\kern 1pt} 43\,000\) nT. If we assume that the properties of the stellar wind of HD 209458 are similar to those of the solar wind, \({{m}_{{{\text{sw}}}}} = 1.92 \times {{10}^{{ - 27}}}\) kg [10]. Calculations show that at the close orbit of the exoplanet, the Alfvén speed \({{V}_{{\text{A}}}}\) is on the order of the stellar wind speed \({{V}_{{{\text{sw}}}}}\) (the orbital motion of the exoplanet does not significantly change the order of magnitude of \({{V}_{{{\text{sw}}}}}\)). Therefore, the Alfvén Mach number \({{M}_{{\text{A}}}}\) is close to unity. Depending on the conditions in the stellar wind, \({{M}_{{\text{A}}}}\) can be greater or less than 1.

In this study, we will simulate the passage of the stellar wind on the orbit of HD 209458b through the Alfvén radius (where \({{M}_{{\text{A}}}} = 1\)) by varying the magnetic field of the stellar wind \({{B}_{{{\text{sw}}}}}\), and investigate the corresponding changes in the structure of the exoplanet’s magnetosphere based on the paraboloid model. It is worth noting that near the magnetopause boundary, the result will be incorrect because the model assumes a priori that the paraboloid magnetopause limits the magnetosphere, which is incorrect in the case of the formation of Alfvén wings. Taking this into account, we will consider the general nature of the changes in the structure of the magnetic field of the exoplanet’s magnetosphere with changes in the Alfvén Mach number.

The paraboloid model of the magnetosphere of the exoplanet [11] is an development of the model of the magnetospheres of magnetic planets in the Solar System [12]. In addition to the planetary magnetic field, it includes large-scale current systems: the tail current system; screening currents at the magnetopause; and partially penetrating IMF with a coefficient \(k \leqslant 1:b = k{{B}_{{{\text{sw}}}}}\). For exoplanets with a magnetodisc (or a powerful ring current), the magnetosphere model also includes a system associated with the m-agnetodisc. Since the distance to the substellar point of the magnetopause for HD 209458b is small (\({{R}_{1}} \sim 2.9{{R}_{{{\text{pl}}}}}\)), and authors [9] did not assume the presence of a magnetodisc for HD 209458b, the latter current system was not included in the model.

For the tail current system in the paraboloid model, a parameter was used, describing the distance from the center of the planet to the inner boundary of the tail current sheet. The magnitude of the magnetic field of the tail current system at a distance \({{R}_{2}} \sim 0.7{{R}_{1}}\) (the inner edge of the tail current sheet located in the equatorial plane of the magnetotail) can be roughly estimated using the formula: \({{B}_{t}} = - 1.2{\kern 1pt} {{B}_{{{\text{pl}}}}}{{({{R}_{{{\text{pl}}}}}{\text{/}}{{R}_{1}})}^{3}}\).

Thus, for the paraboloid model of the magnetosphere of exoplanet HD 209458b, the following parameters were obtained in the first approximation:

In this study, we assume the penetration coefficient of the IMF into the magnetosphere \(k\) to be equal to 1.

Figure 1 shows a cross-section of the magnetosphere of exoplanet HD 209458b for a moderate (relatively weak) IMF. In this case, the Alfvén Mach number is very large, resulting in a comet-like magnetosphere with a bow shock, not shown in the figure. Closed field lines extend from one hemisphere of the ionosphere to the other, crossing the equator. Open field lines cross the magnetopause and extend into the distant tail. At an IMF equal to zero, they do not cross the magnetopause and reach the end of the tail. In the tail lobes, the directions of the magnetic field are opposite in the northern and southern half-spaces; this corresponds to the current in the equatorial tail (neutral current sheet).

Meridional cross-section of the magnetosphere of the exoplanet HD 209458b from day to night. The model parameters are described in the text. The interplanetary magnetic field is presented in the star–magnetosphere coordinate system (\(X\), \(Y\), \(Z\)) with components \(\{ 0,0, - 90\} \) nT corresponding to \({{M}_{{\text{A}}}} > 1\).

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Let us consider at what southward IMF the Alfvén speed becomes approximately \({{V}_{{{\text{sw}}}}} = 400 \times {{10}^{3}}\) m/s, considering that the stellar wind particle concentration \({{n}_{{{\text{sw}}}}} \sim 5 \times {{10}^{9}}\) m^–3 and \({{m}_{{{\text{sw}}}}} = 1.92 \times {{10}^{{ - 27}}}\) kg, V_A = \({{B}_{{{\text{sw}}}}}{\text{/}}{{({{\mu }_{0}}{{n}_{{{\text{sw}}}}}{{m}_{{{\text{sw}}}}})}^{{1/2}}} \sim {{V}_{{{\text{sw}}}}}\), therefore, \(B_{{{\text{sw}}}}^{2}{\text{/}}({{\mu }_{0}}{{n}_{{{\text{sw}}}}}{{m}_{{{\text{sw}}}}}) = \) \(V_{{{\text{sw}}}}^{2}\) or \(B_{{{\text{sw}}}}^{2} = {{\mu }_{0}}{{n}_{{{\text{sw}}}}}{{m}_{{{\text{sw}}}}}V_{{{\text{sw}}}}^{2}\). For the given values, \(B_{{{\text{sw}}}}^{2} = 1{\kern 1pt} {\kern 1pt} 929{\kern 1pt} {\kern 1pt} 211 \times {{10}^{{ - 18}}}\) T\(^{2}\), therefore, \({{B}_{{{\text{sw}}}}} = - 1389\) nT. This value is on the order of the magnetic field of the stellar wind at the orbit of HD 209458b (\( \sim {\kern 1pt} 1440\) nT). If the southward IMF is of this magnitude (IMF: 0, 0, ‒1389 nT), the magnetic field pattern of the magnetosphere will appear as shown in Fig. 2.

Same as Fig. 1, but for IMF: \(\{ 0,0, - 1389\} \) nT, corresponding to \({{M}_{{\text{A}}}} = 1\).

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Let us assume, following Ridley [8], that upon collision of the exoplanet with a CME, the southward IMF increased by an order of magnitude compared to the usual value. In this case, \({{M}_{{\text{A}}}}\) becomes less than unity, and the altered structure of the magnetosphere’s magnetic field takes the form shown in Fig. 3.

Same as Fig. 1, but for IMF: \(\{ 0,0, - 13\,890\} \) nT, corresponding to \({{M}_{{\text{A}}}} < 1\).

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While \({{M}_{{\text{A}}}} > 1\), the paraboloid model of the comet-like magnetosphere of the exoplanet describes the following sources of the magnetospheric magnetic field: the planet’s field, screening currents at the magnetopause, the tail current system, and partially penetrating IMF. In the tail lobes adjacent to each other in the equatorial plane (Fig. 1), the magnetic fields are antiparallel to each other, and their drop corresponds to the current magnitude in the neutral current sheet of the tail.

Upon transitioning to \({{M}_{{\text{A}}}} < 1\) (Figs. 2, 3), these tail lobes spread upward and downward, diverge from each other, and the corresponding equatorial current across the tail disappears along with the \(\theta \)-shaped structure of the tail current system. Moreover, the oscillations of charged particles relative to the plane of the former neutral sheet also disappear. The tail itself becomes very short, consisting of closed field lines, and the tail lobes with open field lines transform into Alfvén wings. At low latitudes, convection from the distant tail is replaced by convection from the near point at the night boundary of the short tail. Reconnections on the day and night sides of the magnetosphere are preserved when the IMF and magnetic moment of the exoplanet dipole are parallel. At reconnection points, particles are accelerated due to magnetic energy; accelerated beams in field-aligned currents cause auroras on the exoplanet (analogous to Ganymede).

Alfvén waves are generated when the flow of plasma is decelerated by an obstacle (the exoplanet). Alfvén wings are caused by the interaction of Alfvén waves with the flowing plasma. Alfvén waves propagate along the IMF in opposite directions at speeds \( \pm {\kern 1pt} {{V}_{{\text{A}}}}\) and simultaneously are carried away by the plasma at a speed \({{V}_{{{\text{sw}}}}}\). As a result, the magnetic field is distorted by the plasma motion, and the field lines are tilted at an angle \(\beta = \arctan (1{\text{/}}{{M}_{{\text{A}}}})\), forming two tubes—Alfvén wings, representing the magnetosphere of the celestial body [8]. If \({{M}_{{\text{A}}}} > 1\), the flow tilts the field lines of the magnetosphere toward the equatorial plane, the two wings converge, merging into an equatorial night magnetosphere, and a stretched tail with a central current sheet is formed. If \({{M}_{{\text{A}}}} < 1\), the tilt of the magnetic field lines toward the equatorial plane sharply increases, the tail sections diverge, and the opposite magnetic fields on both sides of the equator disappear along with the central current sheet.

The change in the structure of the exoplanet’s magnetosphere during the transition from super-Alfvénic to sub-Alfvénic flow regimes has been demonstrated using the example of studying the interaction of the stellar wind from the G0V spectral class star HD 209458 with the hot Jupiter HD 209458b, which presumably has a magnetic field.

Depending on whether the Alfvén Mach number is greater or less than unity, different structures of the magnetic field in the surrounding area of the celestial body and different current systems arise. When \({{M}_{{\text{A}}}} > 1\), a comet-like magnetosphere with a bow shock forms around the exoplanet. When \({{M}_{{\text{A}}}} < 1\), Alfvén wings are formed, extending along the magnetic field lines connected with the central star. At the boundaries of the Alfvén wings (characteristics), the tilt of the magnetic field lines changes. At the same time, the tail current sheet in the equatorial night ma-gnetosphere disappears, and the tail lobes filled with open field lines, forming Alfvén wings, sharply diverge.

Based on the paraboloid model of the exoplanet’s magnetosphere, a transition from one regime to another is demonstrated by changing the magnitude of the IMF parallel to the planet’s magnetic moment.