Interaction of the Exoplanet Hat–P–11b with Stellar Wind

Abstract

We discuss possible existence of a magnetodisk around the exoplanet HAT–P–11b. We used available observations to determine properties of the exoplanet and the stellar wind passing by it and obtained a rough estimate of the size of the planet’s magnetosphere. Comparing our estimate to published results of computations in a 3D electromagnetic relativistic and collisionless particle-in-cell model of the magnetosphere, we found a discrepancy in the magnetosphere size estimated using these two techniques. A possible interpretation of the discrepancy is suggested.

1 INTRODUCTION

In this paper, we consider the system resulting from interaction between the star HAT–P–11 and the exoplanet HAT–P–11b rotating around it. The Neptune-sized exoplanet HAT–P–11b has a low mass, M_pl ∼ 0.08 M_J, where M_J is the mass of Jupiter. The temperature of HAT–P–11b is ∼870 K; its distance from the parent star is d ∼ 0.0465 AU; the exoplanet’s orbit is polar and eccentric [1]. The authors of the cited paper estimate the strength of the equatorial magnetic field of HAT–P–11b, B_pl, to be of the order of 1–5 G, the most probable value being 2.4 G (the terrestrial magnetic field at the equator is ∼0.3 G); the tail length of the exoplanet’s magnetosphere was estimated as 1.8–3.1 AU. It is concluded in [1] that the observed dominant motion of particles in the antistellar direction is related to outflow from the polar region of the exoplanet. This conclusion is based on studying absorption in the C II 133.45 nm and H I Ly $\alpha$ lines during a transit of the exoplanet. The authors of [1] noted that the C II carbon ions could originate only at the exoplanet. They used data from the Hubble Space Telescope (HST) in the far ultraviolet range. HAT–P–11b is one of the first exoplanets with an existing estimate of the magnetic field.

The parent star HAT–P–11 in the constellation Cygnus has the spectral type K2–K4 V. It is an orange dwarf of the Main Sequence (MS) at the distance of ∼38 pc from the Sun, its mass is M_s ∼ 0.8 ${{M}_{ \odot }}$ and its radius, R_s ∼ 0.7 ${{R}_{ \odot }}$, where ${{R}_{ \odot }}$ ∼ 696 000 km is the radius of the Sun. The star’s age is ∼6.5 billion years, its effective temperature is ∼4780 K, its metallicity, [Fe/H] ∼0.31 (the spectral type of the Sun is G2 V and its effective temperature, ∼5770–5780 K [2]). There are two planets orbiting HAT–P–11: HAT–P–11b and HAT–P–11c.¹ The density of the star is estimated as ∼1.8 of the solar de-nsity [3]. The orbital period of the exoplanet HAT‒P–11b is ∼4.9^d. The ratio of the star’s rotation period (29.2^d) to the orbital period of the exoplanet HAT–P–11b is ∼6:1 [3].

Ben-Jaffel et al. [1] claimed that the space size of ionized carbon (C II) depended, in particular, on the magnetic field of HAT–P–11b. These authors concluded that, from their observations, the exoplanet’s magnetosphere should possess a dense plasmasphere and a long tail filled with low-density plasma from the planet’s polar wind. Comparing observations during transits of HAT–P–11b and outside them, they determined three interrelated free parameters: atmospheric metallicity, the exoplanet’s magnetic field, and the length of the magnetospheric tail. They concluded that the metallicity of HAT–P–11b was several times higher than that of the Sun (less than 3 times higher), while the metallicity of the star HAT–P–11 was twice higher than that of the Sun [1]. Using the value of the planet’s magnetic field they found, we will attempt to derive additional information on the magnetosphere of the exoplanet and its interaction with stellar wind.

The magnetic field of HAT–P–11 is unknown, but Morris et al. [3] studied spot activity of the star, interpreted it as an indication of solar-type dynamo. Active latitudes (where star spots mainly appear) were found to be symmetric with respect to the equatorial plane. The mean latitudes are ∼ ±16° [3], while they are ∼ ±15° for the Sun. The stellar activity of HAT–P–11 is higher than that of the Sun. It is assumed in [3] that $\alpha \Omega$ dynamo mechanism is working inside HAT–P–11, like in the solar case. Stellar spots of HAT–P–11 are larger than solar spots. Properties of stellar spots were derived from Kepler photometry during a transit of the exoplanet HAT–P–11b. It turns out that HAT–P–11 is more active than the Sun; in particular, the number of large spots on its surface is larger by a factor of ∼100, with the same valid for the spotted area.

Comparing HAT–P–11 to the most similar stars with known magnetic fields, HD 189733 and the Sun, the authors of [1] were able to estimate the field of HAT–P–11 as 1–2 G (close to the solar value, ∼1 G). These authors also derived the approximate temperature of stellar wind (1.3–1.5) × 10⁶ K; its velocity, 500–600 km s^–1; and density at the distance of the exoplanet, ∼3.3 × 10³ cm^–3. In the current paper, we consider interaction of the stellar wind from the parent star with the magetosphere of the exoplanet HAT–P–11b and compare the resulting structure to the findings from [1].

2 MAGNETOSPHERE OF THE EXOPLANET HAT–P–11b

Ben-Jaffel et al. [1] studied far-ultraviolet (113–146 nm) observations of the system HAT–P–11 obtained with the Hubble Space Telescope (HST). They compared spectra of the star before transits of HAT–P–11b and during the transits. Strong absorption in the blue wings of the C II (133.45 nm) and H I Ly α lines was observed during the transits. From this fact, they made a conclusion on global motion of particles in the anti-stellar direction in the night magnetosphere of the exoplanet. We use the planet’s magnetic field found in [1] in our attempt to derive more information on the magnetosphere of the exoplanet and its interaction with the stellar wind.

It is shown in [1] (Fig. 4d) that the magnetic-field lines found by these authors form a teardrop-shaped magnetosphere of the exoplanet. Such a magnetosphere appears if the distance from the star to the planet (d ≈ 0.0465 AU is our case) exceeds the Alfvén radius in the stellar wind. At the Alfvén radius, the kinetic energy density of the stellar wind equals the density of its magnetic energy, and the Alfvén Mach number is unity. If an exoplanet is within the Alfvén radius, it generates Alfvén wings instead of an ordinary magnetosphere (e.g., [4, 5]). When an exoplanet is outside the Alfvén radius, there appears a teardrop-shaped magnetosphere with a bow shock in front of it, like in the Solar system. All Solar-system planets are in the supersonic and super-Alfvenic solar wind. Alfvén wings are observed in the Solar system in the magnetosphere of Jupiter, when the sub-Alfvenic magnetospheric plasma flows around Jupiter’s satellites Io, Europa, and Ganymede (sometimes also Callisto [6, 7]). It is usually assumed that if a planet orbits its parent star outside the Alfvén radius, the main component of the pressure of stellar wind acting at the dayside magnetopause (boundary of the magnetosphere) is the dynamic pressure, as compared to thermal and magnetic pressure.

3 INTERACTION OF THE STELLAR WIND FROM HAT–P–11 WITH THE EXOPLANET HAT–P–11b

As mentioned above, the shape of the magnetosphere derived in [1] evidences for a super-Alfvenic and supersonic flow of stellar wind at the orbit of HAT–P–11b. Note for comparison that, according to [8], the sonic Mach velocity equals to the stellar wind velocity at the distance ≈5 R_s (≈0.008 AU) for M4.0 dwarfs and at ∼8 R_s (≈0.018 а.е.) for M1.5. Paper [8] studies the influence of the magnetic field of M dwarfs on their planets. The authors noted that the presence of magnetic field in the stellar wind can decrease the size of the exoplanet magnetosphere compared to the case of its absence. They considered only the contribution of the M-dwarf’s magnetic field to the stellar wind pressure, neglecting the dynamic and thermal pressure together with effects of ejections of the star’s coronal mass, and found an upper limit for the distance from the exoplanet center to the substellar point in the magnetosphere, R_ss, since adding any of these terms can only increase the pressure and hence oblateness of the magnetosphere. They demonstrated that the pressure of the magnetic field could be considerable for close-in planets.

The exoplanet we consider here, HAT–P–11b, is close to the parent K-type star, and though, according to [1], the magnetosphere is teardrop-shaped and limited with the bow shock in front of it, the magnetic field of the stellar wind can nevertheless add some contribution to the total pressure.

An expression for the distance from the center of the planet to the substellar magnetopause, based on the pressure balance at the front boundary of the magnetosphere, is presented in [9]:

$$\frac{{{{R}_{{{\text{ss}}}}}}}{{{{R}_{{{\text{pl}}}}}}} = {{\left\{ {\frac{{k_{m}^{2}B_{{{\text{pl}}}}^{2}}}{{2{{\mu }_{0}}({{k}_{{{\text{sw}}}}}{{p}_{{{\text{dynsw}}}}} + B_{{{\text{sw}}}}^{2}{\text{/2}}{{\mu }_{0}} + {{p}_{{{\text{swth}}}}})}}} \right\}}^{{1{\text{/}}6}}},$$

(1)

where k_m = 2.44 is the coefficient increasing the planet’s magnetosphere field at the dayside magnetopause due to screening currents of the magnetopause [10]; k_sw = 0.88 was found for single-atom stellar wind [11]; p_dynsw is the dynamic pressure of stellar wind; B_sw is the magnetic field of the stellar wind; p_swth is the thermal pressure of the stellar wind (all the three stellar-wind components are at the orbit of the exoplanet).

Let us compute the main parameter of the magnetosphere that determines its size, R_ss:

$$k_{m}^{2}B_{{{\text{pl}}}}^{2} = 34.27 \times {{10}^{{ - 8}}}\;{\text{k}}{{{\text{g}}}^{2}}\;{{{\text{A}}}^{{ - 2}}}\;{{{\text{s}}}^{{ - 4}}}.$$

(2)

Nichols and Milan [9] noted that p_swth = n_swthp kT_swthp was usually smaller than other pressure components (here k is Boltsmann constant; n_swthp and T_swthp are the density and temperature of thermal protons in the stellar wind). Figure 4c in [1] gives the density of stellar wind ∼3.3 × 10³ cm^–3 in front of the dayside magnetopause, ∼550 times higher than the stellar wind density at the Earth’s orbit (∼6 cm^–3).

A rough estimate of thermal pressure in the stellar wind from HAT–P–11 is:

$${{p}_{{{\text{swth}}}}} = {{n}_{{{\text{swthp}}}}}k{{T}_{{{\text{swthp}}}}} = 0.64\;{\text{nPa}}{\text{.}}$$

(3)

Here, the temperature of stellar wind near HAT–P–11b is assumed to be 1.4 × 10⁶ K (the mean from (1.3–1.5) × 10⁶ K [1]). Note that 1 Pa is 1 Newton per square meter.

The dynamic pressure of stellar wind, p_dynsw = m_pn_swp $V_{{{\text{swp}}}}^{2}$, was calculated for the stellar wind velocity ∼550 km s^–1 (the mean from 500–600 km s^–1 [1]):

$${{k}_{{{\text{sw}}}}}{{p}_{{{\text{dynsw}}}}} = 1470\;{\text{nPa}}{\text{.}}$$

(4)

If we assume the star’s magnetic field to be mainly dipole, the formulas for its components in the spherical polar coordinates $(r,\theta ,\varphi )$ will be ${{B}_{r}} = 2{{M}_{s}}$ cos θ ${{r}^{{ - 3}}}$, ${{B}_{\theta }} = {{M}_{s}}$ sin θ ${{r}^{{ - 3}}}$, ${{B}_{\varphi }}$ = 0, where M_s is the star’s dipole moment; $\theta$ is co-latitude and $\varphi$ longitude; ${{M}_{s}} = {{B}_{s}}R_{s}^{3}$ (see the site²), B_s being the magnetic field at the stellar equator, which, according to [1], is 1–2 G (in our calculations, we use the mean value, 1.5 G, 1 G = 0.0001 T). Thus, the magnetic field at the orbit of the planet, at the star’s co-latitude $\theta$, is:

$${{B}_{\theta }} = {{B}_{s}}R_{s}^{3}\sin \theta {\kern 1pt} {\kern 1pt} {{d}^{{ - 3}}} = 53\sin \theta \;{\text{nT,}}$$

(5)

$${{B}_{r}} = 2{{B}_{s}}R_{s}^{3}\cos \theta {{d}^{{ - 3}}} = 106\cos \theta \;{\text{nT}}{\text{.}}$$

(6)

To estimate the maximum value of B_sw, we consider the polar region (small θ), since the orbit of the planet HAT–P–11b is polar. Then, ${{B}_{\theta }}$ = 0 and B_r = 106 nT. The upper limit for the magnetic pressure of the stellar wind is:

$$B_{{{\text{sw}}}}^{2}{\text{/2}}{{\mu }_{0}} = 4.47\;{\text{nPa}}{\text{.}}$$

(7)

It follows from here that, in the considered case, the main contribution to the pressure is that from dynamic pressure, k_swp_dynsw = 1470 nPa. The thermal pressure is negligible (0.64 nPa), and the magnetic pressure, 4.47 nPa, is much lower than the dynamic pressure. Thus, the distance to the substellar point, R_ss, can be approximately calculated from Eq. (1) where only the dynamic pressure of the stellar wind is considered (4):

$$\frac{{{{R}_{{{\text{ss}}}}}}}{{{{R}_{{{\text{pl}}}}}}} = {{\left\{ {\frac{{k_{m}^{2}B_{{{\text{pl}}}}^{2}}}{{2{{\mu }_{0}}({{k}_{{{\text{sw}}}}}{{p}_{{{\text{dynsw}}}}})}}} \right\}}^{{1{\text{/}}6}}} \sim 7.$$

(8)

4 DISCUSSION

Here, we compare the results from [1] and our estimates. Ben-Jaffel et al. [1] used HST observations during transits to conclude that the polar wind from the planet HAT–P–11b filled the inner magnetosphere and its tail. The system under consideration being complex, the cited authors note that their results are not definite.

Analyzing Figs. 4c and 4d from [1], where field lines of the magnetic field are shown, we can conclude that the distance from the center of the exoplanet to the substellar magnetopause point, R_ss, is of the order of 20 R_pl, where R_pl is the planet’s radius ∼0.4 R_J, or ∼4.36 R_E, R_E ∼6378 km being the radius of the Earth (R_pl ∼ 2.78 × 10⁹ cm [1] or ∼2.78 × 10⁴ km). At noon, the dipole field structure of the magnetospheric magnetic field is ended at this distance. The distance to the substellar point is not mentioned in [1]. Instead, the authors claim that the front bow shock is located at 20 R_pl.

Our rough estimate gives R_ss ∼7 R_pl and disagrees with the structure shown in [1], Figs. 4c, 4d. Ben-Jaffel et al. [1] determined metallicity of HAT–P–11b from optical/IR HST observations and concluded that this metallicity corresponded to a planet similar to Jupiter rather than to Neptune. We suggest that the magnetosphere of HAT–P–11b can probably contain a ring current or a magnetodisk, like that of Jupiter and Saturn. The magnetic field of this magnetodisk outside it can exceed the dipole planet’s field and considerably increase the size of the magnetosphere and hence R_ss.

It follows from the magnetospheric structure shown in [1], Fig. 4d that the dipole axis of the exoplanet’s magnetic field is parallel to the magnetic field of the stellar wind. Reconnection will provide such a magnetospheric structure only in this case. As the direction of the planet’s magnetic field being not indicated, we do not know it and may arbitrarily choose orientation towards south or north. Correspondingly, the magnetic field of the stellar wind should also be directed to south or north. However, Ben-Jaffel et al. [1] claim that they computed the exoplanet magnetosphere structure only for zero interplanetary magnetic field (IMF), when the exoplanet was in the stellar current layer. A question arises because of this claim: for zero IMF, there should be no open field lines that go from the planet to the interplanetary space besides those from the distant tail’s perpendicular section. Thus, there are no magnetic field lines crossing the magnetopause, in variance with the scheme shown in [1], Fig. 4d.

Besides, Ben-Jaffel et al. [1] report that they determined the angular size of the polar cap as the boundary between open and closed field lines, but, for zero IMF, there should be no open field lines (only those that go in the tail and cross its far perpendicular section, but they are not mentioned in [1]). On the other hand, the caption to Fig. 4d in [1] claims that reconnection on the night side occurs at the distance of 50 R_pl in the tail. This feature is also typical of the case of IMF parallel to the exoplanet’s dipole axis but not for the closed magnetosphere with zero IMF.

5 THE PROBLEM OF THE SIZE OF EXOPLANET’S MAGNETOSPHERE

We see that our estimate gives a too small distance to the front stand-off magnetospheric point, R_ss ∼ 7 R_pl, compared to that derived from observations, ∼20 R_pl [1]. Khodachenko et al. [12] considered, in detail, magnetospheres of close-in giant planets possessing a magnetodisk. For these planets, it is necessary to consider pressure from the magnetodisk’s plasma and its magnetic field, resulting in a considerable increase of the magnetosphere size. Strongly influenced with X-ray and UV radiation from parent stars, atmospheres of close-in giant planets experience heating, ionization, hydrodynamic expansion, and enhanced plasma outflow. This outflow can be very significant and, together with the planet’s rotation, can result in appearance of a magnetodisk or a ring current, like in the case of Jupiter or Saturn [13, 14]. In the case of these gas giants of the Solar system, their satellites, respectively Io and Enceladus, are sources of additional magnetospheric plasma. Influence from parent stars on some close-in giant exoplanets can be so strong that plasma outflow from the atmosphere can be more effective for disk formation than ejections from volcanic satellites. We do not know any details concerning formation of the magnetodisk of HAT–P–11b and thus can only assume its presence due to the physical processes mentioned above. It was noted in [12] that magnetodisk formation changed the magnetosphere size by 40–70%. Magnetic field of Jupiter’s magnetodisk is described in detail in [13, 14]. This field becomes dominant outside the Alfvén radius in the magnetosphere of an exoplanet. At the Alfvén radius, the rigid-body rotation of magnetospheric plasma is disturbed because of radial outflow of plasma under the action of the centrifugal force (see also [15, 16]). As a sequence, there appears a unipolar inductor, it generates field-aligned electric currents that are closed with ionospheric Pedersen currents and perpendicular (radial) currents in the equatorial magnetosphere inside the magnetodisk. If the atmospheric outflow of plasma from the close-in rotating exoplanet is strong, the disk can be strong enough, and it can be extended from the magnetospheric Alfvén radius to the neighborhood of the substellar point, R_ss [12], like in the case of Jupiter. Jupiter’s magnetodisk increases the magnetic field in the subsolar point by a factor of 2.6 [13]. Were there no additional source of plasma (Io) in the magnetosphere of Jupiter, the distance to the subsolar magnetopause point would have been ∼42 R_J. The presence of this source increases R_ss to ∼100 R_J, i.e., by a factor of ∼2.4. If the exoplanet HAT–P–11b possesses a similar volcanic satellite or/and experiences a strong outflow of atmospheric plasma, the increase of R_ss for this exoplanet needed for consistency with measurements (from ∼7 R_pl to ∼20 R_pl) will be by a factor of ∼2.9, of the same order as for Jupiter.

It should be noted that HAT–P–11b is not the only exoplanet with an available magnetic field estimate. Paper [17] presents an estimate of the magnetic moment of the exoplanet HD 209458b from Ly α observations of the HST. Besides, in the assumption that the Alfvén Mach number in the stellar wind is above unity, the authors of [17] found the distance to the point where the flow is terminated with an obstacle and characteristics of stellar wind (density and velocity). Thus, the approach used in our paper can also be used in other cases.

6 CONCLUSIONS

In the present paper, we analyze interaction of the exoplanet with estimated magnetic field, HAT–P–11b, with the stellar wind passing by it. From pressure balance, we find the distance from the exoplanet’s center to the frontal magnetopause point R_ss. For this purpose, we used data from the literature to calculate the dynamic (k_sw p_dynsw = 1470 nPa), thermal (p_swthp = n_swthp kT_swthp = 0.64 nPa), and magnetic ($B_{{{\text{sw}}}}^{2}{\text{/2}}{{\mu }_{0}}$ = 4.47 nPa) pressure at the interplanetary side of the noon magnetopause. For the magnetospheric side of the noon magnetopause, we considered only the magnetic pressure from the planet (with the magnetic field at the equator B_pl = 2.4 G) and the field from screening currents of the magnetopause. The distance to the substellar magnetopause found in this way is R_ss ∼ 7 R_pl, where R_pl is the radius of the planet. This distance is much lower than the value that can be derived from the results of [1], Figs. 4c, 4d, ∼20 R_pl, based on HST observations. From this disagreement, we conclude that the giant planet HAT–P–11b probably has a strong magnetodisk formed due to atmospheric outflow of plasma under the action of the close active K‑type star HAT–P–11. As noted in [12], the presence of such a disk considerably increases the inner total pressure of the magnetosphere, which is able to increase its size. Were there no additional source of plasma (Io) in the magnetosphere of Jupiter, the distance to the subsolar magnetopause point would have been ∼42 R_J. The presence of this source increases R_ss to ∼100 R_J, i.e. by a factor of ∼2.4. If the exoplanet HAT–P–11b possesses a similar volcanic satellite or/and experiences a strong outflow of atmospheric plasma, than the increase of R_ss for it needed for consistency with measurements (from ∼7 R_pl to ∼20 R_pl) would be by a factor of ∼2.9, of the same order as for Jupiter.