A&A 434, 1191-1200 (2005)
DOI: 10.1051/0004-6361:20041680
S. Preusse 1 - A. Kopp1 - J. Büchner 1 - U. Motschmann 2
1 - Max-Planck-Institut für Sonnensystemforschung,
Max-Planck-Str. 2,
37191 Katlenburg-Lindau, Germany
2 -
Institut für Theoretische Physik, Technische Universität
Braunschweig, Mendelssohnstrasse 3, 38106 Braunschweig,
Germany
Received 16 July 2004 / Accepted 26 December 2004
Abstract
Close-in extrasolar planets of Sun-like stars are exposed to
stellar wind conditions that differ considerably from those for planets in the solar
system. Unfortunately, these stellar winds belong to the still unknown parameters of
these planetary systems. On the other hand, they play a crucial role in a number
of star-planet interaction processes that may lead to observable radiation events. In order to lay a foundation
for the investigation of such interaction processes, we estimate stellar wind parameters on the
basis of the solar wind model by Weber & Davis and study the
implications of the stellar magnetic fields. Our results suggest that in contrast
to the solar system planets, some close-in extrasolar planets may be
obstacles in a sub-Alfvénic stellar wind flow. In this case, the stellar wind magnetic pressure is comparable to or even larger than the dynamic flow pressure. We discuss possible consequences of these findings for the wind-exoplanet interactions. Further, we derive upper limit estimates for the energies such stellar winds can deposit in the exoplanetary magnetospheres. We finally discuss the implications the sub-Alfvénic environment may have on the star-planet interaction.
Key words: stars: winds, outflows - stars: magnetic fields - magnetohydrodynamics (MHD) - stars: planetary systems
Among the extrasolar planets detected since 1995 many have been found in close vicinity to their star. OGLE-TR-56 b is an extreme but not unusual example. Its orbital distance of 0.0225 AU will be approx. 20 times closer than Mercury is to the Sun, if the orbital eccentricity is taken into account.
The implications of this extreme vicinity to the stars for the star-planet systems are expected to be manifold. A variety of phenomena resulting from interaction between the star and its close-in planet are expected. Though both components are involved in such interaction, the theories and observations can be sorted according to whether the physical processes are associated with the star or the planet.
From the stellar point of view, gravitational interaction may come into play in terms of spin up by planetary migration towards the star if counteracting processes are neglected as is discussed by Paetzold et al. (2004). Furthermore, it may lead to heating and an increase of stellar activity as pointed out by Cuntz et al. (2000) who also give estimates of the magnetic interaction for several planetary systems. Shkolnik et al. (2003) present observational evidence for a hot spot in the chromosphere of one of these stars, HD 179949, which they attribute to magnetic interaction of the star with its close-in planet. Rubenstein & Schaefer (2000), who discuss superflares on single solar-like stars, suggest that these flares might occur due to a RS CVn binary-like interaction between star and planet.
Strong radiation events are also expected to occur on the planets. Farrell et al. (1999) and Zarka et al. (2001) expect strong radio emission due to the interaction of the stellar wind with planetary magnetospheres. It should be noted that so far the only observational hint of a planetary magnetic field is given by the observations of Shkolnik et al. (2003). Bastian et al. (2000) report on a search for radio emission which, however, was not successful. The possibility of success with state-of-the-art instrumentation is discussed by Rucker (2002).
Paetzold et al. (2004) also studied the influence of the gravitational interplay between star and planet on the rotational evolution of the planets, and Grießmeier et al. (2004) extend these effects to the possible magnetospheric configuration of OGLE-TR-56 b and HD 209458 b. Further, they discuss the possibility of atmospheric erosion due to the stellar wind during the evolution of these planets which is supported by Vidal-Madjar et al. (2004) who observed an escaping atmosphere for the planet HD 209458 b. The first results of modelling of the star-exoplanet magnetosphere interaction numerically with a magnetohydrodynamical description are presented by Ip et al. (2004).
Table 1: Planetary parameters as given by J. Schneider (http://www.obspm.fr/encycl/catalog.html). Listed (in units of Jupiter's mass and radius) are the projected planetary mass, the planetary mass and radius if determined by observation. The orbital semi-major axis and eccentricity are given as well as the inclination. These are followed by the calculated distances in AU and stellar radii. The last column gives the orbital period.
All these models have in common that at some point they need information about the structure of the stellar wind either at the planetary orbit or in between. Up to now, there are no observations available of winds for the stars in question, so that alternative approaches have to be used. Grießmeier et al. (2004) for example use densities and velocities of the solar wind at 1 AU which are well determined due to a long history of measurements. They adopt these as quantities at the planetary orbits and derive changes due to the evolution of the star on the basis of indirect observations of stellar winds from Wood et al. (2002).
In this paper, we attempt to fill this gap using the wind model derived by Weber & Davis (1967). In order to estimate velocities and densities for a range of stellar parameters, some are taken directly from observations while others are varied within intervals as they are less well known. In Sect. 2, we discuss the well-known parameters of the planets and stars of interest. We then introduce the Weber & Davis model with its limits in Sect. 3.1 and discuss the model winds and the influence of magnetic field and stellar rotation period on the stellar wind velocity in Sect. 3.2. With the results from Sect. 3 the stand-off distances and energy deposition onto the magnetospheres are estimated in Sect. 4 before we conclude our study in Sect. 5.
Table 2:
Stellar parameters taken from Allende Prieto & Lambert (1999). The first column gives the name
of the planet followed by the stellar HD and HIP numbers. The other columns
list the apparent visual magnitude, the parallax, the absolute magnitude, the
colour B-V and the stellar mass, radius, bolometric correction and effective
temperature. The rms errors are given below the parameter values except for
the radius. Note that Allende Prieto & Lambert (1999) originally gave the radius as .
Hence, the error stated below the calculated radius is the maximum
deviation of the calculated value obtained by applying the rms error to the
value.
In order to make our derivations as realistic as possible, we use estimates from typical upper and lower limits of the required input parameters based on the diverse observed quantities.
At the beginning of 2004, 120 extrasolar planets in 105 stellar systems, of
which 13 are multiple planetary systems, were listed in J. Schneider's
web-page (http://www.obspm.fr/encycl/catalog.html). From radial
velocity measurements, which is currently the most potent observation method,
the semi-major axis a, the orbital period
,
the eccentricity e and the projected mass
,
where i is the angle of inclination of the planetary orbit with respect to the observer, are available for all planets. The two planets HD 209458 b and OGLE-TR-56 b were also observed in transits across the stellar disk. Since this method allows one to
determine the planetary radius, they are the only planets for which radii
and orbital inclinations and thus the planetary masses
are known from observations.
Almost all stars hosting extrasolar planets are listed in the Hipparcos
Catalogue and approx. 80% are contained in the "Catalogue of Fundamental
Parameters of Stars'' compiled by Allende Prieto & Lambert (1999) hereafter called APL. The
latter derive basic stellar parameters such as stellar mass
and
radius
by isochrone interpolation on the basis of the Hipparcos
Catalogue. For stars with metallicities within 0.4 and 2.5 times the solar
metallicity
they achieve an overall accuracy within
6% for the stellar radii and 8% for the masses. On the basis of these stellar radii, the condition that the star-planet distance d=a(1+e) is less than 10
yields a subset of 12 planets. Table 1 summarises the planetary parameters
and
in Jupiter masses
,
in Jupiter radii
,
a, e, i and
(rounded) as given by Schneider. Our computed distances d in AU and
are listed as well. OGLE-TR-56 was not surveyed by Hipparcos and is not contained in APL. Hence, the stellar mass and radius are taken from Konacki et al. (2003).
The condition of d<10
puts a limit on the star-planet distance of
0.06 AU. Thus, BD-103166 is added to the list for completeness although it
is neither included in the Hipparcos catalogue, nor is its radius specified
in Butler et al. (2000) due to the parallax being unknown.
The stellar parameters, as given by APL are shown in Table 2.
The first column gives for each planet the name of the planet in the first
line, the stellar designation in the Henry Draper Catalogue (HD number) in
the second line and the Hipparcos Catalogue (HIP number) in the third line.
The apparent visual magnitude
,
the parallax plx, the absolute
magnitude
,
the B-V colour as well as the mass, the radius
(calculated from the
values given in the catalogue), the
bolometric correction BC and the effective temperature are given in the
columns. The rms errors are printed below the parameter values except for the
radius, for which the deviation of
obtained by applying the rms error to the
value of Allende Prieto & Lambert (1999) is given.
APL offer the most coherent set of stellar data for the 13 stars of interest.
For comparison, Table 3 lists the stellar masses, radii, ages
and rotation periods
as given by the authors listed beneath
the table. The parameter columns give the minimum and maximum values. Each
column is followed by a list of the individual references, sorted in
ascending order according to the parameter value they provide. Laws et al. (2003)
and Barnes (2001) derive the age with two different approaches each. If
these differ within the paper, they are referred to twice in ascending order
of the parameter value.
Table 3: Summarized stellar parameters taken from the references listed below. The columns show the lower and upper limits of the stellar masses, radii, ages and rotation periods. Each column is followed by a column listing the references, which are sorted in ascending order according to the parameter value the authors give.
A basic hydrodynamic description of the plasma environment of the extrasolar
planets must assume the mass m of the relevant particle species, their
number density n or mass density ,
their temperature T and flow
velocity
.
For a magnetohydrodynamic model we need the magnetic flux
density
.
Which of these parameters can be estimated depends on the
chosen model and on the availability of the input parameters.
A simple hydrodynamic approach for a non-rotating and unmagnetised star was first derived by Parker (1958). Weber & Davis (1967) added rotation and a magnetic field to approach a self-consistent magnetohydrodynamic stellar wind description. As the latter yields the Parker model in the limit of no rotation and no magnetic field, we restrict ourselves to the Weber & Davis model while discussing how both models are related.
Since the temperature variations in the stellar coronae are unknown, the
coronae are assumed to be isothermal. The plasma is taken to be quasineutral,
i.e.
,
were
and
are the
electron and proton number densities, respectively. Hence the pressure p is given by
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For a non-rotating star without a magnetic field the components (2) in Eq. (3) vanish and the equation is equal to Parker's solar wind solution for a purely hydrodynamical expansion (Parker 1958). For rotating stars without magnetic fields the dynamics have been studied previously by e.g. Weidelt (1973) and Mufson & Liszt (1975). If the centrifugal forces due to an increased stellar rotation and magnetic field dominate term (1), Eq. (3) will turn into the fast magnetic rotator approximation as described by Belcher & MacGregor (1976).
Due to solenoidality the magnetic flux is
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If the fluid reaches radial Alfvén velocity Ar, the corresponding
position will be called the Alfvén point .
If the wind velocity
equals the slow or fast magnetosonic wave speed, the bracket in
Eq. (3) will vanish. This defines the two critical points, the
slow and fast magnetosonic points
and
,
which are
necessary for the existence of a stellar wind solution as e.g. shown by
Goldreich & Julian (1970). They are defined as the radial distances at which the
wind reaches the phase velocity of the slow magnetosonic wave
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For the calculation of the radial velocity profile, the Alfvén point
and velocity Ar have to be known a priori. As we are interested in the dependence of the wind parameters on changes of stellar parameters, we define the wind parameters at the base of the corona and search for the resulting Alfvén point
and Alfvén velocity Ar. For this we
define
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For all calculations we use the present day solar mass flux of
1034 s
-1=1.05
108 kg s-1, with m the
mass of the protons, following Mann et al. (1999). The density at the base of the
corona thus varies according to the wind solution for the given parameters.
Hence, increasing the rotation rate while keeping the temperature constant,
the densities at the base of the corona are smaller when the wind speeds are
higher. For the Sun the assumed mass flux is equivalent, with an averaged
wind speed of 425 km s-1 at 1 AU. Combining this with the solar
equatorial rotation period of 25.5 d and a radial magnetic field density of
3.1 nT at 1 AU, the corresponding Weber & Davis solution is found for
a temperature of 0.8132
106 K and a magnetic field strength of
1.435
10-4 T at the base of the corona with
AU
(
). Keeping these values fixed, we calculate the
solution for the set of stars given in Table 3 for rotation
periods of 3 and 30 d. The results for both rotation periods are presented as
functions of the distance from the star in Fig. 1. The hatched
curve indicates the ranges over which the radial velocity profiles vary. The
locations of the planets are indicated by the dotted rectangle that extends
along the whole y-axis. The other two rectangles indicate the locations of
the Alfvén points (dark) and fast magnetosonic points (light), while their
extension along the y-axis shows the range of the corresponding velocities.
The overlap is due to the different stellar parameters. In the solutions the
critical points are clearly distinguishable. For
d the slow
magnetosonic points are located between 0.0239 and 0.0314 AU with critical
velocities of 110 to 112 km s-1 and for
d the critical
velocity of 116 km s-1 is reached between 0.0296 and 0.0449 AU. Two specific velocity profiles are given for
And (thick line) and for
HD 46375 (dash-dotted line). At 1 AU velocities between 406 and
431 km s-1 (Sun 425 km s-1) are reached for
d and 501 and 594 km s-1 (Sun 501 km s-1) for
d,
respectively. For slow rotation rates the fastest wind is reached by the
smallest and lightest star in terms of radius and mass, currently HD 46375,
whereas the fastest by the largest and heaviest, currently
And.
For high rotation rates this is still valid at the location of the planets
but further out, the situation changes. The fastest wind at 1 AU is now
obtained for the largest star, and the slowest wind does not belong to the
smallest and lightest but to the heavier one with the same radius, currently
HD 83443.
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Figure 1:
Stellar wind velocities obtained with the Weber & Davis model for
the stars given in Table 3. The hatched curves indicate the
velocity ranges over which the solutions vary for
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Figure 2:
Radial velocity profiles for ![]() ![]() ![]() ![]() ![]() ![]() |
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In order to study the influence of magnetic field strength and coronal
temperature on the stellar wind velocities, the magnetic flux FB is varied
at the base of the corona, where we assume a magnetic flux density B0 of 1
to 10
10-4 T. The low value of 1
10-4 T is obtained from
with
nT at 1 AU, which was measured e.g. by
Ulysses (see Banaszkiewicz et al. 1998). 10
10-4 T is an approximate
mean value for the field strength at the base of the solar corona in a
coronal hole, following Banaszkiewicz et al. (1998), who used an analytic model
for the coronal magnetic field of the Sun. The difference between both values
results from the additional expansion of magnetic flux tubes emanating from
the coronal hole and then expanding into the equatorial plane.
In Fig. 2 we show the solutions for
d (left
panel) and 3 d (right panel) obtained with the Weber & Davis model for
T=0.5
106 K and 2.0
106 K in comparison to the velocity
profiles obtained in the Parker limit. The lower curves belong to the cool
corona, the upper to the hot one. The magnetic field at the base of the
corona was set to 1
10-4 T (dots) and 10
10-4 T
(triangles). For the long rotation period, the Weber & Davis velocity
profiles are almost equal to those calculated with the Parker model. The
increase of rotation to 3 d shows a strong effect on the velocity profile
while the influence of the magnetic field on the velocity becomes apparent.
The velocity changes are larger for
And, whose radius is larger
than that of HD 46375 and which is the star with the slowest wind in the
Parker limit. The locations of all planets of interest are indicated by the
grey patch.
As we have calculated the velocity profiles for parameters which are
specified either at the stellar surface (T, B0,
)
or at
1 AU (Fm), we obtain the critical points as functions of these parameters
and hence the Alfvén points
as functions of these parameters as
well. In Table 4 the different locations of the Alfvén points
in AU, the corresponding Alfvén velocities Ar and the radial velocities
reached at 1 AU in km s-1 are listed with respect to the parameters
T=[0.5,2.0]
106 K,
B0=[1,10]
10-4 K and
d for the two stars HD 46375 and
And. The values should be considered relative to the planetary locations, which
are 0.041 AU and 0.0597 AU for HD 46375 b and
And b, respectively.
We can see that both planets are located within the Alfvén radius for all
stellar winds that were modelled for their central stars. The velocities at 1 AU
should be compared to the velocities that are obtained
in the Parker limit. The latter yields 310 to 313 km s-1 for
T=0.5
106 K and 759 to 764 km s-1 for T=2.0
106 K.
Table 4:
Alfvén radii
and velocities Ar obtained
with the Weber & Davis model for the two stars HD 46375 and
And.
The stellar winds were modeled for different magnetic field strengths B0at the coronal base, rotation periods
and coronal temperatures T as given in the table. In addition, the wind velocities at 1 AU
are given.
The Alfvén Mach numbers ,
i.e. the ratios of the radial
velocities vr at the planetary location to the radial Alfvén velocities Ar at the same location, are listed in Table 5 for both planets
and the same parameters as above. They are complemented by two additional
Alfvén Mach numbers, which were obtained relative to the planetary
movements around their stars. Such the planets effectively see the wind
moving with the relative velocity of their own movement with respect to the
expanding stellar corona. We thus define an effective velocity
,
where
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Table 5:
Alfvén Mach numbers
at the locations of HD 46375 b and
And b for parallel and antiparallel directions of stellar rotation and planetary movement. The values were derived for different magnetic field strengths B0 at the
coronal base as well as different stellar rotation periods
and
coronal temperatures T as given in the table.
Figure 3 shows the absolute values of the magnetic field
components obtained with the Weber & Davis model for And with
T=2.0
106 K and B0=10
10-4 T for the two rotation periods.
The magnetic field is scaled logarithmically. The radial components are
identical as they are independent of the rotation period and are plotted as
solid line. For the azimuthal component the dashed line is used for a
rotation period of 30 d and the dash-dotted for 3 d. The vertical dotted line
marks the location of the planet.
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Figure 3:
Absolute magnetic field components obtained with the Weber & Davis model for ![]() ![]() ![]() ![]() ![]() |
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Figure 4:
Components of stellar wind pressure for ![]() ![]() ![]() ![]() ![]() ![]() |
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The role of the magnetic field can be assessed by comparing the magnetic
pressure and the dynamic flow pressure. The dynamic flow pressure
can be expressed by
,
where the mass
density
at the planetary orbit is again estimated with Fm=1.05
108 kg s-1. The magnetic pressure
of the stellar
wind is given by
These results indicate that even for weak magnetic fields the magnetic pressure may be comparable to or even higher than the dynamic flow pressure at the planetary location. This should be taken into account if the interaction of the stellar wind with the planet is considered.
In order to estimate the energetics of a possible interaction between stellar wind and planet, we first derive an estimate for the magnetospheric stand-off distance as a characteristic length describing the size of the magnetosphere.
The location of the stand-off distance Rm of the planetary magnetosphere
is defined by the pressure balance between dynamic flow and magnetic pressure
and
of the stellar wind and the magnetic
pressure due to the planetary magnetic field
and the
magnetic field component
,
which is induced by
magnetopause currents
Table 6:
Results for And b and HD 46375 b obtained with the
Weber & Davis model with coronal temperatures T, magnetic field strengths B0 and rotation periods
as given as general parameters. For
the two planets the radial and azimuthal velocity components
and the azimuthal magnetic field components
at the location of the planets are listed. With these, the magnetospheric stand-off distances
and magnetic moments
were estimated,
followed by the power P that can be deposited by the stellar wind into the
magnetosphere. For comparison the stand-off distance and the power input into
the magnetosphere for the same planet located at 1 AU are also given.
Inserting Eqs. (21), (24) and (23) into Eq. (22) yields
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Similar to Perreault & Akasofu (1978) and D'Angelo & Goertz (1979) we calculate the Poynting
flux of the stellar wind by
In order to estimate the energy flux through the cross section of the
planetary magnetosphere given by ,
we assume that the stellar wind
parameters obtained for the equatorial plane with the Weber & Davis model do
not vary perpendicular to it. The multiplication of Poynting flux and
magnetospheric cross section yields
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The detailed results for the planets And b and HD 46375 b are
given in Table 6. At the top the model parameters such as
temperature, magnetic field strength at the stellar surface and rotation
period are given. Beneath, after the name of the planet, the specific
parameters such as radial and azimuthal velocity components and the azimuthal
magnetic field component are given at the location of the planet. These are
followed by the planetary magnetic moments, for which we used the maximum
value obtained as described above. They are given in
,
where
the subscript J stands for Jupiter. With these, the stand-off distances are
obtained, which were used for the estimation of the power that is deposited
in the magnetosphere. Furthermore, the stand-off distance and
the power are given for the same planet, but this time located at 1 AU. Due
to the much higher velocities and lower densities, the stand-off distances
are much larger. On the other hand, the magnetic field strengths decrease
significantly, so that the upper limit of the power input is much smaller.
For the Earth's magnetosphere two processes are discernible in which the
energy is dissipated, see e.g. Baker et al. (1986), Rostoker et al. (1987), Lu et al. (1998)
and Sun et al. (1998). The first is characterised by direct ohmic or convective
dissipation of energy in the ionosphere, the ring current and elsewhere. The
second process is characterised by energy storage in the magnetotail (loading
process) and its sudden dissipation during an unloading process. The time
during which energy is stored in the magnetotail is typically
1 h before it is dissipated in the ionosphere and the ring current. Comparing the results obtained for the same planet at its true orbit and at an assumed Earth-like orbit of 1 AU two to three orders of magnitude are
gained with respect to the energy input. Hence, radiation events due to
energy dissipation may be much more intense than known from the Earth.
We consider the stellar wind conditions for extrasolar planets closer to their star than 10 stellar radii. For these planets, we modelled the stellar wind parameters on the basis of the stellar wind model of Weber & Davis. Due to the lack of exact input parameters we consider limits, i.e. we vary the less well-known parameters within probable parameter ranges. The models suggest that the exoplanets at least for isothermal stellar coronae are most probably not located in the quasi asymptotic wind velocity regime outside the fast magnetosonic point typical for solar system planets. Thus, compared to the solar system and the solar wind at 1 AU, the radial velocity of the stellar winds may be much lower and therefore, the densities of the stellar winds may be much higher then the solar wind at the Earth's orbit.
For slowly rotating stars the Weber & Davis model shows that the
hydrodynamic Parker limit without rotation seems to yield a good first
approximation for the slow rotating stars representing the majority of stars
among the sample with planets within 10 .
In this limit, the
magnetic field does not introduce an additional acceleration of the stellar
wind. However, the radial velocity is changed significantly if the stellar
rotation period is decreased from 30 to 3 d due to the coupling of velocity
and magnetic field.
As a major result, the magnetohydrodynamic approach suggests that the close-in extrasolar planets observed so far may lie within the Alfvén critical radius. Hence, the stellar wind velocities at the planetary orbit may be sub-Alfvénic and the winds behave as if corotating with the star. In such a case, the expansion speed of the corona is smaller than the local Alfvén speed in any place between the planet and the star. Because of this, information about the planet can be carried back to the star with Alfvén waves possibly allowing for a "magnetic communication'' between star and planet. Furthermore, the magnetic pressure may not be neglected in the estimation of the size of the magnetosphere and no bow shocks are formed. In the solar system, Io embedded within the magnetosphere of Jupiter is situated in a similar plasma regime. The resulting Io-Jupiter interaction with Alfvén waves is discussed e.g. by Neubauer (1980). It may be a speculative scenario for the observed chromospheric activity on HD 179949 by Shkolnik et al. (2003). The consequences of a sub-Alfvénic wind velocity on the interaction between stellar wind and possible planetary magnetospheres will be part of our future studies.
We estimated an upper limit for the energy flux deposited in the magnetospheres by the stellar wind for two planets at their true location and at a distance of 1 AU from their star. For the close-in planets the flux is two to three orders of magnitude higher than at the Earth-like orbital distance for the same stellar winds. Hence, we suggest that much more energy will be available for dissipation processes as e.g. aurorae. As these processes will probably depend on the specific shape of the planetary magnetospheres this question is strongly linked to our future studies mentioned above.
We expect the stellar wind regime to differ significantly from what is observed for planets in the solar system at the orbit of close-in extrasolar planets. The possibly sub-Alfvénic velocity regime raises questions about the interaction of the stellar wind with the planet and its influence on the planetary magnetosphere and magnetospheric energetics.
Acknowledgements
The authors acknowledge the fellowship of S. Preusse from the International Max Planck Research School on Physical Processes in the Solar System and Beyond of the Max Planck Society and at the Universities of Braunschweig and Göttingen. We further thank E. Marsch and J.-M. Grießmeier for inspiring discussions and the referee for clarifying and helpful comments.