Stellar coronal magnetic fields and star-planet interaction
aa r X i v : . [ a s t r o - ph . S R ] J un Astronomy&Astrophysicsmanuscript no. 12367 c (cid:13)
ESO 2018November 1, 2018
Stellar coronal magnetic fields and star-planet interaction
A. F. Lanza
INAF-Osservatorio Astrofisico di Catania, Via S. Sofia, 78 – 95123 Catania, Italye-mail: [email protected]
Received ... ; accepted ...
ABSTRACT
Context.
Evidence of magnetic interaction between late-type stars and close-in giant planets is provided by the observations of stellar hot spotsrotating synchronously with the planets and showing an enhancement of chromospheric and X-ray fluxes. Possible photospheric signatures ofsuch an interaction have also been reported.
Aims.
We investigate star-planet interaction in the framework of a magnetic field model of a stellar corona, considering the interaction betweenthe coronal field and that of a planetary magnetosphere moving through the corona. This is motivated, among others, by the di ffi culty ofaccounting for the energy budgets of the interaction phenomena with previous models. Methods.
A linear force-free model is applied to describe the coronal field and study the evolution of its total magnetic energy and relativehelicity according to the boundary conditions at the stellar surface and the e ff ects related to the planetary motion through the corona. Results.
The energy budget of the star-planet interaction is discussed assuming that the planet may trigger a release of the energy of the coronalfield by decreasing its relative helicity. The observed intermittent character of the star-planet interaction is explained by a topological change ofthe stellar coronal field, induced by a variation of its relative helicity. The model predicts the formation of many prominence-like structures inthe case of highly active stars owing to the accumulation of matter evaporated from the planet inside an azimuthal flux rope in the outer corona.Moreover, the model can explain why stars accompanied by close-in planets have a higher X-ray luminosity than those with distant planets.It predicts that the best conditions to detect radio emission from the exoplanets and their host stars are achieved when the field topology ischaracterized by field lines connected to the surface of the star, leading to a chromospheric hot spot rotating synchronously with the planet.
Conclusions.
The main predictions of the model can be verified with present observational techniques, by a simultaneous monitoring of thechromospheric flux and X-ray (or radio) emission, and spectropolarimetric observations of the photospheric magnetic fields.
Key words. stars: planetary systems – stars: activity – stars: late-type – stars: magnetic fields – stars: general
1. Introduction
More than 350 extrasolar giant planets are presently known ,among which ∼
25 percent have a projected orbital semi-major axis smaller than 0.1 AU. Such planets are expectedto interact with their host stars, not only through tides, butalso with other mechanisms, possibly associated with magneticfields. Specifically, reconnection between the coronal field ofthe host star and the magnetic field of the planet is expectedto release heat, produce hydromagnetic waves, and acceler-ated particles that may be conveyed onto the stellar chromo-sphere producing a localized enhanced emission (Cuntz et al.2000; Ip et al. 2004; Preusse et al. 2006; McIvor et al. 2006;Cranmer & Saar 2007). Indeed, Shkolnik et al. (2005, 2008)show that HD 179949 and υ Andromedae have chromospherichot spots that rotate with the orbital periods of their inner plan-ets. Two other stars, HD 189733 and τ Bootis, show evidence ofan excess of chromospheric variability, probably due to flaring,that is modulated with the orbital periods of their respective
Send o ff print requests to : A. F. Lanza See a web catalogue at: http: // exoplanet.eu / planets (cf. also Walker et al. 2008). Modelling such featurescan allow us to obtain information on the magnetic fields of theplanets, although in an indirect way (cf. Shkolnik et al. 2008),which is of great importance to constrain their internal struc-ture and evolution as well as to characterize a possible habit-ability in the case of rocky planets. A recent step forward inthe understanding of the star-planet magnetic interaction (here-inafter SPMI) is the work of Lanza (2008). It applies force-free and non-force-free coronal field models to account for theobserved phase lags between planets and synchronous chro-mospheric hot spots. Moreover, it presents conjectures abouta possible influence of a close-in planet on the hydromagneticdynamo action occurring in its host star. In the present paper,we shall extend that work discussing topology, total energy,and relative magnetic helicity of the coronal field and inves-tigating their role in the e ff ects associated with the motion ofa hot Jupiter inside a stellar corona. We shall propose a newmechanism to account for the energy budget of SPMI and itsintermittent nature, as revealed by the latest observations (seeSect. 2). Moreover, we shall discuss how the di ff erent coronalfield topologies and the presence of a planet embedded in the A. F. Lanza: Stellar magnetic fields and exoplanets corona may a ff ect the formation of prominence-like structuresas well as the X-ray and radio emissions of a star.
2. Observations
Evidence of SPMI was reported by Shkolnik et al. (2003,2005, 2008) who observed chromospheric hot spots rotatingwith the orbital period of the planets in HD 179949 and υ Andinstead of their respective rotation periods. The spots were notlocated at the subplanetary point, i.e., along the line joining thecentre of a star with its planet, but lead the planet by ∼ ◦ in the case of HD 179949 and ∼ ◦ in the case of υ And,respectively. The flux irradiated by the hot spot was greater inthe case of HD 179949, with an excess power of ∼ W, cor-responding to that of a large solar flare. The hot spot synchro-nized with the planet was observed in four out of six seasonsin HD 179949, suggesting that it is not a steady phenomenon,but may have a lifetime not exceeding 300 −
400 days. A sim-ilarly intermittent interaction is suggested by the observationsof υ And, leading Shkolnik et al. (2008) to propose that theseon / o ff SPMI transitions are related to a variation of the con-figuration of the coronal magnetic field of the host stars, pos-sibly connected to their activity cycles. For the planet hostsHD 189733 and HD 73256, Shkolnik et al. (2008) suggest acorrelation of the amplitude of the intranight variability of theCa II K line core flux with the orbital phase of the planet, witha maximum of activity leading the planet by ≈ ◦ in the caseof HD 189733. This may be due to some chromospheric flaringactivity synchronized with the planet.The X-ray flux coming from HD 179949 appears to bemodulated with the orbital period of the planet with a varia-tion of ≈
30 percent (Saar et al. 2007). A statistical analysisof a sample of late-type stars reveals that those hosting a hotJupiter closer than 0.15 AU have, on the average, an X-ray flux ≈ − . τ Bootis by Walker et al. (2008) and in CoRoT-2aand CoRoT-4a by Lanza et al. (2009a), Pagano et al. (2009),and Lanza et al. (2009b), respectively. For τ Boo there is evi-dence of an active region leading the planet by about 70 ◦ whosesignatures have been observed both in the Ca II K line core fluxand in the wide-band optical flux as monitored by the MOST(Microvariability and Oscillation of STars) satellite. The regionshows a rotational modulation of ∼ τ Boo has an average rotationperiod of 3.3 days, synchronized with the orbital period of theplanet, and displays a surface di ff erential rotation comparableto that of the Sun (Catala et al. 2007), it is the constancy of thephase shift between the spot and the planet that gives supportto a possible SPMI in this case. As a matter of fact, the hot spotcan be traced back to 2001, thanks to a previous Ca II K lineflux monitoring.The planet host CoRoT-4a as a spectral type similar to τ Boo and its rotation appears to be synchronized on the av- erage with that of its planet. Modelling two months of unin-terrupted CoRoT observations, Lanza et al. (2009b) found ev-idence of a persistent active region at the subplanetary lon-gitude. Five months of observations of CoRoT-2a revealed amodulation of the total spotted area with a cycle close to tensynodic periods of the hot Jupiter with respect to the stellar ro-tation period (Lanza et al. 2009a), again suggesting some kindof SPMI. Pagano et al. (2009) found that the variance of thestellar flux was modulated in phase with the planetary orbit,with minimum variance at phase 0 . − . . − .
9. Finally, Henry et al. (2002) found evidence ofa photospheric spot in HD 192263 that in two observing sea-sons rotated with the orbital period of its planet for at least threerotation cycles (see Santos et al. 2003).This preliminary evidence of photospheric cool spots syn-chronized with a hot Jupiter is impossible to explain in theframework of a magnetic reconnection model because recon-nection leads to energy release and thus heating of the atmo-sphere. Conversely, it suggests that the planet may a ff ect insome way the stellar dynamo action or the emergence of mag-netic flux, as conjectured by Lanza (2008).Spectropolarimetric observations can be applied to mapphotospheric fields outside starspots and indeed for τ Boo asequence of maps has been obtained that suggests that the mag-netic activity cycle of the star is as short as ∼ f ce = . B MHz, where B is the intensity of the planet’s magneticfield in Gauss. In the case of Jupiter, the maximum of the fluxfalls around 40 MHz. Solar system planets follow a scalinglaw that relates their emitted radio power to the power suppliedby the impinging solar wind at their magnetospheric boundary.Generalizing that scaling law, it is possible to predict exoplan-etary radio emission powers (cf., e.g., Stevens 2005; Zarka2007; Jardine & Collier Cameron 2008). For τ Boo and someother systems, fluxes between 30 and 300 mJy, i.e., within thedetection limits of some of the largest radio telescopes, havebeen predicted for favourable conditions. Nevertheless, no pos-itive detection has been reported yet, not only in the case of τ Boo, observed at several epochs at 74 MHz with upper limitsbetween 135 and 300 mJy (Lazio & Farrell 2007), but also for ǫ Eridani and HD 128311 at 150 MHz (George & Stevens 2007)for which tight upper limits of 10 −
20 mJy were derived, and forHD 189733, for which an upper limit of ∼
80 mJy was reachedover the 307-347 MHz range (Smith et al. 2009). It is hopedthat the next generation of low-frequency radio telescopes canlower those limits by at least one order of magnitude allowingus to clarify whether the missed detections are due to a lack ofsensitivity or a lack of emission from the exoplanets at thosefrequencies. As a matter of fact, if hot Jupiters have magneticfield strengths comparable to that of Jupiter, i.e., B ∼ . B ∼ . . F. Lanza: Stellar magnetic fields and exoplanets 3 power (kinetic and / or magnetic) is significantly smaller thanassumed.
3. Coronal field model
We adopt a spherical polar coordinate frame having its origin atthe baricentre of the host star and the polar axis along the stellarrotation axis. The radial distance from the origin is indicatedwith r , the colatitude measured from the North pole with θ , andthe azimuthal angle with φ . The planet orbit is assumed circularwith a semimajor axis a and lying in the equatorial plane of thestar .Close to a star, the magnetic pressure in the corona is muchgreater than the plasma pressure and the gravitational force,so we can assume that the corona is in a force-free magneto-hydrostatic balance, i.e., the current density J is everywhereparallel to the magnetic field B , viz. J × B =
0. This meansthat ∇ × B = α B , with the force-free parameter α constantalong each field line (Priest 1982). If α is uniform in the stellarcorona, the field is called a linear force-free field and it satisfiesthe vector Helmoltz equation ∇ B + α B =
0. Its solutions inspherical geometry have been studied by, e.g., Chandrasekhar(1956) and Chandrasekhar & Kendall (1957).Linear force-free fields are particularly attractive in viewof their mathematical symplicity and their minimum-energyproperties in a finite domain, as shown by Woltjer (1958).Specifically, in ideal magnetohydrodynamics, the minimum en-ergy state of a magnetic field in a finite domain is a linearforce-free state set according to the boundary conditions andthe constrain posed by the conservation of the magnetic he-licity. In a confined stellar corona, magnetic dissipation is lo-calized within thin current sheets whose global e ff ect is thatof driving the field configuration toward the minimum energystate compatible with the conservation of the total helicity, theso-called Taylor’s state, which is a linear force-free field (see,e.g., Heyvaerts & Priest 1984; Berger 1985, for details). Onlyin large flares the total helicity is not conserved due to strongturbulent dissipation and ejection of magnetized plasma.The total helicity of a confined magnetic structure con-strains its free energy, that is the energy that can be released ina magnetic dissipation process. It is the di ff erence between theinitial energy of the field, which is usually in a non-linear force-free state (i.e., with a non-uniform α ), and the energy of thelinear force-free field satisfying the same boundary conditionsand having the same total helicity (see, e.g., R´egnier & Priest2007). Therefore, a decrease of the total helicity produced by achange of the boundary conditions or a strong magnetic dissi-pation will in general make available more free energy for theheating of the corona.To model the magnetic interaction between the stellar coro-nal field and a close-in planet, we consider only the dipole-likecomponent (i.e., with a radial order n =
1) of the linear force-free solution of Chandrasekhar & Kendall (1957) because ithas the slowest decay with distance from the star and therefore About 70 percent of the hot Jupiters within 0.1 AU from their hoststars have a measured orbital eccentricity lower than 0.05, consistentwith a circular orbit (see http: // exoplanet.eu / ). leads to the strongest interaction. Moreover, since the observa-tions of star-planet interaction show a hot spot rotating with theorbital period of the planet and do not show the periodicity ofstellar rotation, an axisymmetric field (i.e., with an azimuthaldegree m =
0) is a good approximation to describe the interac-tion, as discussed by Lanza (2008).Our linear force-free field can be expressed in the for-mulism of Flyer et al. (2004) as: B = r sin θ " r ∂ A ∂θ ˆ r − ∂ A ∂ r ˆ θ + α A ˆ φ , (1)where A ( r , θ ) is the flux function of the field. Magnetic fieldlines lie over surfaces of constant A ( r , θ ), as can be deducedby noting that B · ∇ A =
0. The flux function for our dipole-like field geometry is A ( r , θ ) = B R g ( q ) sin θ , where 2 B isthe magnetic field intensity at the North pole of the star, R thestar’s radius and the function g ( q ) is defined by: g ( q ) ≡ [ b J − / ( q ) + c J / ( q )] √ q [ b J − / ( q ) + c J / ( q )] √ q , (2)where b and c are free constants, J − / and J / are Besselfunctions of the first kind of order − / /
2, respectively, q ≡ | α | r , and q ≡ | α | R . Making use of Eq. (1), the magneticfield components are: B r = B R r g ( q ) cos θ, B θ = − B | α | R r g ′ ( q ) sin θ, (3) B φ = α B R r g ( q ) sin θ, where g ′ ( q ) ≡ dg / dq . A linear force-free field as given byEqs. (3) extends to infinity with an infinity energy. We considerits restriction to the radial domain q ≤ q ≤ q L , where q L is thefirst zero of g ( q ), in order to model the inner part of the stel-lar corona where magnetic field lines are closed, as discussedin Lanza (2008) (see Chandrasekhar 1956, for the boundaryconditions at r = r L ≡ q L / | α | ).The magnetic field geometry specified by Eqs. (3) dependson two independent parameters, i.e., α and b / c . They canbe derived from the boundary conditions at the stellar photo-sphere, i.e., knowing the magnetic field B ( s ) ( θ, φ ) on the sur-face at r = R . Using the orthogonality properties of the basicpoloidal and toroidal fields (see Chandrasekhar 1961), we find:8 π B R = Z Σ ( R ) B ( s )r cos θ d Σ , π | α | B R g ′ ( q ) = − Z Σ ( R ) B ( s ) θ sin θ d Σ , (4)8 π α B R = − Z Σ ( R ) B ( s ) φ sin θ d Σ , where Σ ( R ) is the spherical surface of radius R , and d Σ = R sin θ d θ d φ .The magnetic energy E of the field confined between thespherical surfaces r = R and r = r L can be found from Eq. (79)in §
40 of Chandrasekhar (1961): E = E p n + q q L [ g ′ ( q L )] − q [ g ′ ( q )] − q o , (5) A. F. Lanza: Stellar magnetic fields and exoplanets where E p ≡ π µ R B is the energy of the potential dipole fieldwith the same radial component at the surface r = R , and µ isthe magnetic permeability. The relative magnetic helicity H R ,as defined by Berger (1985), can be found from his Eq. (19)and is: H R = B R " g ′ ( q ) + π Eq E p | α | α . (6)Note that the field obtained by changing the sign of α has thesame poloidal components B r and B θ , and energy E , while thetoroidal component B φ and the relative helicity H R become op-posite.In the limit q ≪
1, i.e., close to the stellar surface, neglect-ing terms of the order O ( q ), we find that g ( q ) ∼ q / q and g ′ ( q ) ∼ − q / q , independently of b and c . From Eqs. (3), wesee that B r and B θ close to the star are similar to the analogouscomponents of the potential dipole field with the same radialcomponent at the surface.It is also interesting to study the limit α → r L → ∞ and the radial extension of thefield grows without bound. Nevertheless, the field tends to be-come indistinguishable from the corresponding potential fieldand E → E p . In the same limit, the relative magnetic helicitygrows without bound because q → α , E > E p because the potential field has theminimum energy for a given B ( s )r . If we consider all magneticfields with one end of their field lines anchored at r = R andthe other out to infinity, satisfying the same boundary condi-tions of our field at r = R , the field with the lowest possibleenergy is called the Aly field and its energy E Aly = . E p (seeFlyer et al. 2004). We assume that the Aly energy is an upperbound for the energy of our field because it is the lowest energyallowing the field to open up all its lines of force out to infinitydriving a plasma outflow similar to a solar coronal mass ejec-tion.From a topological point of view, the fields obtained fromEqs. (3) can be classified into two classes. If the function g ( q )decreases monotonously in the interval q ≤ q ≤ q L , all fieldlines are anchored at both ends on the boundary r = R . Onthe other hand, if the function has a relative minimum (anda relative maximum) in that interval, the field contains an az-imuthal rope of flux located entirely in the r > R space andrunning around the axis of symmetry. The magnetic field con-figurations considered by Lanza (2008) to model SPMI are ofthe first kind; an example of a field configuration containingan azimuthal flux rope will be discussed in Sect. 4.2. An anal-ogous topological classification holds in the case of the non-linear force-free fields considered by Flyer et al. (2004).Note that the photospheric magnetic field components canbe measured by means of spectropolarimetric techniques if thestar rotates fast enough ( v sin i ≥ −
15 km s − ) as shown inthe case of, e.g., τ Boo by Catala et al. (2007) and Donati et al.(2008). Therefore, Eqs. (4) can be applied to derive the param-eters of the coronal field model and its topology, as we shallshow in Sect. 4.2.
4. Applications
In this Section we shall consider some applications of the abovemodel for the stellar coronal field to the phenomena introducedin Sect. 2.
The power needed to explain the excess flux from a chromo-spheric hot spot or an X-ray emission synchronous with theplanet is of the order of 10 − W. Magnetic reconnectionbetween the stellar coronal field and the planetary field at theboundary of the planetary magnetosphere is not su ffi cient toaccount for such a power, as we show in Sect. 4.1.1. Therefore,we propose an interaction mechanism that may be capable ofsustaining that level of power in Sect. 4.1.2. The first mechanism proposed to account for the energy budgetof SPMI is reconnection between the planetary and the stel-lar magnetic fields at the boundary of the planetary magneto-sphere. This boundary is characterized by a balance betweenthe magnetic pressure of the coronal field and that of the plan-etary field. The ram pressure is negligible because the planetis inside the region where the stellar wind speed is subalfvenic(cf., e.g., Preusse et al. 2005), and the orbital velocity of theplanet is about one order of magnitude smaller than the Alfvenvelocity (Lanza 2008). Therefore, assuming a planetary fieldwith a dipole geometry, the radius of the planetary magneto-sphere R m , measured from the centre of the planet, is given by: R m = R pl " B ( a , π ) B pl − , (7)where R pl is the radius of the planet, B ( a , π ) is the coronal fieldof the star on the stellar equatorial plane at r = a , and B pl is themagnetic field strength at the poles of the planet.We specialize our considerations for the SPMI model ofHD 179949 developed by Lanza (2008) that assumes b / c = − . α = − . R − to explain the phase lag between thechromospheric hot spot and the planet. The magnetic field in-tensity vs. the distance from the star is plotted in Fig. 1. Thefield strength decreases as that of a potential dipole field closeto the star, i.e., where g ( q ) ∼ q / q (cf. Sect. 3), then it decreasesmore slowly for r > (6 − R because the decrease of g ( q ) be-comes less steep (cf., e.g., Fig. 3).At the distance of the planet, i.e., a = . R , the field isreduced by a factor of ∼
300 with respect to its value at thestellar surface. Assuming a mean field at the surface B =
10 G(cf. Donati et al. 2008), the field at the boundary of the plane-tary magnetosphere is B ( a , π ) ∼ .
03 G. Adopting a planetaryfield B pl = R m ∼ . R pl = . × m. The power dissipated by magnetic reconnectioncan be estimated as: P d ≃ γ πµ [ B ( a , π R v rel , (8) . F. Lanza: Stellar magnetic fields and exoplanets 5 where 0 < γ < v rel is the relative velocity between the planet and the stel-lar coronal field. Adopting the parameters for HD 179949 and γ = .
5, we find P d ≃ . × W, which is insu ffi cient byat least a factor of 10 . Since P d ∼ B / B / , to explain theobserved power the stellar surface field should be increased to B ∼
180 G, which is too high in view of the mean values mea-sured at the surface of τ Boo (Donati et al. 2008) that is a fasterrotator than HD 179949, but with a comparable X-ray lumi-nosity (Kashyap et al. 2008). Christensen et al. (2009) suggestthat the surface fields of hot Jupiters may be up to 5 −
10 timesstronger than that of Jupiter. However, if we adopt B pl =
75 Gand consider the B / dependence, we obtain a power increaseby only a factor of 6. To solve the energy problem, it is important to note that the en-ergy released at the reconnection site is only a fraction of theenergy that is actually available. The emergence of magneticflux from the convection zone and the photospheric motionslead to a continuous accumulation of energy in the coronalfield, independently of the presence of a planet. Our conjec-ture is that a close-in planet triggers a release of such an ac-cumulated energy, modulating the chromospheric and coronalheating with the orbital period of the planet.As a matter of fact, the reconnection between the coronaland the planetary field lines has remarkable consequences forthe topological structure of the coronal field. The final con-figuration of the coronal field after reconnection is in generala non-linear force-free one, i.e., α is no longer spatially uni-form. This happens because the field must simultaneously sat-isfy the boundary conditions at the photosphere, which imposethe value of α close to the star, and those at the reconnectionsite close to the planet which in general will not be compati-ble with a uniform value of α . Since α must be constant alonga given field line, the field re-arranges itself into a configura-tion with di ff erent values of α along di ff erent field lines to sat-isfy the boundary conditions. We know from Woltjer theoremthat the energy of the field in such a state is greater than thatof the linear force-free field with the same magnetic helicityand boundary conditions at the stellar surface. In other words,some energy can be released if the field makes a transition tothis linear force-free state, restoring its unperturbed configura-tion. Such a transition can be stimulated by the fact that thehelicity in the region where the field has reconnected is lowerthan the initial helicity because reconnection processes lead toa steady dissipation of helicity. Considering the volume V ( t )occupied at any given time t by the plasma where reconnectionhas just occurred, the rate of helicity change inside V ( t ) is givenby Eq. (13) of Heyvaerts & Priest (1984): DH R Dt = Z S ( V ) ( A · v rel )( B · dS ) + Z S ( V ) A × J σ · dS − Z V B · J σ dV , (9)where S ( V ) is the surface bounding the volume V ( t ), A is thevector potential of the magnetic field, viz. ∇ × A = B , and σ is the electric conductivity of the plasma. We assume that thereconnected field is in a force-free state, thus µ J = α B , andchoose a gauge transformation for the vector potential to have A = α − B . Since v rel · B ∼ DH R Dt ≃ − h α − i µ Z V J σ dV , (10)where h α − i is some average value of α − over the volume V ( t ).Eq. (10) shows that the absolute value of the helicity decreasessteadily inside the volume V ( t ) following the motion of the re-connected region through the stellar corona.We conjecture that once a magnetic energy release is trig-gered by such a decrease of helicity in the reconnection vol-ume, it extends to the whole flux tube connecting the planetarymagnetosphere with the stellar surface and proceeds faster andfaster thanks to a positive feed-back between energy releaseand helicity dissipation. In other words, the fast and localizedenergy release produces a turbulent plasma which in turn en-hances turbulent dissipation of magnetic energy and helicitygiving rise to a self-sustained process.An order-of-magnitude estimate of the helicity dissipationrate can be based on Eq. (10), now considering turbulent dissi-pation; it can be recast in the approximate form: DH R Dt ∼ µδ P d , (11)where δ is the lengthscale of variation of the magnetic fieldassociated with the current density, i.e., J ∼ B / ( δµ ), and P d is the dissipated power. We assume that the largest turbulentlengthscale is δ ∼ | α | − = . R − = . × m for R = × m. Adopting the maximum dissipated power suggested by theobservations, i.e., P d = W, we find: DH R Dt ∼ . × T m s − . (12)In the above derivation we have guessed the dissipatedpower from the available observations. For the mechanism tobe at least plausible, we need to show that it is possible tosustain that level of dissipated power. This implies a compu-tation of the energy stored in the non-linear force-free field thatis produced by the reconnection process, which is beyond thecapability of our simple linear model. Nevertheless, we mayestimate the energy available by means of a simplified argu-ment that is by no means rigorous, but has the advantage ofusing our model for linear force-free configurations. Its resultsshould be regarded only as an illustration of the plausibility ofthe proposed mechanism, deferring a more detailed and rigor-ous treatment to future studies.To compute the available energy, we assume that the recon-nection between the coronal and planetary fields produces adissipation of the helicity of the initial field configuration, i.e.,that unperturbed by the planet. Again, we specify our consid-erations for the case of HD 179949, adopting the linear force-free model of Lanza (2008). Therefore, we fix α = − . R − and b / c = − . E / E p = . | H R | = . B R (cf. Eqs. 5 and 6). The boundary conditions at A. F. Lanza: Stellar magnetic fields and exoplanets the stellar surface specify the values of g ′ ( q ) and α throughEqs. (4). The value of α depends on the radial and azimuthalfield components at the surface and must be regarded as fixed.Conversely, the value of b / c is pratically independent of thevalue of g ′ ( q ) for b / c < − .
0. This is illustrated in Fig. 2 andis discussed further in Sect. 4.2. In other words, the ratio b / c is only marginally constrained by the boundary conditions andcan be assumed to vary in order to decrease the total helicity ofthe field. The minimum helicity is obtained for b / c → −∞ ,which gives g ′ ( q ) = − . b / c = − . g ′ ( q ) = − . E / E p = . | H R | = . B R ,i.e., an helicity variation ∆ H R = . B R with respect tothe initial state. Assuming B =
10 G and R = × m, the transition from the initial state to this final state re-leases an energy ∆ E = . E p = . × J. A crucialpoint is the timescale for the release of ∆ E because it deter-mines the power available to explain the observed phenomena.It depends on the time scale for the dissipation of the helic-ity and can be computed from the helicity dissipation rate as τ hd ∼ ∆ H R / ( dH R / dt ) ∼ . × s, where we made use ofEq. (12). Therefore, the available maximum power, estimatedfrom ∆ E /τ hd , turns out to be ∼ . × W which is of theright order of magnitude to account for the chromospheric hotspot and the enhancement of X-ray flux.It is important to note that our argument provides us onlywith a lower limit for the energy that can be released becausewe approximate the process as a transition between two lin-ear force-free fields. As a matter of fact, when the field is in anon-linear force-free state, its energy is greater than the energyof the initially unperturbed linear force-free field, allowing thesystem to release more energy than we have estimated above.This can explain the modulation of the emitted flux with theorbital phase of the planet if only a portion of the stellar coronasurrounding the flux tube connecting the planet with the star isinvolved in the energy release process at any given time.The initial field configuration needs to be restored on a timescale comparable with the orbital period of the planet to achievea quasi-stationary situation in agreement with the observations.An order of magnitude estimate of the helicity flux comingfrom the photospheric motions can be obtained from Eq. (11)of Heyvaerts & Priest (1984) as: dH R dt ≈ π R B v e α − , (13)where v e is the velocity of magnetic flux emergence at the pho-tosphere, that we can take as a fraction, say 0.1, of the con-vective velocity, yielding v e =
150 m s − . The timescale forrestoring the helicity of the above magnetic configuration is: τ hr ≡ ∆ H R dH R / dt ∼ . × s , (14)where dH R / dt comes now from Eq. (13). It is comparable tothe fastest helicity dissipation timescale applied above and issignificantly shorter (i.e., ≈
10 percent) than the orbital periodof the planet. The magnetic energy flux associated with the he-licity build up can be estimated as: ∼ π R B v e /µ ∼ . × Fig. 1.
The modulus of the magnetic field on the equatorialplane of the star normalized to its value at the surface vs. the ra-dial distance from the star for b / c = − . α = − . R − (solid line). For comparison, we plot also the case of a potentialfield (dotted line) and of a field decreasing as r − (dashed line).W, which can account for the observed energy release in thecorona.It is interesting to note that the greater the magnetic fieldof a hot Jupiter, the more e ffi cient the triggering of helicitydissipation in a stellar corona by the planet itself because thedissipated power scales as B / in Eqs. (8) and (10). However,even if the planetary magnetic field is negligible, we still ex-pect some dissipation by the currents induced in the planetaryconductive interior by its motion through the stellar coronalfield (Laine et al. 2008) or by the currents associated with theAlfven waves excited by the motion of the planet through thestellar wind (Preusse et al. 2005, 2006). The corresponding he-licity dissipation rate in Eq. (10) is expected to be at least twoorders of magnitude smaller than when the planet has a field B pl of 5 −
10 G, essentially because the current dissipation is con-fined within a much smaller volume. Therefore, the e ffi ciencyof the proposed mechanism is likely to be significantly reducedwhen the planetary field vanishes. The fact that chromospheric hot spots rotating sychronouslywith the planetary orbit have not always been observed inHD 179949 and υ And (Shkolnik et al. 2008) can be interpretedby assuming that their activity cycles are very short, i.e., ofthe order of 1 − τ Boo (Donati et al. 2008; Fares et al. 2009). . F. Lanza: Stellar magnetic fields and exoplanets 7
Fig. 2.
The first derivative of the function g at the surface of thestar (i.e., r = R ) vs. b / c for α = . R − .Moreover, Cranmer & Saar (2007) and Shkolnik et al. (2009)build a statistical model for SPMI, based on the observationsof the solar photospheric field along cycle 22 and its extrapo-lation to the corona by means of a potential field model, whichpredicts an intermittent interaction. The relative durations ofthe on and o ff phases depend on the phase of the solar cycleand are in general agreement with the few available observa-tions reported in Sect. 2. Nevertheless, another interpretation ispossible which does not require a global reversal of the mag-netic field or a complex geometry of the coronal field, but onlya change of the photospheric boundary conditions that inducesa change of the global topology of the coronal field.We shall explore the e ff ect of a change of the boundaryconditions on the force-free configuration adopted by Lanza(2008) to model SPMI in the case of HD 179949. It assumes α = − . R − and b / c = − .
1. The mean of the merid-ional field component over the surface of the star B ( s ) θ in ther.h.s. of the second of Eqs. (4) fixes the value of g ′ ( q ) fromwhich we can derive the model parameter b / c . A plot of g ′ ( q ) vs. b / c is given in Fig. 2 for α = − . R − and showsthat there is a large interval of b / c where the variation of g ′ ( q ) is very small. In other words, a modest variation of themeridional field component produces a remarkable variation of b / c . On the other hand, to change significantly the param-eter α we need a remarkable change of the axisymmetric az-imuthal field component over the surface of the star that canbe achieved only on timescales comparable with the stellar ac-tivity cycle. Therefore, the dependence on the boundary condi-tions suggests to explore the modification of the field topologyfor a variable b / c helding α fixed.The plots of the radial function g ( q ) and its first deriva-tive g ′ ( q ) vs. q are shown in Fig. 3 for several values of b / c and α = − . R − . For ( b / c ) < − . g ismonotonously decreasing and all the field lines have both ends Fig. 3.
Upper panel : The radial function g vs. q for α = − . R − and di ff erent values of b / c as indicated by thedi ff erent linestyles, i.e., solid: b / c = − . − . − .
5; dash-dotted: − . − . Lower panel:
The first derivative g ′ vs. q for α = . R − anddi ff erent values of b / c , according to the same linestyle codingadopted in the upper panel.anchored onto the photosphere. A sketch of the meridional sec-tion of the field lines is given in Fig. 1 of Lanza (2008) for b / c = − .
1. On the other hand, for ( b / c ) ≥ − . g ′ vanishes at two points within the interval q < q < q L , and the field develops an azimuthal rope of fluxcentred around the maximum of g ( q ) and whose inner radiuscoincides with the minimum of g ( q ). The dependences of themagnetic field energy E , outer radius r L , and absolute valueof the relative helicity | H R | on b / c for α = − . R − areillustrated in Fig. 4. For − . ≤ b / c ≤ − .
3, the energy,outer radius, and absolute value of the relative helicity all in-crease very slowly with the parameter, thus we restrict the plotto the interval showing the steepest variations. The parametercorresponding to the Aly energy is b / c = − .
107 and it ismarked by a vertically dashed line. A meridional section of thefield lines when the field has the Aly energy is plotted in Fig. 5and shows how the azimuthal flux rope has extended to occupymost of the available volume, while the domain with field lines
A. F. Lanza: Stellar magnetic fields and exoplanets
Fig. 4.
Upper panel:
The magnetic field energy as a frac-tion of the corresponding potential field energy vs. b / c for α = − . R − . The Aly energy limit is marked by the hori-zontal dotted line while the vertically dashed line marks thecorresponding value of b / c = − .
107 and has been reportedinto all the other plots.
Middle panel:
The outer radial limitof the field vs. b / c for α = − . R − . Lower panel: theabsolute value of the relative magnetic helicity vs. b / c for α = − . R − .connected to the surface of the star has been squeezed below aradial distance ∼ R .We have illustrated the dependences of g ′ , field energyand helicity on the parameter b / c for the particular case of α = − . R − , but their qualitative behaviours are the samealso for di ff erent values of α . Specifically, we have explorednumerically the dependence of E / E p and | H R | in the rectangle(0 . R − ≤ | α | ≤ . R − ) × ( − . ≤ b / c ≤ − . b / c for any fixed value of α .In Sect. 4.1.2, we have assumed that the helicity of thestellar corona is determined by a dynamical balance betweenthe opposite contributions of the emerging magnetic fields andphotospheric motions that build it up, and the orbital motionof the planet that triggers a continuous dissipation of helic-ity and magnetic energy in the corona. Therefore, the photo-spheric boundary conditions that fix the value of the helicity inour model can be regarded as a result of those processes thatrule the helicity balance of the stellar corona.To explain the transition between states with and with-out a chromospheric hot spots, we assume that the helicity ofthe coronal field and the corresponding boundary conditionscan vary on a timescale shorter than the stellar activity cycle.Specifically, the flux of helicity into the corona and the merid-ional component of the surface field are expected to vary asa result of the fluctuations characterizing turbulent hydromag-netic dynamos (cf., e.g., Brandenburg & Subramanian 2005).As a consequence, b / c varies, spanning a certain range of Fig. 5.
Meridional section of the magnetic field lines for α = − . R − and b / c = − .
107 which corresponds to the Alyenergy limit. The distance from the rotation axis is indicatedby s = r sin θ , while the distance from the equatorial plane is z = r cos θ , with R being the radius of the star.values. If such a range is large enough, the value of the pa-rameter can sometimes cross the thresold b / c = − . ff phasesdepend on the statistical distribution of the fluctuations of themeridional component of the surface field which is presentlyunknown. However, an on / o ff transition can take place on atime scale as short as 10 − s assuming the helicity fluxesand dissipation rates estimated in Sect. 4.1.2.The phase lag ∆ φ between the planet and the chromo-spheric hot spot depends on b / c for a fixed α (see Lanza2008, for the method to compute ∆ φ ). We plot such a de-pendence in Fig. 6. When the field has no flux rope, i.e., b / c < − . ∆ φ depends only slightly on b / c varying onlyby ± ◦ around the mean observed value of ∼ ◦ , even forvalues of b / c as small as − .
0, i.e., well beyond the rangeconsidered to model the observations of HD 179949. The vari-ation of ∆ φ becomes steep only when the field is very close todevelop a flux rope. The inner radius of the flux rope reachesthe distance of the planet ( r / R = .
72) when b / c = − . . F. Lanza: Stellar magnetic fields and exoplanets 9 Fig. 6.
The phase lag between the planet and the synchronouschromospheric hot spot vs. b / c for α = . R − .We assume that the transition from, say, b / c = − . − . ∆ φ given the limited duty cycles of present ground-based observations.By increasing b / c , the energy of the field grows up to theAly limit where an instability opening the field lines may pos-sibly be triggered. The magnetic field intensity vs. the distancefrom the star is plotted in Fig. 7 for a model with α = − . R − and the Aly energy. Following a rapid initial decrease, closelysimilar to that of a potential field, the field stays almost constantfor 5 ≤ ( r / R ) ≤
14, and then decreases slowly far away fromthe star. This is due to the slow decrease of the field intensitywith radial distance inside the flux rope. A first consequenceis a remarkable increase of the power dissipated at the bound-ary of the planetary magnetosphere, as given by Eq. (8). In thecase of HD 179949, the field intensity at a = . R is ∼ P d by a factor of ∼
4. Nevertheless, the released en-ergy cannot reach the chromosphere and no hot spot is formedin the present case. If a balance is eventually reached betweenthe helicity flux from the photosphere and the dissipation in thecorona, the flux rope configuration may become stationary. Itsend may occur either by an increase of the energy above theAly limit which may open up the field lines, or by a suddendecrease of the helicity flux which will change the topology ofthe field into one with all field lines connected to the stellarsurface, thus resuming a chromospheric hot spot.The explanation suggested above for the intermittent be-haviour of SPMI can be tested by measuring the direction of B ( s ) θ , i.e., the meridional field component at the stellar surface,which is possible by means of spectropolarimetric techniques(cf. Moutou et al. 2007; Donati et al. 2008) if the star rotatesfast enough ( v sin i ≥ −
15 km s − ). If the on / o ff SPMI tran-sition is not associated with a reversal of B ( s ) θ , the present ex- Fig. 7.
The modulus of the magnetic field on the equatorialplane of the star normalized to its value at the surface vs.the radial distance from the star for b / c = − .
107 and α = − . R − (solid line). For comparison, we plot also thecase of a potential field (dotted line) and of a field decreasingas r − (dashed line).planation gains support. In principle, by measuring the surfacecomponents of the field and the angle ∆ φ with su ffi cient ac-curacy, it is possible to estimate b / c , thus constraining themodel parameters during the SPMI on phases. Specifically, wecan use the first and the third of Eqs. (4) to find B and α , andthen fix the range of b / c that reproduces the observed ∆ φ (cf.,e.g., Fig. 6). The second of Eqs. (4) can be used as an indepen-dent check of the accuracy of the linear force-free assumptionbecause the resulting value of g ′ ( q ) should be close to − An interesting property of the flux rope topology is the pos-sibility of storing matter in the stellar corona. The evapora-tion of hot Jupiters under the action of the ionizing radia-tion of their host stars originates a flow of cool plasma at atemperature of ∼ K that escapes from the planetary at-mospheres into the stellar coronae (cf., e.g., Ehrenreich et al.2008; Murray-Clay et al. 2009). The flux rope geometry keepsthe evaporating plasma confined into a torus in the equatorialplane of the star, thus making its detection easier in the case oftransiting hot Jupiters (Vidal-Madjar et al. 2003).Considering a moderately active star and assuming a life-time of ∼
300 days for the flux rope configuration accord-ing to the duration of the o ff phases of chromospheric inter-action, the evaporated mass is of the order of 5 × kg (cf.,e.g., Murray-Clay et al. 2009). Such a material may eventuallycondense in the form of prominence-like structures around theminimum of the gravitational potential inside the flux rope, that is in the equatorial plane of the star, possibly closer to the starthan the hot Jupiter (cf. Fig. 5). It is unlikely that all the evap-orated matter collects into a single condensation. The thermalinstability of the plasma is expected to lead to many conden-sations with typical lengthscales of 10 − m, comparablein order of magnitude to those observed in, e.g., solar promi-nences (Field 1965). Moreover, a high degree of inhomogene-ity is expected inside each condensation, in analogy with thefilamentary structure of solar prominences, because the pres-sure scale height at a temperature of ∼ K is only of theorder of 10 − m.Assuming a mass of 1 M ⊙ for the star and a radius R = R ⊙ ,the gravitational potential energy of the whole prominence ma-terial is ∼ × J, for a mean radial distance of 7 R . It is only1 percent of the magnetic energy of the coronal field, thereforeit is not expected to perturb appreciably the field geometry.On the other hand, in the case of a very active star, i.e., hav-ing an X-ray luminosity 2 − ∼ − J.This can significantly perturb the field, whose configurationcan no longer be assumed force-free. A non-force-free modelmay be applicable, such as that of Neukirch (1995), discussedby Lanza (2008). It gives rise to the same field topologies ofthe present force-free model because it is obtained from a lin-ear force-free model by means of a linear transformation of theindependent variable q of the radial function g ( q ) (cf. Lanza2008). A large amount of prominence material may help to sta-bilize the flux rope configuration, because lifting up the promi-nences to open or change the geometry of the magnetic fieldlines requires a comparable additional amount of energy. Thismay imply that the flux rope topology is the most stable in thecase of very active stars, leading to a lower probability of ob-serving a chromospheric hot spot synchronized with the planet.Prominence-like structures have indeed been detectedaround some young and highly active dwarf stars, throughthe absorption transients migrating across their H α , Ca IIH & K and Mg II h & k emission line profiles (e.g.,Collier Cameron & Robinson 1989). Typical masses are in therange (2 − × kg for the young ( ≈
50 Myr), single K0dwarf AB Dor (Collier Cameron et al. 1990). We expect a re-markable increase of the number of such phenomena in highlyactive stars hosting evaporating hot Jupiters. Assuming a life-time of ∼
300 days for the flux rope configuration, a total evap-orated mass of 2 × kg can lead to the formation of ∼ − α line profile is between5 and 20 percent, with typical equivalent widths of the ab-sorption features between 50 and 500 mÅ, as reported byCollier Cameron & Robinson (1989) for AB Dor, whose coro-nal parameters can be considered fairly typical of those of veryactive stars.Finally, note that a prominence does not form at the dis-tance of the planet when the field topology does not containa flux rope because in that case all the evaporated matter willfall onto the star, except when the star rotates so fast that thecentrifugal force reverses the e ff ective gravity at the top of theloops interconnecting the planet with the star. In our model, all the magnetic field lines are closed, so thecorona consists of closed loops. However, if we adopt a morerealistic model, such as the non-linear force-free model ofFlyer et al. (2004), it is possible to have open field lines whichmay account for the configuration observed in solar coronalholes. Also in those non-linear models we can have a coronalfield with a large azimuthal flux rope which a ff ects the topol-ogy of the coronal field lines close to the star. From a qualitativepoint of view, we can refer to Fig. 5 showing that all the mag-netic structures at low and intermediate latitudes must have aclosed configuration with a top height lower than ∼ − R be-cause they must lie below the flux rope. This implies that openfield configurations, akin solar coronal holes, are not allowedat low latitudes when a sizeable flux rope has developed in theouter corona. Since closed magnetic configurations are charac-terized by X-ray fluxes up to ∼ −
100 times greater thancoronal holes, an azimuthal flux rope configuration is expectedto be associated with a greater X-ray luminosity of the star thanin the case when all magnetic field lines are connected to thephotosphere. Of course, this is independent of the presence ofa planet.When a star is accompanied by a close-in planet, it mayincrease the dissipation of magnetic helicity and energy, asconjectured in Sect. 4.1.2. According to Kashyap et al. (2008),stars with a distant planet have an average X-ray luminos-ity L X ∼ × W, while stars with a hot Jupiter have L X ∼ . × W. Considering the model introduced inSect. 4.1.2, the dissipated power is proportional to B , where B is the photospheric magnetic field (see Sect. 3). This holds trueboth for a field topology with all field lines connected to thephotosphere and with an azimuthal flux rope. We find that forboth field topologies an average photospheric field B ∼
30 Gis su ffi cient to account for the enhancement of X-ray luminos-ity in stars with hot Jupiters. Since our model underestimatesthe free energy available in real non-linear force-free fields,such a value of B should be regarded as an upper limit, thusour estimate agrees well with the available observations (e.g.,Moutou et al. 2007). . F. Lanza: Stellar magnetic fields and exoplanets 11 The frequency of radio emission from exoplanets depends onthe planetary magnetic fields (Zarka 2007). According to thescaling laws adopted by Grießmeier et al. (2004), the magneticmoments of hot Jupiters are expected to be one order of mag-nitude smaller than that of Jupiter owing to tidal synchroniza-tion between their rotation and orbital motion. Adopting a sur-face field of 10 percent of Jupiter, i.e., 1.4 G, the cyclotronmaser emission should peak at ∼ ff (Jardine & Collier Cameron 2008). The situation is much morefavourable if the magnetic fields of hot Jupiters are indeed asstrong as predicted by the models of Christensen et al. (2009)in which the planetary dynamo is powered by the internal con-vective motions and the field intensity is pratically independentof the rotation rate of the planet. Assuming a field intensity atthe surface of 25 G, the cyclotron maser emission peaks at fre-quencies around 70 MHz.In addition to the emission from the poles of the planet, weexpect radio emission from the stellar corona. When the fieldlines are connected to the stellar surface, electrons acceleratedat the reconnection sites travel down to the star producing emis-sions up to the GHz range from localized regions above photo-spheric spots with fields of 10 − G. However, this is notpossible when the field has a topology with an azimuthal fluxrope. In this case the coronal field intensity is of the order of10 − G and the emission peaks at very low frequencies, compa-rable to or below the plasma frequency, i.e., it is self-absorbedbefore escaping from the emitting region. In this case, we ex-pect detectable radio emission only from the poles of the planet,i.e., from a highly localized region. The beaming of cyclotron-maser emission may additionally decrease the power towardthe observer thus explaining the lack of detection at frequen-cies of 50 −
100 MHz.In conclusion, we expect that the best chances of detectingradio emission from exoplanets with the current instrumenta-tion can be achieved when observing systems with a chromo-spheric hot spot synchronized with the planet. In this case, afraction of the electrons accelerated at the reconnection sitesinside the long loop connecting the star with the planet may bedriven to the polar regions of the planet and to the stellar sur-face increasing the irradiated power. In this case, the emissionis expected to be strongly modulated with the orbital period ofthe planet.
5. Discussion
We have presented a model for the coronal magnetic field oflate-type stars that allows us to investigate the magnetic inter-action between a star and a close-in exoplanet. The same modelwas used by Lanza (2008) to explain the observed phase lagsbetween chromospheric hot spots attributed to SPMI and theplanets. Now, we have discussed the energy budget of SPMIand its intermittency on the base of that model. We suggest thatthe coronal field evolution in a star hosting a hot Jupiter is ruledby a dynamical balance between the helicity coming up into thecorona from the photosphere and that dissipated by the recon- nection events triggered by the orbital motion of the planet. Atany given time, most of the energy is dissipated in the loop con-necting the planet with the stellar surface and in its neighbourmagnetic structures rather than at the boundary of the planetarymagnetosphere.This scenario is quite di ff erent from that characteristic ofstars with distant planets, like our Sun. In that case the accu-mulation of magnetic helicity and energy in the corona leadsto an instability of closed field structures that erupt as coro-nal mass ejections (CMEs) eliminating the excess of helicity(Zhang & Low 2005; Zhang et al. 2006). In the present model,in addition to the CME mechanism, the interaction betweenthe coronal field and the planetary magnetosphere takes part inthe helicity dissipation process. In principle, one expects thata large flare that extends over most of the stellar corona maysometimes occur thanks to the capability of the planet to triggera large-scale helicity dissipation process. If most of the coronalhelicity is dissipated during such an event, the maximum avail-able energy can approach the di ff erence between the Aly limitand the energy of the potential field with the same radial com-ponent at the photosphere, i.e., ∆ E max = . E p . For the case ofHD 179949 with an assumed photospheric field of B =
10 G,we have ∆ E max = . × J. Assuming a photospheric meanfield B =
40 G and R = .
75 R ⊙ , as suggested by the observa-tions of the K dwarf HD 189733 by Moutou et al. (2007), weget ∆ E max = . × J. Such large energies may explain thesuperflares observed in some dwarf stars, giving support to aconjecture by Rubenstein & Schaefer (2000).Processes like CMEs, i.e., capable of reducing the helic-ity of the stellar field, are required for the operation of a stel-lar hydromagnetic dynamo (Blackman & Brandenburg 2003;Brandenburg & Subramanian 2005). Therefore, a hot Jupitermay help the star to get rid of the helicity generated by its dy-namo in the convection zone, increasing dynamo e ffi ciency andthe overall level of magnetic activity. This may ultimately ex-plain the greater X-ray luminosity of stars with hot Jupiters.Moreover, shorter activity cycles may be expected (cf., e.g., § τ Boo (Fares et al. 2009).The modulation of the helicity loss with the orbital periodof the planet might account for the photospheric cool spots thatappear to rotate synchronously with the planet, as conjecturedby Lanza (2008). These spots are related to the emergence ofmagnetic flux from the convection zone that may contributeto the formation of hot spots in the chromosphere and in thecorona by reconnecting with pre-existing fields, thus contribut-ing to the energy budget of SPMI.Our model predicts a field intensity that decreases slowerthan that of a potential field far away from the star. In the case ofthe flux rope topology, the field decreases even slower than r − in the outer part of the closed corona. This may enhance the dy-namical coupling between the star and the planet, as suggestedby some preliminary computations of the angular momentumexchange between CoRoT-4a and its hot Jupiter having an or-bital semimajor axis a ∼ . R , where R is the stellar radius.They indicate that the rotation of the outer convection zone ofthe star may have been synchronized with the orbital motion of the planet if the surface field B ≥
10 G and the age of the sys-tem is ≥ . ∼ . J , Lanza et al. 2009b).A limitation of the present approach is the use of a linearforce-free model for the stellar coronal field. This mainly af-fects our energy estimates, while the main topological featuresof our linear model are shared by non-linear models (cf., e.g.,Flyer et al. 2004; Zhang et al. 2006). The latter can be usefulto treat the contribution of the outer coronal fields because theycan provide us with field configurations that extend to the in-finity with a finite magnetic energy. However, given the muchgreater mathematical complexity of non-linear force-free mod-els, the present treatment is preferable for a first description ofthe most relevant physical e ff ects.Non-linear models can be useful also to investigate thestability of coronal magnetohydrostatic configurations. In theframework of the adopted model, stability is warranted byWoltjer theorem because a linear force-free configuration isthe minimum-energy state for given total helicity and bound-ary conditions (Berger 1985). However, a real coronal field thatis in a non-linear force-free state may become unstable beforereaching the Aly energy limit by, e.g., kink modes, when it de-velops an azimuthal flux rope.
6. Conclusions
We have further investigated the model proposed by Lanza(2008) to interpret the observations of star-planet magnetic in-teraction. A linear force-free model of the stellar coronal fieldhas been applied to address the energy budget of the interac-tion and to understand its intermittency. We propose that themagnetic helicity budget plays a fundamental role in the inter-action. An hot Jupiter contributes to this budget by increasingthe helicity dissipation that triggers an additional magnetic en-ergy release in the stellar corona.The transition between phases with and without a chromo-spheric hot spot rotating synchronously with the planet is inter-preted as a consequence of a topological change of the coronalfield induced by an accumulation of helicity. Its timescale de-pends on the mechanisms ruling the helicity fluctuations in thestellar hydromagnetic dynamo and the helicity budget of thestellar corona which are presently poorly known. However, atimescale as short as 10 − s could in principle be possible(cf. Sect. 4.2).The model can be tested in the case of su ffi ciently rapidlyrotating stars, such as τ Boo or HD 189733, by combining spec-tropolarimetric measurements of the stellar photospheric fieldswith Ca II K line observations to determine the phase lag be-tween a planet-induced hot spot and the planet. Such simul-taneous measurements can in principle constrain the topologyof the field by allowing us to estimate the parameters of theforce-free field model as detailed at the end of Sect. 4.2.The present model also bears interesting consequences forthe coronal emissions of stars hosting hot Jupiters. It may ex-plain why stars with a close-in giant planet have, on the aver-age, a higher X-ray luminosity than stars with a distant planet. Moreover, it suggests that the best chances to detect radio emis-sion from hot Jupiters or their host stars are found in systemsshowing a chromospheric hot spot rotating synchronously withthe planet.We have also investigated the consequences of the di ff erentfield topologies for the confinement and the storage of the mat-ter evaporated from a planetary atmosphere under the action ofthe radiation from the host star. When the field has an azimuthalrope of flux encircling the star, the evaporated matter can nei-ther escape nor fall onto the star and is expected to condense inthe outer corona forming several prominence-like structures. Itcan in principle be detected in the case of rapidly rotating andhighly active stars through the observations of transient absorp-tion features moving across the profile of their chromosphericemission lines.Simultaneous optical, X-ray and radio observations canprove the association between a chromospheric hot spot in-duced by a hot Jupiter and the X-ray and radio emission en-hancements expected on the basis of our model because theenergy is mainly released in the corona of the star and thenconveyed along magnetic field lines to heat the lower chro-mospheric layers. Most of the coronal energy should be re-leased within one stellar radius, where the magnetic field isstronger. Therefore, no large phase lags are expected betweenthe chromospheric enhancement and the X-ray and radio en-hancements varying in phase with the orbital motion of theplanet. On the other hand, when there is no signature of SMPIin the chromosphere, we expect that also the modulation ofthe X-ray flux with the orbital motion of the planet is sig-nificantly reduced. No detectable radio emission is expectedin this case, except when the planet has a polar field of atleast 20 −
30 G. Finally, we expect to observe the signaturesof several prominence-like condensations in the coronae ofrapidly rotating ( v sin i ≥ −
50 km s − ), highly active stars( L X ∼ − W) hosting transiting hot Jupiters, if ourassumption that coronal flux ropes have a typical lifetime of200 −
300 days is true.
Acknowledgements.
The author wishes to thank an anonymousReferee for a careful reading of the manuscript and valuable com-ments. AFL is grateful to Drs. P. Barge, S. Dieters, C. Moutou andI. Pagano for drawing his attention to the interesting problem of star-planet magnetic interaction and for interesting discussions. This workhas been partially supported by the Italian Space Agency (ASI) un-der contract ASI / INAF I / / /
0, work package 3170. Active starresearch and exoplanetary studies at INAF-Catania AstrophysicalObservatory and the Department of Physics and Astronomy of CataniaUniversity is funded by MIUR (
Ministero dell’Istruzione, Universit`ae Ricerca ), and by
Regione Siciliana , whose financial support is grate-fully acknowledged. This research has made use of the ADS-CDSdatabases, operated at the CDS, Strasbourg, France.
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List of Objects ‘ υ Andromedae’ on page 1‘HD 189733’ on page 1‘ τ Bootis’ on page 1‘HD 179949’ on page 2‘ υ And’ on page 2‘HD 179949’ on page 2‘ υ And’ on page 2‘HD 179949’ on page 2‘HD 189733’ on page 2‘HD 73256’ on page 2‘HD 189733’ on page 2‘HD 179949’ on page 2‘ τ Bootis’ on page 2‘ τ Boo’ on page 2‘CoRoT-4a’ on page 2‘CoRoT-2a’ on page 2‘HD 192263’ on page 2‘ ǫ Eridani’ on page 2‘HD 128311’ on page 2‘HD 189733’ on page 2‘HD 179949’ on page 4 ‘ τ Boo’ on page 5‘HD 179949’ on page 5‘HD 179949’ on page 9‘HD 189733’ on page 11‘ ττ