Mon.Not.R.Astron.Soc.000,1–13(2010) Printed7January2011 (MNLATEXstylefilev2.2) Powerful Winds from Low-Mass Stars: V374 Peg A. A. Vidotto1⋆, M. Jardine1, M. Opher2, J. F. Donati3 and T. I. Gombosi4 1SUPA, School of Physics and Astronomy, Universityof St Andrews, North Haugh, St Andrews, KY16 9SS, UK 1 2George Mason University,4400 UniversityDrive, Fairfax, VA, 22030-4444, USA 1 3LATT- CNRS/Universit´e de Toulouse, 14 Av. E. Belin, Toulouse, F-31400, France 0 4Universityof Michigan, 1517 Space Research Building, AnnArbor, MI, 48109-2143, USA 2 n a Accepted .Received;inoriginalform J 6 ABSTRACT ] The M dwarf V374 Peg (M4) is believed to lie near the theoretical mass threshold R for fully convective interiors.Its rapidrotation(P =0.44 days)along with its intense S magneticfieldpointtowardmagneto-centrifugalaccelerationofacoronalwind.Inthis . h work,weinvestigatethestructureofthe coronalwindofV374Pegbymeansofthree- p dimensionalmagnetohydrodynamical(MHD)numericalsimulations.Forthefirsttime, - anobservationallyderivedsurfacemagneticfieldmapisimplementedinMHDmodels o of stellar winds for a low mass star. By self-consistently taking into considerationthe r t interaction of the outflowing wind with the magnetic field and vice versa, we show s a that the wind of V374 Peg deviates greatly from a low-velocity, low-mass-loss rate [ solar-type wind. We have found general scaling relations for the terminal velocities, mass-lossrates,andspin-downtimesofhighlymagnetizedMdwarfs.Inparticular,for 2 V374 Peg, our models show that terminal velocities across a range of stellar latitudes v 2 reachu∞ ≃(1500–2300)n1−21/2 km s−1,wheren12 isthecoronalwindbasedensityin 6 units of 1012 cm−3, while the mass-loss rates are about M˙ ≃4×10−10n112/2 M⊙ yr−1. 7 Wealsoevaluatetheangular-momentumlossofV374Peg,whichpresentsarotational 4 brakingtimescaleτ ≃28n−1/2 Myr.Comparedtoobservationallyderivedvaluesfrom . 12 0 perioddistributionsofstarsinopenclusters,thissuggeststhatV374Pegmayhavelow 1 coronalbasedensities(.1011cm−3).WeshowthatthewindrampressureofV374Peg 0 is about 5 orders of magnitude larger than for the solar wind. Never the less, a small 1 planetarymagneticfieldintensity(∼0.1G)isabletoshieldaplanetorbitingat1AU : v against the erosive effects of the stellar wind. However, planets orbiting inside the i habitable zone of V374 Peg, where the wind ram pressure is higher, might be facing X a more significant atmospheric erosion. In that case, higher planetary magnetic fields r of, at least, about half the magnetic field intensity of Jupiter, are required to protect a the planet’s atmosphere. Key words: MHD – methods: numerical – stars: individual (V374 Peg) – stars: low-mass – stars: magnetic field – stars: winds, outflows 1 INTRODUCTION dMstarslieontheboundarybetweenfullyconvectivestars (spectral types later than ∼M4) and stars with a radiative M dwarf (dM) stars comprise the most abundant group of core (spectral types earlier than ∼M4). The latter are be- stars in the Galaxy. Due to their small masses (∼ 0.06 – lieved to host magnetic fields generated by solar-type dy- 0.8 M⊙, Scalo et al. 2007), they have long main-sequence namos, where the layer between the radiative core and the lifetimes, which exceed the Hubble time. They are low- convective outer shell (the tachocline) produces rotational luminosityobjectsandbecauseofthat,theirhabitablezone, shear,which,inconjunctionwithconvectivemotions,isable whereliquidwatermaybefound,isbelievedtolieveryclose toamplifyandmaintainsurfacemagneticfields.Thegenera- tothestar(Kasting et al.1993).ThismakesdMstarsprime tionofmagneticfieldsinfullyconvectivestarsisstillamat- targets in searches for terrestrial habitable planets. ter of debate though. Because such stars lack a tachocline, From the point of view of stellar evolutionary theory, the dynamo mechanism should be different from a solar- type dynamo. Also, their nearly solid-body rotation indi- cates that the magnetic fields should cause a quenching of ⋆ E-mail:[email protected] 2 A. A. Vidotto et al. thesurfacedifferentialrotation,aningredientbelievedtobe Estimates of mass-loss rates from dM stars vary essential in field generation. considerably. It has been suggested that the coronal Although details of the dynamo mechanism may still winds of dM stars, despite of being very tenuous, pos- be unknown, the change in the stellar internal condi- sess mass-loss rates (M˙) that can considerably exceed tions appears to be reflect in the shift in the magnetic the solar value (M˙⊙ ≃ 2 × 10−14 M⊙ yr−1) by fac- field large-scale topology around spectral type M4. The tors of 10 to 104 (Mullan et al. 1992; Badalyan & Livshits observation of surface magnetic field distributions sug- 1992; Lim & White 1996; van den Oord & Doyle 1997; gests that early-dM stars host weak large-scale fields with Wargelin & Drake 2001). Based on radio observations from dominantly toroidal and non-axisymmetric poloidal con- IRAS,VLAandJCMTdata,Mullan et al.(1992)estimated figurations (Donati et al. 2008), while mid-dM stars host mass-loss rates for active dM stars (in particular, YZ CMi, strong, mainly axi-symmetric large-scale poloidal fields Gl 644, Gl 873) of a few times 10−10 M⊙ yr−1. This re- (Morin et al. 2008a). sultwas contradictedbyLim & White(1996),whoclaimed This topology shift should produce a corresponding that the active M dwarfs YZ CMi (dM4.5e) and AD Leo change in the coronal structure of the star, and, in partic- (dM3.5e) should presentM˙ .10−13 M⊙ yr−1,andlater on ular, in coronal tracers such as X-ray and radio emissions. by van den Oord & Doyle (1997), who suggested that the However,boththeX-rayluminosityL aswellasthequies- mass-loss rates of dM stars should be at maximum a cou- X cent radio luminosity Lrad do not present sharp transitions ple of M˙ . 10−12 M⊙ yr−1. van den Oord & Doyle (1997) across the M4 boundary. Lrad, for instance, remains rela- pointed out that both YZ CMi and AD Leo are consid- tively the same for all the spectral subtypes of dM stars, ered young stars (ages of about 0.5 – 1 Gyr), and that while L is roughly proportional to the bolometric lumi- at their early life, M˙ should have been about one order X nosityLbol,presentingasaturationlimitofLX/Lbol ∼10−3 of magnitude larger: M˙ ∼ (1.1 – 3.6) ×10−11 M⊙ yr−1. out tospectral typesM7(Delfosse et al. 1998;Berger et al. EvenforthecloseststartotheSolarSystem,ProximaCen- 2010). As a consequence, a correlation between radio and tauri (dM5.5e), estimates of mass-loss rates are rather con- X-ray emissions is observed for almost all the spectral sub- troversial. Wargelin & Drake (2002) suggest an upper limit types of dM stars, with a sharp transition being seen only of M˙ . 3×10−13 M⊙ yr−1 and Lim et al. (1996) suggest around spectral type M7 (Gu¨del et al. 1993; Berger 2006; M˙ . 7×10−12(u∞/300 km s−1) M⊙ yr−1 (where u∞ is Berger et al.2010).Becauseofsuchacorrelation,itisoften the wind terminal velocity), although Wood et al. (2001) suggested that the origins of both X-ray emission, which is claim an upper limit of M˙ . 4×10−15 M⊙ yr−1, 5 times believedtotracethehotplasmacontainedinmagneticloop below the value of the solar wind mass-loss rate. Accord- structures, and radio emission, which is believed to be pro- ing to Zendejas, Segura, & Raga (2010), if Proxima Cen is ducedbygyro-synchrotronradiation,arecommonandmost about 1 – 4 Gyr old, the “young” Proxima Cen should probably dueto thecoronal magnetic fields. have had considerably larger mass-loss rates M˙ ∼ (4 – Regarding rotation, the evolution of dM stars can be 64)×10−11 M⊙ yr−1. inferred from observations of open clusters at different ages (Irwin et al. 2006; Scholz & Eisl¨offel 2007; Hartman et al. 2009; Collier Cameron et al. 2009; Meibom et al. 2009). In In this work, we investigate the coronal wind of a young (. 700 Myr) open clusters, dM stars still present specific fully-convective dM star, V374 Peg, for which highrotationrates,whichsuggeststhatangularmomentum observed surface magnetic maps have been acquired losses at the early main-sequence phase are negligible for (Donatiet al. 2006; Morin et al. 2008b). For this, we them (Irwin & Bouvier 2009). However, as the cluster ages usethree-dimensional(3D)magnetohydrodynamics(MHD) (& 700 Myr), the number of rapidly rotating dM stars de- simulations based on our previous models developed for creases,implyingthatthereshouldexistamechanismofan- solar-like stars (Vidotto et al. 2009a) and weak-lined T gular momentum removal that acts on time-scales of a few Tauri stars (Vidottoet al. 2009b, 2010). For the first time, hundred million years (Scholz & Eisl¨offel 2007). For solar- anobservationallyderivedsurfacemagneticfieldmapisim- like main sequence stars, the magnetised stellar wind is be- plemented in MHD models of stellar winds for a low-mass lievedtospindownthestarbycarryingawaystellarangular star.V374Pegisasuitablecaseformodellingasafirststep, momentum.Ithasbeenobservationallyestablishedthatthe because its surface magnetic distribution is close to poten- angularvelocity rateΩ0 forsolar-like starsvariesasafunc- tial, which implies that the adopted boundary conditions tion of age t as Ω0 ∝ t−1/2 (Skumanich 1972). However, it match the observed map closely. We self-consistently take seems that the empirical Skumanich’s law is not valid for into consideration the interaction of the outflowing wind low-mass stars,suggesting thatasolar-typewind (i.e.,with with the magnetic field and vice-versa. Hence, from the in- lowvelocitiesandmass-loss rates)cannotreproducethero- terplaybetweenmagneticforcesandwindforces,weareable tational evolution of fully-convectivestars. todeterminetheconfigurationofthemagneticfieldandthe The existence of hot coronae, rapid rotation, and high structureof the coronal winds. levelsofmagneticactivityindMstarssuggeststhepresence of winds with an enhanced mass loss as compared to the solarwind.However,thelow-density,opticallythinwindsof This paper is organised as follows. In Section 2, we thesestarspreventstheobservationoftraditional mass-loss present the details of the stellar wind model we use to de- signatures, such as P Cygni profiles. The still unobserved scribethewindofV374Peg.Section3presentstheresultsof highmass-lossratesfromdMstarscouldbeabletodisperse our simulations. In Section 4, we present the discussions of debris discs, explaining why discs around dM stars older ourwork,includingimplicationsofthewindonhabitability than &10 Myr are scarcely found (Plavchan et al. 2005). of planets. Section 5 presents theconclusion of our work. Powerful Winds from Low-Mass Stars: V374 Peg 3 2 THE NUMERICAL STELLAR WIND MODEL a grid for the simulations of the wind of V374 Peg that is more refined around the star, which is placed at the centre Observations of the solar corona during eclipses have re- oftheCartesiangrid.Thegridaxesx,y,andz extendfrom vealed the presence of closed magnetic loops, extending −20R∗to20R∗,whereR∗isthestellarradius.Afterapply- out to several solar radii. Beside the closed loops, re- ing 8 levels of refinement, the more refined zone of the grid gions of open magnetic field lines form coronal holes, from (for r<5 R∗) has cells with a resolution of 7.8×10−2 R∗. wherethefast solar wind originates. Based ontheobserved The inner boundary of the system is the base of the solar configuration, bi-component coronal models, where wind at radius r = R∗, where fixed boundary conditions closed and open magnetic field line regions coexist, have are adopted. The outer boundaries at the edges of the grid been used to explain the coronal structure of the Sun and haveoutflowconditions,i.e.,azerogradientissettoallthe other stars (Pneuman & Kopp 1971; Mestel & Spruit 1987; primary variables (u, B,p, and ρ). Keppens& Goedbloed 1999, 2000; Ud-Doula& Owocki V374 Peg has mass M∗ = 0.28 M⊙, radius R∗ = 2002; Matt & Pudritz 2008; Vidotto et al. 2009a,b, 2010). 0.34 R⊙ and is rotating with negligible differential rotation In these models, closed field structures form a belt around (i.e., as a solid body) with a period of rotation P0 =0.44 d theequator of thestar, while open field structures give rise (Morin et al. 2008b). We consider that its axis of rotation to the fast solar/stellar wind. We adopt such wind models liesinthez-direction.Thesimulationsareinitialised witha to simulate the yet unobservedwind of V374 Peg. 1Dhydrodynamicalwindforafullyionisedplasmaofhydro- To perform the simulations, we use the 3D MHD nu- gen.Immersedinthiswindweconsideramagneticfieldan- mericalcodeBATS-R-USdevelopedatUniversityofMichi- choredonthestellarsurfacethathaseitheradipolargeom- gan(Powell et al.1999).BATS-R-USsolvestheMHDequa- etry (§2.1) or a geometry derived from extrapolations from tions in the finite-volume scheme. It has currently several observed surface magnetic maps (§2.2) using the potential- approximate Riemann solvers available, although originally field source surface (PFSS) method. The MHD solution is usedtheupwindRoe-typescheme.Thetemporalintegration evolvedintimefromtheinitialmagneticfieldconfiguration is performed using either an explicit, implicit or combined to a fully self-consistent solution. We do not adopt fixed explicit-implicit time stepping scheme (details are found in topologies for either the magnetic field or for the wind, as T´oth et al.2006).Weperform ourcalculation inanexplicit both thewind andmagnetic field lines are allowed to inter- scheme, where a two-level Runge-Kutta scheme is used. act with each other. Here, we summarise the characteristics of our model. More details can be found in Vidotto et al. (2010). BATS-R-US solves the ideal MHD equations, that in 2.1 Dipolar Surface Field theconservative form are given by Because V374 Peg presents a surface magnetic field distri- ∂ρ +∇·(ρu)=0, (1) bution that is mainly dipolar, a dipolar field geometry was ∂t initially used for our simulations. This allows comparisons ∂(ρu) B2 BB with the previous works of Vidotto et al. (2009a,b, 2010) +∇· ρuu+ p+ I− =ρg, (2) ∂t (cid:20) (cid:18) 8π(cid:19) 4π (cid:21) andwitharealisticmagneticfielddistributionatthestellar surface.Insuchcases,theinitialmagneticfieldconfiguration ∂B +∇·(uB−Bu)=0, (3) is described in spherical coordinates (r,θ,ϕ) by ∂t ∂ε B2 (u·B)B B= B0R∗3 cosθ,1sinθ,0 . (6) +∇· u ε+p+ − =ρg·u, (4) r3 (cid:18) 2 (cid:19) ∂t (cid:20) (cid:18) 8π(cid:19) 4π (cid:21) To be compatible to the maximum intensity derived in the where ρ is the mass density, u the plasma velocity, B the surfacemaps,weadoptedamaximumintensityofthemag- magneticfield,pthegaspressure,gthegravitational accel- eration due to the central body, and ε is the total energy netic field of B0 = 1660 G, which is evaluated at the magnetic pole. Here, we assume that the dipole moment is density given by aligned with the rotation axis of the star. Figure 1a shows ρu2 p B2 theinitial configuration of themagnetic field lines. ε= + + . (5) 2 γ−1 8π We consider an ideal gas, so p = nk T, where k is the B B 2.2 Observed Surface Field Boltzmann constant, T is the temperature, n=ρ/(µm ) is p thewinddensity,µm isthemeanmassoftheparticle,and Intheremainingcases,weimplementedamorerealisticdis- p γ is the heating parameter (polytropic index). In practice, tributionofmagneticfieldsatthesurfaceofthestar.Thisis γ represents the energy balance of the wind, for which the the first time an observed magnetogram has been included details are unknown. In our simulations, we adopt γ = 1.2 in a MHD wind model for a low-mass star. Magnetic maps and µ=0.5. for V374 Peg have been presented in Donati et al. (2006); BATS-R-US uses a Cartesian computational domain Morin et al.(2008b),andarederivedfrom dataacquiredin that is block-based, where blocks can be either created or 2005 Aug,2005 Sep,and2006Aug.Asafirststep,wehave destroyedasthecomputationproceeds.Todothat,anadap- opted to implement the maps derived from the particular tivemesh refinement(AMR)techniqueisused intheMHD set of observations made in 2005 Aug. Figure 2 shows the algorithm of BATS-R-US. Although we opted not to mod- radial, meridionaland azimuthalcomponentsofthesurface ify our grid resolution during the numerical computation, magneticfieldofV374Pegobtainedthroughextrapolations we used the AMR capabilities of BATS-R-US to construct of the surface map using the PFSS technique, as described 4 A. A. Vidotto et al. Figure1.(a)Initialconfigurationofmagneticfieldlinesforadipoleand(b)fortheextrapolationofthesurfacemapusingthepotential- fieldsourcesurface(PFSS)technique,wherethedashedcircumferencerepresentsthepositionofthesourcesurface.(c)Finalconfiguration oftheofmagneticfieldlinesaftertheself-consistentinteractionwiththestellarwindconsideringadipolarsurfacedistributionofmagnetic fieldsand(d)consideringtheobservationallyderivedmagneticmap.Notethatpanels(b)and(d)areshownfordifferentstellarrotation phases. inJardine et al.(2002).InthePFSSmodel,thestellarwind alwaysbemet.However,theadvantageofthePFSSmethod plasma is not included directly, but its effects on the mag- overtheMHDmodelsreliesonitssimplicity:itissimplerto neticfield(andvice-versa)areincorporated throughthein- implement and requires much less computer resources than clusionofthesourcesurface.Suchasurface,forinstance,al- MHD models. In our model, we use the magnetic field con- tersthenumberofopenmagneticfieldlines,throughwhere figuration derived by the PFSS method as initial condition astellar windcould escape.PFSSmethodsareusuallycrit- and boundary condition at thesurface of the star. Wenote icisedbecausetheirbasicassumptions(themagneticfieldis that the surface maps presented in Figure 2 matches the apotentialfieldandthesourcesurfaceisspherical)maynot observed maps closely. Powerful Winds from Low-Mass Stars: V374 Peg 5 maximum intensity of about 1660 G, i.e., 3 orders of mag- nitudelarger than theSun. Thewindtemperatureanddensityareless constrained for V374 Peg. We, therefore, adopt values representative of dM stars. dM stars are believed to host coronae with a high-temperature plasma ∼ 107 K in conjunction with a low-temperature one 2 – 3×106 K (Schmitt et al. 1990; Giampapa et al. 1996). In our models, we adopt a temper- ature at the base of the coronal wind of T0 = 2×106 K or 107 K. These coronal temperatures are about the same order of magnitude as the solar coronal temperatures of 1.56×106 K. Coronal densities inferred from X-ray observations of dM stars suggest densities ranging from 1010 cm−3 to 5× 1012 cm−3 (Ness et al. 2002,2004).Therefore, we adopt,at the base of the coronal wind, densities in the range 1010 – 1012 cm−3. Compared to the solar coronal density of about ∼2×108 cm−3, coronal densities inferred for dM stars are about 2 – 4 orders of magnitude larger than for the solar corona. The density, along with the magnetic field, are key pa- rameters in defining the magnetic field configuration of the stellarwindanditsvelocityprofile(Vidottoet al.2009a,b). Together, they define the plasma-β, defined by the ratio of thermaltomagneticenergydensities.Therefore,atthebase of thecoronal wind of V374 Peg, β0 = Bn02k/B(8Tπ0) ≃2.5×10−5n10T6, (7) 0 where the index “0” means the variable is evaluated at the base of the coronal wind, n10 = n0/(1010 cm−3) and T6 = T0/(2 × 106 K). For n10 = 1 and T6 = 1, β0 is about5ordersofmagnitudesmallerthanforthesolarwind (Pneuman & Kopp1971).ThisimpliesthatthewindsofdM Figure 2. The three components of the surface magnetic field stars are highly magnetised and, therefore, are expected to ofV374Pegobtainedthroughextrapolations ofthesurfacemap using the PFSS technique. Maps were derived by Donati etal. differfrom solar-type winds. (2006);Morinetal.(2008b)andextrapolationsweredoneasde- Table 1 presents the parameters adopted for the set of scribedinJardineetal.(2002). simulations we performed. The suffixes ‘Dip’ and ‘Map’ at the case name (first column) stand for the distribution of the surface magnetic field adopted: dipolar or from a mag- Ameridionalcutoftheinitialconfigurationofthemag- neticmap,respectively.The dipolar cases form theSet 1of netic field lines is presented in Figure 1b. The source sur- simulationswhilethecaseswithmagnetogramsformtheSet face (dashed circumference) is chosen to lie at rSS = 5 R∗, 2. where beyond that, the magnetic field is considered to be We were able to find a MHD solution for the wind for purely radial. We can see that the extrapolated field is in- allthesimulationsweran,showingthatitispossibletode- deed mainly dipolar, slightly inclined with respect to the velop coronal wind models with a realistic distribution of rotationalaxisofthestar(z-axis).Wenotethatthesurface magnetic field. In general, MHD wind models are studied ofthestarthatoccupiesco-latitudes&120o isneverinview undertheassumptionofsimplisticmagneticfieldconfigura- as the star rotates and so the magnetic field there can not tions, especially when in pursuit of an analytical solution. bereconstructed reliably. Therefore,thestudyofamagnetisedcoronalwindwherean observed magnetic field distribution is considered has long been awaited. Our work also sheds some light on the yet unobservedwinds from dM stars. 3 RESULTS Our results show that cases where all parameters are Our simulations require a set of input parameters for the the same, except for the surface magnetic field configura- wind.Unfortunately,someofthemarepoorlyconstrainedby tion (e.g., 1Dip/1Map, or 2Dip/2Map, and so on) present observations.ForV374Peg,themagneticfieldisthebetter- similar general wind characteristics and magnetic field con- constrained parameter. Wehave, therefore, implemented in figurations.Thewindvelocitiesareapproximatelythesame our previous models (Vidotto et al. 2010) surface magnetic forcases3Dipand3Map,althoughthedetailsofthevelocity maps derived by data acquired in 2005 Aug (Donati et al. profilesmaydiffer.Overall,oursolutionsdifferconsiderably 2006). These observations show that V374 Peg hosts an in- fromthesolarwindsolution,wherealow-velocitywind(ter- tense, mainly axi-symmetrical dipolar magnetic field, with minalvelocitiesofu∞,⊙ ≃400–800kms−1)withlowmass- 6 A. A. Vidotto et al. Table1.Adoptedparametersforthesetsofsimulations.Thecolumnsare,respectively:thecasename,thedensityn0 andtemperature T0 at the base of the coronal wind (r = R∗), the plasma-β parameter evaluated at R∗ [Eq. 7], the mass loss rate M˙, the angular momentumlossrateJ˙,andthetime-scaleforrotationalbrakingτ.Thecasenamesarelabelledeither“Dip”or“Map”toindicatethat thedistributionofthesurfacemagneticfieldis,respectively,dipolarorfromanobservedmagnetogram. Case n0 T0 β0 M˙ J˙ τ (cm−3) (MK) (10−11 M⊙ yr−1) (1033 ergs−1) (Myr) Set1:Dipole 1Dip 1010 2 2.52 10−5 3.4 2.3 280 2Dip 1011 2 2.52×10−4 12 7.7 84 3Dip 1012 2 2.52×10−3 41 26 24 4Dip 1011 10 1.26×10−3 25 8.3 78 × Set2:Map 1Map 1010 2 2.52 10−5 4.2 3.4 180 2Map 1011 2 2.52×10−4 14 7.6 84 3Map 1012 2 2.52×10−3 50 32 17 4Map 1011 10 1.26×10−3 26 9.1 48 × loss rate(M˙⊙ ≃2×10−14 M⊙ yr−1)isfound.Terminalve- crease in the density leads to winds with lower velocities. locities ofourmodelsarearoundu∞ ≃1500 –2300 kms−1 Simulations 1Dip, 2Dip, and 3Dip are in a very low-β forcases wheren0 =1012 cm−3,increasing asn0 decreases, regime, where magnetic effects completely override thermal whilethemass-lossratesareaboutM˙ ≈4×10−10 M⊙ yr−1 and kinematic effects of the wind. The equation of motion increasing as n0 increases. A comparison of terminal ve- in steady state reduces to locities derived from simpler wind models is presented in j×B Appendix A. We note that, based on more simplistic wind ρ(u·∇)u=−∇p+ρg+ , (9) c models,suchasWeber & Davis(1967),inthefastmagnetic rotatorlimit,windterminalvelocitiesof≃3320km s−1 are wherej=c(∇×B)/(4π)isthecurrentdensity.Forsimplic- expectedforawindmass-lossrateofabout10−11 M⊙ yr−1. ity, let us take the radial component of previous equation Figures1cand1dpresentthefinalconfigurationofthemag- and integrate it. Neglecting terms with p and ρg relative netic field lines, where we note the similarities of the field to the much larger magnetic term, the previous equation is topologyforcaseswithadipolardistributionofsurfacemag- rewritten as neticflux(Fig.1c)andwithadistributiongivenbyobserved |j×B| magnetic maps (Fig. 1d). As a result of the rotation of the u2 ∼ rdr. (10) r Z cρ star, the apparent positions of the magnetic poles in the initial (Fig. 1b) and in the final magnetic field configura- This shows that in a domain where p ≪ B2 (very low- tions(Fig. 1d)are differentbecause theplotsrepresent two β regime) and gravitational forces can be neglected, the different stellar phases of rotation. squaredvelocityofthewindisinverselyproportionaltothe Morin et al. (2010) suggested that the magnetic field density [Eq. (8)]. structure of very low-mass stars may switch from a strong, AttheAlfv´ensurface,thepoloidalwindvelocityequals simplefieldconfigurationtoaweak,complexfieldovertime. thelocal Alfv´en velocity If this is the case for V374 Peg and its magnetic field is in such a transient state, its wind characteristics as well as u2 = Bp2,A. (11) mass-lossandangularmomentum-lossratesareexpectedto p,A 4πρA change along with its magnetic field structure. As u2 ∝ 1/ρ for cases 1Dip, 2Dip, and 3Dip [Eq. (8)], In thenext sections, we present more details of our re- p,A A the location of the Alfv´en surface will only depend on the sultsdivided in twosets definedbytheconfiguration of the intensityB ofthemagneticfieldattheAlfv´ensurface.As surface magnetic distribution adopted. p,A thesecasespossessthesamemagneticfieldatthebaseofthe coronalwind,andbecausetheconfigurationofthemagnetic field lines is very similar, the magnetic field at the Alfv´en 3.1 Set 1: Dipolar Field surfaceshouldbeapproximatelythesame.Thisimpliesthat, Set 1 considers a magnetic surface distribution of a dipolar despite the increase in the base density, the location of the field. By comparing cases 1Dip, 2Dip, and 3Dip, i.e., where Alfv´en surface is similar for cases 1Dip, 2Dip, and 3Dip. only the base coronal density n0 was varied, we found that ThisisdifferenttowhatwasfoundinVidotto et al.(2009a), thepoloidal velocity of thewind scales approximately as where an increase in the coronal density modified the loca- tion of the Alfv´en surface. This Alfv´en surface saturation 1 u2p ∝ n0 for a given B0. (8) hvearpypleonws-bβerceaguimsees.iTmhuelaretfioonres,1EDqi.p(,82)Dshipo,uladndbe3tDreipataerdewinitha This qualitatively agrees with previous results caution, as under different β regimes (for example, when it (Vidotto et al. 2009a), where it was found that an in- approaches β0 ∼ 1), it becomes invalid. Figure 3a presents Powerful Winds from Low-Mass Stars: V374 Peg 7 Figure 3. Meridional cut of scaled poloidal wind velocity u¯p = up/√n12 profile (a) for cases with a dipolar field surface distribution 1Dip,2Dip,and3Dip,and(b)4Dip,(c)forcaseswhereamagneticsurfacemapwasused1Map,2Map,and3Map,and(d)4Map.Black linesrepresentthemagneticfieldconfiguration, andwhitelineisthelocationoftheAlfv´ensurface. thescaledwindvelocityprofileu¯ forcases1Dip,2Dip,and mass-loss rates for cases 1Dip, 2Dip, and 3Dip are M˙ ≈ p 3Dip, 4×10−10n11/22 M⊙ yr−1. Figure 3b shows the scaled poloidal velocity profile u¯ u¯p =upn11/22, (12) forcase4Dip.Thiscaseconsidersadifferenttemperatureapt where n12 = n0/(1012 cm−3). We note that the wind ter- thebaseofthecorona(107 Kasopposedto2×106 K),and, minal velocity is u∞ ≈ (1500 – 2300)n−121/2 km s−1, where dbiedcanuostefiofndthaant,ahnaaslaytliacragleerxβp0re(sTsiaobnler1el)a.tFinogr tvheilsocciatsye,awnde therangeofvelocitiesreferstodifferentwindlatitudes(low- temperature.TheAlfv´ensurfacelocation andconfiguration wind velocity near the equator, high-wind velocity around ofmagneticfieldlinesaresimilartotheotherdipolarcases, thepoles). butthescaled windvelocityu¯ issmallerthantheprevious p windE(qMu˙a∝tioρnu(8))sahlosuoldimspclaileesatshatthemass-loss rateofthe cases: u∞ ≈ (800 – 1800)n−121/2 km s−1. The lower velocity r observedincase4Diphappensbecauseofitshigherβ0.The M˙ ∝ρu ∝n1/2, (13) mass-loss rate for case 4Dip is M˙ ≃2.5×10−10 M⊙ yr−1. r 0 which means that, despite the fact that the wind velocity of case 3Dip is 10 times smaller than case 1Dip [Eq. (8)], 3.2 Set 2: Observationally Derived Magnetic Map its mass-loss rate is one order of magnitude larger than for case1Dip [Eq.(13)].This hasimplications ontheefficiency Theresults of thesimulations where themagnetic field dis- of angular momentum loss, as will be shown in §4.1. The tribution at the surface of the star is set according to ob- 8 A. A. Vidotto et al. served magnetic maps is presented in a similar fashion as an age estimate of t ∼ 0.7 – 0.8 Gyr. The calibration from done in §3.1: the profile of the scaled velocity u¯ [Eq. (12)] West et al.(2009)forM4dwarfs resultsintheconsiderably p for cases 1Map, 2Map, and 3Map is shown in Figure 3c, older age of t ∼ 5.4 Gyr, using LHα and Lbol as given by while for case 4Map it is presented in Figure 3d. Because Delfosse et al. (1998). Because age estimates are still loose themagneticfieldinthelowerhemisphereofthestarisnot and imprecise, its comparison with τ is not able to restrict reliably reconstructed (co-latitudes &120o of thesurface of ourmodel parameters. the star are not observed), a high-velocity wind develops Observations of the rotation evolution of dM stars in there.Although this featureis local and does not affect the openclustersatdifferentages,ontheotherhand,providea remaining parts of the grid other than radially away from waytoconstrainthetime-scaleτ fortheangular-momentum thestellarsurface,itisanartifactofourmethodandshould loss.Ithasbeensuggestedthatτ ∼200Myror,mostlylikely not be taken into consideration (e.g., see the accumulation 400 – 800 Myr, (Scholz & Eisl¨offel 2007) for dM stars. of magnetic field lines near the low-hemisphere of the star Angular momentum of the star is carried away by the in Figure 1b). The wind velocities of Set 2 are consistent stellar wind. Because in someof oursimulations thereisno with the wind velocities of Set 1, shown in §3.1. This can axi-symmetry,thetorqueJ˙ on thestarhasx,y andz com- be seen by comparing Fig. 3a and 3c, and 3b and 3d. We ponents. Here, we are interested only on the z-component, foundsimilarvaluesofwindterminalvelocities(u∞ ≈[1300 as it is the only one responsible for the rotational brak- – 2100]n−1/2 km s−1 for cases 1Map, 2Map, and 3Map, ing (because the angular velocity of the star points in the 12 and u∞ ≈ [850 – 1600]n−121/2 km s−1 for case 4Map) and z-direction). The z-component of the angular momentum mass-loss rates (M˙ ≈ 4 × 10−10n112/2 M⊙ yr−1 for cases carried by thewind is (Mestel & Selley 1970) 1Map, 2Map, and 3Map, and M˙ ≃ 2.6×10−10 M⊙ yr−1 J˙ = αˆz× r×ρ(V+αˆz×r)dV forcase4Map).Thissimilarity occursbecausetheobserved (cid:20) Z A(cid:21) magnetic map does not deviate significantly from the dipo- VA z B2 lar field adopted in §3.1. Future works will consider other + p+ (r×nˆ) dS starswithmorecomplexsurfacemagneticfielddistributions, ZSA(cid:18) 8π(cid:19) z A where more significant deviations are expected. + [r×(αˆz×r)] ρV·nˆdS , (14) Tests with different positions for the source surface Z z A SA showed that final configuration of the wind does not differ where V = u−αˆz×r is the velocity vector in the frame from what is presented in this paper. rotating with angular velocity αˆz, ˆz is the unit vector that points in the z-direction, S is the Alfv´en surface that de- A limitsthevolumeV ,andnˆisthenormalunitvectortothe A 4 DISCUSSION Alfv´ensurface1.ThefirsttermontherightofEq.(14)does not contribute to the z-component torque and is therefore 4.1 Constraining Model Parameters: Angular null. The second term disappears in the case of a spheri- Momentum Loss calAlfv´en surface. It isalso nullin theaxi-symmetriccases One way to restrict the parameters of our model is by consideredhere(i.e.,thedipolarcases),butitisnon-nullin comparing the derived values for u∞ or M˙ with ob- thecaseswhereasurfacemagneticmapisconsideredandit served values. However, traditional measurements of mass- becomesrelativelymoreimportantforthecases withlarger loss rates or wind terminal velocities have not yet been adopted β0. The third term is the dominant term in our precisely obtained for dM stars. Estimates of mass-loss simulations. rates from dM stars, for instance, are rather controver- We can estimate the time-scale for rotational braking sial, ranging from 10 to 104 times the solar wind mass- as τ =J/J˙, where J is the angular momentum of the star. loss rate (e.g., Mullan et al. 1992; Badalyan & Livshits If we assume a spherical star with a uniform density, then 1992; Lim & White 1996; van den Oord & Doyle 1997; J =2/5M∗R∗2Ω0 and thetime-scale is Wargelin & Drake 2001), but predictions for sub-solar val- 9×1036 M∗ 1 d R∗ 2 uesexistaswell(Wood et al.2001).Mass-lossratesderived τ ≃ Myr, (15) from our wind models are M˙ ≈ 4×10−10n11/22 M⊙ yr−1 J˙ (cid:18)M⊙(cid:19)(cid:18)P0(cid:19)(cid:18)R⊙(cid:19) for cases 1Dip/1Map, 2Dip/2Map, and 3Dip/3Map, and where P0 = 2π/Ω0 is the rotational period of the star. For M˙ ≃ 2.5×10−10 M⊙ yr−1 for case 4Dip/4Map. Such val- V374 Peg, thisresults in uesareinaccordancewithseveralestimates.Unfortunately, 6.45×1035 they do not allow us to constrain the input parameters of τ ≃ Myr. (16) J˙ our model, in particular, thedensity at thecoronal base. Another possible way to restrict our model parame- Because J˙ depends on the mass flux crossing a given ters is to compare the time-scale for rotational braking τ surface, i.e., on the mass-loss rate of the wind M˙, from with the age t of V374 Peg. Because V374 Peg is a fast Eq. (13), we have a rough scaling relation between J˙ and rotator, we expect that τ ≫ t. However, there exist only M˙ for cases 1Dip/1Map, 2Dip/2Map, and 3Dip/3Map loose ageestimatesforV374Peg.Delfosse et al.(1998)and J˙∝M˙ ∝n1/2, (17) Mohanty & Basri (2003) estimated the kinematic age of 0 V374 Peg as “young disc”, which roughly corresponds to t . 3 Gyr. Using the calibration of LX versus age pro- 1 ThecomputationofthelocalunitvectornˆnormaltotheAlfv´en vided by Catala´n et al. (2009), for the observed value of surfacewas done withthe routine developed by F. Alouani Bibi LX = 1028.44 erg s−1 (Delfosse et al. 1998), it results in (Opheretal.2009). Powerful Winds from Low-Mass Stars: V374 Peg 9 whichimpliesinatime-scale[Eq.(16)]forrotationalbraking that scales as τ ∝n−1/2. (18) 0 For cases 1Dip, 2Dip, and 3Dip, this implies that τ ≃ 28n−1/2 Myr, while for cases 1Map, 2Map, and 3Map, 12 τ ≃ 18n−1/2 Myr. In either case, τ is well below the es- 12 timated solar spin-down time τ⊙ ≃ 7 Gyr (Weber& Davis 1967). Table 1 presents themass and angular momentum loss rates, and the time-scale for rotational braking calculated for all simulations, where we verify the approximate scal- ing given by Eqs. (13), (17), and (18). Comparing to the observationally derived rotational braking time-scales of a couple of hundreds of Myr for dM stars is open clusters (Scholz & Eisl¨offel 2007), we tend to rule out cases with larger coronal base densities (i.e., n0 & 1011 cm−3). Ac- cordingtothiscomparison,themostplausiblewinddensity Figure 4. Isosurface of Br /B = 0.97 calculated for the MHD | | model (case 3Map). Color represents the radial velocity of the is the one used for models 1Dip/1Map. Such a density is wind. alsoabletoreproducetypicalemissionmeasuresofdMstars (EM≈1051 cm−3) and comparatively (with the remaining cases) smaller mass-loss rates and higher wind velocities. ityofthestarwillnotbechanged,andconsequentlyneither Becauseofitshighmass-lossrate,V374Pegisexpected thestellar mass-loss/angular momentum-loss rates. to lose about 0.01 M⊙ (i.e., ∼ 3.5% of its mass) during a time-scale τ. If the star persists on losing this amount of 4.2 Departure from Potential Field mass during a longer period of time, this could have signif- icant effects on its evolution. However, when the star ages Here, we compare the results of our MHD modelling of the andboth itsrotation aswell asthestellar surface magnetic wind of V374 Peg with the output of the PFSS method, field intensity are reduced, its mass-loss rate is expected to whichwasusedastheinitialconfigurationforthemagnetic become less intense and therefore present a less significant field lines in Set 2 of our simulations. Such a configuration effect on the evolution of thestar. is shown in Fig. 1b. The spherical source surface is located There is also great interest in low-mass stars at at r = rSS = 5 R∗ and represented by the dashed circum- young ages, when accretion discs are still observed. Accre- ference. The field is considered to be everywhere potential, tion rates observed for pre-main sequence low mass stars and its surface distribution derived from observed surface (Herczeg & Hillenbrand 2008) are comparable to the wind magnetic maps. Above r =rSS, the PFSS method assumes rates that we derived here. Although it is not straightfor- thatthemagneticfieldispurelyradial,decayingasr−2.Ide- ward to extend our models to these very early phases, we ally,wewould liketocomparethesizeofthesourcesurface notethat thehigh wind rates we predict could haveimpor- derived from the MHD modelling. However, this compar- tant consequences for the evolution of low-mass stars and ison is not straightforward, as there is no surface beyond their discs. whichthemagneticfieldlinesarepurelyradialintheMHD With the inclusion of an observed distribution of sur- model.Toovercomethisdifficulty,Figure4showsanisosur- face magnetic field, we head towards a more realistic mod- face of |B |/B = 0.97, which represents the surface where r ellingofmagnetisedcoronalwinds.Nevertheless,ourmodel 97 percent of the magnetic field is contained in the radial presents limitations, such as the neglect of a detailed en- component (Riley et al. 2006). The first immediate conclu- ergy balance. Instead, we consider a polytropic relation be- sion from the comparison between this isosurface and the tween pressure and density (or temperature) parametrised source surface of the PFSS method is the lack of sphericity through γ in the derivation of the energy equation Eq. (4). of the former. Moreover, the size of the surface is consid- Oncethemagneticfielddistributionisset,thethermalpres- erably smaller than the radius rSS = 5 R∗ adopted in the sureadjustsitself inordertoprovideadistributionofheat- PFSS method. In our MHD model, the average size of the ing/cooling that is able to support the MHD solution ob- isosurface ishrSSi=2.6R∗,extendingtoward thepolesout tained (Low & Tsinganos 1986). If the wind of V374 Peg is to 1.9 R∗ and toward the equator out to 2.9 R∗. We note able to cool down, e.g., by radiative cooling not considered that thissize is verysimilar to theadopted size of 2.5 solar in our models, the terminal velocities of the wind could be radiifromPFSSmethodsofthesolarcoronalmagneticfield. considerably smaller. Depending on where in the wind en- The stored magnetic energy contained in the potential ergy deposition (or removal) occurs, the wind velocity may fieldisintheloweststate,i.e.,itisthetheminimumvalueof change, without affecting the mass-loss rates. For instance, energy that the magnetic field lines can store. In the MHD ifasubstantialcoolingoccursabovetheAlfv´ensurface, the wind case, excess energy is contained in the magnetic field velocity profile of the wind from that point outwards will lines due to stresses imposed by the wind. To quantify the beaffected. As theinformation of what is happeningabove departure of the MHD solution from the potential field so- theAlfv´en point cannot betransmitted tothesub-Alfv´enic lution,weevaluatethestoredmagneticenergyineachcase. region,thewinddensityandvelocityprofilesintheproxim- Definingthemean magnetic energy as 10 A. A. Vidotto et al. creationanddevelopmentoflife.Therampressurep ofthe w wind that impacts on a planet orbiting V374 Peg is p =ρu2=m˙u r−2, (21) w r r where m˙ = ρu r2. In the asymptotic limit of the wind ve- r locity, for large distances from thestar, pw,∞ =m˙u∞r−2 ≃ M4˙πur∞2 . (22) UsingourscalingrelationsforV374Peg(cases1Dip/1Map, 2Dip/2Map,3Dip/3Map) M˙ ≃4×10−10n11/22 M⊙ yr−1 and u∞ ≃(1500 – 2300)n−121/2 km s−1,we have (3 to 4.6)×1023 pw,∞ ≃ r2 dyn cm−2, (23) where the range of values exists due to different latitudi- nal wind velocities. We note that pw,∞ does not depend on the density assumed for our wind models of V374 Peg. Therefore,evenanincreaseinthewindmass-loss rate(e.g., through an increase in the wind base density) would not Figure 5.Currentdensity. haveany furthereffect on an orbitingplanet, becausethere is an associated decrease in the wind velocity. For thesolar wind B2dV hB2i= RVV dV , (19) pw,∞,⊙ = M˙4⊙πur∞2,⊙ ≃ 4×r12018 dyncm−2, (24) R whereV isagivenvolume,wecalculatedtheratiof between the energy contained in the MHD solution and the one in where we have used M˙⊙ =2×10−14M⊙ yr−1 and u∞,⊙ = 400 km s−1. Therefore, the wind of V374 Peg has a ram thePFSS solution pressurethatisabout5ordersofmagnitudelargerthanfor f = hB2iMHD. (20) thesolar wind. The immediate question we ask ourselves is hB2iPFSS how strong the planetary magnetic field should be in other toshield theplanet’satmospherefrom theerosiveeffectsof For a volume contained between r = 1 and 5 R∗, f = 1.5, thewind.Pressurebalancebetweenthewind totalpressure implyingthattheMHDsolutionhasstoredabout50percent andtheplanettotalpressurerequiresthat,atadistancer moremagneticenergythanthepotentialfieldsolution.The M from theplanet, same evaluation from r = 1 to 2 R∗ results in f = 1.3 and from r = 2 to 5 R∗ results in f = 2.8. We conclude that pw + Bw2 +p= Bp2,rM +p , (25) closer to the star, the MHD solution deviates little from 2 4π 4π p the potential field solution. However, the departure from a where p , B and p are the local wind ram pressure, mag- w w potentialfieldbecomesmoreimportantfartheroutfromthe netic field and thermal pressure, and B and p are the star. p,rM p local planetary magnetic field intensity and thermal pres- The existence of currents in the MHD solution also il- sure. Far from the central star, pw → pw,∞, while Bw and lustrates the departure of the magnetic field configuration pbecomenegligible. Assumingthat theplanetarymagnetic fromapotentialone.IncontrasttothePFSSmethod,which fieldisdipolar,B =B (R /r )3,whereB istheequa- assumes null currents everywhere (j ∝ ∇×B = 0), in the p,rM p p M p torialplanetarymagneticfieldintensityatitssurfaceandR p MHDwindmodelling,currentdensitiesarisenaturally.Fig- is the planetary radius. A minimum magnetic shielding of ure5 shows theprofileof current densities, which is similar the planet requires that r > R . Therefore, for a planet M p in all the simulations of the wind of V374 Peg. This simi- orbiting V374 Peg at r = 1 AU, the minimum planetary larity occurs because, in theideal MHD,j only dependson magnetic field required is theintensityand configuration of themagneticfield,whose characteristics are similar in our simulations. The current Bp,min=(2πpw,∞)1/2 ∼0.1 G, (26) densityismaximumnearthestaranditdecreaseswithdis- where we assumed that p is negligible. A significant p p p tance. A more intense current sheet is seen separating the would only increase the limit obtained before. Therefore, closed-field line region and theopen-field line region. for a planet orbiting far from the star (∼ 1 AU), a mag- netic field of only 0.1 G would be sufficient to prevent atmospheric erosion caused by the wind. In the asymp- 4.3 Effects on Habitability totic limit of the wind, Equations (22) and (26) show that The concept of habitable zones relies on thelocation where Bp,min≃0.1[(1 AU)/r] G. liquid water may be found, which is essentially determined However, it should be noted that for planets orbiting by the luminosity of the star. Because dM stars are low- closer to the star, pw > pw,∞ and both the magnetic and luminosityobjects,thehabitablezoneisbelievedtolienear thermal wind pressures in Equation (25) may not be negli- the star. However, a strong wind is able to erode the at- gible. Therefore, the minimum planetary magnetic field for mosphere of a planet, removing an important shield to the shielding is