Modelling the Hidden Magnetic Field of Low-Mass Stars

3.7 MB·

by  P. Lang

Mon.Not.R.Astron.Soc.000,1–11(2014) Printed21January2014 (MNLATEXstylefilev2.2)
Modelling the Hidden Magnetic Field of Low-Mass Stars
P. Lang1∗, M. Jardine1, J. Morin2,3,4, J-F. Donati5, S. Jeffers2,6, A. A. Vidotto1, R. Fares1
1.SUPA,SchoolofPhysics&Astronomy,UniversityofStAndrews,NorthHaugh,StAndrews,KY169SS,UK
2.Institutfu¨rAstrophysik,Friedrich-Hund-platz1,370777Go¨ttingen
3.DublinInstituteforAdvancedStudies,SchoolofCosmicPhysics,31FitzwilliamPlace,Dublin2,Ireland
4
4.LUPM,UniversitéMontpellier2,CNRS,PlaceEugèneBataillon,34095,Montpellier,France
1
5.IRAP-UMR5277,CNR&Univ.deToulouse,14Av.E.Belin,F-31400Toulouse,France
0
6.UniversityofUtrecht,P.O.Box80000,3508TA,Utrecht,TheNetherlands
2

n
a
J
Accepted2014January14.Received2014January9;inoriginalform2013August26
8
1

ABSTRACT
]
R
Zeeman-Dopplerimagingisaspectropolarimetrictechniquethatisusedtomapthelarge-
S
scalesurfacemagneticfieldsofstars.Thesemapsinturnareusedtostudythestructureofthe
.
h stars’coronaeandwinds.Thismethod,however,missesanysmall-scalemagneticfluxwhose
p polarisation signatures cancel out. Measurements of Zeeman broadening show that a large
-
percentageofthesurfacemagneticfluxmaybeneglectedinthisway.Inthispaperweassess
o
r the impact of this ‘missing flux’ on the predicted coronal structure and the possible rates of
t spindownduetothestellarwind.Todothiswecreateamodelforthesmall-scalefieldand
s
a add this to the Zeeman-Doppler maps of the magnetic fields of a sample of 12 M dwarfs.
[ We extrapolate this combined field and determine the structure of a hydrostatic, isothermal

corona. The addition of small-scale surface field produces a carpet of low-lying magnetic
1
v loopsthatcoversmostofthesurface,includingthestellarequivalentofsolar‘coronalholes’
5 wherethelarge-scalefieldisopenedupbythestellarwindandhencewouldbeX-raydark.We
4 showthatthetrendoftheX-rayemissionmeasurewithrotationrate(theso-called‘activity-
5 rotationrelation’)isunaffectedbytheadditionofsmall-scalefield,whenscaledwithrespect
4 tothelarge-scalefieldofeachstar.Theadditionofsmall-scalefieldincreasesthesurfaceflux;
. however, the large-scale open flux that governs the loss of mass and angular momentum in
1
0 thewindremainsunaffected.Weconcludethatspin-downtimesandmasslossratescalculated
4 fromsurfacemagnetogramsareunlikelytobesignificantlyinfluencedbytheneglectofsmall-
1 scalefield.
:
v Keywords: stars:magneticfield,stars:low-mass,stars:coronae,stars:activity,X-rays:stars
i
X
r
a
1 INTRODUCTION anypotentialorganismsontheseplanets.Thismakesitvitaltoin-
vestigatehowthestructureandevolutionofthemagneticfield,both
Mdwarfs,muchsmaller,dimmerandcoolerthanstarslikeourSun, large-scaleandsmall-scale,canaffectcoronalproperties.
arebyfarthemostcommontypeofstarinourgalaxy.Thestudy
ofthesestarshasremainedlimitedduetotheirfaintnessandinthe
Time-resolvedspectropolarimetricobservationsofastarcan
pastitwaspresumedthatMdwarfswereunlikelytohostdetectable
be analysed by means of Zeeman Doppler Imaging (ZDI, Semel
habitableplanets.Morerecentlyhowever,theadvantagesofsearch-
1989,Donatietal.2006)inordertoreconstructamapofthevec-
ing for habitable planets around M dwarfs have been recognised.
tormagneticfieldonthestellarsurface.ZDIreliesonthefactthat
Forexample,thehabitablezoneiscloserandsoitiseasiertofind duetothecombinationofthepropertiesoftheZeemaneffecte.g.,
planetsbyradialvelocitysearches.Despitetheadvantagesofde-
rotation-induced Doppler and rotational modulation, a strong re-
tectingplanetsaroundthesestars,Mdwarfshavebeenshowntobe
lation exists between the distribution of the magnetic field at the
extremely magnetically active which may have significant effects
surfaceofastarandtherotationalevolutionofpolarisationinspec-
onanyplanetarysystem.Forexample,intensemagneticfields,stel-
trallinesduringastellarrotation.However,severallimitationsex-
larflares,UVandX-rayemissionandthepowerfulstellarwinds
ist:inparticular,withthesolutionbeingnon-unique,amaximum
(Vidottoetal.2013)mayaffectplanetaryatmospheresaswellas
entropy criterion has to be used, and due to the mutual cancella-
tionofpolarisedsignalsoriginatingfromneighbouringregionsof
oppositepolarities,themapshavealimitedspatialresolutionand,
∗ E-mail:[email protected] therefore,systematicallymissmagneticfluxcorrespondingtomag-
(cid:13)c 2014RAS
2 P.Langetal
neticfieldsorganisedonsmallspatialscales.Theactualresolution
ismostlydrivenbytherotationalvelocityofthestarprojectedon
theobserver’sline-of-sight(vsini):thehigherthevsini,thehigher
theresolution.Inaddition,fortheinclinationofthestellarrotation
axiswithrespecttotheline-of-sightdifferingfrom90◦,apartof
thestarisnevervisible.Therefore,inthatregionthereisnocon-
straintonthemagneticfield,exceptthatgloballyithastosatisfy
thenull-divergenceconstraint.
Studies based on spectropolarimetric observations and ZDI
haveprovidedthefirstinformationonthestructureofthesurface
magnetic fields of M dwarfs. In particular, partly-convective M Figure1.Radialmagneticsurfacemapforthemodelofsmallscalefield
dwarfshavebeenshowntohostlarge-scalemagneticfieldswhich covering62%ofthestellarsurface.|Bmax|is500Gwithanunsignedsurface
arenon-axisymmetricandfeatureastrongtoroidalcomponent(Do- fluxvalueφsurface≈1024Mx.
natietal.2008),whereasthoseclosetothelimitoffull-convection
havebeenshowntohostmuchstrongerlarge-scalefielddominated
byamainlyaxisymmetricpoloidalcomponent(e.g.,Donatietal. were created using the Doppler imaging code ‘Doppler Tomog-
2006,Morinetal.2008,Morinetal.2008).However,thesestud- raphy of Stars’ (DoTS) and all spots were modelled, following
iesdonotconstrainthesmall-scalefieldcomponentofthemagnetic Solanki (1999), with circular umbral areas and a ratio of umbral
fieldsoflow-massstars.Inparallel,studiesbasedontheanalysisof topenumbralareaof1:3.
theZeemanbroadeninginunpolarisedspectroscopyprovidecom- We use the spot brightness to allocate field strengths to the
plementaryinformation:themeasureofthedisc-averagedmagnetic centreoftheactiveregionsandallowthefieldstrengthtofall-offin
fieldincludingthecontributionsofboththelarge-scaleandsmall- aGaussian-likedistribution,totheedgeofeachspot,i.e.,
scalecomponents.Reiners&Basri(2009)compiledmeasurements
of mean magnetic flux from Stokes I (total intensity) and Stokes Bsrs= ΣBmax e−x22 , (1)
V (thefractionaldegreeofcircularpolarisation)parametersfora brightness
selection of partially-convective and fully-convective M dwarfs. where Bss represents the field strength in the small-scale field,
r
They find that the fraction of magnetic flux visible in Stokes V Σbrightness isthespotbrightness,Bmax isthearbitrarilychosenmax-
is a small percentage of the total flux measured in Stokes I. This imum field strength, and x is the distance from the centre of the
meansthatalargeportionofthemagneticfluxstoredinmagnetic spot.Wenoteherethatahigherspotbrightnessindicatesalower
fieldsisinvisibletoStokesV.Onepossibleexplanationisthatthe fieldstrengthvalue.
majorityofmagneticfluxonMdwarfsisgroupedintosmallstruc- Weimposeaconditionthatthesmall-scalefieldmustbesmall
turesdistributedoverthestellarsurface,wheredifferentpolarities enough not to be detected in ZDI i.e., invisible in Stokes V. The
cancel each other out in Stokes V. More specifically, Reiners & typical area over which the circular polarisation cancels out e.g.,
Basri(2009)findthatalthoughforthelower-massfully-convective theareaoverwhichthesignedmagneticfluxcancelsorthetypi-
stars,themeanmagneticfluxdoesnotsignificantlydifferfrompar- caldistancebetweentwospotsofoppositepolarities,corresponds
tiallyconvectivestars(Reiners&Basri2007),thefractionofthe toabout12◦,forarapidlyrotatingstarwithvsini ≈40km/se.g.,
totalmagneticfluxdetectedinStokesVisdifferentforpartially- V374Peg.Thisconditionmeansthatanyactiveregionmusthave
convectiveandfully-convectivestars:6%and14%respectively. adiameterlessthanthetypicalZDIresolutioni.e.,<5◦.Oursyn-
The aim of this paper is to determine the influence that this theticmapsassumespotswithradii(cid:54)1◦.
small scale field might have on the stellar coronae. We create a Toensurethesmall-scalefieldisevenlydistributed overthe
modelforsmall-scalefieldandaddittothereconstructedsurface entiresurfaceofthestar,wekeepthespotcoverageconstant.We,
radialmapsforastellarsampleof12MDwarfs(Donatietal.2008; therefore,findthatanappropriateparametertovaryinthemodelis
Morinetal.2008)thatspanthefully-convectiveboundary.Bycom- Bmax.TakingintoaccountthemagnitudeofthefielddetectedinZDI
parisonwiththeobservedlarge-scalemagneticfieldstructure,we foroursampleofpartly-convectiveandfullyconvectiveMDwarfs,
investigatetheeffectthissmall-scalefieldhasonthegeometryof we (1) fix the value of Bmax to be either ±500G or ±1000G; and
the extrapolated 3D magnetic field and subsequent coronal prop- (2)setthevalueofBmax suchthatthelarge-scalefieldcontributes
erties,suchasopenflux,coronalextent,X-rayemissionmeasure betweenapproximately6%and14%ofthetotalfieldrespectively,
andcoronaldensity.Weapproachthisintwoways:(1)byincor- asindicatedinReiners&Basri(2009).Thevaluesfortheaverage
poratingsmall-scalefieldthathasthesamesurfacedistributionand radialfluxineachcasecanbefoundinTable(2).
magnitudeontoeachstarinthesample;and(2)usingtheresultsof The surface magnetic radial map for the small-scale field is
Reiners&Basri(2009),weaddinapercentageamountofsmall- showninFig.(1)wherethespotdistributioncoversapproximately
scalefieldsuchthatthelarge-scalefieldcontributesonly6%and 62% of a model star. Fig. (1) represents the case where Bmax =
14% of the total magnetic field, for the partially-convective and 500G.Thesimulatedmagneticradialmapsforthesmall-scalefield
fully-convectivestarswithinthesamplerespectively. are added to the reconstructed radial maps obtained through ZDI
andnewsurfacemapswithbothlarge-scaleandsmall-scalefield
arecreatedforeachMdwarfi.e.,B =Bss+Bls.
Total r r
2 MODELLINGANDINCORPORATINGTHE
SMALL-SCALEFIELD
2.2 TheCoronalField
2.1 TheSurfaceField
The magnetic field is extrapolated above the stellar surface us-
To create small-scale field on the stellar surface we use the syn- ingthepotential-fieldsourcesurface(PFSS)method(Altschuler&
thesised spot brightness maps of Barnes et al. (2011). The spots Newkirk1969),wherethemagneticfieldisassumedtobecurrent-
(cid:13)c 2014RAS,MNRAS000,1–11
000 3
Dwarfs,109−1012cm−3(Nessetal.2002,2004).Typicalvaluesfor
logκ=[−5:−7]
We assume that the pressure varies along each field line ac-
cordingto
(cid:82) g·Bds
p= poe |B| , (5)
asdescribedbyJardineetal.(2002)andGregoryetal.(2006).Ex-
pandingthe(dimensionless)componentofgravityalongthefield
line(g·B),wehave
 (cid:16) (cid:17) 
Figure2.3Dcoronalextrapolationofthesmall-scalefieldshowninFig. p=κB2oexp(cid:90) −rφ2g +φcrsin(cid:113)2Bθ2B+rB+2(+φcBr2sinθcosθ)Bθds ,(6)
1.Coloursarescaledtothemaximumandminimumvaluesofthesurface r θ φ
radialmagneticfieldcomponent. Wherer=R/R andtheratiosofcentrifugal(φ )andgravitational
(cid:63) c
(φ )tothermalenergyaregivenby
g
(cid:32) (cid:33)
(ωR )2
free (∇× B = 0) and divergence free (∇· B = 0). In a format φc=me k T(cid:63) (7)
B
similartoJardineetal.(1999)thecomponentsforthecoronalmag-
(cid:32) (cid:33)
neticfieldaredeterminedfromthesolutiontoLaplace’sequation φ =m GM(cid:63) , (8)
∇2ψ=0,whereψisthescalarpotential: g e R(cid:63)kBT
(cid:88)N (cid:88)l whereR(cid:63)isthestellarradius,M(cid:63)isthestellarmass,ωisthestellar
B =− [la rl−1−(l+1)b r−(l+2)]P (cosθ)eimφ (2) rotation rate, k is the Boltzmann constant,G is the gravitational
r lm lm lm B
l=1 m=1 constantandmeistheelectronmass.
Toensurethatonlyregionsoftheclosedstellarcoronacon-
(cid:88)N (cid:88)l d
B =− [a rl−1+b r−(l+2)] P (cosθ)eimφ (3) tributetowardstheemissionmeasure,thegaspressurealongopen
θ lm lm dθ lm
l=1 m=1 field lines is taken to be zero. In addition to this, if there is any
over-pressure along the designated closed loops i.e. gas pressure
B =−(cid:88)N (cid:88)l [a rl−1+b r−(l+2)]P (cosθ) im eimφ (4) (p=2nekT)(cid:62)magneticpressure(pB = B2/2µ),thenthepressure
φ lm lm lm sinθ atthatgridpointisalsosettozero.
l=1 m=1
Assumingthegasisopticallythin,theX-rayemissionmea-
with B,B ,B representing the radial, meridional and azimuthal
r θ φ surevarieswithdensity,i.e.
componentsofthemagneticfield,respectively,P representsthe
lm (cid:90)
associatedLegendrepolynomials,almandblmaretheamplitudesof EM(r)= n2dV . (9)
e
thesphericalharmonics,listhesphericalharmonicdegree,misthe
orderorazimuthalnumberandr=R/R(cid:63). Thetemperature(T=2×106K),sourcesurface(Rss=2.5R∗)
Toextrapolatethe3Dcoronalfieldanddeterminetheampli- and constant of proportionality κ (10−6) are kept constant in this
tudeofthesphericalharmonics,almandblm,weapplytwobound- paper(formoredetailsseeLangetal.(2012)).
aryconditions.TheupperconditionisthatattheSourceSurface,
R (Schatten et al. 1969), the field opens and is purely radial
ss
(B = B = 0), while the lower boundary condition imposes the
θ φ
3 RESULTS
observedradialfield.Wechoosethesolarvalueforthesourcesur-
faceat2.5R∗.Thecodeusedtoextrapolatethefieldisamodified 3.1 FieldStructure
versionoftheglobaldiffusionmodeldevelopedbyvanBallegooi-
With the addition of small-scale field B drops more rapidly with
jenetal.(1998).
height,closetothestellarsurface.Assuch,wedonotfindanygreat
The extrapolated 3D small-scale field is shown in Fig. (2),
changeinthelarge-scalefieldstructure.ThisisevidentfromFig.
wherethefieldlinesremainclosedandclosetothestellarsurface.
(3)whichshowstheradialfieldatboththestellarsurfaceandthe
Thisextrapolationdemonstratesthatthesmall-scalefieldproduces
sourcesurface.Fig.(3a)&(3c)whichrepresentthelarge-scaleand
a‘carpet’oflow-lyingloopsacrossthesurface.
large- + small-scale field at the stellar surface respectively, show
verydifferenttopologies;however,whenthisisextrapolatedoutto
thesourcesurface(Fig.(3b)&(3d))thetopologiesaresimilar.We
2.3 X-rayEmissionModel
concludefromthisthatthemagneticpressurefallsoffwithradius
The structure of the magnetic field is influenced by the strength morequicklywiththeadditionofsmall-scalefieldleavingonlythe
of the surface field and the distribution of plasma. Following the large-scalecomponentsnearthesourcesurface.
modelinLangetal.(2012),thedensitystructurecanbeestimated Fig.(4)showsthecoronalmagneticfieldof;(1)thelarge-scale
fortheextrapolatedcoronabyassumingtheplasmaishydrostatic field extrapolated from the reconstructed radial maps (left-hand
and isothermal and that the gas pressure at the stellar surface is column); and (2) the large- + small-scale field, where the small-
proportionaltothemagneticpressure(p =κB2).κisaconstantof scalefieldisscaledaccordingtoReiners&Basri(2009),suchthat
o o
proportionalityrelatingthebasegaspressure p tomagneticpres- B =6%B ,forpartly-convectiveMdwarfsandB =14%B ,
o ls Total ls Total
sureB throughthemagneticconstant2µ.Thevalueofκischosen forfully-convectiveMdwarfs(right-handcolumn).Acomparison
o
suchthatthecoronaldensitiesliewithintheobservedrangeforM oftheextrapolationsinboththeleft-andright-handcolumnsfrom
(cid:13)c 2014RAS,MNRAS000,1–11
4 P.Langetal
(a)Large-scaleradialfieldatstellarsurface (b)Large-scaleradialfieldatsourcesurface
(c)Large-+small-scaleradialfieldatstellarsurface (d)Large-+small-scaleradialfieldatsourcesurface
Figure3.Upper:Large-scale(ZDI)reconstructedradialfieldmapsforGJ49at(a)thestellarsurface,and(b)thesourcesurface.Lower:ZDI+small-scale
radialfieldmapforGJ49at(c)thestellarsurface,and(d)thesourcesurface.Small-scalefieldisscaledaccordingtotheresultsofReiners&Basri(2009)
suchthatBls=6%BTotal,forpartly-convectiveMdwarfs.
Fig.(4)showthelocationofcoronalholeswherethestellarwind For comparison with our previous work (Lang et al. 2012)
isemittedandtheangleofthemagneticdipoleaxisfromtherota- conducted on the field visible only to ZDI, we keep the model
tionpoleremainlargelyunchangedonthemajorityofthestarsin parameters e.g., the temperature (T = 2×106K), source surface
thesample.However,theextrapolationsofDTVir,OTSerandEQ (R =2.5R )andκ=10−6,constant.Keepingκconstantresultsin
ss ∗
PegAshowsmallchangesinthestructureoftheclosedlarge-scale anincreaseinpressureandcoronaldensityduetotheincreasein
field. thesurfaceflux.Thecoronaldensities(showninTable1)nowlie
atthehigherendoftheacceptedrange:109−1012cm−3(Nessetal.
2002,2004),asopposedtotheirpreviousvalueswhichwereatthe
3.2 Activity-RotationRelation lowerend.Thevalueofκcouldbealteredinsuchawaytoreduce
thecoronaldensitiesbacktothevaluescalculatedfortheZDImaps
TheX-rayluminosityhasbeenshowntocorrelatewellwitheither
andinturnreducethemagnitudeoftheX-rayemissionmeasure.
rotationalvelocityorRossbynumberRo(theratioofthestellarro-
tationperiod,P,totheconvectiveturnovertime,τ )(e.g.,Pizzolato ComparisonoftheresultsofFig.(5)andFig.(6)indicatethat
c
et al. 2003a,b; Jeffries et al. 2011); in general, L /L increases theadditionofthesamesmall-scalefieldtoeachstarremovesthe
X bol
and then saturates (L /L ≈ 10−3; (Delfosse et al. 1998)) with activity-rotationrelation;however,scalingthesmall-scalefieldto
increasingrotationratXe.Tbhoilsbehaviourisattributedtocoronalsat- the large-scale, e.g., Bls = 6%BTotal, for the partly-convective M
uration(Vilhu&Walter1987;Staufferetal.1994)andoccursatRo dwarfs and Bls = 14%BTotal, for the fully convective M dwarfs,
≈0.1.ForasampleofMdwarfs,Langetal.(2012))reproducethe recoverstherelation.Thiswouldsuggestthatthesmall-scalefield
saturationoftheX-rayemissionmeasureforthelarge-scalefield hasthesamedependenceonrotationperiodasthelarge-scalefield.
detectedthroughZDI.TheseresultsareshowninFigs.(5)and(6) Inourpreviouswork(Langetal.2012),therotationalmod-
asblacksymbols. ulation of the X-ray emission measure for the stellar sample did
Withtheadditionofsmall-scalefield,thecorrelationbetween not demonstrate any trend with Rossby number and could not be
theX-rayemissionmeasureandRossbynumberchangesdepend- used as an indicator of field topology due to too many contribut-
ingontheamountofsmall-scalefluxadded.Forcase(1),shown ingfactorse.g.,theangleofstellarinclinationandtheangleofthe
inFig.(5),theriseandsaturationoftheX-rayemissionmeasure magneticdipoleaxisfromtherotationpole.Wefindthatwiththe
isnotasprominentwiththeadditionof500Gofsmall-scaleflux additionofsmall-scalefield,bothwiththesamesurfacedistribu-
(red symbols) and is no longer evident for 1000G of small-scale tionandmagnitude,andscaledwithrespecttothelarge-scalefield,
flux(bluesymbols).Thismeansthataddingthesamesurfacedis- thisisstillthecase.
tributionofsmall-scalefieldtoeachstardestroystherelationship With the addition of small-scale field, the magnitude of the
between magnetic flux and Rossby number. If we now consider rotation modulation of the X-ray Emission Measure changes for
case(2)whereB = 6%B ,forthepartly-convectiveMdwarfs eachstarasaresultofchangesinsurfacefieldstrength(SeeTable
ls Total
and B = 14%B ,forthefullyconvectiveMdwarfsshownin (1)). For OT Ser, an early-M Dwarf, the change in the rotational
ls Total
Fig.(6)theriseandsaturationoftheX-rayemissionwithRossby modulationbetweenthelarge-scalefieldandtheadditionofsmall-
numberisonceagainapparent.ThemagnitudeoftheX-rayemis- scalefluxisnearly30%,whereasforGJ49,alsoanearly-Mdwarf,
sionhasincreasedbyapproximately5ordersofmagnitudedueto thechangeinthemodulationis1%.Sincewedonotfindanygreat
theincreaseinflux(Table1). changeinthelarge-scalefieldstructurewiththeadditionofsmall-
(cid:13)c 2014RAS,MNRAS000,1–11
000 5
(a)GJ182(07):0.75M(cid:12); Large-scale (b)Large-+small-scalefield
field
(c) DT Vir (08): 0.59M(cid:12); Large-scale (d)Large-+small-scalefield
field
(e) DS Leo (08): 0.58M(cid:12); Large-scale (f)Large-+small-scalefield
field
(g) GJ 49 (07) : 0.57M(cid:12); Large-scale (h)Large-+small-scalefield
field
Figure4.Column(1)showsthe3DcoronalextrapolationforthereconstructedsurfaceradialmapsforoursampleofMdwarfs.Column(2)istheextrapolation
forcase(2)withthecombinationofthesmall-scalefieldscaledwithrespecttothelarge-scalefielde.g.,Bls =6%BTotal,forthepartly-convectiveMdwarfs
andBls=14%BTotal,forthefullyconvectiveMdwarfs.
(cid:13)c 2014RAS,MNRAS000,1–11
6 P.Langetal
(a) OT Ser (08): 0.55M(cid:12); Large-scale (b)Large-+small-scalefield
field
(c)CEBoo(08):0.48M(cid:12); Large-scale (d)Large-+small-scalefield
field
(e)ADLeo(07):0.42M(cid:12); Large-scale (f)Large-+small-scalefield
field
(g)EQPegA(06):0.39M(cid:12); Large-scale (h)Large-+small-scalefield
field
Figure4–continued
(cid:13)c 2014RAS,MNRAS000,1–11
000 7
(a) EV Lac (06): 0.32M(cid:12); Large-scale (b)Large-+small-scalefield
field
(c)YZCMi(07):0.31M(cid:12); Large-scale (d)Large-+small-scalefield
field
(e)V374Peg(05/06):0.28M(cid:12); Large- (f)Large-+small-scalefield
scalefield
(g)EQPegB(06):0.25M(cid:12); Large-scale (h)Large-+small-scalefield
field
Figure4–continued
(cid:13)c 2014RAS,MNRAS000,1–11
8 P.Langetal
Table1.ResultsforthecoronalpropertiesforthesampleofMdwarfs.Thepredictedvaluesforthelogarithmicemissionmeasure(bothmagnitude,logEM,
androtationalmodulation,RotMod)andlogarithmiccoronaldensity,logne,fortheobservedlarge-scalefield,arefromLangetal.(2012).Thelogarithmic
emissionmeasure(bothmagnitude,logEM,androtationalmodulation,RotMod)andlogarithmiccoronaldensity,logneforthesimulatedlarge-scale+small
scalefieldatBmax=500GandBmax=1000GaswellasforBls=6%BTotalforpartly-convectivestarsandBls=14%BTotalforfullyconvectivestars,arefrom
thiswork.
Large-scale +500G +1000G +%
Star SpType logEM RotMod logne logEM RotMod logne logEM RotMod logne logEM RotMod logne
(cm−3) % (cm−3) cm−3 % (cm−3) cm−3 % (cm−3) cm−3 % (cm−3)
GJ182 M0.5 50.3 12 8.6 51.2 31 9.2 52.1 31 9.8 55.4 25 11.0
DTVir M0.5 50.9 3 9.3 50.4 21 9.1 51.6 10 9.8 54.8 20 11.3
DSLeo M0 48.4 10 7.9 49.9 30 9.1 50.8 43 9.9 53.6 18 10.7
GJ49 M1.5 46.8 18 7.1 50.0 24 9.2 50.9 26 9.8 52.0 19 9.9
OTSer M1.5 50.7 49 9.2 50.8 24 9.3 51.6 13 9.8 54.2 20 11.3
CEBoo M2.5 49.5 2 8.5 50.2 4 9.2 51.3 10 9.7 54.9 20 11.4
ADLeo M3 50.2 3 8.9 50.6 0.4 9.4 50.6 0.4 9.4 54.5 10 11.7
EQPegA M3.5 51.4 51 9.7 51.5 44 9.7 51.8 48 10.0 54.9 27 12.0
EVLac M3.5 52.0 12 10.1 52.1 4 10.2 52.2 10 10.3 55.4 6 12.2
YZCMi M4 50.2 10 10.3 52.4 10 10.4 52.4 9 10.4 55.7 28 12.1
V374Peg M4 52.6 27 10.3 52.9 22 10.4 52.8 24 10.4 56.6 22 12.4
EQPegB M4.5 51.1 14 9.6 51.3 12 9.8 51.6 21 10.1 54.3 22 11.8
Figure 5. X-ray Emission Measure as a function of Rossby number for Figure6.X-rayEmissionMeasureasafunctionofRossbynumberforboth
boththelarge-scalefield(blacksymbols)andthesimulatedsmall-scale+ thelarge-scalefield(blacksymbols)andthesimulatedsmall-scale+large-
lSayrmgeb-oslcsa:leasfiteerldisk(Bsmraexpr=es5e0n0tGparretdlysycmonbvoelcst,ivBemadxw=ar1fs0,00MG>blu0e.4syMm(cid:12)b,oalns)d. sdcwaalerffis,elMd(>pu0rp.4leMs(cid:12)y,mabnodlsd)i.amSyomnbdorles:praessteernitskfuslrlyepcroensevnetctpivaretldywcaornfvs,ecMtiv(cid:54)e
diamondrepresentfullyconvectivedwarfs,M(cid:54)0.4M(cid:12). 0.4M(cid:12).
maystillbeacarpetofsmall-scalefield,whichcouldcontributeto
scalefield,thechangeinthemagnitudeoftherotationalmodulation
poweringthestellarwind(e.g.,Nishizukaetal.(2011)).
couldbearesultofthesmall-scalefieldproducinglow-lyingsmall
Thestellarwindisresponsibleforangularmomentumlossand
closedfieldregionswhichcarpetthestellarsurfaceincludingthe
influencesthestellarspindowntime.Toinvestigatetheeffectthe
areaswherethelarge-scalefieldisopen.
small-scalefieldhasontheoverallcoronalstructure,weexamine
bothitsinfluenceonthetotalmagneticfluxatthesurfaceofthestar
andalsothetotalopenflux.Weanalysethegeometryofthefield
3.3 OpenFluxandSpinDown
bypredictingandcomparingtheopenfluxtoobservedsurfaceflux
CoronalstructureisimportantasitdeterminestheX-rayemission valuesobtainedfromtheoverallcombinationofland m modes,
fromregionsofclosedmagneticfieldbutalsoareaswherethemag- (cid:82)
Φ R2 |B(R ,θ,φ)|dΩ
nlaertgice-fiscealdleismoapgenneatincdfitehledss,tetlhlaerrwaningdefoofrmfiesl.dFsotrresntagrtshwspitrhesweenatkoenr ΦSOurpfeance = Rs2∗s(cid:82) |Brr(R∗ss,θ,φ)|dΩ , (10)
thestarhasbeenalteredbytheadditionofsmall-scalefield.This
haslittleeffectonthegeometryofthelarge-scalefield(Fig.4).The whereΩisthesolidangle.Wenotethisgivesalowerlimittothe
trueopenfluxasonsomefieldlinesthegaspressuremayexceed
dipoleaxisandthelocationandextentoftheopenfieldregionsare
themagneticpressure.
largelyunchanged.Thisistobeexpectedasthesmall-scalefieldis
Addinginsmall-scalefieldincreasesthesurfaceflux.Fig.(7a)
distributedaxisymmetricallyoverthesurfacesoithasnopreferred
showsthisincreaseexpressedas
direction.Thissuggeststhatthelatitudesfromwhichastellarwind
couldbelaunchedwouldnotbeaffectedbythepresenceofsmall- ∆φ = (φSurface)large+small −1 . (11)
scalefield.Withinregionswherethelarge-scalefieldisopenthere Surface (φ )
Surface large
(cid:13)c 2014RAS,MNRAS000,1–11
000 9
a) b)
Figure7.Thepercentagechangein(a)thesurfacefluxand(b)theopenfluxasafunctionoftheobservedsurfaceflux,duetotheadditionofsmall-scalefield.
SymbolsareasofFig5.
andonlyasinglefluxestimateispossible.Assumingthatallofthe
surfacefluxiscontainedinonesinglemode,forexampleadipole,
canhoweverleadtoanoverestimateoftheamountofopenflux.As
discussedinLangetal.(2012),theopenfluxforanysinglemode
issimplyrelatedtothesurfacefluxas:
Φ (2l+1)(Rss)l+1
Open = R∗ . (13)
ΦSurface l+(l+1)(RRs∗s)2l+1
Figure (8) shows that when small-scale field is added there is an
increaseinsurfacefluxbutthepredictedopenfluxisstillatleastan
orderofmagnitudesmallerthanitwouldbehadweonlyconsidered
thedipolemodes.Therefore,theangularmomentumloss,J˙,dueto
thestellarwind,whichisdeterminedbytheamountofopenflux
i.e.aWeber-Davismodel(Weber&Davis1967),givenby
J˙∝Φ2 , (14)
Open
Figure8.Comparisonofthemagnitudeoftheminimumpredictedopen
is influenced by the topology of the field and would be overesti-
fluxasafunctionoftheobservedsurfacefluxwithandwithoutsmall-scale
mated by at least 2 orders of magnitude. This is also true for the
field.Thedashedlineshowsthepredictedopenfluxofapuredipole.Four
starswhichspanthespectralrangeofoursamplehavebeenchosentoshow masslossrate,
thatwhenproducingamodelforthestellarcoronaarangeoflandmmodes M˙ ∝Φ , (15)
mustbeconsideredtoreproducethecorrectcoronalstructure.Symbolsare Open
asofFig5. whichcouldbeoverestimatedbyanorderofmagnitudeifthetopol-
ogyisover-simplified.Weconcludefromthisresultthatwhenpro-
ducingamodelforthestellarcorona,orthestellarwind,arangeof
Thefractionalincreaseinsurfacefluxisclearlygreatestforthose
landmmodesmustbeconsideredtoreproducethecorrect,more
starswhoselarge-scalesurfacefluxislowest.
complex,coronalstructure.
The addition of the small-scale field also results in a slight
change(10%to40%)intheopenflux(Fig.7b)butshowsnopref-
erence between partly-convective and fully-convective stars.. The
4 SUMMARY
fractionfortheopenfluxisgivenby
We have created a model for small-scale field using synthesised
∆φ = (φOpen)large+small −1 . (12) spot distribution maps. We allocate field strengths in a Gaussian
Open (φ )
Open large distribution from the centre of the spot by either (1) fixing the
This result is in keeping with the increase in surface flux, value of B to be either ±500G or ±1000G; or (2) setting the
max
whichwouldincreasethemagneticpressure(Eq.5).Wenotethat value of B such that the large-scale field contributes only 6%
max
the magnitude of the open flux is dependent on the chosen value of the total field for partly-convective M dwarfs and 14% of the
forthesourcesurface(asR =→∞,Φ →0)buttheeffectof total field for fully-convective M dwarfs, as indicated in Reiners
ss Open
addinginsmall-scalefieldisthesameforallvaluesofthesource &Basri(2009).Wehaveincorporatedtheradialsurfacemappro-
surface.Asthereislittlechangeintheopenfluxwiththeaddition duced by this model into the reconstructed maps of the observed
ofsmall-scalefield,wewouldnotexpecttheangularmomentumor radial magnetic field at the stellar surface for a sample of early-
masslosstobesignificantlyaffected. to-midMdwarfsandextrapolatedtheir3Dcoronalmagneticfield
For many stars a full surface magnetic map is not available usingthePotentialFieldSourceSurfacemethod.
(cid:13)c 2014RAS,MNRAS000,1–11
10 P.Langetal
Table2.Valuesformass,radius,Rossbynumberand(cid:104)BV(cid:105)(theaveragelarge-scalemagneticfluxderivedfromspectropolarimetricmeasurements),arefrom
Donatietal.(2008);Morinetal.(2008).Valuesof (cid:104)BV(cid:105) forGJ182,DTVir,CeBoo,ADLeo,EVLacandYZCMiarefromReiners&Basri(2009),while
(cid:104)BI(cid:105) (cid:68) (cid:69)
valuesforDSLeo,GJ49,OTSerEQPegA,V374Peg,andEQPegBareestimates(depictedbye)basedonReiners&Basri(2009).Valuesfor Bls
r
(theaveragelarge-scale[ls]radialflux),(cid:104)Bss(cid:105)(theaveragesmall-scale[ss]flux,scaledwithrespecttothelarge-scalefield)and(cid:104)Bls+ss(cid:105)(theaveragelarge-+
small-scaleflux)aswellasφOpen(theopenflux)andφSurface(thesurfaceflux)valuesarefromthiswork.
500G 1000G
Star Mass Ro (cid:104)BV(cid:105) (cid:68)Blrs(cid:69) (cid:68)Blrs+ss(cid:69) (cid:68)Blrs+ss(cid:69) (cid:104)(cid:104)BBVI(cid:105)(cid:105) (cid:104)Bss(cid:105) (cid:68)BlRs+ss(cid:69) |βlMs| |βlMs| φlOspen φlOs+pesns φlSsurface φlSsu+rsfsace
(M(cid:12)) (10−2) (kG) (kG) (kG) (kG) (%) (kG) (kG) (◦) (◦) 1023Mx 1023Mx 1025Mx 1025Mx
GJ182 (07) 0.75 17.4 0.17 0.10 0.19 0.21 6 1.8 1.8 41.1 41.4 2.9 3.3 3.0 4.6
(07) 0.59 9.2 0.15 0.08 0.18 0.20 5 1.5 1.5 83.6 84.0 1.0 1.2 0.1 7.4
DTVir
(08) - - 0.15 0.12 0.21 0.22 5 2.3 2.3 20.0 13.0 0.5 1.2 0.1 2.3
(07) 0.58 43.8 0.10 0.04 0.16 0.18 5e 0.75 0.76 41.1 38.8 0.5 0.3 0.05 0.8
DSLeo
(08) - - 0.9 0.03 0.16 0.18 5e 0.60 0.3 42.9 42.6 0.3 0.3 0.04 0.6
GJ49 (07) 0.57 56.4 0.3 0.02 0.15 0.18 6e 0.30 0.30 10.9 2.3 0.30 0.30 0.03 0.30
OTSer (08) 0.55 9.70 0.14 0.13 0.19 0.22 6e 1.9 0.8 12.1 12.8 1.0 4.8 0.09 4.0
CEBoo (08) 0.48 35.0 0.10 0.12 0.20 0.23 6 1.8 1.8 7.4 6.3 1.2 1.3 0.1 1.3
(07) 0.42 4.7 0.19 0.21 0.27 0.30 7 2.8 2.9 4.5 4.5 1.5 1.6 0.1 1.6
ADLeo
(08) - - 0.18 0.21 0.27 0.29 7 2.8 2.9 7.3 7.3 1.5 1.6 0.1 1.6
EQPegA (06) 0.39 2.0 0.48 0.39 0.42 0.44 10e 3.8 3.8 25.9 25.9 2.4 1.5 0.2 1.7
(06) 0.32 6.8 0.57 0.63 0.66 0.67 13 3.8 3.9 45.8 45.8 2.8 2.4 0.2 1.3
EVLac
(07) - - 0.49 0.57 0.59 0.61 13 3.8 3.9 43.2 43.3 2.6 1.7 0.2 0.96
(07) 0.31 4.2 0.56 0.73 0.74 0.76 14 3.8 3.9 24.8 24.8 3.1 2.4 0.3 1.3
YZCMi
(08) - - 0.55 0.66 0.69 0.70 14 3.9 4.0 12.1 11.1 3.0 1.8 0.3 0.92
V374Peg (05) 0.28 0.6 0.78 1.0 1.1 1.1 14e 5.9 6.0 9.3 8.0 4.4 2.0 0.4 1.7
EQPegB (06) 0.25 0.5 0.45 0.49 0.51 0.53 14e 3.3 3.4 6.5 6.4 1.7 1.0 0.1 0.8
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ACKNOWLEDGEMENTS
P., Jardine M. M., Collier Cameron A., Albert L., Manset N.,
PLacknowledgessupportfromanSTFCstudentship.JM,AVand DintransB.,ChabrierG.,ValentiJ.A.,2008,MNRAS,384,77
RF acknowledge support from fellowships of the Alexander von MorinJ.,DonatiJ.-F.,PetitP.,DelfosseX.,ForveilleT.,AlbertL.,
Humboldtfoundation,theRoyalAstronomicalSocietyandSTFC, AurièreM.,CabanacR.,DintransB.,FaresR.,GastineT.,Jar-
respectively.Theauthorswouldliketothanktherefereeforapar- dineM.M.,LignièresF.,PaletouF.,RamirezVelezJ.C.,Théado
ticularlythoroughanddetailedreport. S.,2008,MNRAS,390,567
(cid:13)c 2014RAS,MNRAS000,1–11