Mem. S.A.It. Vol. 78, 271c© SAIt 2007 Memorie della

Magnetic flux emergence in fast rotating stars

V. Holzwarth

Max-Planck-Institut fur Sonnensystemforschung, Max-Planck-Str. 2, 37191 Katlenburg-Lindau, Germany; e-mail: holzwarth@mps.mpg.de

Abstract. Fast rotating cool stars are characterised by high magnetic activity levels andfrequently show dark spots up to polar latitudes. Their distinctive surface distributions ofmagnetic flux are investigated in the context of the solar-stellar connection by applying thesolar flux eruption and surface flux transport models to stars with different rotation rates,mass, and evolutionary stage. The rise of magnetic flux tubes through the convection zone isprimarily buoyancy-driven, though their evolution can be strongly affected by the Coriolisforce. The poleward deflection of the tube’s trajectory increases with the stellar rotation rate,which provides an explanation for magnetic flux eruption at high latitudes. The formationof proper polar spots likely requires the assistance of meridional flows both before and afterthe eruption of magnetic flux on the stellar surface. Since small radiative cores support theeruption of flux tubes at high latitudes, low-mass pre-main sequence stars are predicted toshow high mean latitudes of flux emergence. In addition to flux eruption at high latitudes,main sequence components of close binary systems show spot distributions which are non-uniform in longitude. Yet these ‘preferred longitudes’ of flux eruption are expected to vanishbeyond a certain post-main sequence evolutionary stage.

Key words. stars: magnetic activity – stars: rotation – stars: pre-main sequence – binaries:close – magnetohydrodynamics (MHD)

1. Introduction

The magnetic activity of cool stars has a cru-cial impact on their rotational evolution andon their appearance in different wavelengthranges. Detailed observations of solar activ-ity phenomena lead to models for the cyclicre-generation of magnetic fields through self-sustained dynamo processes inside the convec-tion zone, which are based on the interaction ofconvective motions and (differential) rotation(??,and references therein). Yet a comprehen-sive understanding of the sub-surface origin ofmagnetic flux is only possible when the het-

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erogeneity of other cool stars is taken into ac-count as well, since the large ranges of stellarrotation rates, stellar masses, and evolutionarystages provide the testbed required to verify (orfalsify) current theories.

Photometric and spectro-polarimetric ob-servations allow for the reconstruction ofstellar surface brightness and surface mag-netic field distributions (e.g. ??,and referencestherein). The surface maps frequently showdistinctive differences compared to the solarcase like huge spots at high latitudes. In thecontext of the solar-stellar connection, thesecharacteristic phenomena are investigated inthe framework of solar flux transport mod-els. The present review focuses on the storage,

272 V. Holzwarth: Magnetic flux emergence in fast rotating stars

transport, and eruption of magnetic flux in theconvective envelope of fast rotating cool stars.

2. Magnetic flux of fast rotating stars

Zeeman-broadening of magnetic sensitivelines provides a measure for the surface-averaged magnetic flux (flux density times fill-ing factor) of cool stars, which follows roughlya power law, Φ ∝ ΩnΦ , with a rate of increasenΦ ' 1.2 (?); Ω is the stellar rotation rate.Excluding targets which might be in the regimeof saturated dynamo operation, ?) suggest ahigher value, nΦ ' 2.8. In contrast, observa-tions of young open stellar clusters of differentage indicate a spin-down of stellar rotation dueto magnetic braking, which is consistent with alinear increase of the open magnetic flux (e.g.?,and references therein). The combination ofan empirical relationship between unsignedmagnetic flux and coronal X-ray emission (?)with empirical activity-rotation-relations (e.g.?) implies an intermediate value nΦ ∼ 2.

In the course of its 11-year activity cy-cle, the total spot coverage of the Sun mayreach 0.5% of the visible hemisphere. In con-trast, the spot coverage of rapidly rotating starscan be over two orders of magnitude larger(?). Whereas sunspots appear within an equa-torial belt between about ±35o, fast rotatingstars frequently show huge spots at high andpolar latitudes as well (e.g. ?????; see also ?,and references therein). Furthermore, spectro-polarimetric Zeeman-Doppler imaging obser-vations indicate that the magnetic field patternof rapidly rotating stars is characterised by asignificant mixture of polarities, which is incontrast to the unipolar field around the solarpoles (?).

3. The Solar Paradigm

The magnetic field permeating the solar atmo-sphere is expected to originate from the bottomof the convection zone. The field is amplified inthe tachocline and stored in the stably stratifiedovershoot region at the interface to the radia-tive core (??). When the field strength is largerthan a critical value, perturbations lead to the

Fig. 1. Rise of a magnetic flux loop inside the con-vection zone, from the onset of the instability (left)to its eruption at the stellar surface (right). The fluxtube radius is shown 5× magnified.

formation of rising flux loops (Fig. 1), whicheventually emerge at the surface (???).

After the dynamical disconnection from itssub-surface roots (??), the surface magneticflux feature follows the differential rotation andthe meridional flow in the photosphere. Duringits transport toward the pole, surface magneticflux merges and annihilates with ambient fluxfeatures and eventually dissolves through dif-fusion and through the convective turnover ofsupergranular motions (???).

The decapitated flux tube below the sur-face disintegrates in magneto-convective mo-tions and may be transported by meridionalcirculations and through convective pumpingback to the bottom of the convection zone forrecurrent amplification (?).

Calculations based on the solar paradigmand carried out in the framework of the thinflux tube approximation (?) predict criticalfield strengths, eruption latitudes, tilt angles,and proper motions of spots which are in agree-ment with observed properties of emergingbipolar spot groups on the Sun (????).

4. Flux tubes in fast rotating stars

Theoretical investigations of magnetic fluxeruption in cool stars are based on analyses ofthe equilibrium, stability, and rise of flux tubesfor different stellar rotation rates, masses, andevolutionary stages.

Equilibrium properties The magnetic fluxtubes are assumed to be initially situated in-side the overshoot region, stored in mechani-

V. Holzwarth: Magnetic flux emergence in fast rotating stars 273

curvature force

equator

buoyancy flowmeridional

pole

drag

Coriolis force

Fig. 2. Force balance of a magnetic flux ring in me-chanical equilibrium in the absence (black arrows)and in the presence (gray arrows) of an equatorwardmeridional flow.

cal equilibrium parallel to the equatorial plane(??). The flux ring is in pressure equilib-rium with its environment and, in the absenceof meridional circulations, non-buoyant. Themagnetic tension force pointing toward theaxis of rotation is balanced by the Coriolisforce (Fig. 2), which is caused by an internalprograde flow along the flux ring.

A relative motion between a flux tube andits environment perpendicular to the tube’s axisgives rise to a hydrodynamic drag. In the pres-ence of an equatorward meridional flow, the re-sulting drag is balanced by assuming a buoyantflux tube with a somewhat lower internal flowvelocity (??)

Stability properties In the case of the Sun,magnetic flux tubes in the overshoot region are(linearly) stable against perturbations for fieldstrengths . 105 G (??). Beyond that criticalvalue, buoyancy-driven instabilities (?) lead tothe onset of rising flux loops with character-istic growth times of less then a few hundreddays. Fast stellar rotation increases the stabilityof a flux ring, since its enhanced angular mo-mentum hampers perturbations perpendicularto the rotation axis. For otherwise unchangedequilibrium conditions, higher field strengthsare required to obtain buoyancy-driven insta-bilities with comparable growth times (Fig. 3).

10 100magnetic field strength B [104 G]

0

20

40

60

80

latit

ude

λ [o ]

5d10d15d20d25d

Fig. 3. Buoyancy-driven instability with a growthtime of 100 d for pre-main sequence stars (M =

1 M,R = 1.4 R, t = 4.7 Myr) with different ro-tation periods.

Eruption properties The eruption latitudeof magnetic flux loops is mainly determinedby the ratio between magnetic buoyancy andCoriolis force. Magnetic buoyancy is sustainedthrough a net downflow of plasma inside theflux tube from its crest into the lower segmentsremaining in the overshoot region. The down-flow and the density contrast increase with themagnetic field strength. If the rise is dominatedby magnetic buoyancy, the trajectory will beradial and the eruption latitude similar to theinitial latitude of the flux ring in the overshootregion. The azimuthal flow velocity of plasmawithin a rising flux loop decreases, owing tothe (quasi-)conservation of angular momen-tum. The associated decrease of the Coriolisforce reduces the outward directed net forceperpendicular to the rotation axis and entailsa deflection of the loop’s trajectory to higherlatitudes. The dependence of the Coriolis forceon the stellar rotation rate causes the polewarddeflection (of flux tubes with comparable fieldstrength and equilibrium position) to be largerin more rapidly rotating stars (Fig. 4, left).

The deflection mechanism applies to eachtube segment as well. The downflow velocityin the leading leg (relative to the direction ofrotation) is larger than the upflow velocity inthe following leg, so that the former is lessdeflected toward the pole than the latter. Theasymmetric deflection of the two legs causes atwist of the rising loop and a tilt of the emerg-

274 V. Holzwarth: Magnetic flux emergence in fast rotating stars

Fig. 4. Poleward deflection (left) and tilt angle(right) of erupting flux tubes. All flux tubes startwith the same initial conditions (B0 = 2 ·105 G, λ0 =

5o, r0 = 5.07 · 1010 cm).

5 10 15 20 25rotation period, P [d]

200

400

600

800

1000

1200

erup

tion

time,

t [d

]

Fig. 5. Eruption times of magnetic flux tubes. Thesolar-like stellar structure and initial equilibriumconditions (B0 = 2 · 105 G, λ0 = 5o, r = 5.07 ·1010 cm) are the same for all flux tubes.

ing bipolar spot group at the surface with re-spect to the East-West direction (Fig. 4, right).

Both the poleward deflection and the tiltangle of emerging bipoles depend on the ratiobetween buoyancy and Coriolis force, i.e. erup-tion timescale and rotation period, respectively.The eruption times of magnetic flux tubes withthe same initial field strength and equilibriumposition increase with the rotation rate (Fig. 5),which confirms the influence of the enhancedangular momentum on the sub-surface evolu-tion of magnetic flux indicated by the linearstability analysis.

5. Formation of polar spots

Poleward deflection In fast rotating stars,the magnetic field strengths required for the

0.0 0.2 0.4 0.6 0.8 1.0x/R*

0.0

0.2

0.4

0.6

0.8

1.0

y/R

*

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4x/R*

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

y/R

*

Fig. 6. Flux loop trajectories in a 1 M-main se-quence star (left) and a pre-main sequence star(right) of age 4.7 Myr. The initial field strengths areB0 = 22·104 G in the former case and B0 = 30·104 Gin the latter; the stellar rotation period is P = 6 d.

onset of buoyancy-driven instabilities are sig-nificantly higher than in the solar case, whichwould imply a dominance of magnetic buoy-ancy and radial trajectories. Yet higher ini-tial field strengths entail higher internal flowvelocities to achieve mechanical equilibrium,which in conjunction with the large rotationrate increase the Coriolis force considerably.The resulting strong poleward deflection of ris-ing flux loops (Fig. 6, left) provides an expla-nation for the occurrence of flux eruption athigh latitudes (??).

Meridional circulation Since the polewarddeflection decreases for larger initial latitudes,the formation of polar spots through bona fideflux eruption would require the presence (andpossibly generation) of large amounts of mag-netic flux at very high latitudes in the lowerpart of the convection zone. Although this pos-sibility can a priori not be ruled out, it is morelikely that the formation of polar spots is sup-ported by an additional poleward transport ofmagnetic flux through meridional flows.

The attempt to simulate the formation ofpolar spots on the basis of a solar surface fluxtransport model with a 30× larger flux emer-gence rate generates unipolar magnetic flux atthe poles (Fig. 7, left), which disagrees with themixture of polarities observed on rapidly rotat-ing stars (??). The additional assumption of alarger latitudinal range of flux emergence anda fast poleward meridional flow yields an in-termingling of polarities (Fig. 7, right), which

V. Holzwarth: Magnetic flux emergence in fast rotating stars 275

Fig. 7. Surface distributions of the radial magneticfield, assuming solar-like surface transport proper-ties. The assumption of a 30 times solar flux erup-tion rate (left) yields a unipolar field at high lati-tudes, whereas the additional assumption of a largerlatitudinal range of flux eruption and a fast pole-ward meridional flow yields a mixture of polarities(right). From ?).

0.0 0.2 0.4 0.6 0.8 1.0x/R*

0.0

0.2

0.4

0.6

0.8

1.0

y/R

*

10 m/s 75 m/s100 m/s

Fig. 8. Trajectories of rising flux loops with ini-tial field strength 15 · 104 G and initial tube radius100 km. The stellar rotation period is 6 d. The merid-ional flow is poleward at the surface and equator-ward at the bottom of the convection zone. From ?).

is in qualitative agreement with observations(?). The assumed meridional flow velocities(& 100 m/s) are significantly higher than in thesolar case (11 m/s).

Strong meridional circulations increase thepre-eruptive poleward deflection of rising fluxtubes (Fig. 8). The deflection is strongest forflux tubes originating from mid latitudes withlow field strengths and small cross sections,

Fig. 9. Stellar butterfly diagram in the presence ofa meridional circulation with 100 m/s flow velocity.From ?).

which renders the wings of predicted stellarbutterfly diagrams distinctively convex (Fig.9). The enhanced pre-eruptive poleward deflec-tion explains the required larger range of fluxemergence latitudes (?).

Influence of stellar structure The marginalcase of poleward deflection corresponds to therise of a flux ring parallel to the stellar ro-tation axis (e.g. ?), which enables flux emer-gence at high latitudes, depending on the rela-tive size of the radiative core. This dependenceon the stellar structure makes rapidly rotatingpre-main sequence stars optimal candidates forpolar spots (??).

The poleward deflection is supported bythe stratification of the convection zone ofyoung stars, which is characterised by largerpressure scales heights and lower superadia-baticities than in a main sequence star (Fig.10). The smaller magnetic buoyancy enhancesthe influence of the Coriolis force, so that thepoleward deflection is increased (Fig. 6, right).The larger stellar radii and deeper convectionzones of pre-main sequence stars also implylonger rise times, during which the polewarddeflection cumulates to higher eruption lati-tudes.

6. Distributions of flux eruption

Young stars Given the influence of the stel-lar structure and stratification on the formation

276 V. Holzwarth: Magnetic flux emergence in fast rotating stars

0.6 0.8 1.0 1.2 1.4radius r [RO •]

107

108

109

1010

1011

pres

sure

sca

le h

eigh

t Hp

[cm

] (so

lid)

10-10

10-8

10-6

10-4

10-2

100

supe

radi

abat

icity

|δ| (

dash

ed)

Fig. 10. Pressure scale height (solid) and superadi-abaticity, δ = ∇ − ∇ad, in the convection zone of a4.7 Myr old pre-main sequence star (thick lines) anda solar-like main sequence star (thin lines); to theleft of the vertical dotted lines the stratification issubadiabatic (i.e. δ < 0).

Fig. 11. Latitudinal probability distributions of fluxeruption in stars of different stellar mass on thezero-age main sequence (top) and shortly after theHayashi phase (bottom). From ?).

of polar spots, the mean latitude of magneticflux eruption is predicted to increase for starsof lower mass and of earlier evolutionary stage(??). Comparing pre-main sequence stars ofsimilar evolutionary stage, latitudinal probabil-ity distributions based on simulations of erupt-ing flux tubes show that for lower mass starsthe mean flux eruption latitude is very high al-ready for rotation periods of a week, whereasfor higher mass stars flux emergence can be ex-pected to proceed at low and intermediate lati-tudes for rotation rates in the saturated regime(Fig. 11, top). For even earlier evolutionarystages, magnetic flux emergence can still occur

Fig. 12. Surface distributions of erupting flux tubesin a main sequence component (left) and a post-mainsequence component (right) of a binary system withtwo 1 M-stars and a 2 d rotation period; the crossmarks the direction to the companion star. From ?).

down to low latitudes through a different erup-tion mechanism (Fig. 11, bottom). When therelative size of the radiative core is very small,unstable magnetic flux rings slip at the bottomof the convection zone to the pole and detachfrom the overshoot region. If the instability isaxial symmetric, the flux ring can rise along therotation axis and erupt at the pole; if the insta-bility is non-axisymmetric, the flux tube driftstoward the equator and eventually emerges atlower latitudes (?). The predicted eruption ofmagnetic flux down to low latitudes even in thecase of rapid rotation is in agreement with ob-servations.

Binary stars Close binaries with cool stel-lar components like RS CVn- or BY Dra-systems show strong magnetic activity, sincetidal interactions maintain high rotation ratesagainst magnetic braking. The presence of thecompanion star gives rise to tidal forces anda deformation of the stellar structure, which inthe lowest order of approximation is π-periodicin longitude. The tidal effects modify the equi-librium, stability, and eruption properties ofmagnetic flux tubes (??). Albeit the tidal per-turbations are rather small, their resonant in-teraction with double-looped (i.e. roughly π-periodic) flux tubes result in considerable non-uniformities in the surface probability distri-butions of magnetic flux eruption (Fig. 12,left). The orientation of the resulting π-periodic‘preferred longitudes’ of flux eruption depends

V. Holzwarth: Magnetic flux emergence in fast rotating stars 277

on the initial field strength and latitude of theoriginal flux ring in the overshoot region.

Beyond a certain post-main sequence evo-lutionary stage the different stellar structureand stratification change the stability proper-ties of flux rings. Double-loop flux tubes are nolonger the dominant mode of flux eruption butsuperseded by single-loop flux tubes (?). Sincethe latter are incongruent with the π-periodicityof the tidal interaction, they are less suscep-tible to the presence of the companion star.The resulting surface probability distributionis almost axial symmetric, showing hardly anysigns of preferred longitudes (Fig. 12, right).Owing to the rapid rotation, flux eruption re-mains to occur at high latitudes.

7. Summary and Conclusion

High-latitude spots on fast rotating stars are as-cribed to the combined pre-eruptive and post-eruptive poleward transport of magnetic fluxoriginating from the bottom of the convec-tion zone. The pre-eruptive poleward deflec-tion and tilt of emerging spot groups is mainlydetermined through the ratio between mag-netic buoyancy and Coriolis force, which de-

pend on the stellar structure and rotation rate,respectively. Since the deflection typically in-creases with increasing stellar rotation rate anddecreasing size of the radiative core, rapidlyrotating pre-main sequence stars are expectedto have high mean latitudes of flux eruption,though low latitude spots are still possible.

The identification of characteristic activ-ity features on fast rotating stars may providethe key for our understanding of essential dy-namo processes. Solar activity models describ-ing the pre-eruptive and post-eruptive transportof magnetic flux have demonstrated their ap-plicability in the investigations of stellar activ-ity signatures and their dependence on stellarstructure and rotation rate. Yet there are ac-tivity phenomena unaccounted for by the cur-rent flux eruption model, for example, regard-ing the high activity levels of fully convectivelow-mass stars (e.g. ?) or the tentative signs forpreferred longitudes on some single stars in-cluding the ‘flip-flop’ phenomenon (???).

Acknowledgements. The author thanks the organis-ers for the invitation to present this paper, and Drs.T. Granzer and D. H. Mackay for providing severalimages.