Field linkage and magnetic helicity density
K. Lund, M. Jardine, A. J. B. Russell, J.-F. Donati, R. Fares, C. P. Folsom
MMNRAS , 1–8 (2020) Preprint 23 February 2021 Compiled using MNRAS L A TEX style file v3.0
Field linkage and magnetic helicity density
K. Lund , M. Jardine , A. J. B. Russell , J.-F. Donati , R. Fares , C. P. Folsom ,S. V. Jeffers , S. C. Marsden , J. Morin , P. Petit and V. See SUPA, School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews, KY16 9SS, UK School of Science & Engineering, University of Dundee, Nethergate, Dundee DD1 4HN, UK IRAP, Université de Toulouse, CNRS, UPS, CNES, 14 Avenue Edouard Belin, 31400, Toulouse, France Physics Department, United Arab Emirates University, P.O. Box 15551, Al-Ain, United Arab Emirates Institut für Astrophysik, Universität Göttingen, Friedrich-Hund-Platz 1, D-37077 Göttingen, Germany University of Southern Queensland, Centre for Astrophysics, Toowoomba, QLD, 4350, Australia LUPM, Université de Montpellier, CNRS, Place Eugène Bataillon, F-34095 Montpellier, France University of Exeter, Department of Physics & Astronomy, Stocker Road, Devon, Exeter, EX4 4QL, UK
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
The helicity of a magnetic field is a fundamental property that is conserved in ideal MHD. It can be explored in the stellar contextby mapping large-scale magnetic fields across stellar surfaces using Zeeman-Doppler imaging. A recent study of 51 stars in themass range 0.1-1.34 M (cid:12) showed that the photospheric magnetic helicity density follows a single power law when plotted againstthe toroidal field energy, but splits into two branches when plotted against the poloidal field energy. These two branches dividestars above and below ∼ (cid:12) . We present here a novel method of visualising the helicity density in terms of the linkage of thetoroidal and poloidal fields that are mapped across the stellar surface. This approach allows us to classify the field linkages thatprovide the helicity density for stars of different masses and rotation rates. We find that stars on the lower-mass branch tend tohave toroidal fields that are non-axisymmetric and so link through regions of positive and negative poloidal field. A lower-massstar may have the same helicity density as a higher-mass star, despite having a stronger poloidal field. Lower-mass stars aretherefore less efficient at generating large-scale helicity. Key words: stars: magnetic field – methods: analytical
Magnetic helicity is a fundamental property of magnetic fields thatmeasures the amount of linkage and twist of field lines within agiven volume. Since it is exactly conserved in ideal MHD and highlyconserved for high magnetic Reynolds numbers in general (Woltjer1958; Taylor 1974), helicity is an important factor when attemptingto understand how magnetic fields are generated and evolve (e.g.Brandenburg & Subramanian 2005; Chatterjee et al. 2011; Pipinet al. 2019). Until recently, this could only be measured for the Sun(e.g. reviews by Démoulin 2007; Démoulin & Pariat 2009). We can,however, now map all three components of the large-scale magneticfield at the surfaces of stars using the spectropolarimetric techniqueof Zeeman-Doppler imaging (Semel 1989).These magnetic field maps now exist for a large enough sample ofstars that trends with stellar mass and rotation period have becomeapparent (Donati & Landstreet 2009). In particular, it appears thatmagnetic fields show different strengths and topologies in the massranges above and below ∼ (cid:12) , which is believed to correspond tothe onset of the transition from partially to fully convective interiors.Rapidly-rotating stars in the mass range above ∼ (cid:12) tend tohave fields that are predominantly toroidal (Donati et al. 2008a). Thestronger the toroidal field, the more likely it is to be axisymmetric(See et al. 2015). In the mass range below ∼ (cid:12) , stars show predominantly axisymmetric poloidal fields. For the lowest masses,however, a bimodal behaviour is found, such that stars may havestrong, predominantly axisymmetric poloidal fields, or much weaker,non-axisymmetric poloidal fields (Donati et al. 2008a; Morin et al.2008b; Donati & Landstreet 2009; Morin et al. 2010).This difference in magnetic fields in stars that are partially or fullyconvective is also apparent in their photospheric helicity densities.Using observations of 51 stars, Lund et al. (2020) found that thehelicity density scales with the toroidal energy according to |(cid:104) ℎ (cid:105)|∝ (cid:104) B tor2 (cid:105) . ± . . The scaling with the poloidal energy is morecomplex, however, revealing two groups with different behaviours.Specifically, stars less massive than ∼ (cid:12) appear to have anexcess of poloidal energy when compared to more massive stars withsimilar helicity densities. It appears that stars with different internalstructures and different total magnetic energies may nonetheless gen-erate magnetic fields with the same helicity density at their surfaces.The aim of this paper is to explore the nature of this division and thetypes of flux linkage that support the measured helicity densities. Inorder to do that, we have developed a novel method of visualising thelinkages of different field components across the surfaces of stars. © a r X i v : . [ a s t r o - ph . S R ] F e b K. Lund et al.
For the purposes of this paper the stellar magnetic fields discussedwill be decomposed into their poloidal and toroidal components: 𝑩 = 𝑩 pol + 𝑩 tor . The poloidal and toroidal fields can be expressedin a general form as (see Appendix III of Chandrasekhar 1961): 𝑩 pol = ∇ × [ ∇ × [ Φ ˆ 𝑟 ]] , (1) 𝑩 tor = ∇ × [ Ψ ˆ 𝑟 ] . (2)In a spherical coordinate system the scalars Φ and Ψ take the form: Φ = 𝑆 ( 𝑟 ) 𝑐 𝑙𝑚 𝑃 𝑙𝑚 𝑒 𝑖𝑚𝜙 , (3) Ψ = 𝑇 ( 𝑟 ) 𝑐 𝑙𝑚 𝑃 𝑙𝑚 𝑒 𝑖𝑚𝜙 . (4) 𝑃 𝑙𝑚 ≡ 𝑃 𝑙𝑚 ( cos 𝜃 ) is the associated Legendre polynomial of mode 𝑙 and order 𝑚 and 𝑐 𝑙𝑚 ≡ √︄ 𝑙 + 𝜋 ( 𝑙 − 𝑚 ) ! ( 𝑙 + 𝑚 ) ! (5)is a normalisation constant. 𝑆 ( 𝑟 ) and 𝑇 ( 𝑟 ) are functions describingthe radial behaviour of the magnetic field components. Determiningthe complete form of these functions from observations of stellarmagnetic fields is impossible, however values can be obtained atstellar surfaces ( 𝑟 = 𝑅 ★ ).The Zeeman-Doppler imaging technique (Semel 1989) describesthe large-scale (low 𝑙 modes) magnetic fields at the surfaces of starsin terms of 𝛼 𝑙𝑚 , 𝛽 𝑙𝑚 , and 𝛾 𝑙𝑚 coefficients (e.g. Donati et al. 2006;Vidotto 2016): 𝑩 pol ( 𝜃, 𝜙 ) = ∑︁ 𝑙𝑚 𝛼 𝑙𝑚 𝑐 𝑙𝑚 𝑃 𝑙𝑚 𝑒 𝑖𝑚𝜙 ˆ 𝑟 + ∑︁ 𝑙𝑚 𝛽 𝑙𝑚 ( 𝑙 + ) 𝑐 𝑙𝑚 d 𝑃 𝑙𝑚 d 𝜃 𝑒 𝑖𝑚𝜙 ˆ 𝜃 + ∑︁ 𝑙𝑚 𝛽 𝑙𝑚 𝑖𝑚 ( 𝑙 + ) sin 𝜃 𝑐 𝑙𝑚 𝑃 𝑙𝑚 𝑒 𝑖𝑚𝜙 ˆ 𝜙, (6) 𝑩 tor ( 𝜃, 𝜙 ) = ∑︁ 𝑙𝑚 𝛾 𝑙𝑚 𝑖𝑚 ( 𝑙 + ) sin 𝜃 𝑐 𝑙𝑚 𝑃 𝑙𝑚 𝑒 𝑖𝑚𝜙 ˆ 𝜃 − ∑︁ 𝑙𝑚 𝛾 𝑙𝑚 ( 𝑙 + ) 𝑐 𝑙𝑚 d 𝑃 𝑙𝑚 d 𝜃 𝑒 𝑖𝑚𝜙 ˆ 𝜙. (7)These expressions are consistent with the general form of the poloidaland toroidal fields evaluated at the stellar surface when 𝑆 ( 𝑅 ★ ) = 𝛼 𝑙𝑚 𝑅 ★ 𝑙 ( 𝑙 + ) , d 𝑆 ( 𝑟 ) d 𝑟 (cid:12)(cid:12)(cid:12) 𝑟 = 𝑅 ★ = 𝛽 𝑙𝑚 𝑅 ★ ( 𝑙 + ) (8)and 𝑇 ( 𝑅 ★ ) = 𝛾 𝑙𝑚 𝑅 ★ ( 𝑙 + ) . (9)Given the surface magnetic field, magnetic energies are estimated We use a right-handed spherical coordinate system where a positive radialfield component points out of the star, the 𝜃 component is positive point-ing from North to South and the 𝜙 component is positive in the clockwisedirection as viewed from the South pole. Figure 1.
The cartoon shows four different combinations of symmetries(axisymmetric and non-axisymmetric relative to the rotation axis) of thepoloidal (red arrows) and toroidal (blue arrows) fields. Poloidal field linesthat link with the toroidal field are represented by solid lines, the ones that donot are dashed. by calculating the mean squared magnetic flux density (cid:104) 𝐵 (cid:105) . Forinstance, in the case of the poloidal energy : (cid:104) 𝐵 (cid:105) = Ω ∫ 𝑩 pol · 𝑩 pol d Ω . Accordingly, the fraction of axisymmetric poloidal magneticfield energy is given by : (cid:104) 𝐵 , m = (cid:105)/(cid:104) 𝐵 (cid:105) . The toroidal energyand axisymmetry fraction are calculated analogously. Magnetic helicity can be defined as 𝐻 = ∫ 𝑨 · 𝑩 dV (Woltjer 1958),where 𝑨 is a vector potential corresponding to the magnetic field 𝑩 . As magnetic helicity is a quantity measuring the linkage of fieldswithin a volume , our surface magnetic fields limit us to evaluating themagnetic helicity density ℎ = 𝑨 · 𝑩 . The separation of the magneticfield into its poloidal and toroidal components is particularly usefulin this regard. It dispenses with the need to invoke a gauge (Berger &Hornig 2018) since the usual gauge field (the corresponding potentialfield with the same boundary flux) has zero helicity. In addition, in aspherical coordinate system, toroidal field lines lie purely on spher-ical surfaces while poloidal field lines pass through these surfaces.This makes visualising the linkage of field lines straightforward. Fig.1 illustrates how poloidal fields lines (shown red) that pass throughthe stellar surface may link through loops of toroidal field (shownblue) that lie on the stellar surface. It is notable that in these exam-ples, only some fraction of the poloidal field links with the toroidalfield line shown. When calculating the mean squared magnetic flux density we integrate overa full sphere, dividing by the solid angle of
Ω = 𝜋 . We define axisymmetric as 𝑚 =000
Ω = 𝜋 . We define axisymmetric as 𝑚 =000 , 1–8 (2020) ield linkage and magnetic helicity density r^ D Y B tor r^(a) (b) Figure 2. (a) Contours of the function Ψ lying on the surface of the star areshown in blue. (b) This shows an enlarged version, illustrating that 𝑩 tor = ∇ Ψ × ˆ 𝑟 . Interpreting magnetic helicity as the linking of poloidal andtoroidal fields (Berger 1985; Berger & Hornig 2018) allows the he-licity density to be calculated for any stellar magnetic map given onlyits 𝛼 𝑙𝑚 and 𝛾 𝑙𝑚 coefficients and the stellar radius 𝑅 ★ (Lund et al.2020): ℎ ( 𝜃, 𝜙 ) = Re (cid:18) ∑︁ 𝑙𝑚 ∑︁ 𝑙 (cid:48) 𝑚 (cid:48) 𝛼 𝑙𝑚 𝛾 𝑙 (cid:48) 𝑚 (cid:48) 𝑅 ★ ( 𝑙 (cid:48) + ) 𝑙 ( 𝑙 + ) 𝑐 𝑙𝑚 𝑐 𝑙 (cid:48) 𝑚 (cid:48) 𝑒 𝑖𝜙 ( 𝑚 + 𝑚 (cid:48) ) (cid:18) 𝑃 𝑙𝑚 𝑃 𝑙 (cid:48) 𝑚 (cid:48) (cid:18) 𝑙 ( 𝑙 + ) − 𝑚𝑚 (cid:48) sin 𝜃 (cid:19) + d 𝑃 𝑙𝑚 d 𝜃 d 𝑃 𝑙 (cid:48) 𝑚 (cid:48) d 𝜃 (cid:19)(cid:19) . (10)The magnetic helicity density, as expressed in Eq. 10, depends onthe 𝛼 𝑙𝑚 and 𝛾 𝑙𝑚 coefficients, but not the 𝛽 𝑙𝑚 coefficients found inthe 𝜃 and 𝜙 components of the poloidal field. This is because onlythe radial component of the poloidal field ( 𝑩 pol , r ) passes through thespherical surfaces containing the toroidal field.When comparing different magnetic maps, e.g. for different stars, itis often useful to summarise the overall helicity with a single number.For this purpose, we consider an average value across the hemispherefacing the observer. We note that typically only one hemisphere isfully visible as part of the star never comes into view. Furthermore,we take the absolute value of the averaged helicity density as weare interested in comparing magnitudes, not signs. This absoluteaverage helicity density (| (cid:104) ℎ (cid:105) |) will for simplicity be referred to asthe “helicity density” for the remainder of this paper.It is possible to visualise the linkage of poloidal and toroidal fieldsthat results in helicity density at the stellar surface through mapsshowing the strength of 𝑩 pol , r (calculated from the 𝛼 𝑙𝑚 according toEq. 6) with the field lines of 𝑩 tor superimposed. Expanding Eq. 2 as 𝑩 tor = ∇ Ψ × ˆ 𝑟 (11)shows that the contours of Ψ correspond to the field lines of 𝑩 tor (seeFig. 2). In particular, at the stellar surface, Ψ = 𝛾 𝑙𝑚 𝑅 ★ ( 𝑙 + ) 𝑐 𝑙𝑚 𝑃 𝑙𝑚 𝑒 𝑖𝑚𝜙 , (12)from Eqs. 4 and 9. Our sample of stellar magnetic maps are all created using Zeeman-Doppler imaging. They describe the magnetic fields of 51 different B (G ) | h | ( M x c m ) GJ182 WXUMaGJ1245BGJ49 R o ss b y nu m b e r Figure 3.
Absolute helicity density averaged across a single hemisphereversus the mean squared poloidal magnetic flux density ( 𝑙 ≤ 𝑀 ★ > 0.5 M (cid:12) anddiamonds represent 𝑀 ★ ≤ (cid:12) . When there are multiple measurements forthe same star these are connected by lines. The thick grey lines show the bestfit of | (cid:104) ℎ (cid:105) | = (cid:104) 𝐵 (cid:105) 𝛼 𝛽 for stars in the two mass groups; 𝛼 = . ± . , 𝛽 = . ± .
13 for 𝑀 ★ > 0.5 M (cid:12) (circles) and 𝛼 = . ± . , 𝛽 = . ± .
74 for 𝑀 ★ ≤ (cid:12) (diamonds). The stars outlined are shown inFig. 4. stars, 15 of which are represented by multiple maps. The stars rangein spectral type from F to M, and in mass from 0.1-1.34 M (cid:12) . Detailsof each star/map are provided in Table 1, along with the calculatedhelicity densities and magnetic energy components. For the sakeof a fair comparison every magnetic map is evaluated to the sameresolution, which means every calculation is performed up to thesame 𝑙 -mode ( 𝑙 ≤
4) even when higher modes are available (Lundet al. 2020).
The very well-defined dependence of helicity density on the toroidalfield of |(cid:104) ℎ (cid:105)| ∝ (cid:104) B tor2 (cid:105) . ± . was shown in Lund et al. (2020).One enduring puzzle, however, is that the dependence on the poloidalfield revealed two branches, as shown in Fig. 3. The higher-massbranch ( 𝑀 ★ > (cid:12) ) follows | (cid:104) ℎ (cid:105) | = (cid:104) 𝐵 (cid:105) . ± . . ± . and the lower-mass branch ( 𝑀 ★ ≤ (cid:12) ) follows | (cid:104) ℎ (cid:105) | = (cid:104) 𝐵 (cid:105) . ± . . ± . . When fitting these power laws the sam-ple is split specifically at 0.5 M (cid:12) because a number of magneticproperties, including helicity density, have been shown to changebehaviour across this value (Donati et al. 2008b; Morin et al. 2008b,2010; See et al. 2015; Lund et al. 2020). Fig. 3 shows that the lowestmass stars have higher poloidal energies than higher-mass stars withthe same helicity density. The lowest mass stars in our sample alsotypically have the lowest Rossby numbers, as indicated by the coloursin the plot. MNRAS , 1–8 (2020)
K. Lund et al. M (cid:803) < 0.5 M ⊙ M (cid:803) > 0.5 M ⊙ Larger | ⟨ h ⟩ |Smaller | ⟨ h ⟩ | Figure 4.
A visualisation of the linkage of the dipole ( 𝑙 =
1) poloidal and toroidal field components of GJ 182, WX UMa (2008), GJ 49 and GJ 1245B (2006).The colour shows the strength of the radial magnetic (poloidal) field, and the black contours represent the toroidal magnetic field lines. The heavy black contourseparates regions of positive (solid) and negative (dashed) toroidal field. These examples correspond roughly to the four classes shown in Fig. 1.
It appears from Fig. 3 that the lower-mass fully-convective starshave excess poloidal field that does not contribute to their helic-ity density. In order to explore the distribution of the poloidal andtoroidal fields on these two branches and to determine their contribu-tion to the helicity density, we plot maps showing how their poloidaland toroidal fields link. As an example, Fig. 4 presents maps for GJ182, WX UMa, GJ 49 and GJ 1245B. These stars are highlighted inFig. 3 and represent two pairs of stars with approximately the samehelicity density, and thus similar toroidal energies. Each pair consistsof one star from the higher-mass branch and one star from the lower-mass branch, and the two pairs are distinguished by the magnitude oftheir helicity densities (GJ 182 and WX UMa (top row) have higherhelicity densities than GJ 49 and GJ 1245B). To illustrate field link-ages at the largest scale the maps show the dipole ( 𝑙 =
1) mode. Wenote that these four stars fall into the four categories shown in Fig. 1.The clearest trend to emerge is in the toroidal field. It is notable thatthe stars on the higher-mass branch, GJ 182 and GJ 49 (left columnof Fig. 4), both have fairly axisymmetric toroidal fields, whilst thestars on the lower mass branch, WX UMa and GJ 1245B, have non-axisymmetric toroidal fields. We can quantify this trend by plottingthe fraction of the toroidal field energy that is held in axisymmetricmodes as a function of the ratio of toroidal to poloidal energy (seeFig. 5). Lower-mass stars tend to have toroidal fields that are non-axisymmetric and magnetic energy budgets that are dominated bythe poloidal field.Lund et al. (2020) showed that stars with a similar helicity densityhave toroidal fields of similar strengths. As can be seen from Fig. 4,however, these toroidal fields may have different symmetries if the B / B T o r o i d a l a x i s y mm e t r y f r a c t i o n GJ182WXUMa GJ1245B GJ49 R o ss b y nu m b e r Figure 5.
Axisymmetric toroidal energy as a fraction of total toroidal en-ergy versus the ratio of the mean squared toroidal to poloidal magnetic fluxdensities ( 𝑙 ≤000
Axisymmetric toroidal energy as a fraction of total toroidal en-ergy versus the ratio of the mean squared toroidal to poloidal magnetic fluxdensities ( 𝑙 ≤000 , 1–8 (2020) ield linkage and magnetic helicity density Stellar mass (M ) h GJ182WXUMaGJ1245B GJ49 R o ss b y nu m b e r Figure 6.
The helicity energy fraction ( 𝑙 ≤ ℎ ≡ | (cid:104) ℎ (cid:105) |/ (cid:104) 𝑅 ★ 𝐵 (cid:105) ,versus stellar mass. The symbols are the same as in Fig. 3. stars lie on two different branches. The fact that these different sym-metries are able to produce the same helicity density is because thepoloidal field, although typically stronger on the lower-mass branch,can also be of different symmetry to the toroidal field. In both rows ofFig. 4, the transition from the higher-mass branch to the lower-massbranch is accompanied by an increase in the strength of the poloidalfield. At the same time, for the lower mass stars, toroidal field linesenclose regions of both positive and negative poloidal field such thatthe total linked flux is relatively small. The difference in magneticfield topology roughly cancels the difference in field strength, suchthat the helicity is very similar within each pair, although it is arrivedat very differently.To separate the effects of field strength and geometry Fig. 6 plotsstellar mass against the “helicity energy fraction”; ˜ ℎ ≡ | (cid:104) ℎ (cid:105) |/ (cid:104) 𝑅 ★ 𝐵 (cid:105) .Dividing the helicity density by the mean squared magnetic fluxdensity and stellar radius results in a dimensionless helicity densitynormalised by magnetic field strength. Fig. 6 shows an even spread oflow helicity energy fractions across the entire range of stellar masses,however only the higher mass stars ( 𝑀 ★ > (cid:12) ) exceed a fractionof 0.35. This illustrates that even though the lower-mass stars areamong those whose dynamos are most efficient at injecting magneticenergy into the largest spatial scales that ZDI is able to detect andmap (e.g. Morin et al. 2008b), they are apparently less efficient atgenerating helicity at these largest scales. This is most likely becauseof the inefficient linking between their poloidal and toroidal magneticfield components.Inefficient linking can be arrived at in more than one way. Forthe higher-helicity pair in Fig. 4 (GJ 182, WX UMa), the lower-mass star has a strongly axisymmetric poloidal field, and since thetoroidal field is non-axisymmetric, this combination produces inef-ficient linking. In the case of the lower-helicity pair (GJ 49 and GJ1245B), the lower-mass star’s poloidal and toroidal fields are bothstrongly non-axisymmetric; nonetheless, they still offset each anotherby approximately 90 degrees, which again gives inefficient linking.We can place these trends in a broader context by showing how Rotation period (d) S t e ll a r m a ss ( M ) GJ182WXUMaGJ1245B GJ49
Helicity density10 Mx cm Mx cm Mx cm Mx cm Mx cm Mx cm T o r o i d a l a x i s y mm e t r y f r a c t i o n Rotation period (d) S t e ll a r m a ss ( M ) GJ182WXUMaGJ1245B GJ49 P o l o i d a l a x i s y mm e t r y f r a c t i o n Figure 7.
Helicity density of the large-scale magnetic fields ( 𝑙 ≤
4) of ourstellar sample shown according to stellar mass and rotation period. The dashedlines split the sample at 0.5 M (cid:12) . The size of the symbol indicates relativestrength of the helicity density. The colour corresponds to the fractionalaxisymmetry of the toroidal ( top ) and poloidal ( bottom ) magnetic energies.The stars outlined are shown in Fig. 4. the axisymmetry and helicity density varies across the stellar mass-rotation period plane. This is shown in Fig. 7 which also showsseparately the variation of the toroidal and poloidal axisymmetryfractions. High-helicity stars exist in both mass ranges. The declinein the axisymmetry of the toroidal fields with decreasing mass isapparent, but the trends in poloidal axisymmetry are more complex.The reasons for this, and the potential role of magnetic cycles, arenot yet clear.Fig. 7 also demonstrates very clearly the differences in the strengthand structure of the magnetic field that is possible for stars in thebimodal regime, which includes the stars of lowest mass and shortestperiod. Two of our example stars (WX Uma and GJ 1245B) lie in thisregime. They have similar masses (0.1 and 0.12 M (cid:12) respectively) androtation periods (0.78 and 0.71 days, respectively) but their magneticfields and the helicity densities they support are quite different.
MNRAS , 1–8 (2020)
K. Lund et al.
Helicity measures the linkage within a field. By studying the linkageof the poloidal and toroidal components of stellar magnetic fields,we can learn about the underlying dynamo processes generating thefield, and thereby the form of the magnetic cycles that might takeplace and the evolution of stellar fields as stars spin down over theirmain sequence lifetimes. This study reveals that stars in differentmass ranges, which may be either fully or partially convective, gen-erate their helicity through different forms of toroidal/poloidal fieldlinkage.For partially convective stars (those that lie on the higher-massbranch in Fig. 3) the toroidal fields are mainly axisymmetric. Anincrease in rotation rate (or a decrease in Rossby number) generallyleads to increased helicity density. For fully-convective stars (thosethat lie on the lower-mass branch in Fig. 3) the toroidal fields aremainly non-axisymmetric. For the lowest-mass stars, a bimodal be-haviour is present in the form and strength of the magnetic field(Morin et al. 2011) which may be weak with a non-axisymmetricpoloidal field, or strong with an axisymmetric poloidal field. Thesetwo types correspond to low and high helicity densities respectively.This may have implications for the possibility that this representsa bimodality in the dynamo operating in this regime (Morin et al.2011).Stars can evolve from one mass-branch to the other if their fieldstructure changes, for instance as a result of their internal structurechanging from mainly convective to mainly radiative at a very youngage, or if they transition from one bimodal dynamo mode to the other.Furthermore, stars can evolve along each branch as their rotation ratesdecay with age. Their field linkages can also evolve on much shortertimescales due to magnetic cycles. It is notable in Fig. 5 that wherethere are multiple observations of a star, taken at different times,these typically follow the trend that an increase in the ratio of 𝐵 to 𝐵 leads to an increase in the axisymmetry of the toroidal field. Asimilar behaviour is seen in Fig. 3 where for each star with multipleobservations, these all lie within the scatter about the best-fit line.It is not clear what causes the non-axisymmetry of the toroidalfields in the lowest-mass stars. Their deep-seated convection mayproduce bipoles that emerge through the stellar surface with ran-domised axial tilts, leading to a lack of axisymmetry in the toroidalfield. Their low surface differential rotation may also reduce theshearing of bipoles and hence the diffusive cancellation of poloidalfield that results. Both of these processes, however, occur at lengthscales well below what can be resolved by these Zeeman-Dopplerfield measurements. The field characteristics that are most robustlyrecovered by Zeeman-Doppler imaging (field axisymmetry and theratio of poloidal to toroidal field (Lehmann et al. 2019)) are nonethe-less the very ones that underpin the helicity density.In summary, we find that lowest-mass stars tend to be inefficientat generating helicity on the largest scales, given their magnetic en-ergy. The helicity density at a stellar surface depends not only onthe stellar radius and the strengths of the individual poloidal andtoroidal field components (see Eq. 10) but also on their spatial dis-tribution relative to each other . The fraction of the poloidal flux thatlinks with the toroidal flux is maximised when the axes of symmetryof the two fields align perfectly. The bottom row of Figure 1 illus-trates two ways in which such an alignment is possible: where bothfields are axisymmetric (left) and where both are non-axisymmetric(right). Conversely, if the poloidal and toroidal axes of symmetry areorthogonal to each other there is no linkage, and consequently nohelicity. Different orientations with different amounts of field linkingcan nonetheless result in the same helicity density due to the depen- dence on field strengths. In short, to achieve a full understanding ofthe source of the helicity density at a stellar surface, it is not enoughsimply to look at, for example, the radius and field strength, it isalso necessary to produce a surface map showing the field linkage.It is only through a combination of all these components that a clearpicture can be formed. ACKNOWLEDGEMENTS
The authors would like to thank the referee for thoughtful and con-structive comments. MJ and KL acknowledge support from Scienceand Technology Facilities Council (STFC) consolidated grant num-ber ST/R000824/1. VS acknowledges funding from the EuropeanResearch Council (ERC) under the European Unions Horizon 2020research and innovation programme (grant agreement no. 682393,AWESoMeStars). JFD acknowledges funding from the European Re-search Council (ERC) under the H2020 research and innovation pro-gramme (grant agreement 740651 NewWorlds). RF acknowledgesfunding from United Arab Emirates University (UAEU) startup grantnumber G00003269.
DATA AVAILABILITY
The data used to make Figures 3, 5 and 7 are in Table 1. Archivaldata underpinning the plots is available at polarbase ( http://polarbase.irap.omp.eu ). REFERENCES
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Our stellar sample, with the four stars we focus on in this paper highlighted in bold (GJ 182, GJ 49, GJ 1245B and WX UMa). From left to rightthe columns show: star name, mass, radius, rotation period,Rossby number, absolute helicity density averaged across the visible hemisphere, (cid:104) 𝐵 (cid:105) , poloidalaxisymmetric magnetic energy as a fraction of poloidal energy, (cid:104) 𝐵 (cid:105) , toroidal axisymmetric magnetic energy as a fraction of toroidal energy, 𝑙 max andobservation epoch. The helicity density and the energies are all calculated for 𝑙 ≤
4. References for the stellar parameters are given in the last column, wherereferences to the papers where the magnetic maps were published are in italic. A more comprehensive table of parameters for these stars can be found in Vidottoet al. (2014).Star ID 𝑀 ★ 𝑅 ★ 𝑃 rot 𝑅 𝑜 | (cid:104) ℎ (cid:105) | (cid:104) 𝐵 (cid:105) Pol (cid:104) 𝐵 (cid:105) Tor 𝑙 max Obs. Ref.(M (cid:12) ) (R (cid:12) ) (d) (Mx cm − ) (G ) Axi (G ) Axi epoch Solar-like stars
HD 3651 0.88 0.88 44.0 1.916 1.82E+11 1.49E+01 0.87 4.96E-01 0.98 10 – 1, 2, HD 9986 1.02 1.04 22.4 1.621 7.16E+09 4.71E-01 0.50 3.43E-02 0.94 10 – 1, 2, HD 10476 0.82 0.82 35.2 0.576 6.77E+09 6.23E+00 0.00 5.69E-01 0.40 10 – 1, 4, 2, HD 20630 1.03 0.95 9.00 0.593 2.09E+13 2.61E+02 0.32 4.56E+02 0.90 10 Oct 2012 1, 5, 2, HD 22049 0.86 0.77 11.76 0.366 7.19E+11 1.16E+02 0.70 3.35E+00 0.23 10 - 1, 5, 2, HD 39587 1.03 1.05 5.136 0.295 1.95E+12 1.45E+02 0.07 2.04E+02 0.84 10 – 1, 5, 2, HD 56124 1.03 1.01 20.7 1.307 2.43E+11 5.22E+00 0.90 9.32E-01 0.91 10 – 1, 2, HD 72905 1 1 5.227 0.272 1.39E+13 1.43E+02 0.15 8.70E+02 0.97 10 – 1, 5, 2, HD 73350 1.04 0.98 12.3 0.777 3.54E+11 8.99E+01 0.00 8.91E+01 0.90 10 – 1, 8, 2, HD 75332 1.21 1.24 3.870 >1.105 9.88E+11 4.58E+01 0.80 3.96E+00 0.39 15 – 1, 5, 2, HD 78366 1.34 1.03 11.4 >2.781 2.39E+12 2.11E+02 0.94 8.36E+00 0.29 10 2008 1, 2, . . . . . . . . . . . . . . . 4.27E+11 4.83E+01 0.06 7.91E+00 0.54 . . . 2010 . . .. . . . . . . . . . . . . . . 4.29E+11 2.18E+01 0.77 2.25E+00 0.30 . . . 2011 . . .HD 101501 0.85 0.9 17.04 0.663 3.02E+12 1.28E+02 0.26 5.68E+01 0.80 10 – 1, 5, 2, HD 131156A 0.90 0.80 6.25 0.256 2.11E+13 1.48E+03 0.30 4.12E+03 0.93 10 Aug2007 10, 2, . . . . . . . . . . . . . . . 5.24E+12 5.58E+02 0.59 3.00E+02 0.62 . . . Feb 2008 . . .. . . . . . . . . . . . . . . 6.64E+12 4.91E+02 0.07 4.54E+02 0.81 . . . Jun 2009 . . .. . . . . . . . . . . . . . . 4.05E+12 3.89E+02 0.08 1.91E+02 0.37 . . . Jan 2010 . . .. . . . . . . . . . . . . . . 1.40E+13 2.73E+02 0.34 2.03E+02 0.95 . . . Jun 2010 . . .. . . . . . . . . . . . . . . 6.60E+13 7.18E+02 0.73 1.12E+03 0.96 . . . Aug2010 . . .. . . . . . . . . . . . . . . 2.91E+13 5.25E+02 0.27 1.77E+03 0.97 . . . Jan 2011 . . .HD 131156B 0.66 0.55 11.1 0.611 5.59E+12 2.60E+02 0.25 1.25E+02 0.81 10 – 10, 2, HD 146233 0.98 1.02 22.7 1.324 8.87E+08 1.90E+00 0.09 1.48E-02 0.05 10 Aug2007 12, 2, HD 166435 1.04 0.99 4.2 0.259 5.12E+12 2.53E+02 0.50 1.70E+02 0.79 10 – 1, 2, HD 175726 1.06 1.06 4.0 0.272 1.71E+12 6.65E+01 0.18 2.65E+01 0.80 10 – 1, 13, 2, HD 190771 0.96 0.98 8.80 0.453 5.02E+12 8.50E+01 0.35 1.53E+02 0.98 10 2007 9, 2, HD 201091A 0.66 0.62 34.1 0.786 8.29E+10 3.02E+01 0.03 3.57E+00 0.42 10 – 1, 5, 2, HD 206860 1.1 1.04 4.6 0.388 2.77E+13 3.03E+02 0.49 3.71E+02 0.94 10 – 1, 2, Young suns
AB Dor 0.76 1.00 0.5 0.026 2.87E+14 3.34E+04 0.14 1.21E+04 0.56 25 Dec 2001 16, 17, 2, . . . . . . . . . . . . . . . 6.80E+13 2.97E+04 0.09 5.92E+03 0.50 . . . Dec 2002 . . . .BD-16351 0.9 0.88 3.21 0.14 + . − . HII 296 0.9 0.93 2.61 0.13 + . − . HII 739 1.15 1.07 1.58 0.25 + . − . HIP 12545 0.95 1.07 4.83 0.14 + . − . HIP 76768 0.80 0.85 3.70 0.09 + . − . TYC 0486-4943-1 0.75 0.69 3.75 0.13 + . − . TYC 5164-567-1 0.90 0.89 4.68 0.19 + . − . TYC 6349-0200-1 0.85 0.96 3.41 0.07 + . − . TYC 6878-0195-1 1.17 1.37 5.70 0.10 + . − . Hot Jupiter Hosts 𝜏 Boo 1.34 1.42 3 >0.732 1.52E+11 1.61E+00 0.48 1.54E+00 0.86 5 Jun 2006 20, 21, 2, . . . . . . . . . . . . . . . 3.11E+11 9.42E+00 0.59 1.80E+00 0.65 8 Jun 2007 20, 21, 2, . . . . . . . . . . . . . . . 7.83E+10 3.10E+00 0.13 6.16E+00 0.91 . . . Jan 2008 20, 2, . . . . . . . . . . . . . . . 7.76E+10 3.12E+00 0.30 5.48E-01 0.45 . . . Jun 2008 20, 2, . . . . . . . . . . . . . . . 2.53E+10 2.31E+00 0.62 2.12E-01 0.36 . . . Jul 2008 20, 2, . . . . . . . . . . . . . . . 1.71E+11 3.75E+00 0.56 1.18E+00 0.63 . . . Jun 2009 20, 21, 2, HD 73256 1.05 0.89 14 0.962 3.28E+11 4.30E+01 0.03 1.18E+01 0.78 4 Jan 2008 25, 2, HD 102195 0.87 0.82 12.3 0.473 2.43E+12 7.09E+01 0.23 9.30E+01 0.88 4 Jan 2008 26, 27, 2, MNRAS , 1–8 (2020)
K. Lund et al.
Table 1. -continuedStar ID 𝑀 ★ 𝑅 ★ 𝑃 rot 𝑅 𝑜 | (cid:104) ℎ (cid:105) | (cid:104) 𝐵 (cid:105) Pol (cid:104) 𝐵 (cid:105) Tor 𝑙 max Obs. Ref.(M (cid:12) ) (R (cid:12) ) (d) (Mx cm − ) (G ) Axi (G ) Axi epochHD 130322 0.79 0.83 26.1 0.782 1.20E+11 5.40E+00 0.58 1.03E+00 0.96 4 Jan 2008 20, 28, 29, 2, HD 179949 1.21 1.19 7.6 >1.726 5.82E+10 5.15E+00 0.57 1.17E+00 0.81 6 Jun 2007 12, 2, HD 189733 0.82 0.76 12.5 0.403 5.26E+12 1.73E+02 0.30 2.73E+02 0.91 5 Jun 2007 31, 2, . . . . . . . . . . . . . . . 1.05E+13 2.72E+02 0.17 9.50E+02 0.96 5 Jul 2008 . . . M dwarf stars
GJ 569A 0.48 0.43 14.7 <0.288 1.57E+14 1.38E+04 0.96 6.75E+02 1.00 5 Jan 2008 2, DS Leo 0.58 0.52 14 <0.267 1.76E+14 2.22E+03 0.58 1.01E+04 0.99 5 Jan 2007 2, . . . . . . . . . . . . . . . 8.76E+13 2.09E+03 0.15 8.31E+03 0.94 . . . Dec 2007 . . . GJ 182 0.75 0.82 4.35 0.054 5.62E+14 1.09E+04 0.17 2.65E+04 0.90 8 Jan 2007 2, GJ 49 0.57 0.51 18.6 <0.352 2.15E+13 4.19E+02 0.67 4.51E+02 1.00 5 Jul 2007 2, GJ 494A 0.59 0.53 2.85 0.092 6.79E+13 9.99E+03 0.12 1.70E+04 0.91 8 2007 2, . . . . . . . . . . . . . . . 1.95E+14 1.09E+04 0.27 1.44E+04 0.89 . . . 2008 . . .GJ 388 0.42 0.38 2.24 0.047 9.77E+13 4.45E+04 0.97 3.97E+02 0.25 8 2007 2, . . . . . . . . . . . . . . . 1.11E+14 4.33E+04 0.92 1.78E+03 0.08 . . . 2008 . . .EQ Peg A 0.39 0.35 1.06 0.02 4.38E+14 1.81E+05 0.71 2.41E+04 0.29 4 Aug 2006 2, EQ Peg B 0.25 0.25 0.4 0.005 4.25E+14 2.14E+05 0.95 2.70E+03 0.42 8 Aug 2006 2, GJ 873 0.32 0.3 4.37 0.068 1.33E+15 3.45E+05 0.28 3.05E+04 0.61 8 2006 2, . . . . . . . . . . . . . . . 2.12E+14 2.82E+05 0.30 4.12E+03 0.20 . . . 2007 . . .GJ 9520 0.55 0.49 3.4 0.097 1.76E+14 1.63E+04 0.85 4.44E+03 0.87 8 2007 2, . . . . . . . . . . . . . . . 2.38E+14 1.33E+04 0.63 7.56E+03 0.91 . . . 2008 . . .V374 Peg 0.28 0.28 0.45 0.006 1.29E+14 5.30E+05 0.82 1.43E+04 0.01 10 2005 34, 2, . . . . . . . . . . . . . . . 8.11E+13 3.86E+05 0.81 1.12E+04 0.01 . . . 2006 . . .GJ 1111 0.1 0.11 0.46 0.0059 1.84E+13 1.29E+04 0.79 6.54E+02 0.68 6 2007 2, . . . . . . . . . . . . . . . 9.22E+12 5.43E+03 0.31 1.26E+03 0.88 . . . 2008 . . .. . . . . . . . . . . . . . . 1.57E+13 4.47E+03 0.66 1.81E+03 0.79 . . . 2009 . . .GJ 1156 0.14 0.16 0.49 0.0081 1.39E+12 4.38E+03 0.02 3.74E+02 0.19 6 2007 2, . . . . . . . . . . . . . . . 2.55E+13 1.44E+04 0.12 1.69E+03 0.58 . . . 2008 . . .. . . . . . . . . . . . . . . 3.86E+11 1.04E+04 0.01 4.88E+02 0.05 . . . 2009 . . . GJ 1245B 0.12 0.14 0.71 0.011 2.57E+13 3.34E+04 0.06 5.30E+03 0.37 4 2006 2, . . . . . . . . . . . . . . . 4.89E+12 4.68E+03 0.13 4.86E+02 0.30 . . . 2008 . . . WX UMa 0.1 0.12 0.78 0.01 . . . . . . . . . . . . . . . . . . . . .1: Marsden et al. (2014); 2: Vidotto et al. (2014); 3: Petit et al. (in prep); 4: Saar & Brandenburg (1999); 5: Hempelmann et al. (2016); 6: do Nascimento et al.(2016); 7: Jeffers et al. (2014); 8: Petit et al. (2008); 9: Morgenthaler et al. (2011); 10: Fernandes et al. (1998); 11: Jeffers et al. (in prep); 12: Valenti & Fischer(2005); 13: Mosser et al. (2009); 14: Boro Saikia et al. (2016); 15: Boro Saikia et al. (2015); 16: Maggio et al. (2000); 17: Innis et al. (1988); 18: Donati et al.(2003); 19: Folsom et al. (2016); 20: Takeda et al. (2007); 21: Fares et al. (2009); 22: Catala et al. (2007); 23: Donati et al. (2008a); 24: Fares et al. (2013); 25:Udry et al. (2003); 26: Melo et al. (2007); 27: Ge et al. (2006); 28: Udry et al. (2000); 29: Simpson et al. (2010); 30: Fares et al. (2012); 31: Bouchy et al.(2005); 32: Fares et al. (2010); 33: Donati et al. (2008b); 34: Morin et al. (2008b); 35: Morin et al. (2008a); 36: Morin et al. (2010)Jeffers S. V., Petit P., Marsden S. C., Morin J., Donati J.-F., Folsom C. P.,2014, A&A, 569, A79Lehmann L. T., Hussain G. A. J., Jardine M. M., Mackay D. H., Vidotto A. A.,2019, MNRAS, 483, 5246Lund K., et al., 2020, MNRAS, 493, 1003Maggio A., Pallavicini R., Reale F., Tagliaferri G., 2000, A&A, 356, 627Marsden S. C., et al., 2014, MNRAS, 444, 3517Melo C., et al., 2007, A&A, 467, 721Morgenthaler A., Petit P., Morin J., Aurière M., Dintrans B., Konstantinova-Antova R., Marsden S., 2011, Astronomische Nachrichten, 332, 866Morin J., et al., 2008a, MNRAS, 384, 77Morin J., et al., 2008b, MNRAS, 390, 567Morin J., Donati J. F., Petit P., Delfosse X., Forveille T., Jardine M. M., 2010,MNRAS, 407, 2269Morin J., Dormy E., Schrinner M., Donati J. F., 2011, MNRAS, 418, L133Mosser B., et al., 2009, A&A, 506, 33Petit P., et al., 2008, MNRAS, 388, 80Pipin V. V., Pevtsov A. A., Liu Y., Kosovichev A. G., 2019, The AstrophysicalJournal, 877, L36Saar S. H., Brandenburg A., 1999, ApJ, 524, 295See V., et al., 2015, MNRAS, 453, 4301Semel M., 1989, A&A, 225, 456Simpson E. K., Baliunas S. L., Henry G. W., Watson C. A., 2010, MNRAS,408, 1666 Takeda G., Ford E. B., Sills A., Rasio F. A., Fischer D. A., Valenti J. A., 2007,ApJS, 168, 297Taylor J. B., 1974, Physical Review Letters, 33, 1139Udry S., et al., 2000, A&A, 356, 590Udry S., et al., 2003, A&A, 407, 679Valenti J. A., Fischer D. A., 2005, ApJS, 159, 141Vidotto A. A., 2016, MNRAS, 459, 1533Vidotto A. A., et al., 2014, MNRAS, 441, 2361Woltjer L., 1958, Proceedings of the National Academy of Science, 44, 489do Nascimento J. D. J., et al., 2016, ApJ, 820, L15This paper has been typeset from a TEX/L A TEX file prepared by the author.MNRAS000