6 Geometries of Stellar Magnetic Fields

Stellar magnetic fields are vector fields. The total strength and energy contained in a stellar magnetic field are probably characteristic of the overall dynamo efficiency and resulting activity, and probably determine the rules of magnetic braking. The geometry of stellar fields adds information that is crucial for our understanding of these effects. The difficulties measuring both, total field strength and geometry, were discussed in the sections above.

As a starting point, again, we can take a look at the Sun. The surface-averaged flux density on the Sun is much lower than on many other stars, but we have a better view on it. Figure 22View Image show a recent visualization of the Sun’s magnetic field during the eruption happening on August 1, 2010. As in stellar magnetic field reconstructions, this visualization rests on model assumptions and leaves some room for fields not captured by the applied methods. Nevertheless, the picture is tremendously rich in details revealing an enormously complex structure of the solar magnetic field.

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Figure 22: Three-color composite of EUV images of the Sun obtained by the Solar Dynamics Observatory during the Great Eruption of August 1, 2010. White lines show a model of the Sun’s complex magnetic field based on an extrapolation for the full-sphere magnetic field (from Schrijver and Title, 2011).

A remarkable feature of the Sun’s magnetic field are the large magnetic loops visible in Figure 22View Image. Such loops also occur in images of the solar upper atmosphere where the plasma seems to follow the magnetic field. If the Sun was observed as a star, what part of its magnetic field would we be able to see? It was mentioned before that the Sun’s magnetic field would probably be too small to be detected at any rate. In integrated light, Zeeman broadening would be too small by one or two orders of magnitude to produce any detectable signal. Linear polarization would probably remain undetectable, too. In circular polarization, the situation is more difficult. Measurements of polarization from net magnetic fields on the order of one or even a tenth of a Gauss were reported for some stars, and this may be within the range of an observable net field of the Sun at a given moment. However, the information about field geometry from such a measurement alone would certainly be very limited.

More information is available if a Doppler Image from a star with a stronger field can be obtained. Such an image takes into account all the net field snapshots visible at different rotational phases, which greatly enhances the detectability of tangled fields. An overview about the current picture of magnetic field geometries in low-mass stars, in particular among stars of spectral type M, was given by Donati and Landstreet (2009Jump To The Next Citation Point). The powerful methods of Least Squares Deconvolution and Zeeman Doppler Imaging have provided a wealth of Doppler Images showing very different pictures of stellar magnetic field geometries. A particularly interesting example are low-mass stars of spectral class M; not only are there many Doppler Images of M stars, this spectral range is also of particular interest for our understanding of the solar and stellar dynamos as pointed out earlier in this review.

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Figure 23: Properties of the large-scale magnetic geometries of cool stars (Donati, 2011Jump To The Next Citation Point) as a function of rotation period and stellar mass. Symbol size indicates magnetic densities with the smallest symbols corresponding to mean large-scale field strengths of 3 G and the largest symbols to 1.5 kG. Symbol shapes depict different degrees of axisymmetry of the reconstructed magnetic field (from decagons for purely axisymmetric fields to sharp stars for purely non-axisymmetric fields). Colors illustrate field configuration (dark blue for purely toroidal fields, dark red for purely poloidal fields, intermediate colors for intermediate configurations). Full, dashed, and dash-dot lines trace lines of equal Rossby number Ro = 1, 0.1, and 0.01, respectively (from Donati, 2011Jump To The Next Citation Point, reproduced by permission of Cambridge University Press).

Morin et al. (2010Jump To The Next Citation Point) summarized the results from Zeeman Doppler Imaging currently available in M-type stars. Including the results of Morin et al. (2010), Figure 23View Image shows properties of the large-scale magnetic geometries of cool stars from Donati (2011) in a visualization of magnetic field geometries as a function of mass and rotation period. Many of the stars follow the trend of stronger average fields in less massive and more rapidly rotating stars (Donati and Landstreet, 2009Jump To The Next Citation Point). These more active stars have field geometries that seem to be more axisymmetric and predominantly poloidal. This leads to the suggestion that rapidly rotating low-mass stars tend to produce strong, axisymmetric, and poloidal fields. Whether the reason for such a trend is due to rotation, mass (radius), or structural differences in the interior of the stars, is unknown. In any case, a more axisymmetric and poloidal field geometry is not what one expects from the general picture of magnetic dynamos; distributed dynamos in fully convective stars should not be able to produce strong fields that are more symmetric and poloidal than fields in sun-like stars in which the dynamo operating at the tachocline is believed to produce a rather organized global field. The trend towards stronger and more organized fields in low-mass stars is challenged by a number of very-low mass (M ∼ 0.1 M ⊙) rapid rotators (P ∼ 1 d) exhibiting rather weak fields and geometries with a low degree of axisymmetry: a number of very-low-mass stars produce fields with entirely different geometries and field strengths (lower left in Figure 23View Image).

It is well known that early-M dwarfs (M0 – M3) in the field are generally much less active and slower-rotating than later, fully convective M stars (e.g., Delfosse et al., 1998; Reiners and Basri, 2008; Reiners et al., 2012). Early-M dwarfs appear to suffer much more severe rotational braking so that their activity lifetime is shorter than in later M dwarfs (West et al., 2008), and this can be explained by the severe change in radius and its consequences to angular momentum loss (Reiners and Mohanty, 2011Jump To The Next Citation Point). Do magnetic fields suffer significant change around spectral type M3/M4? In total field strength, visible to Stokes I, no change is detected; differences between field strengths are consistent with the assumption that flux generation is ruled by Rossby number (or rotation period) on both sides of the threshold to fully convective stars. On the other hand, Doppler Images show that differences between sun-like and early-M type stars on one side and very-low-mass stars on the other are enormous. If we assume that these differences are real, low-mass stars must be able to somehow generate fields of structure radically different from fields in sun-like stars. This would probably imply either a small scale dynamo mechanism capable of generating fields with very different global properties, or the co-existence of different dynamo mechanisms in fully convective stars.

It is important to realize that at spectral type M3/M4, severe changes happen also in more basic parameters of these stars, and the reason for a change for example in braking timescales seems to be much more fundamental than magnetic field geometry. For example, from spectral type M2 to M5, radius and mass diminish by more than a factor of two, which is enough to cause the observed differences in rotation and activity (Reiners and Mohanty, 2011). In other words, less effective magnetic braking in fully convective stars does not require a change in field geometry. To what extent such changes may also influence the detectability of magnetic fields, in particular of small-scale magnetic structures, is a question that is important for our understanding of stellar dynamos and, in particular, for the differences between dynamos in fully and partially convective stars.

We have seen in Section 2 that the fraction of the magnetic flux detected in the currently available Zeeman Doppler Images (from Stokes V) may be substantially lower than one, due to cancellation effects or the weak-field approximation. This fraction can be determined if the field is also visible in Stokes I, where the full field is measurable. The typical average field strength of a few hundred Gauss, as detected in Doppler Images, is much lower than average field strengths of magnetically active stars observed in Stokes I that are a typically few kG. We can compare the field measurements in Stokes I and V for stars contained in Tables 2 and 5. This comparison is shown in Figure 24View Image, which is an update of Figure 2 in Reiners and Basri (2009Jump To The Next Citation Point).

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Figure 24: Measurements of M dwarf magnetic fields from Stokes I and Stokes V. Top panel: Average magnetic field – Open symbols: measurements from Stokes I; Filled symbols: measurements from Stokes V. Center panel: Ratio between Stokes V and Stokes I measurements. Bottom panel: Ratio between magnetic energies detected in Stokes V and Stokes I. Circles show objects more massive than 0.4 M ⊙, stars show objects less massive than that.

Figure 24View Image shows the average magnetic fields from Stokes I and V, their ratios, and the ratios of magnetic energies as a function of Rossby number and stellar mass. In the top panel, the measurements are shown directly, the center panel shows the ratio between the average magnetic fields ⟨BV ⟩∕⟨BI ⟩. For the majority of stars, the ratio is on the order of ten percent or less, which means that < 10% of the full magnetic field is detected in the Stokes V map. In other words, more than 90% of the field detected in Stokes I is invisible to this method. As discussed above, this is probably a consequence of cancellation between field components of different polarity. One very interesting case with a very high value of ⟨BV ⟩∕⟨BI ⟩ is the M6 star WX Uma, which has an average field of approximately 1 kG in Stokes V (Gl 51 shows an even higher field but has not yet been investigated with the Stokes I method).

A second observable that comes with the Stokes V maps is average squared magnetic field, 2 ⟨B ⟩, which is proportional to the magnetic energy of the star. Under some basic assumptions, this value can be approximated from the Stokes I measurement, too (see Reiners and Basri, 2009Jump To The Next Citation Point). The ratio between approximate magnetic energies detected in Stokes V and I is shown in the bottom panel of Figure 24View Image, it is between 0.3 and 15% for the stars considered. In contrast to the conclusions suggested in Donati et al. (2008b) and Reiners and Basri (2009), evidence for a change in magnetic geometries at the boundary between partial and complete convection is not very obvious when the latest results are included. Four of the late-M dwarfs have ratios ⟨BV ⟩∕ ⟨BI ⟩ below 10% while earlier results suggested that more flux is detectably in Stokes V in fully convective stars. On the other hand, the ratio of detectable magnetic energies stays rather high in this regime (≥ 2%), which may reflect an influence of the convective nature of the star. An important question, however, is why the five low-mass stars with < 0.2 M ⊙ show a relatively high fraction of detected magnetic energy, while they show such a low fraction of detected field strength? This may well be an effect of different magnetic geometries but cannot be clearly identified at this point.

In a typical Zeeman Doppler Image of a low-mass star, about 90% of the magnetic field and much more than 90% of the magnetic energy remains undetected. It is a challenging task to derive global properties for a field of which only a small fraction is visible. Small-scaled structures like sunspots are in principle difficult to detect in Stokes V measurements, but it is believed that the method can well reproduce the large-scale structures. Nevertheless, in field geometries as complex as shown in Figure 22View Image, it is not immediately clear which part can be reconstructed by a given observation, and which part cannot. Our understanding of magnetic field geometries and magnetic dynamos, both in the Sun and other stars, will therefore depend on whether it will be possible to characterize the properties of the remaining 90% magnetic flux on stars other than the Sun.

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