Zeeman effect

2.1 Zeeman effect

2.1.1 Absorption lines in a magnetic field

In this section, I will give a brief introduction on the basics of the Zeeman effect. More comprehensive discussions of the Zeeman effect and equations to calculate Zeeman splitting in stellar atomic absorption lines can be found, e.g., in Condon and Shortley (1963 ), Beckers (1969 ), Saar (1988 ), Landstreet (1992 ), Mestel and Landstreet (2005 ), and Donati and Landstreet (2009 ).

An atomic or molecular absorption or emission line is excited if electrons make a transition from one energy level to another. The energy of each level is altered in the presence of a magnetic field according to the vector product between the spin ( $S$ ) and orbital angular momenta ( $L$ ) of the electron, and the magnetic field vector (so-called LS coupling). Each energy level with total angular momentum quantum number $J$ splits into ( $2J + 1$ ) states of energy with different magnetic quantum numbers $M$ . The difference between subsequent states of energy is proportional to $Bg$ with $B$ the magnetic field and $g$ the Landé factor, the latter being a function of the energy level’s orbital and spin angular momentum quantum numbers,

A dipole transition between two energy levels must obey the selection rule $ΔM = − 1,0,+1$ , hence there are generally three groups of transitions between two energy levels. Spectral lines with $ΔM = 0$ are called $π$ components, spectral lines with $ΔM = − 1$ or $+1$ are called $σblue$ and $σred$ -components, respectively. Since orbital and spin angular momentum quantum numbers can be different between the two energy levels, the Landé-factors of both levels can be different, and transitions between energy levels are not only a function of $M$ but depend on the Landé-factors of the energy states. The result is that each component consists of a group of transitions.

An often used quantity in the characterization of Zeeman splitting is the so-called effective Landé-factor, which is the average displacement of the group of $σ$ -components with respect to line center. The effective Landé factor $g$ of a transition is a combination of Landé values of the two energy levels involved (Beckers, 1969 ),

In principle, the effective Landé factor can be calculated from the energy level’s individual Landé factors and Equation (2

). For many transitions, however, LS-coupling is a poor approximation of the real situation leading to large errors in the calculation of individual $gi$ values. In such cases, it can be more appropriate to measure $gi$ in laboratory experiments (e.g., Reader and Sugar, 1975 ) and use Equation (2

) to obtain more useful empirical effective Landé factors (Landi Degl’Innocenti, 1982 ; Solanki and Stenflo, 1985 ).

In summary, in the presence of a magnetic field, the transition energies of the $σ$ -components are shifted according to the sensitivity of the transition (the Landé-factor $g$ ) and the strength of the magnetic field $B$ . The energy perturbation depends on the energy level’s quantum numbers, condensed in $g$ , and the magnetic field, $B$ . If we measure the energy shift in terms of wavelength shift $Δ λ$ or as Doppler displacement $Δv$ , the perturbation becomes a function of the initial wavelength of the transition, $λ0$ . The wavelength displacement of the $σ$ components is

with $Δ λ$ in mÅ, $λ0$ in µm, and $B$ in kG. The average velocity displacement of the spectral line components can then be written as

with $B$ in kG, $λ0$ in µm, and $Δv$ in km s^–1. The typical Zeeman velocity displacement of a spectral line at visual wavelengths in the presence of a kG-field is on the order of 1 km s^–1, which is somewhat smaller than the typical resolving power of a high-resolution spectrograph and the intrinsic line-width of stellar absorption lines. The Doppler displacement is proportional to the wavelength $λ0$ , which facilitates the detection of Zeeman splitting at infrared wavelengths compared to measurements in the visual.

2.1.2 Polarization of Zeeman components

The three groups of Zeeman components, $σblue$ , $σred$ , and $π$ , are characterized by different magnetic moments, which means that the three Zeeman components have distinct polarization states. Furthermore, the actual intensity and polarization seen by an observer depends on the angle between the line of sight and the magnetic field at the atom. Figure 2 shows a simplified scheme of the splitting (left panel) and of the different observable polarization states (right) of the $π$ and $σ$ components: The $π$ component is not shifted in energy, it is always linearly polarized but is not observed if the line of sight is parallel to the magnetic field vector. The $σ$ components, on the other hand, are shifted according to the formulae above. They can be observed linearly or circularly polarized depending on the observer’s view. If the line of sight is parallel (longitudinal field) to the magnetic field vector, both components are circularly polarized but in opposite directions. If the line of sight is perpendicular (transverse field) to the magnetic field, the $σ$ -components are linearly polarized in the direction perpendicular to the polarization of the $π$ -component.

It is important to realize that measurements of longitudinal and transerve fields as seen in circular and linear polarization, and also field measurements from unpolarized light are usually not identical to the real surface magnetic field because of the measuring principles discussed here. For the following, I will speak of longitudinal fields if magnetic fields are derived from circular polarization. This includes results from Stokes V magnetic maps, which combine observational information of longitudinal fields visible at different epochs.

Figure 2: Schematic view of Zeeman splitting. (a) The upper level in the example is split into three levels producing three spectral lines that are separated. (b) Polarization of the $π$ and $σ$ components.

2.1.3 The Stokes vectors

For the characterization of a magnetic field, the measurement of intensity in different polarization states can be of great advantage. This is immediately clear from Figure 2 b since the different Zeeman components are polarized in a characteristic fashion. A commonly used system are the Stokes components I, Q, U, and V (Stokes, 1852 ) defined in the following sense:

Stokes I is just the integrated (unpolarized) light. Stokes Q and U measure the two directions of linear polarization, and Stokes V measures circular polarization. Note that Stokes Q and U are the differences between two linearly polarized beams with perpendicular directions of polarization. The components measured in Stokes U are rotated by 45° with respect to the components measured in Stokes Q; directions of Q and U are not defined in an absolute sense but require the definition of a frame of reference, in which polarization is measured. For a very readable introduction to Stokes vectors and alternative forms the reader is referred to Tinbergen (1996 ).

Stokes vectors are useful in astronomy because perpendicular circular and linear polarization states can be measured with relatively straightforward instrumentation. The representation of astronomical polarization measurements is usually done in terms of Stokes vectors. The problem we are concerned with, however, is under what circumstances the magnetic field of a star can be recovered from measurements of the Stokes vectors. If solar magnetic fields are a good example for other stellar magnetic fields, we can expect that they often show up in groups of different polarity. This is a problem to measurements in Stokes V because equal amounts of polarized light with opposite polarization simply cancel out and become invisible. In Stokes Q and U, on the other hand, regions with magnetic field vectors pointing into directions perpendicular to each other will cancel. Thus, magnetism on a star with one magnetic region on the eastern limb and another one of an identical field strength and geometry on the northern limb will show no linear polarization and be invisible to Stokes Q and U measurements.

In reality, stellar magnetic fields can be expected to be very complicated structures with a continuum of field strengths and orientations. Therefore, we can in general not expect to resolve classical splitting patterns with three groups of Zeeman components, because magnetic field strengths and, thus, the velocity displacement of the $σ$ -components will be continuous. Finally, the visible surface of a star is not a flat disk but one half of a sphere and even the academic case of a completely radial (non-potential) field has vector components that would be observed under viewing angles between 0° and 90°. It is, therefore, a formidable task to measure magnetic fields and their geometries in stars that are spatially entirely unresolved.

For a first estimate of the signals that may be expected from Stokes measurements in sun-like stars, we can take a look at sunspot data. In the center of a sunspot with B = 2200 G, the polarization of spectral lines at around 630 nm is on the order of 10% in Stokes Q and U, and about 20% in Stokes V if measured at very high spectral resolving power (R = 200 000) (Lites, 2000 ). Observations of spatially unresolved sun-like stars will obviously not reach that level because of canceling effects and lower spectral resolution. Piskunov and Kochukhov (2002 ) calculated the four Stokes parameters in profiles at $λ$ = 615 nm for a star with a dipolar magnetic field configuration with a field strength of 8 kG at a spectral resolving power of R = 100 000. This model star is probably not very similar to a low-mass star, but the example nicely shows what level of polarization can be expected in an extreme case of a very strong and organized magnetic field. The signal from disk-integrated observations reaches maximum values of 0.5% in Stokes Q and U, and 5% in Stokes V. Thus, if magnetic fields are to be detected in all four Stokes parameters, extremely high data quality is required. At visual wavelengths, linear polarization observations must have signal-to-noise ratios of roughly 1000 while circular polarization perhaps is relaxed by a factor 10, approximately. For similar magnetic field measurements in sun-like stars, the requirements are likely higher by at least one order of magnitude.

A series of very high quality stellar measurements in all four Stokes vectors of magnetic Ap and Bp stars was presented by Wade et al. (2000 ). Again, no comparable measurements are available for sun-like stars, but hotter stars with very strong fields may serve as guideline for our expectations of polarization in cool star observations. Wade et al. (2000 ) show that in magnetic Ap and Bp stars, circular polarization detected in strong, magnetically sensitive lines is typically around 1 – 2 × 10^–2 while linear polarization is a factor 10 – 20 lower. Typical fields in cool stars are probably much weaker so that polarization can be expected to be a lot weaker than this, too. Recently, Kochukhov et al. (2011 ) presented the first detection of linearly polarized spectra in cool stars. In an active K-dwarf, they detected circular polarization at a level of 5 × 10^–5, and linear polarization of roughly a factor of 10 weaker.

In order to demonstrate the visibility of magnetic fields in the Stokes parameters and the canceling effects of spot groups with opposite polarization (as observed on the Sun), Figures 3 and 4 show some basic simulations of line profiles from magnetic regions in the Stokes parameters. The left panel visualizes the “geometry”, which is actually not a geometry of some real stellar magnetic field, but nothing else than two areas of radial magnetic fields put on a flat surface, where the spherical shape of a star has not been taken into account in this example. It should be emphasized that on a real, spherical star, cancellation effects would not be as obvious as in this simplistic example and some net flux would usually remain. This is even more important in the case of rotating stars where opposite polarization of magnetic regions with different (local) radial velocities would not lead to complete flux cancellation.

The magnetic regions in this toy model as shown in Figure 3 are observed with the field direction perpendicular to the line of sight, i.e., observations of the transverse field. Figure 4 shows the same cases for observations of magnetic regions observed with the field direction parallel to the line of sight, i.e., the longitudinal field. In the first case of transverse field orientation, no circular polarization is visible at all. For longitudinal field observations, linear polarization is invisible. Spectral resolving power is set to R = 100 000, the model line is a fictitious Fe line at rest wavelength 600 nm, thermally broadened according to a temperature of 4000 K. Broadening due to turbulence and rotation are not taken into account.

Figure 3: Three examples of simplified field geometries and their signals in Stokes I, V, and Q or U if the field is perpendicular to the line of sight (transverse field). Blue, green, and red lines show the line profiles of individual Zeeman components $σblue$ , $π$ , and $σred$ , respectively. The black line is the sum of the three, that means the line that will be observable. In the Stokes I panel, the magenta line shows how the line would appear with zero magnetic field. Rotational Doppler effects are ignored in these examples.

Figure 4: Three examples of simplified field geometries and their signals in Stokes I, V, and Q or U if the field is tangential to the line of sight (longitudinal field). Colored lines like in Figure 3

. Rotational Doppler effects are ignored in these examples.

The first example in the top row of both figures is a simple magnetic field region with only one polarity; total field strength and signed “net” field both are 1000 G in this example. Stokes I exhibits very little broadening that is difficult to detect. There is a difference between the two directions of observation since for the longitudinal field (Figure 4 ), the $π$ component does not appear. Linear polarization of the transverse field (Figure 3 ) and circular polarization of the longitudinal field (Figure 4 ) are on the order of 10%. Note that the direction of the Stokes V signal indicates the orientation of the magnetic field vector. In the second row, two magnetic regions, each with only half the size as in the first example are observed. Both regions have the same absolute field strength and area, but opposite polarity. The Stokes I signal is identical to the first example (the individual components are weaker but there are twice as many). The same is true for the linear polarization signal in Stokes Q and U because a shift in the polarization direction of 180° has the same signal as the original one. The signal in Stokes V, however, entirely vanishes (for all observing angles) because the net (signed) field of this configuration is exactly zero; any field strength in this canceling configuration is invisible to Stokes V. The last row in Figures 3 and 4 show the case of two magnetic areas with slightly different sizes; the total field is still 1000 G, but here the net field is 100 G. Again, Stokes I, and Q and U are the same as in the examples above. Because of the non-vanishing net field, the amplitude in Stokes V is now different from zero at ca. 1%. In Figure 3 , linear polarization always provides a strong signal because opposite magnetic field polarities do not cancel out. In a situation with two spots located at relative polarization of 90° to each other, linear polarization would completely cancel, too. As mentioned above and will be discussed again in Section 2.1.7, Doppler shifts on a rotating star add valuable signal to polarimetric measurements. Since real stars have usually non-zero rotation, most cool stars show non-zero magnetic features visible in polarimetric measurements.

2.1.4 Reconstruction of stellar magnetic fields from Stokes vectors

The few examples shown in Figures 3 and 4 demonstrate the principal sensitivity of the four Stokes vectors to magnetic fields and their configurations. In spatially resolved regions on the solar surface, measurements of polarization provide relatively well-defined information on the magnetic field (at least if compared to the case in other stars). In other stars, however, we do not quite know what kinds of fields to expect. The average flux density on the Sun is only on the order of a few G and remains undetectable in observations of integrated solar light. Slowly rotating stars of a comparable activity level probably have fields as weak as the solar one. On the other hand, the magnetic geometry of more rapidly rotating and, hence, more active stars is entirely unknown and may not be very similar to the solar case.

A major difficulty in measuring stellar Zeeman splitting is the small value of $Δv$ compared to other broadening agents like intrinsic temperature and pressure broadening, and rotational broadening. In a kG-magnetic field, typical splitting at optical wavelengths is of the order 1 km s^–1, which is well below intrinsic line-widths of several km s^–1 and also below the spectral resolving power of typical high-resolution spectrographs. Thus, individual components of a spectral line can normally not be resolved even if the star only had one well-defined magnetic field component. Real stars, however, can be expected to harbor a magnetic field distribution that is much more complex than this. Thus, even if spectral lines were intrinsically very narrow and spectral resolving power infinitely high, we would expect the Zeeman-broadened lines to look smeared out since in our observations we integrate over all magnetic field components on the entire visible hemisphere.

Stellar activity manifests itself in magnetic regions that can be darker than the quiet photosphere (e.g., spots) or brighter (e.g., faculae). The contribution of a surface region to an observed spectral line depends on its intensity contrast and local opacity while average field densities in active regions like spots or faculae are known to be systematically different from each other. This implies that regions of different field strenghts are systematically weighted in their contribution to the observed Zeeman pattern, and that the choice of diagnostic is very important for the field density measured.

Another point that becomes immediately clear is that the geometric interpretation of Zeeman splitting on an unresolved stellar disk can be arbitrarily complex, no matter if polarized or unpolarized light is used. In addition to the ambiguity between magnetic field strength and the fraction of the star being occupied with magnetic fields (which includes our ignorance about the number and distribution of magnetic components), the signature of a magnetic field region in stellar spectra depends on the angle between the magnetic field lines and the line of sight. In reality, a continuous distribution of angles can be expected because field lines are probably bent on the stellar surface, and because the stellar surface is spherical. As a result, even geometrically relatively simple field distributions will lead to highly complex splitting patterns. If the star is rotating at significant speed, as most active stars probably do, that pattern again depends a lot on the time a star is observed. This, in turn, can be utilized to reconstruct the geometry of the magnetic field by observing the variation of the observed spectra with rotation.

There are two basically quite different ways to gather information about stellar magnetic fields:

Measure the integrated scalar, unsigned magnetic field (Stokes I).
Measure the magnetic vector field.

The most promising way, clearly, to obtain information about the magnetic field is to determine simultaneously the integrated field and its vector components. Observationally, however, there are important differences between measurements in Stokes I (integrated flux measurements) and measurements in polarized light, so that in practice both parts are often done separately.

Integrated field measurements

The value of the integrated magnetic field strength can be derived from observations in Stokes I. Such observations can be carried out with every high-resolution spectrograph and do not require polarization optics. Stokes I measurements are sensitive to the entire magnetic field on the star, independent of field geometry and canceling effects. A simultaneous measurement of Stokes I is, therefore, always helpful in order to determine the fraction of a magnetic field that may be invisible to polarized light measurements.

Unfortunately, in a measurement of Zeeman splitting in Stokes I one faces the difficulty to disentangle the effect of Zeeman broadening from all other broadening agents. This requires precise knowledge of the spectral line appearance in the absence of a magnetic field. This task requires extremely good knowledge about spectral line formation, velocity fields, and the temperature distribution on the star. Signatures of cool spots or differential rotation, for example, can be very similar to Zeeman splitting patterns in integrated starlight. The amplitude of Zeeman spitting due to a strong magnetic field (e.g., 1000 G) is very subtle in sun-like stars observed at visual wavelengths because intrinsic line width, surface velocity, and typical instrumental resolution are of the same order as Zeeman broadening. This implies that the detection of magnetic fields lower than $∼$ 1 kG is extremely difficult at visual wavelengths (see Section 3.1). Thus, stellar Stokes I measurements are typically not sensitive to magnetic fields lower than a few hundred Gauss. The degeneracy between Zeeman splitting and other broadening agents is lifted at longer wavelengths, hence infrared observations have much higher sensitivity to magnetic fields. Unfortunately, only very few high-resolution infrared spectrographs exist today but more and more measurements are being reported (Section 3.1).

The Zeeman splitting pattern in surface-integrated starlight is the sum of Stokes I patterns from the entire stellar surface. The absorption line from a star is very different from a sunspot observation in which individual components from relatively well-defined magnetic regions can be visible. The line broadening pattern in Stokes I depends on the magnetic field strength of the individual components, the strongest fields are visible in the components responsible for the widest line wings. The fractional area of the surface filled with magnetic fields (filling factor) and the weight of individual surface features in the final line profile are parameters that are hidden in the line profile shape and are degenerate with respect to each other. The information on the field distribution and the contribution of individual magnetic areas is, therefore, very limited in observations of Stokes I alone. Another limitation of Stokes I measurements became visible in observations of the solar magnetic field using the Hanle effect (see above). These measurements revealed that the Sun harbors a field that is not of 10 G but more of 100 G strength. It is unclear whether a similar difference (either in absolute or relative units) would also appear if stars with much higher field strengths are observed, but it clearly shows that Stokes I measurements have difficulties capturing the entire magnetic flux but can mainly provide a lower limit.

Reconstruction of the magnetic vector field

Observations in Stokes V, Q, or U are sensitive to the magnetic field vector, not only to the unsigned field. This provides information about the direction of the magnetic field that is not accessible to Stokes I measurements. The signal of a non-polarized spectral line is zero in Stokes V, Q, and U. This means that the problem of disentangling Zeeman splitting from other line broadening mechanisms does not exist, and the method is much more sensitive to small field values (1 G and below). A problem is, however, that the signal seen in polarized light is only the “net” magnetic field; regions of opposite polarity cancel out in Stokes V and magnetic fields at 90° orientation cancel out in Stokes Q and U. Therefore, depending on what observing technique is used, an arbitrary large magnetic field may be hidden on the stellar surface without any signal in Stokes V or Stokes Q and U alone. The problem is more severe for circular polarization because the $π$ components are not detected here.

It has been shown that the magnetic field distribution of a star can be reconstructed in great parts from simultaneous observations of all four Stokes parameters (Kochukhov and Piskunov, 2002 ; Kochukhov et al., 2010 ). Successful reconstruction requires that the star is observed over an entire rotation period for two reasons: 1) to reconstruct the surface field hemisphere, the star needs to be seen from different sides (note that if the star is seen under high inclination angles, the invisible part close to the hidden pole always remains undetectable); 2) at different phases, the angles between the magnetic field lines and the line of sight vary with the result that field components that may have canceled when observed at disk center, can become visible when observed close to the limb. The spatial resolution of magnetic field reconstructions depends on the frequency of observations during stellar rotation and on intrinsic line broadening (all Stokes components are subject to line broadening). Typically, a resolution element has a size of ten or several ten degrees on the stellar surface. Kochukhov and Piskunov (2002 ) showed that using only a subset of Stokes vectors leads to ambiguities that should be interpreted with great caution. Unfortunately, measurements of linear polarization are extremely challenging in cool stars because of the low polarization signal so that typically only Stokes I and (sometimes) Stokes V are available (see Section 3.2.1). Zeeman broadening in Stokes I is very subtle at least at visual wavelengths where most available spectrographs operate, and Stokes V, Q, and U measurements are both difficult to acquire and exhibiting subtle Zeeman signals. The observational difficulties obtaining all four Stokes components led to the practice that in cool stars in the past usually either Stokes I or Stokes V alone were investigated.

2.1.5 Field, flux, and filling factor

In general, a stellar surface may be covered with a homogenous field of one particular field strength, or it can be covered with several magnetic areas of different field strength. One example is a surface of which 50% is covered with a field of strength B. If the other 50% of the surface has no magnetic field, the average field is Bf = B/2 with filling factor f = 0.5. An important consequence of the fact that individual Zeeman-components are usually not resolved is the degeneracy between magnetic field B and filling factor f . A strong magnetic field covering a small portion of the star looks similar to a weaker field covering a larger portion of the star. An often used way around this ambiguity is to specify the value Bf , i.e., the product of the magnetic field and the filling factor; if more than one magnetic component is considered, Bf is the weighted sum over all components. Products of B with some power of f , for example Bf ^0.5 or Bf ^0.8 are often considered because they seem to be better defined by observations (see Gray, 1984 ; Saar, 1988 ; Valenti et al., 1995 ). One important point to observe is that Bf is often called the “flux” – because it is the product of a magnetic field and an area – but it has the unit of a magnetic field. In fact, the term flux is very misleading since: 1) with f specifying a relative fraction of the stellar surface, Bf is really the average flux density that is identical to the average unsigned magnetic field on the visible stellar surface, i.e., $Bf ≡ ⟨B ⟩$ ; and 2) the total magnetic flux of two stars with the same values of Bf can be extremely different according to their radii because the actual flux is proportional to the radius squared, $ℱ ∝ Bf r2$ . As a consequence, the value Bf will be much lower in a young, contracting star compared to an older (smaller) one if flux is conserved.

A related source of confusion is the difference between the signed magnetic field (or flux), and the unsigned values or the square of the fields (used to calculate magnetic energy). With Stokes I, both polarities produce the same signal and the total unsigned flux is measured. This implies that Stokes I carries only partial information about field geometry, but it also means that Stokes I always probes the entire magnetic flux of the star (see above). On the other hand, Stokes V can provide information on the sign of the magnetic fields, but this comes with the serious caveat that opposite magnetic fields cancel out and can become invisible to the Stokes V signal. Thus, results on Bf from Stokes V measurements can be much lower than Stokes I measurements.

2.1.6 Equivalent widths

Shifting of the $σ$ -components to either side of the line center leads to broadening of the spectral line and, in general, to a flattening of the line core (see Figure 3 ). An interesting effect can be used to measure magnetic fields if lines that are saturated are used , i.e., lines that have equivalent widths smaller than the sum of the individual $π$ - and $σ$ -components. If such a saturated line is split in the presence of a magnetic field, the core depth of the line will remain at approximately constant level while the line grows wider (see Figure 5 ). As a result, the equivalent width of a saturated, magnetically sensitive line will grow with magnetic field strength.

Figure 5: The net polarization of a weak line is zero, and its equivalent width remains constant if a magnetic field is applied. In contrast, the net polarization of a saturated line in a transverse magnetic field is nonzero, and the equivalent width of a saturated line becomes larger in a magnetic field (from Mullan and Bell, 1976

, after Leroy, 1962

; reproduced by permission of the AAS).

Basri et al. (1992 ) introduced a method to detect cool star magnetic fields searching for enhanced equivalent widths of Zeeman-sensitive absorption lines. As in other work searching for Zeeman splitting in Stokes I observations, they carefully modeled polarized line transfer and compared the appearance of Zeeman sensitive to Zeeman insensitive lines. The advantage of the equivalent width method is that equivalent widths are more easily measured than the subtle differences in line shape, in other words, information from several spectral bins within one spectral line is extracted into one number that can be measured more accurately. Nevertheless, the method cannot lift degeneracies between magnetic field strength (times filling factor) and other features like starspots or uncertainties in the model atmosphere; the equivalent width method can only make existing differences in the lines easier detectable.

The variation of line equivalent widths can be monitored over time. If one assumes that variations occur because of varying visible magnetic field strength, spectroscopic time series can be used to obtain information about the surface distribution of co-rotating magnetic regions (see also next section). This method was used for example by Saar et al. (1992 , 1994c ) for Stokes I magnetic surface imaging.

2.1.7 Doppler Imaging

In addition to measuring the average magnetic field on a star, signed or unsigned, the Doppler shift of individual features carries information about the geometry of the stellar surface. Doppler Imaging exploits the correspondence between wavelength position across a rotationally broadened spectral line and spatial position across the stellar disk to reconstruct surface maps of rotating stars (Vogt and Penrod, 1983 ); the method goes back to work by Deutsch (1958 ), Falk and Wehlau (1974 ), and Goncharskii et al. (1977 ). Spatial resolution of the maps depends on the rotation velocity of the star and the sampling frequency at which spectra are taken, among other factors. It has been used very successfully to reconstruct temperature maps of cool stars (see, e.g., Strassmeier, 2002 ) and abundance maps of hotter stars (e.g., Kochukhov et al., 2004 ). Zeeman Doppler Imaging (ZDI) follows the same approach but investigating polarized light (Semel, 1989 ). As the star is observed at different phases, the magnetic field vectors are observed under different projection angles leading to characteristic signatures in polarized light; field components that may be invisible at one phase can have large Stokes parameters at other phases.

Figure 6: (a) Surface image consisting of two magnetic spots with 8 kG radial field of opposite polarity, and (b) reconstructions involving all four Stokes parameters and (c) involving only Stokes I and V (from Kochukhov and Piskunov, 2002

Figure 7: Removed figure post-publication due to copyright restrictions. Springer did not grant permission to reuse material “in a work to be published on an Open Access Website”. Field reconstructions shown in a flattened polar projection with parallels drawn as concentric circles every 30° down to a latitude of –30°. Bold circle and central dot denote equator and visible pole, respectivey. Black and white code field intensities of 1000 G and –1000 G. Reconstructions of a synthetic dipole field (left panel) are shown assuming unconstrained field structure (center panel) and linear combination of force-free fields (right panel) (from Donati, 2001

Update

Figure 8: Six spot star simulations for a star observed under an inclination angle of i = 30°. The data set includes Stokes V profiles at 10 evenly spaced phases. The original images is shown in the left two columns. The next two columns show the optimal reconstruction followed by reconstructions with noise levels increased to 5 × 10^–5 (S/N = 20 000) and 1.25 × 10^–4 (S/N = 8000) (from Donati and Brown, 1997

Two fundamental issues for Doppler Imaging techniques are that DI assumes the field not to be evolving, and that temperatures of magnetic regions are not generally known. The assumption of non-evolving fields is questionable given the high level of activity and rate of flaring of these stars, but we have only little information on characteristic timescales and evolution patters. Also, temperatures of stellar active regions are poorly known in stars other than the Sun, but regions of higher (lower) temperature add more (less) flux to the observed spectra than the quiet stellar photosphere.

The approaches to construct Doppler Images can be very different. It has been shown that relatively simple magnetic geometries can be reconstructed using all four Stokes parameters simultaneously and calculating magnetic radiative transfer. An example from Piskunov and Kochukhov (2002 ) is shown in Figure 6 , another one from Donati (2001 ) is reproduced in Figure 7 , and a third example from Donati and Brown (1997 ) is shown in Figure 8 . There is an extensive literature on the applicability of ZDI that goes far beyond the scope of this review. For detailed information, the reader is referred to Donati and Landstreet (2009 ), Kochukhov and Piskunov (2002 ), and Donati (2001 ) and references therein. As a few examples, Figure 6 shows a reconstruction of a star with two magnetic spots (Kochukhov and Piskunov, 2002 ), Figure 7 show reconstructions of a large-scale dipolar configurations (Donati, 2001 ) using different assumptions on the field structure, and Figure 8 shows a configuration with two relatively large spots (Donati and Brown, 1997 ).

In cool stars, no Zeeman Doppler Image from all four Stokes parameters exists today, but may become achievable with high-resolution spectro-polarimeters like PEPSI (Strassmeier et al., 2004 ). Because the signal in Stokes I is extremely weak at visual wavelengths and for magnetic fields much weaker than several kG (as used for example in Figure 6 ), even using only Stokes I and V together is usually not an option in cool stars (see also next section). Effects of using Stokes I and V, or Stokes V alone are shown in the examples in Figure 6 and 8 . Neglecting Stokes Q and U leads to an underestimate of the area covered by the magnetic spots at low latitudes and to strong crosstalk from the radial to the meridional field map while no crosstalk appears from the radial to the azimuthal maps (Kochukhov and Piskunov, 2002 ). Donati and Brown (1997 ), using examples with two large spots, show that imaging in Stokes V suffers essentially from crosstalk between low-latitude radial and meridional field features at low inclinations, but otherwise reasonably well recovers the input field structure. They also demonstrate how reconstructions deteriorate when data quality is lower (Figure 8 ). Another example addressing the crosstalk issue is given by Donati (2001 ) using examples of a large magnetic spot and dipolar magnetic field configurations.

Obviously, ZDI is a powerful method that can be used to recover useful information on stellar magnetic field configurations. While it is undisputable that pure large-scale fields are more easily observable than small-scale field components, and that crucial information about the large-scale surface magnetic field can be recovered, it is not entirely clear what part of a more complex field geometry is reconstructed under realistic conditions in low-mass stars (including cool spots and hot emission regions, small spot groups, and temporal evolution). A very practical limitation for the Doppler Imaging technique in cool stars is that extremely high signal-to-noise ratios are required in polarized light in order to measure the subtle signatures of net polarization. Simply integrating over long times in order to collect enough photons is not applicable because individual exposures for Doppler Imaging must be kept short enough so that adequate spatial resolution can be achieved. One way out is to use bigger telescopes, another is to cleverly co-add the information contained in the many spectral lines that all contain similar information from the star; this can be done with a technique called Least Squares Deconvolution.

2.1.8 Least Squares Deconvolution

The basic idea of Doppler Imaging is to translate line profile variations into a map of the stellar surface. The information of the surface itself is contained in every spectral line, but each line is sampled with relatively high noise in the spectroscopic data. If one assumes that line formation is similar in all lines, the full spectrum can be described as a convolution between a broadening function characteristic of the stellar surface at a given rotational velocity, and the spectrum of the star as it would look if the star was not rotating. Least Squares Deconvolution (LSD, developed by Semel, 1989 and Donati et al., 1997 ) is the inverse process: assuming a non-broadened intrinsic spectrum of the star, one searches for the broadening function that must be convolved with this intrinsic function so that the result of the convolution provides the best match to the observed data. Donati et al. (1997 ) treat the observed spectrum as the convolution of the broadening function with a set of weighted “delta” functions located at the wavelengths taken from a spectral line list. Reiners and Schmitt (2003b ) used a similar approach but iteratively optimizing the weights of individual lines so that the fit to the spectrum is improved.

In its simplest incarnation, LSD can provide the broadening function that is inherent in all spectral lines, and using many lines can boost the signal-to-noise ratio of the derived broadening function with respect to individual lines. Furthermore, line blending can be treated very effectively. LSD can provide an accurate measure of the broadening profile inherent to all spectral lines if one makes the assumption that the broadened template spectrum captures all differences between the lines used (e.g. Reiners and Schmitt, 2003a ). This implies that lines are not allowed to follow different broadening patterns or line formation processes (Sennhauser and Berdyugina, 2010 ). As a consequence, lines with different Landé factors following different broadening patters cannot be used to derive a broadening profile that can be interpreted as the broadening profile inherent in each line. If the broadening patterns of individual spectral lines differ, however, LSD can still be used to determine an average broadening function from many lines. As an approximation for Zeeman broadening, average Landé $g$ values are sometimes assumed to derive an average Zeeman broadening profile in Stokes I (e.g., Morin et al., 2008 ). The interpretability of these signatures is limited (Sennhauser and Berdyugina, 2010 ) but can still allow a useful mapping of the stellar surface.

For polarized light, Donati et al. (1997 ) show an elegant way how LSD can be used to extract mean broadening profiles from circular polarization in Stokes V data, and Wade et al. (2000 ) extend this formalism to linear polarization. A crucial step is to apply the so-called weak-field approximation (see Unno, 1956 ; Stenflo, 1994 ): if Zeeman splitting is much smaller than the Doppler width of spectral lines, the following equations hold for every line $i$ :

V(v) ∝ giB ∂I(v-), (5 ) ∂v 2 2∂2I (v) Q(v) ∝ giB ----2--, (6 ) ∂v

with $V$ and $Q$ the Stokes parameters, $gi$ the Landé factor for line $i$ , $B$ the magnetic field, and $v$ the Doppler velocity. Thus, under the weak-field assumption, polarized spectra can be written as a convolution between an average line profile ( $∂I(v) ∂v$ or $∂2I(v2) ∂v$ ) and a line list in which each line is weighted by its Landé factor. The amplitude of the deconvolved broadening function in is proportional to $B$ and $2 B$ in $V$ and $Q$ , respectively. In the weak-field approximation (together with the weak-line approximation; Sennhauser and Berdyugina, 2010

), all line profiles have identical shape and only differ in intensity, which allows the use of a linear multi-line approach like LSD, which makes interpretation of the derived profile relatively straightforward. If fields are strong enough so that Stokes V splitting patterns significantly differ in shape between different lines, or if several lines are saturated, the meaning of the derived function becomes less obvious. Several other methods that overcome these limitations like Principal Component Analysis (PCA; Martínez González et al., 2008 ) or Zeeman Component Decomposition (ZCD; Sennhauser and Berdyugina, 2010 ) were developed during the last years.

As was mentioned several times already, polarization signals from integrated observations of cool stars are so small that usually they cannot be detected in individual spectral lines with current instrumentation. If the weak-field approximation is used, it is difficult to assess how the reconstruction of magnetic fields is affected, in particular together with ZDI. Donati and Brown (1997 ) point out that the weak field approximation is in principle no longer valid for field strengths above 1.2 kG, but the authors claim that in special cases the weak field approximation can adequately describe Stokes V profiles up to 5 kG (see also Donati and Collier Cameron, 1997 ). In summary, it appears not obvious that algorithms applying the weak field approximation are sensitive to (and can correctly interpret) the signatures of fields much larger than 1 kG. A potential consequence could be that they are not only insensitive to average fields above kG-strength, but would also systematically miss spatially small magnetic components with fields of this strength, as for example large spots similar to the largest sunspots.