MAGNETIC FIELD TOPOLOGY IN LOW-MASS STARS: SPECTROPOLARIMETRIC OBSERVATIONS OF M DWARFS^*

All the six targets have spectral types from M0V to M4.5V and their physical parameters are listed in Table 1. From our mass estimates (see Table 1 and references therein), three M dwarfs (KP Tau, G 164-31, and 2E 4498) have masses estimated below 0.35 M_☉ (or absolute magnitudes fainter than M_K = 6.62) and are expected to be fully convective according to standard models. Two early-M dwarfs (Gl 890 and LHS 473) are partially convective. The absolute magnitude of the unresolved binary system Gl 896Bab is M_K = 7.28, which gives M_K > 7.28 for each companion. Both components are therefore expected to be fully convective.

2.1.1. Gl 890 (HK Aqr), M0V

2.1.2. LHS 473 (GJ 752A), M2.5V

2.1.3. KP Tau (Wolf 1246, RX J0339.4+2457), M3V

2.1.4. G 164-31 (Gl 490B), M4V

2.1.5. Gl 896B (EQ Peg B), M4.5V

2.1.6. 2E 4498 (RX J2137.6+0137), M4.5V

No spectral type estimate of this dwarf is available in the literature. From our unpolarized (Stokes I) spectra obtained with ESPaDOnS (see Section 3.1), using a spectral index versus spectral type relationships for M dwarfs (e.g., Martín et al. 1999), we estimate its spectral type as M4.5 ± 0.5, giving it an absolute magnitude M_I ≈ 10.1 (e.g., Leggett 1992). Comparison with the DENIS apparent magnitude I = 10.38 (Epchtein 1997) then gives a distance of ∼11.4 pc, assuming no reddening. The source shows coronal activity in X-ray emission (0.1–2.4 keV) with a measured flux f_X = 4.52 × 10⁻¹² erg s⁻¹ cm⁻² and 20 cm radio emissions (see Brinkmann et al. 2000 and references therein). At the source distance, we derive logL_X = 28.85. The radius of an M4.5 dwarf is about 0.3 R_☉ (Chabrier & Baraffe 1997), resulting in a maximum rotational period of 7.5 hr at v sin i = 55 km s⁻¹ (Mochnacki et al. 2002). We detected strong circular polarization in this dwarf (see Figure 1). Since its rotation period is quite short, our detection makes the source (M_⋆ ∼ 0.2 M_☉) an excellent target for future mapping of the magnetic field on a fully convective star.

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We observed the six M dwarfs described above with the Canada–France–Hawaii Telescope's ESPaDOnS high-resolution spectrograph (R = 65,000; Donati 2003), which provides a wavelength coverage of 370–1000 nm. The spectropolarimetric mode was used to provide unpolarized (Stokes I) and circularly polarized (Stokes V) spectra. Among the six targets, G 164-31 is a particularly interesting source since it is an analog of V374 Peg (G 188-38), which is a rapidly rotating M4 dwarf with v sin i ≈ 37 km s⁻¹ showing a large-scale poloidal field (Donati et al. 2006a). G 164-31 is therefore also an excellent target on which to study the field topology in fully convective stars.

Each observation consists of four exposures taken at different polarimeter configurations and combined together to filter out spurious polarization signatures to first order (Donati et al. 1997). Exposure times were computed using the exposure time calculator for ESPaDOnS to obtain signal-to-noise ratios (S/Ns) above 100 at the wavelengths of interest. This signal-to-noise value is typical of what is needed to detect the circular polarization in M dwarfs (e.g., Phan-Bao et al. 2006). In the case of G 164-31, we monitored the star for a total of 20 hr from 2008 March 23 to 26 with 33 observations, aiming to cover at least one full rotation period of the star. The full observing logs are given in Table 2.

Data reduction was performed using Libre-ESpRIT (Donati et al. 1997) utilizing the principles of optimal extraction as given in Horne (1986). As the Zeeman signature is typically very weak, several methods (Semel et al. 2009; Martínez González et al. 2008; Donati et al. 1997) have been used in the past to increase the S/N of Zeeman signatures. In this work, the least-squares deconvolution (LSD; Donati et al. 1997) multi-line analysis procedure was applied to the spectra to extract the polarization signal from a large number of photospheric atomic lines⁸ and compute a mean Zeeman signature corresponding to an average photospheric profile centered at 700 nm. Both mean Stokes I and V profiles are computed for all collected spectra. Figure 1 shows the LSD Stokes I and V profiles of KP Tau, Gl 896B, and 2E 4498 in which we detect Zeeman signatures. The profiles of G 164-31 are shown in Figures 2 and 3. Using the basic equations listed in (Wade et al. 2000; see also Brown & Landstreet 1981; Donati et al. 1997), we compute the net longitudinal magnetic field strength, B_z, from our LSD Stokes I and V profiles. In the case of G 164-31, its time series of Stokes I and Stokes V profiles are used in the following section to reconstruct the field topology on this M4 dwarf.

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Table 1 lists our measurements of B_z for all targets. We detected strong Zeeman signatures in KP Tau, G164-31, Gl 896B, and 2E 4498 but we did not detect them in the two early-M dwarfs Gl 890 and LHS 473. The photospheric field strength is estimated from LSD Stokes I and V profiles, and the chromospheric field is computed from the Stokes profiles of only the Hα line.

For G 164-31, we find that the Zeeman signatures repeat after 0.540 ± 0.006 days (12.96 ± 0.14 hr), which is in the expected range of its rotation period. We thus conclude that the rotation period of this dwarf is 12.96 hr. We note that this value is slightly greater than the estimated maximum rotation period of 12.1 hr for this star. The difference is possibly due to G 164-31 having a larger radius than the expected value from theory. Matching unpolarized (Stokes I) spectra of G 164-31 with that of slowly rotating references requires a projected rotation velocity of v sin i = 41 ± 1 km s⁻¹. Our value is greater than one of Fleming et al. (1989), v sin i = 34 ± 8.5 km s⁻¹, though both measurements are consistent to within the uncertainties. Dues to its smaller uncertainty, we adopt our v sin i = 41 km s⁻¹ for G 164-31. From P_rot and v sin i, we derive R sin i = 0.44 ± 0.02 R_☉. Our R sin i is ∼30% larger than the estimated stellar radius (see Section 2.1.4). It is unlikely that i exactly equals 90°, and the results obtained in the imaging process described below are similar for 60° ⩽ i ⩽ 80°. We thus set i = 70°, which results in the G 164-31 radius being ∼38% larger than the typical value for M dwarfs of the same luminosity. This is likely due to the fact that G 164-31 is a young object belonging to a moving cluster identified by Orlov et al. (1995). We refer the reader to Section 5 for further discussion.

We monitored G 164-31 for 20 hr spread over three successive nights, aiming for coverage over a full rotational cycle. The true period (12.96 hr) is a bit longer than our expected maximum value (12.1 hr). With this period, our observations cover only half of the rotational phase. In other words, only half of the stellar surface was monitored. A few of the Stokes I and V LSD profiles are discarded due to the contamination of moonlight or flaring events. Figure 2 and 3 show Stokes I and V LSD profiles of G 164-31.

Cool spots on the surface of a rapidly rotating star produce distortions in the stellar spectral lines. If the star is rotating fast enough (at least 20–30 km s⁻¹ for late-type dwarfs; Vogt et al. 1987), the shape of the star's spectral line profiles is dominated by rotational Doppler broadening. In this case, there is a strong correlation between the position of any distortion within a line profile and the position of the corresponding spot on the stellar surface. A spectrum of the rapid rotator is a one-dimensional image that is resolved in the direction perpendicular to the stellar rotation axis and the light of sight. By observing the star at different rotation phases, a two-dimensional image of the spot distribution can be reconstructed using the so-called Doppler imaging (DI) technique (see Vogt et al. 1987 and references therein). In the imaging process, the stellar surface is divided into a grid of 1000 elementary cells. Using the spot-occupancy model of Cameron (1992), the local line profile at each grid point of the surface is described as a linear combination of two reference profiles, one representing the quiet photosphere and the other for cool spots. Both reference profiles are assumed to be equal and only differ by their relative continuum levels. For the assumed reference profile, we can use either the LSD profile of the very slowly rotating inactive M4 dwarf Gl 402 or a simple Gaussian profile with similar full width at half-maximum and equivalent width. Both options yield very similar results, indicating that the exact shape of the assumed local profiles has very minor impact on the reconstructed images, as long as the rotational velocity of the star is much larger than the local profile. Each element on the surface is quantified by the local fraction occupied by spots. The spot occupancy ranges from 0 (no spots) to 1 (complete spot coverage). The maximum entropy image reconstruction technique (Skilling & Bryan 1984; Vogt 1980) is used to find the image having the smoothest variation (or in some sense the simplest image), giving the least contrast between spots and the quiet photosphere.

The field is described as the sum of a poloidal and a toroidal component, both expressed as spherical harmonics expansions (Jardine et al. 1999). For a given set of the complex coefficients of the spherical harmonic expansions, one can produce a corresponding topology (or a Stokes V data set) at the stellar surface. The code uses the maximum entropy image reconstruction technique, in which entropy (i.e., quantifying the amount of reconstructed information) is calculated from the coefficients of the spherical harmonics expansions. The imaging process starts from a null magnetic field and iteratively adjusts the spherical harmonic coefficients in order to match the synthetic Stokes V profile to the observed LSD one. The code uses a multidirectional search in the image space until the required maximum entropy image is obtained. This corresponds to an optimal field topology that reproduces the data at a given χ² level, i.e., usually down to noise level. Since the inversion problem is partially ill posed, we use the entropy function to select the magnetic field map with lowest information content among all those reproducing the data equally well.

To calculate the synthetic Stokes V profiles for a given field topology, the surface was again divided into a grid of 1000 elementary surface cells. In each cell, the three components (radial, azimuthal, and meridional in spherical coordinates) of the vector field are estimated directly from the spherical harmonic expansion. The code uses the analytic solutions (Landolfi & Landi Degl'Innocenti 1982) of Unno–Rachkovsky's radiative transfer equations to calculate the contribution to the Stokes V profiles of all visible cells at each observed rotation phase. The free parameters in the Unno–Rachkovsky equations are obtained by fitting the LSD Stokes I profile of a very slowly rotating and weakly active star with a similar spectral type (e.g., Gl 402). To obtain the best fit (in both amplitude and width) between the synthetic and observed Stokes V profiles, the code introduces a filling factor f_c, which represents the fractional amount of circular flux being constant over the whole stellar surface, and speculates that large-scale fields in M dwarfs can be structured on a small scale (Donati et al. 2008). The circularly polarized flux from each cell is thus f_cV_loc, where V_loc is the Stokes V profile derived from the analytic solutions of the Unno–Rachkovsky equations.

Differential rotation is also implemented in the calculation process of the synthetic Stokes V profiles for yielding the best fits to the observation. For this purpose, we assume the rotation rate varies with latitude, θ, following a solar-like differential rotation law as Ω(θ) = Ω_eq − dΩ sin²θ, where Ω_eq is the angular rotation rate at the equator and dΩ is the difference in angular rotation rate between the equator and the pole. Differential rotation is detected when χ² of the fit to the data shows a well-defined minimum in the range of Ω_eq and dΩ values. In the case of G 164-31, we did not obtain a clear minimum in the explored Ω_eq–dΩ range. This indicates that our data are not suitable for measuring differential rotation given the fairly simple (and low amplitude) rotational modulation of the Stokes V profiles we observe. This is mostly due to the fact that the field distribution lacks well-defined features at different latitudes from which differential rotation can be estimated. Based on the previous observations of analogs of G 164-31 (Donati et al. 2006a; Morin et al. 2008), we therefore assumed that this star rotates as a rigid body while performing the field mapping. The obtained map of G 164-31 is shown in Figure 4.

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Figure 4 presents our maps of the spot occupancy and the magnetic field at the stellar surface of G 164-31. For the half stellar surface that was not observed, the field topology has been derived using a spherical harmonic expansion based on the collected Stokes V data set. Our brightness map shows a low contrast spot at the pole of the star. The magnetic field morphology on G 164-31 shows that the radial field component is dominant, which appears similar to what is seen in V374 Peg. The azimuthal and meridional components are negligible.

The average large-scale flux is 680 G and the poloidal field dominates with 99% of the energy content. Most (95%) of this poloidal field is stored in the mode corresponding to a dipole aligned with the rotation axis (spherical harmonics degree l = 1 and order m = 0). One should note that due to the high v sin i of G 164-31, for any value of the filling factor 0 < f_c < 1, we find similar results. Hence, we arbitrarily set f_c = 1. We also examine the binarity of G 164-31. Based on the LSD Stokes I profiles, we find an average radial velocity value of −6.7 ± 0.2 km s⁻¹. No significant variation in radial velocity is found. Therefore, our current data do not reveal any hint of an additional close-in component around G 164-31.

We did not detect Zeeman signatures in Gl 890 and LHS 473. For the latter case, the non-detection in both the photosphere and the chromosphere indicates that LHS 473 is inactive, likely due to its low v sin i, and this result is consistent with previous observations showing its low coronal activity level. However, in the fast rotating case of Gl 890, the non-detection suggests that the field topology on this star may be different from what has been observed in more slowly rotating early-M dwarfs (Donati et al. 2008), whose fields are large-scale and dominantly toroidal producing Stokes V signatures detectable at any time. Since the detections of X-ray and radio emission indicate that Gl 890 is magnetically active, phase-resolved spectropolarimetric observations of this star are therefore needed to clarify its magnetic field topology.

For the three remaining M dwarfs, all these fast rotators show Zeeman signatures, and they are particularly strong in Gl 896B and 2E 4498 (Figure 1). This result indicates the presence of strong, large-scale fields on these stars. Since KP Tau (0.31 M_☉) and 2E 4498 (0.21 M_☉) are fully convective, they are good targets for studying the field morphology by carrying out spectropolarimetric and simultaneous multiple-wavelength observations (e.g., Berger et al. 2009). We note that Gl 896Bab is an unresolved binary. Therefore, its Zeeman signature is contributed by two components if both are magnetically active.

The G 164-31 magnetic field topology, which is mostly poloidal with most of the magnetic energy concentrated within the lowest order axisymmetric modes, is very similar to what has been observed in V374 Peg (Donati et al. 2006a) and other mid-M dwarfs (Morin et al. 2008). The average large-scale magnetic flux on G 164-31 is 0.68 kG, which is significantly smaller than the previous measurements of few kilogauss magnetic fluxes in active mid-M dwarfs using synthetic spectrum fitting techniques (e.g., Johns-Krull & Valenti 1996). This is possibly due to the ZDI technique being sensitive to only large-scale and simple structures, while Zeeman signatures from magnetic regions with complicated or small-scale topology may cancel each other in circularly polarized spectra whereas these signatures add up in unpolarized spectra. To explore this "missing" magnetic flux, we used the synthetic spectrum fitting technique described in Johns-Krull & Valenti (1996) to measure the mean magnetic flux on the surface of G 164-31. Briefly, spectra in the wavelength interval around the Zeeman-sensitive Fe I line at 8468.40 Å are analyzed. In stars as cool as G 164-31, this line is actually blended with a Ti i line at 8468.47 Å which is similar in strength and also Zeeman sensitive. The spectrum synthesis models both lines plus numerous TiO lines in this wavelength region. Because of the ubiquitous TiO, spectral changes due to magnetic fields are seen more clearly in the ratio of active to inactive line profiles. Therefore, spectra of G 164-31 were divided by inactive references. In our case, we used two inactive reference stars GJ 725B (or LHS 59, M3.5V) and GJ 876 (or LHS 530, M4V) which are meant to be identical in all respects (T_eff, log g, [M/H], etc.) to G 164-31 except for the field. Line profiles were synthesized with and without magnetic fields (see Johns-Krull & Valenti 1996 and references therein) and the ratio of active to inactive profiles was fitted by adjusting the magnetic field strength, B, and its filling factor, f, in the model spectrum of G 164-31. The line profiles are generated assuming a uniform radial field everywhere with a uniform filling factor.

The best fits were obtained with Bf = 3.1 kG and Bf = 3.3 kG for GJ 725B and GJ 876. The fit to the observations using GJ 876 as the inactive reference star is shown in Figure 5. Due to the high v sin i of G 164-31, we cannot measure B and f separately. The model fit in the line core is not as good as we would like, and is only slightly improved using GJ 725B as the reference star. We note though that if the radius of G 164-31 is indeed larger than expected due to youth (see below), its gravity is likely somewhat lower than that of either inactive comparison star. Our spectrum synthesis indicates that the 8468 Å feature should actually weaken somewhat at lower gravity, which would result in the discrepancy in the core getting worse by ∼0.005 in the ratio. A possible way to correct for this would be to fit a distribution of magnetic field strengths on the stellar surface as has been done for M dwarfs (Johns-Krull & Valenti 2000) and on T Tauri stars (e.g., Johns-Krull et al. 1999). The current fit (Figure 5) attempts to use a single field strength to fit both the wings and the core, whereas a distribution of magnetic fields gives more flexibility in fitting the entire profile and can result in bigger changes in the line equivalent width than produced when using only a single field value. We therefore adopt Bf = 3.2 ± 0.4 kG for G 164-31 as the best estimate from our single field fitting described above, noting that this may be a slight underestimate. This value for Bf is larger than the one measured from the ZDI technique by a factor of 4.7. Our result is consistent with the previous measurements from the literature (e.g., EV Lac; Johns-Krull & Valenti 1996; Morin et al. 2008), suggesting that in active mid-M dwarfs a significant part of the magnetic energy is stored in small-scale structures. As found by Reiners & Basri (2009), it appears that a large majority of the magnetic energy is stored in the small-scale field of G 164-31. Apparently, a successful dynamo model should be able to produce both large-scale poloidal fields and small-scale features in rapidly rotating mid-M dwarfs.

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To study chromospheric activity, we computed the Hα emission equivalent widths for each exposure. Our Hα emission light curve is shown in Figure 6. One should keep in mind that since each observation to obtain a Stokes V profile consists of four exposures, measuring Hα emission at individual exposures therefore increases time resolution of the light curve. Figure 6 also indicates that the Hα equivalent width variations are likely sinusoidal and they are modulated by the stellar rotation with P_rot = 0.54 days determined from the Stokes V profiles. This effect has also been observed in late-M and brown dwarfs (see Berger et al. 2009 and references therein). Flaring events were seen strongly in the last sequences of the first observing night and moderately in the middle sequences of the second night. These events were also observed in the Hβ emission profiles. While the likely rotational modulation and nonzero minimum of the Hα light curve imply either a large-scale (poloidal or toroidal) chromospheric field or a localized concentration of small-scale magnetic activity, it is reasonable to assume that the chromospheric field in G 164-31 has both a large-scale poloidal component and small-scale structures as observed in the photospheric field. This field configuration probably dominates in both the photosphere and the chromosphere of the star. We note that simultaneous observations at multiple wavelengths of the source will place more constraints on the chromospheric and coronal field morphology (e.g., Berger et al. 2009).

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It is very interesting to note that the radius of G 164-31 is at least 30% larger than that estimated from standard models for M dwarfs at the same luminosity (see Section 2.1.4). There are two possible interpretations of this discrepancy. First, the magnetic field effects of reduced convective efficiency due to fast rotation and large field strengths, and/or to spot coverage as proposed by Chabrier et al. (2007) might yield a cooler effective temperature T_eff and thus a larger radius in low-mass stars, keeping their luminosity unchanged (L ∝ T⁴_effR²). Our spot mapping (Figure 4) shows the fraction of the stellar surface covered by the low contrast spot is relatively small in G 164-31. However, one should note that the imaging technique is only sensitive to spots or spot groups with sizes comparable to our resolution element. We therefore cannot rule out the possibility of unresolved small spots spread everywhere on the stellar surface. Hence, we cannot conclude whether the effect due to spot coverage is significant in this star or not. Observational results (see Ribas 2006; Morales et al. 2008 and references therein) suggest that these effects might yield a ∼10%–15% larger radius in G 164-31, therefore the discrepancy in radius of over 30% cannot be explained by only those effects. Second, we explore the possibility that G 164-31 is a young M4 dwarf, its radius thus larger than one computed for old M dwarfs. We note that the K-band absolute magnitude of G 164-31, M_K = 6.73, is 0.67 mag brighter than the average value for M4 dwarfs,⁹ supporting the young nature of the star. From a literature search, the binary system G 164-31 (Gl 490B)+G 164-32 (Gl 490A) indeed belongs to a moving cluster previously identified by (Orlov et al. 1995; for a review on young nearby stars, see Zuckerman & Song 2004). One should note that the primary component G 164-32 also exhibits high coronal X-ray emission with log(L_X/L_bol)=−3.23 (Fleming et al. 1989) and chromospheric Hα emission (Stauffer & Hartmann 1986), indicating activity typical in young early-M dwarfs. We measured an upper limit for the Li λ6708 equivalent width of 17 mÅ for G 164-31. This gives the star an age older than 10 Myr since early-M dwarfs (⩾M4) are expected to completely deplete their lithium within ∼10 Myr. We suggest that G 164-31 is an analog of the M4 dwarf HIP 112312A at 23.6 pc (ESA 1997), which is a member of the ∼12 Myr old β Pictoris moving group (Song et al. 2002). To estimate the mass and age of G 164-31, we used the Chabrier et al. (2007) and Baraffe et al. (1998) theoretical models with constraints of M_K = 6.73, I − J = 1.61, and the deduced radius R_⋆ = 0.47 R_☉ of G 164-31, and we also considered the magnetic field effects that might yield about a 10%–15% larger radius. We then derived M_⋆ ∼ 0.15 M_☉ at an age of about 25–30 Myr, making G 164-31 the least massive M dwarf whose magnetic field has been mapped to date. More observations are needed to precisely determine the fundamental parameters of this moving cluster.