Online material
Appendix A: Observations and data reduction
We collected linear polarimetry images of TVLM 513−46 using the Andalucía Faint Object Spectrograph and Camera (ALFOSC) mounted on the 2.56 m Nordic Optical Telescope (NOT) on 2013 May 18 and 19. ALFOSC has a 2048 × 2048 E2V detector with a pixel size of 0.̋19. The target was monitored in the I band (λ_{c} = 810 nm) over ~3.5 and ~4.2 h during the first and second nights, and in the R band (λ_{c} = 631 nm) over ~1 h during the second night. We thus covered ~4 rotation cycles in the I band and half a cycle in the R band. Sky conditions were clear and seeing varied in the interval 1.̋1−1.̋5. Observations were carried out at air masses ranging from 1.05 through 2.34.
The linear polarimetry observing mode of ALFOSC consists of a halfwave plate and a calcite block, which provides simultaneous images of the ordinary and the extraordinary beams separated by 15′′. The total unvignetted field of view is 140′′ in diameter, which we rotated by 103° east of north to align TVLM 513−46 and a nearby, bright reference star along the yaxis of the detector, as indicated in Fig. A.1. Individual exposure times were 50 s (I, first night) and 100 s (R and I, second night) per position of the half wave plate (0°, 22.̊5, 45°, and 67.̊5). On May 18, polarimetric images were taken at two nod positions on the detector separated vertically by 10′′ for a proper sky subtraction. This nodding pattern was not applied on the following night since TVLM 513−46 is detected well above the sky contribution with a S/N of ~220 and ~300 (I band) in 50 s and 100 s individual integrations, respectively (S/N is measured at the peak of the energy distribution of the source images). One polarimetric cycle was completed every 6.2 min (I, first night) and 9.6 min (R and I, second night), including overheads. This allowed us to sample one rotation of TVLM 513−46 using a minimum of 12 and a maximum of 19 linear polarimetry measurements. Raw images were biassubtracted and flatfieldcorrected using data acquired during dawn and dusk and routines within IRAF^{1}. We also observed nonpolarized stars (GJ 838.4 and WD 1615−154, Fossati et al. 2007) and one polarized source (Hilter 997, Whittet et al. 1992) by employing the same instrumental configuration and doing so on the same observing dates as the target. This allowed us to check for the efficiency of the instrument and to set an upper limit on the instrumental polarization.
Fig. A.1
ALFOSC linear polarimetric Iband image of TVLM 513−46. The circle indicates the unvignetted area. 

Open with DEXTER 
Fig. A.2
Stokes q − u plane for the R and Iband measurements of TVLM 513−46, taken during first and second night of observations. Instrumental polarization is shown with a white circle. The typical uncertainties in q and u are indicated. 

Open with DEXTER 
Fig. A.3
Top: LombScargle periodograms of the intensity (left) and polarimetric (right) light curves of TVLM 513−46 (red) and the reference star (black). Bottom: relative probabilities of the explored periods using a Bayesian formalism for the generalized LombScargle periodogram (see text). 

Open with DEXTER 
We derived the normalized Stokes parameters q and u, the degree of linear polarization (P), and the polarization vibration angle (Θ) using the fluxratio method and equations 1−4 from Zapatero Osorio et al. (2005). The Stokes parameters and the angle Θ were properly corrected for the rotation of the field of view. Fluxes of all ordinary and extraordinary images were extracted by defining circular apertures ranging between 0.5 and 6 times the size of the fullwidthathalfmaximum (FWHM) and using the PHOT package within IRAF. Sky annuli had inner radii ranging from 3.5 × to 6 × FWHM and widths of 1.5 × FWHM. The detailed procedure is described in MilesPáez et al. (2013). Our final q, u, P, and Θ values result from averaging the aperture photometry in the interval 2−4 × FWHM; the errors associated with q and u are defined as the standard deviations of the measurements in the selected range of apertures; the errors in P are computed as the quadratic sum of the q and u quoted uncertainties plus the uncertainty introduced by a possible instrumental linear polarization (see below); and errors in Θ are determined following the equations given in MilesPáez et al. (2013) and Wardle & Kronberg (1974). We caution that the expression for the Θ uncertainty is valid for P/σ_{P} ≥ 3, where σ_{P} is the error in linear polarimetry. Our linear polarimetric data of TVLM 513−46 have a typical uncertainty of ±0.46% (R) and ±0.35% (I) in P, and ±9° in Θ.
Using the observations of the zeropolarized standard stars, we checked that the instrumental polarization lies below 0.31% and 0.27% in the R and Ibands. The polarized standard star allowed us to determine the zero point correction for the polarization vibration angle to be Θ_{°,R} = 0.̊8 ± 2.̊8 and Θ_{°,I} = 1.̊3 ± 3.̊0, which agrees with the values tabulated in the ALFOSC manual^{2}. From now on, we shall use the debiased linear polarization degree, p^{∗}, defined as (A.1)which accounts for an overestimation of the polarization signal at low S/N or small values of P/σ_{P}(Wardle & Kronberg 1974). We set p^{∗} = P if σ_{P} ≥ P. In Figs.2, 3, and A.4, these polarimetric data are displayed with an associated arrow indicating that the true polarization index most likely lies between 0% and the symbol position. We note that the dispersion of consecutive Θ measurements is of the order of the uncertainty in the polarization vibration angle when P/σ_{P}> 2.7, which is close to the regime where the calculation of Θ has statistical significance (Wardle & Kronberg 1974). This provides support to our method for deriving the error bars associated with q, u, and P or p^{∗}. The individual Stokes parameters values for the R and I filters are shown in the q − u plane of Fig. A.2.
Fig. A.4
R and I band differential intensity (top), linear polarimetry (middle), and polarization vibration angle (bottom) curves folded in phase using a periodicity of 1.9798 h. Symbols as in Fig. A.2. In the top panel, each data point stands for the average of four individual measurements and its associated dispersion. In the bottom panel, Θ is plotted for P/σ_{P} ≥ 2.7. Two phases are presented for clarity. 

Open with DEXTER 
In addition to the linear polarimetric evolution, we retrieved the R and Iband intensity curves of TVLM 513−46 by means of differential photometry. The star 2MASS J15011008+2250069 (2.05 mag brighter in I and located near our target) acted as the reference/comparison source. This is the only bright star in the ALFOSC field of view that was usable for the differential photometry technique (Fig. A.1). Because of its lower luminosity, the error bars in the differential intensity curves are dominated by the photon noise of TVLM 513−46. The typically 1.0 × FWHMaperture fluxes of the ordinary and extraordinary images of TVLM 513−46 and its reference star were extracted with the VAPHOT package (Deeg & Doyle 2001), which works under the IRAF environment and is optimized for differential light curves. To build the intensity curves, ordinary and extraordinary fluxes were
combined per individual frame, which provides a time sampling four times higher than the cadence of the linear polarimetry data. We estimated the uncertainties in the differential light curves to be ±3.4 mmag (I band) for the first night photometry, and ±1.6 mmag (I) and ±9 mmag (R) for the second night data. We checked that the measured linear polarization degree and the differential intensity light curves do not correlate with airmass.
The LombScargle periodograms of the I band intensity and polarimetric light curves of TVLM 513−46 computed following Lomb (1976) and Scargle (1982) are illustrated in the top panels of Fig. A.3. The periodogram of the polarimetric light curve of the reference star is also included in the figure. We explored frequencies between ω = 2π/T and ω = 2N_{o}/T with a spacing of about 1 /T(Horne & Baliunas 1986), where N_{o} is the number of data points of the light curves (296 for the intensity curve, and 74 for the polarimetric curve), and T is the total time coverage of the data (26.4 h). We thus surveyed periods in the range 0.3−26.4 h using 433 and 88 independent frequencies for the intensity and polarimetric data, respectively. The confidence levels of the 2h peaks were estimated using the falsealarmprobability (1−FAP) at 99% (intensity light curve) and 83% (polarimetry). The bottom panels of Fig. A.3 depict the relative probabilities according to the Bayesian formalism described in Mortier et al. (2015).
© ESO, 2015