KEPLER MONITORING OF AN L DWARF I. THE PHOTOMETRIC PERIOD AND WHITE LIGHT FLARES

W1906+40 was observed with Kepler Director's Discretionary Time (Program GO30101) from 2011 June 28 to 2012 June 27 and with Guest Observer time (Program GO40004) from 2012 June 28 to 2012 October 3. Its parameters are listed in Table 1. It was assigned a new name for each quarter: KIC 100003560 (Quarter 10), KIC 100003605 (Quarter 11), KIC 100003905 (Quarter 12), KIC 100004035 (Quarter 13), and KIC 100004076 (Quarter 14). The target was observed in long-cadence mode (Jenkins et al. 2010b), providing 30 minute observations, for the full time period and also in 1 minute short-cadence mode (Gilliland et al. 2010) during Quarter 14. For each observation, the Kepler pipeline (Jenkins et al. 2010a) provides the pixel data, the time (t = BJD − 2454833.0, or mission days), the aperture flux (SAP_Flux), the uncertainty, and a corrected aperture flux (PDCSAP_Flux), which is meant to correct instrumental drifts (Stumpe et al. 2012). After excluding data flagged as bad (Quality Flag >0), there are 18,372 long-cadence measurements. The pipeline-estimated noise levels are 0.5%. The spacecraft is rotated between quarters and the focus is a compromise, so W1906+40 was observed on four different detectors with considerably different point-spread functions (PSFs). The Kepler pipeline therefore selects different apertures for each quarter, which in some cases we can improve upon. For Quarters 10 and 14, we adopt the PDCSAP_Flux photometry, which used an aperture of 9 pixels, because we found no significant improvement with smaller apertures. However, the default pipeline analysis used larger apertures for Quarters 11–13, which resulted in noisier photometry than Quarters 10 and 14. Furthermore, Quarter 13's PDCSAP_Flux is very low (<100 counts s⁻¹), which we found is due to two negative (bad) pixels included in the aperture. Smaller apertures also reduce the possibility of contamination by nearby background stars. We therefore adopted smaller apertures of 6 pixels for Quarters 11 and 12 and 4 pixels for Quarter 13 using the PyKE tool kepextract.⁸ Instrumental drifts for these three quarters were removed with third-order polynomial fits. Kepler photometry is not calibrated in any absolute sense since such calibration is not needed to meet the primary mission goals, so we divided each quarter's data by the median count rate for that quarter.

The Kepler data show a strong peak in the periodogram at a period of 0.3702 days. We emphasize that this period is present in each quarter, and in both the pipeline photometry and our revised (smaller aperture) photometry. This 8.9 hr period is consistent with the rotation period distribution deduced for early L dwarfs by their vsin i measurements (Reiners & Basri 2008). We fit a sine curve to all five quarters of long-cadence data with a Markov Chain Monte Carlo (MCMC) analysis and find the period is 0.370152 ± 0.000002 days with an amplitude of 0.00763 ± 0.00006 (8.3 mmag). The peak-to-peak variation is twice this (1.5%, or 16 mmag). Fitting each quarter separately, then averaging, yields a period of 0.37015 days with a standard deviation of 0.00005 days. The full phased dataset with sine curve is shown in Figure 1. Excluding the small number of outliers, whose nature are discussed in Section 4, the standard deviation about the fit is 0.65%, or 7 mmag. We adopt this as the noise level of the measurements, as it is only a slight increase over the expected value (0.5%) and there are other possible noise sources in the Kepler instrument (Caldwell et al. 2010), though real intrinsic variations may contribute. Figure 2 shows four days of Kepler data as an example of unphased data including a possible flare (Section 4.2); our radio observations (Section 2.3) were in this time period. The Kepler filter response curve is shown in Figure 3 along with the estimated relative count rate at each wavelength due to the L1 dwarf photosphere; possible contributions from flares are discussed in Section 4.2.

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W1906+40 was observed with the Gemini Multi-Object Spectrograph (Hook et al. 2004) on the Gemini North Telescope (Program GN-2012A-Q-37). The R831 grating, OG515 order blocking filter, and 10 slit were used to yield wavelength coverage 6276–8393 Å, with two small gaps between detectors that were interpolated over. We obtained sixteen 600 s exposures on UT Date 2012 July 24, 25 on 2012 July 29, and 6 on 2012 August 8. The flux calibration star BD+28 4211 was observed on 2012 July 29. The data were bias-subtracted, flat-fielded, wavelength-calibrated, extracted and flux-calibrated in the standard way using the Gemini gmos package in IRAF. A small correction was then applied to bring the spectra in agreement with the i-band magnitude.

We compared the Gemini spectra to the spectral type standards defined by Kirkpatrick et al. (1999) and find that W1906+40 is spectral type L1 (Figure 4.) It is not a low gravity L dwarf (Cruz et al. 2009). We see no evidence for lithium absorption in any of the spectra. The L1 optical spectral type agrees with the L1 near-infrared spectral type derived in the discovery report. Hα is variable on all nights but detected in emission in all spectra. On July 24 the Hα emission equivalent width increased from 0.5 Å at the beginning of observations to 7.5 Å at the end. On 2012 July 29, the Hα emission begins at 2.8 Å, but is then dominated by flares, discussed further in Section 4.1, as seen in Figure 4. On August 8, the first five spectra have an equivalent width of 2.5–3.0 Å, but the last increases to 6.5 Å. The average non-flare Hα equivalent width is 4 Å. The flares on July 29 overwhelm the small periodic signal, so we defer any attempt to identify small photospheric changes on the other nights to a future paper when additional simultaneous ground-based and Kepler data are available.

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W1906+40 was observed in clear and dry conditions on 2011 September 10 (UT) with the Keck II NIRSPEC near-infrared echelle spectrograph (McLean et al. 2000), as part of an ongoing search for radial velocity variables among nearby L dwarfs (Burgasser et al. 2012). The source was observed using the high-dispersion mode, N7 filter and 0432 × 12'' slit to obtain 2.00–2.39 μm spectra over orders 32–38 with λ/Δλ = 20,000 (Δv = 15 km s⁻¹) and dispersion of 0.315 Å pixel⁻¹. Two 120 s exposures were obtained in two nods separated by 7'' along the slit. We also observed the A0 V calibrator HD 192538 (V = 6.47) in the same setup for telluric correction and flux calibration, and internal quartz flatfield and NeArXeKr arc lamps for pixel response and wavelength calibration, respectively.

Spectral data were extracted from the images using the REDSPEC package,⁹ and we focused our analysis on order 33 which samples the strong CO band around 2.3 μm (Blake et al. 2008). We used REDSPEC and associated routines to perform pixel-response calibration, background subtraction, order identification, image rectification, and extraction of raw counts for both target and A0 V star. We also obtained an initial linear wavelength calibration using the arc lamp images, but performed no telluric or flux calibration. We then used these data to perform a forward-modeling analysis similar to that described in Blake et al. (2010). Our full methodology will be presented in a future paper, but in brief we explored a parameterized model of the raw spectrum

$$\begin{equation} S[p] = \left\lbrace (M[p(\lambda, {\rm RV})]{\star }K){\times }T[p(\lambda)]^{\alpha }{\times }C[p]\right\rbrace {\star }{\rm LSF} \end{equation} \tag{ 1 }$$

through MCMC methods. Here, M is a synthetic model chosen from the BT-Settl set of Allard et al. (2011), spanning 1800 K ⩽ T_eff ⩽ 2500 K in 100 K steps and 4.5 ⩽ log g ⩽ 5.5 (cgs) in 0.5 dex steps, and sampled at a resolution of 300,000 (Δv = 1 km s⁻¹); p(λ) is the conversion function from wavelength to pixel, modeled as a third-order polynomial; RV is the barycentric radial velocity of the star; K is the rotational broadening kernel defined by Gray (1992) which depends on vsin i; T[λ] is the telluric absorption spectrum, adopted from the solar absorption spectral atlas of Livingston & Wallace (1991); α scales T[λ] to match the observed telluric absorption; C is the flux calibration function, modeled as a second-order polynomial; LSF is the spectrograph line spread function, modeled as a Gaussian with width σ_G; and ⋆ denotes convolution. To facilitate rapid exploration of this 11-component parameter space, we performed the rotation convolution on all of the models in advance, sampling 5 km s⁻¹ ⩽ vsin i ⩽ 100 km s⁻¹ in 1 km s⁻¹ steps. Parameters were initiated by first fitting the A0 V observation; we then fit the W1906+40 spectrum using MCMC chains of length 1000 for each of the spectral models deployed (T_eff and log g), following standard procedures (e.g., Ford 2005) and using χ² as our cost function.

Figure 5 shows our best-fit model to the data, with T_eff = 2300 K, log g = 5.0 dex, heliocentric radial velocity of −22.5 km s⁻¹ (assuming a barycentric motion of −10.7 km s⁻¹) and vsin i = 11 km s⁻¹. The fit is good (reduced χ² = 2.9), and the strong blend of both stellar and telluric absorption features provides a robust framework for constraining both radial and rotational velocities. To determine estimates of each, we marginalized all of the chains for each parameter p using a probability function based on the F-test statistic

$$\begin{equation} \langle {p}\rangle = \sum \limits_i^{\rm all chains}{p_i\wp _i}, \end{equation} \tag{ 2 }$$

$$\begin{equation} \sigma ^2_p = \left(\sum \limits_i^{\rm all chains}{p_i^2\wp _i}\right) - \langle {p}\rangle ^2, \end{equation} \tag{ 3 }$$

where the probability function ℘ for each chain was determined from the F-test probability distribution function

$$\begin{equation} \wp _i \propto 1-F\left(\chi ^2_i/{\rm min}(\lbrace \chi ^2\rbrace),\nu,\nu\right) \end{equation} \tag{ 4 }$$

(see Burgasser et al. 2010) with min({χ²}) being the minimum χ² (best-fitting model), ν the degrees of freedom (# pixels − # parameters), and enforcing ∑℘_i = 1. From this calculation we determined RV = −22.6 ± 0.4 km s⁻¹ and vsin i = 11.2 ± 2.2 km s⁻¹ for W1906+40. The fitted T_eff is consistent with other L dwarf studies (see the reviews of Chabrier & Baraffe 2000; Kirkpatrick 2005), so we estimate the uncertainty in T_eff as ±75 K.

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The parallax and proper motion were obtained with the ASTROCAM (Fischer et al. 2003) imager at the 1.55 m telescope of the U.S. Naval Observatory, using the astrometric observing and reduction principles described in Vrba et al. (2004). Observations in J-band were obtained on 27 nights over the course of 1.23 yr. A reference frame employing 16 stars with a range of apparent J magnitude of 11.8–14.6 was employed. The conversion from relative to absolute parallax was based on using both Two Micron All Sky Survey (2MASS) and SDSS photometry to determine a mean distance of the reference frame stars of 2.20 mas. We caution that this is a preliminary parallax since, even with an excellent reference frame and a location in the sky giving a significant parallax factor in declination, an observational time baseline of 1.23 yr is not sufficient to completely separate parallax and proper motion. This object will be on the USNO infrared parallax program for the next 2–3 yr to establish a final parallax. This distance ($16.35^{+0.36}_{-0.34}$ pc) is in excellent agreement with that derived in the discovery paper (16.6 ± 1.9 pc) assuming the object to be single. If there is a unresolved companion, it is therefore unlikely to contribute significantly to the Kepler photometry. As shown in the Kirkpatrick (2005) review, an L1 dwarf like W1906+40 is most likely a hydrogen-burning star with age ≳ 300 Myr. The observed tangential velocity is greater than the median (30 km s⁻¹) for L1 dwarfs (Faherty et al. 2009), supporting an old age and stellar status. Both integrating the W1906+40 near-infrared spectrum and optical to mid-infrared photometry and comparison to other L1 dwarfs (Golimowski et al. 2004) gives BC_K = 3.22 ± 0.06, and therefore log L/L_☉ = −3.67 ± 0.03. Overall, our optical and near-infrared observations show that W1906+40 is a typical L1 dwarf, though the Hα and rotation rate places it in the more chromospherically active and slowly rotating half of the population. If the L dwarf rotation–age relation suggested by Reiners & Basri (2008) is correct, W1906+40 would be older than 5 Gyr.

W1906+40 was observed with the Karl G. Jansky Very Large Array (VLA)¹⁰ for 3 hr starting at 2012 February 23.46 UTC (Project 12A-088). The total observing bandwidth was 2048 MHz, with two subbands centered at 5000 and 7100 MHz and divided into 512 channels each. Standard calibration observations were obtained, with 3C 286 serving as the flux density scale and bandpass calibrator and the quasar J1845+4007 (4° from W1906+40) serving as the phase reference source. The total integration time on W1906+40 was 120 minutes.

The VLA data were reduced using standard procedures in the CASA software system (McMullin et al. 2007). Radiofrequency interference was flagged manually. After calibration, a deep image of 2048 × 2048 pixels, each 1'' × 1'', was created. The imaging used multi-frequency synthesis (Sault & Wieringa 1994) and CASA's multi-frequency CLEAN algorithm with two spectral Taylor series terms for each CLEAN component; this approach models both the flux and spectral index of each source. The rms residual of the deconvolution process around W1906+40 was 2.9 μJy bm⁻¹.

W1906+40 is detected at the 8σ level with flux density 23.0 ± 4.1 μJy at a mean observing frequency of 6.05 GHz. This corresponds to νL_ν = (4.5 ± 0.9) × 10²² erg s⁻¹ with spectral index (S_ν∝ν^α) α = −2.0 ± 1.4. The mean time of the observation is 2012 February 23 12:40:50 UTC which equals Kepler mission time 1148.026. To place the radio coverage in context, the time of the radio observations is marked on the corresponding Kepler photometry in Figure 2; the mean time of the radio observations corresponds to phase 0.91 in Figure 1, close to the minimum in the Kepler light curve. We searched for radio variability in W1906+40 by using the deep CLEAN component model to subtract all other detectable sources from the visibility data, then plotting the real component of the mean residual visibility as a function of time, rephasing the data to the location of W1906+40. No evidence for radio variability was seen. We imaged the LL and RR polarization components separately and found no evidence for significant circular polarization of the emission from W1906+40. W1906+40 is just the sixth L/T dwarf detected in quiescent radio emission (Berger 2002, 2006; Berger et al. 2009; Burgasser et al. 2013; Williams et al. 2013).

The W1906+40 discovery paper was based on the Preliminary Wide Field Infrared Explorer (WISE; Wright et al. 2010) data release. It appears as source WISE J190648.47+401106.8 in the final WISE All-Sky Source Catalog. The source was scanned 28 times over 1.9 days in 2010 April as part of the main WISE mission and again 34 times over 4.3 days in 2010 October in the Post-Cryo mission. We find that in the WISE All-Sky Single Exposure (L1b) Source Table and WISE Preliminary Post-Cryo Single Exposure (L1b) Source Tables, W1 is consistent between and within the two epochs, but the source apparently brightened from W2 = 11.23 ± 0.02 in 2010 April to a mean W2 = 11.12 ± 0.02 in October (with individual uncertainties ±0.03 mag). The October data also shows more scatter in W2 than expected, but higher precision mid-infrared observations would be needed to confirm variability. The WISE data are more than 680 periods before the beginning of the Kepler data.

The radius and inclination of W1906+40 are important parameters in interpreting the Kepler light curve. We can constrain them both using our observations. First, the radius is a function of the observed distance (and thereby the luminosity) and our effective temperature fit:

$$\begin{equation} R = 0.90 \pm 0.03 R_J \left(\frac{2300\, {\rm K}}{T_{{\rm eff}}}\right)^2 \left(\frac{d}{16.35\, {\rm pc}}\right). \end{equation} \tag{ 5 }$$

The uncertainty of 3% includes the uncertainty in the bolometric correction and photometry but not parallax or temperature uncertainty. Second, assuming the photometric period is the rotation period, we can also constrain the radius and inclination with our spectroscopic vsin i:

$$\begin{equation} R \sin i= 0.80 R_J \left({P}\over {0.37015 \, {\rm d\rm ays}}\right) \left({v \sin i}\over {11.2 \ {\rm km\ s^{-1}}}\right). \end{equation} \tag{ 6 }$$

The uncertainty in this value is 20% due to the vsin i uncertainty. Combining this constraint with Equation (5) suggests i ≈ 60°, but a range of inclinations are possible. Since the period is so strongly constrained, we can infer the posterior probability using three observations for the data (luminosity, effective temperature, and vsin i) and a three parameter model (luminosity, radius, and sin i). Our priors are flat in luminosity and radius, but proportional to sin i on geometric grounds. Using an MCMC chain of one million steps, and discarding a burn-in of 1000 steps, we find the posterior probability distribution shown in Figure 6. The inferred radius is R = 0.92 ± 0.07R_J. At the 95% confidence level, we find that sin i > 0.59, with the median value sin i = 0.83 and most likely value sin i = 0.90. The inferred radius is consistent with theoretical predictions of 0.8 ≲ R ≲ 1.0 R_J (Chabrier et al. 2000; Burrows et al. 2011).

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The most striking aspect of the Kepler light curve is that it has a consistent phase for over a year. Furthermore, while we cannot rule out deviations from a sine function or small changes in the shape, it appears that the light curve has no flat portion. The simplest explanation is that the rotating dwarf has a single feature (darker or brighter than the surrounding photosphere) that is always in view. To explore this possibility, we model the first year of data using the circular spot equations of Dorren (1987), which we implemented in Python with an MCMC code. We assumed linear limb darkening coefficients of 0.9 based on the predictions of Claret (1998) and Claret & Bloemen (2011). Many solutions are possible, and generally we can find single dark spot models with χ² similar to the sine models. Some possible solutions with completely dark spots with sin i consistent with Figure 6 and are listed in Table 2. Solutions with i ≈ 80° can be found, but they require a larger spot. The different light curve during Quarter 13 could be explained in large part in the single circular dark spot model by increasing the spot size (to ∼13° for i = 60°), or by lowering the latitude (to ∼70° for i = 60°). While attributing a long-lived feature to a single "spot" is attractive, it could also be modeled in other ways, such as multiple large dark spots at lower latitudes. An intriguing alternative spot model that does not favor large circumpolar spots was presented for active dwarfs by Alekseev & Gershberg (1996). In this "zonal model," it is assumed that a large number of small spots together form large dark bands symmetric around the equator. This naturally produces sinusoidal light curve even at sin i ≈ 1 without requiring us to be viewing polar spots.

The physical cause of the putative large spot at high latitude (or even multiple spots at low latitudes) might be a magnetic starspot (i.e., a region of cooler photosphere due to magnetic fields) or an cloud structure. Jupiter's Great Red Spot would produce a strong periodic signal at 9.9 hr even in unresolved data (Gelino & Marley 2000), though a single spot on W1906+40 would be at much higher latitude. Gelino et al. (2002) comment that small holes in a cloud layer would form bright (I-band) spots "similar to the '5 μm hot spots' of Jupiter." These might be distributed according to the zonal model. Harding et al. (2011) argue for a "magnetically driven auroral process" to produce optical variations in some ultracool dwarfs. We cannot determine whether the features are brighter or darker than the photosphere with the single-band Kepler data, so all of these possibilities are plausible. It is worth considering whether chromospheric emission line variations would explain the Kepler periods. Varying Hα emission from 0 to 10 Å equivalent width, even with associated bluer Balmer lines, would only produce a signal of ∼0.0015%, an order of magnitude too small to explain our data.

The stability of the period and amplitude makes a striking contrast with the L dwarf observations discussed in the Introduction. Besides the stable 8.9 hr period, we find no evidence of other periods, transient or not, in the data. To search for other periods of hours to days, we have calculated periodograms for every 3, 5, and 10 day period and found no other significant period in any of them. The long-term drifts we removed were typically 1%, and only reached 6% during Quarter 11, and these are consistent with the drifts in bright stars corrected by the Presearch Data conditioning (PDC) pipeline process. Although intrinsic slow changes in W1906+40 on timescales of months might be removed, it is likely that a strong signal, like the ∼30 day period M dwarf GJ 4099 (KIC 414293; Basri et al. 2011), would still be detected. Variations on timescales less than a week that are ≳ 1% would be detected. We conclude that the various "masking" periods and non-periodic variations reported in other L dwarfs at amplitudes >2%, and attributed to condensate cloud inhomogeneities, are not present in W1906+40. On the other hand, variations ≲ 1% can be plausibly be attributed to noise, but could also be due to real variations. (The data shown in Figure 2 are suggestive and typical.)

If the short-lived variability observed on non-active ultracool dwarfs is associated with condensate clouds, perhaps the long-lived periodic feature on this active L dwarf is associated with different, possibly magnetic, phenomena. A number of studies (Berger et al. 2005, 2009; Hallinan et al. 2006, 2008; McLean et al. 2011) have suggested that some ultracool dwarfs have stable magnetic large-scale topologies, like dipoles or quadrupoles. This could result in a large high-latitude feature. Hussain (2002) reviewed starspot lifetimes: while starspots on single, young main-sequence stars have typical lifetimes of a month, other types of starspots can have lifetimes of years or decades. Furthermore, an active longitude may remain more heavily spotted for years, even as individual spots come and go on shorter timescales. Of course, as Lane et al. (2007) remark, a cooler starspot on an L dwarf might have different cloud properties than the normal photosphere.

After the first 40 minutes of Gemini data on 2012 July 29 show relatively quiescent Hα emission, the subsequent spectra are dominated by a series of flares characterized by strong emission lines and a blue continuum. The first flare is in the 10 minute exposure beginning at UT 08:29:28, followed by a more powerful "main" flare at UT 10:14:15 (shown in Figure 4). Both of these flares are detected in the Kepler short-cadence photometry: the first flare peaks at mission time 1304.8597 and the main flare peaks at mission time 1304.9353. The first three main flare spectra, with the quiescent L dwarf photosphere subtracted, are shown in Figure 7 showing decay of the blue continuum. The atomic emission lines as a function of time throughout the night are shown in Figure 8 along with the simultaneous Kepler short-cadence photometry.¹¹ The Kepler light curve is in good agreement with the appearance of the spectroscopic white light continuum. Even with multi-wavelength data at high time resolution (see Gershberg 2005), flare spectra have proven to be challenging to interpret. Most notably, this L1 dwarf is able to produce a white light flare that is similar to that observed in M dwarf flare stars. Indeed, the flare spectrum is very similar to Fuhrmeister et al.'s (2008) flare in the M6 dwarf CN Leo (Wolf 359), including the white light, broad Hα, and atomic emission lines. The flare also resembles the most powerful observed flares in M7 (Schmidt et al. 2007) and M9 (Liebert et al. 1999) dwarfs, demonstrating continuity in the flare properties across the M/L transition. The white light and emission line light curves are qualitatively very similar to the 1984 March 28 AD Leo flare described by Houdebine (1992) as "impulsive/gradual."

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The main (1304.9395) flare shows both atomic emission lines and continuum emission with a blue slope. The higher time resolution photometry shows that the flare began 4 minutes into the Gemini UT 10:14:15 exposure, and reached its Kepler maximum in the final 2 minutes. The "white light" flare continuum spectrum is consistent with an 8000 ± 2000 K blackbody. In Allred et al.'s (2006) radiative hydrodynamic simulations, an electron beam injected at the top of a magnetic loop heats the chromosphere, producing emission lines in general agreement with observed flares, but does not penetrate to the denser photosphere, and white light emission is not produced. As Kowalski et al. (2010) discuss in the case of a flare on the M4.5 dwarf YZ CMi, evidently the W1906+40 photosphere must somehow be heated to produce ∼8000 K continuum emission. This is also seen in the older models of Cram & Woods (1982), where white light continuum emission requires moving the temperature minimum deeper and heating the photosphere (see their Models 4–6). Free-free emission from a hot (T ≈ 10⁷ K) plasma is also consistent with our white light flare spectrum but is disfavored by Hawley & Fisher's (1992) analysis of M dwarf flares. The Kepler photometry and our calibration (Section 4.2, Table 3) indicates that the total (included extrapolated to ultraviolet) flare energy release during the Gemini 10:14:15 UT exposure was ∼5 × 10³¹ erg. The photometric light curve is typical of those observed in UV Ceti (M dwarf) flare stars as summarized by Gershberg (2005): it is asymmetric, with a rapid rise, an initial fast decrease, and a subsequent slow decay which began when the flare was at ∼0.2 of the maximum. The flare lasts 106 minutes in the Kepler photometry. Using the calibration discussed in detail in Section 4.2, the observed flare energy through the Kepler filter during this time is 6.4 × 10³¹ erg, which may be extrapolated to ∼1.6 × 10³² erg for the full ultraviolet/optical range.

The white light in the main flare is accompanied by strong, very broad Hα emission, especially evident in the top spectrum of Figure 7. Most stellar flares do not show much Hα broadening (Houdebine 1992). If modeled with two Gaussians, as sometimes done for M dwarf flares (Eason et al. 1992; Fuhrmeister et al. 2008), the narrower Gaussian component has FWHM  ∼  6 Å and the broader Gaussian component has FWHM  ∼  25 Å. Very broad Hβ and bluer Balmer flare lines are usually attributed to Stark broadening, but Eason et al. (1992) argued in the case of a UV Ceti flare that broad Hα is better explained by turbulent motions. Fuhrmeister et al. (2008) also interpret broad Hα flare emission in the M6 dwarf CN Leo as due to turbulent motions, in addition to Stark broadening, possibly due to "a chromospheric prominence that is lifted during the flare onset and then raining down during the decay phase." (Their observations have higher time and spectral resolution.) This interpretation could also be applied to the W1906+40 flare. Zirin & Tanaka (1973) noted broad (FWHM 12 Å) Hα emission from the kernels of a great solar flare; these are also responsible for the white light emission and show turbulent motions (Zirin 1988). Detailed modeling should be pursued, however, as very broad Hα emission wings are caused by Stark broadening in some Cram & Woods (1982) models along with the white light (Models 5 and 6), while Kowalski et al. (2011, 2012) found broad Hα absorption associated with a white light flare on YZ CMi reminiscent of an A star.

As the blue continuum and broad Hα decays, the Hα line stays strong but narrows. Atomic emission lines from He i (6678 Å, 7065 Å, 7281 Å) and O i (7774 Å) are easily visible. These emission lines remain strong for 2 hr (Figure 8). One hour after the main flare at mission times 1304.97–1304.99, Hα strengthens again, perhaps due to another flare, creating a double peaked time series profile, though only a small increase in the spectroscopic blue continuum and Kepler photometry occur. Subtracting the quiescent photospheric spectrum, we also identify emission lines from neutral K and Na which fill in the core of the photospheric absorption features. These spectra are similar to red M dwarf flare spectra (Fuhrmeister et al. 2008), and are understood as the result of the increase in electron density and heating of the chromosphere (Fuhrmeister et al. 2010). The much longer duration of the emission lines is not surprising, and is consistent with solar flares (Zirin 1988) and the statistics of SDSS spectra of M dwarf flares (Kruse et al. 2010; Hilton et al. 2010). The flare luminosity for the first 100 minutes in units of 10²⁸ erg is 480, 13, and 3.5 in Hα, He i (6678 Å), and O i (7774 Å). During the time period covered by the first three Gemini spectra (Figure 7), 80% of the Kepler luminosity was emitted but only 60% of the O i, 50% of the He i and 40% of the Hα luminosities.

The first (1304.8597) flare was also a powerful one in its own right. In the first Gemini spectrum of the first flare, Hα strengthens to equivalent width 30 Å, which we can correct to 42 Å given the timing of the Kepler photometry. In the next 10 minute exposure, Hα has increased to 55 Å. When the quiescent photosphere is subtracted, a blue continuum is present in the first exposure and the flare is clearly detected in the Kepler photometry. We are unable to say if this flare, which would otherwise be the strongest spectroscopically observed L dwarf flare, is a physically related to the main flare almost 2 hr (a fifth of a rotation period) later. Overall, the complex time evolution, with the first flare 2 hr before and a secondary flare 1 hr after the main flare, suggests the emergence and reconfiguration of a complex, multi-loop magnetic structure. This may be consistent with the L dwarf flare theory sketched out by Mohanty et al. (2002) in which flux tubes generated in the warmer interior rise rapidly into the cooler atmosphere, but in any case the flares appear very similar to flares on hotter stars, suggesting similar mechanisms. Although probable flares in early L dwarfs have been noted before (Liebert et al. 2003; Schmidt et al. 2007; Reiners & Basri 2008), they were seen in the Hα line only. The July 29 flares are both stronger in Hα than in those flares, in addition to showing other atomic emission lines and white light.

There are a total of 21 candidate white light flares in the Quarter 14 short-cadence data that peak at least 10% above the average W1906+40 count rate and are elevated for two or more measurements (Table 4). Kepler time series photometry of these flares are shown in Figure 9 The strongest candidate flares include data points flagged as likely cosmic rays by the Kepler pipeline, but we believe this is incorrect. The sharp rise of the two July 29 flares are also flagged as possible cosmic rays (see the discussion in Jenkins et al. 2010b), but the Gemini spectroscopy proves they are flares. The centroid of the main flare peak shifts by 0.4 pixels not because it is a off-center cosmic ray, but because the positions are wavelength dependent, and (blue) flare photons outnumber (red) photospheric photons more than three to one. As the flare decays the centroid returns to normal. We therefore accept flare-like light curves even if the sharp rise triggered the cosmic ray flag.

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It can be useful to discuss flares in terms of energies instead of relative counts. Walkowicz et al. (2011) and Maehara et al. (2012) have estimated the energy of white light flares on G, K and M dwarfs observed by Kepler by approximating the flare as a 10,000 K blackbody. The Kepler mission filter is similar to g, r, and i combined (Koch et al. 2010): as illustrated in Figure 3, the L dwarf photosphere contributes mainly at the very reddest wavelengths, but hot flares will contribute through the entire filter. Indeed, the effective wavelength for the Kepler photometry for W1906+40 is 7990 Å rather than ∼6200 Å for hot sources. In Table 3, we present energy calculations of hypothetical flares that have the same observed count rate as W1906+40, assuming the flares radiate isotropically. We use the i magnitude, the distance derived in Section 2.2, and the L1 dwarf spectrum shown in Figure 3 to estimate that the L dwarf luminosity is 1.4 × 10²⁸ erg s⁻¹ in the wavelength range 4370–8360 Å. The blackbody energies for equal count rates, 2.1 × 10²⁸ erg s⁻¹, are 50% higher because the average energy per photon is larger. Thus, to double the observed Kepler count rate, one needs a flare of 1.3 × 10³⁰ erg (over that wavelength range) during a short-cadence exposure and 3.8 × 10³¹ erg during a long-cadence exposure. Most of the flare energy will be emitted at shorter optical and ultraviolet wavelengths. The corresponding bolometric luminosities for the blackbodies are 4.4–6.4 × 10²⁸ erg s⁻¹. Real flares are not blackbodies, so we also estimated the total luminosities by consulting Table 6 of Hawley & Pettersen (1991), which gives energies from 1200 Å to 8045 Å for the impulsive and gradual phases of great flare of 1984 April 12 on AD Leo. The results are close to the blackbody approximations.

Seven of the flares last at least an hour. Most rise to their peak value within a few minutes, and drop to half their peak flux in 1–3 minutes. We list the rise time to the peak in Table 4) as t_r, the time to drop to half the flux as t_1/2, and the total length of the flare as t_L. The flare at mission day 1298.5401 is an exception with, a 23 minute rise time and slow decay. Overall, both the typical shape and the variety of the light curves is consistent with stellar flares (Gershberg 2005). We also list the peak long-cadence count rate in Table 4; most short-cadence flares are enhancements of 2% or less in the 30 minute data and therefore would not be significant detections without the short-cadence data. The cosmic ray flagging may have a significant effect on real flares in the long-cadence data: the long-cadence flux for the Gemini-confirmed 1304.9353 flare is half that we obtain by integrating the short-cadence flux, and the discrepancy is due to the removal of "cosmic rays" in the long-cadence pipeline. We have searched the Quarters 10–13 data for other candidate flares; however, flares are easily confused with cosmic rays, and we have concluded that we cannot reliably distinguish between flares and cosmic rays using the long-cadence data. We nevertheless mention two interesting candidates. At mission time 1148.407, the target increases by 48%, decaying to 10% in the next exposure; this would represent a flare of energy ∼1.4 × 10³² erg. (This is the time period also plotted in Figure 2.) These points are not flagged as cosmic rays and the pixel data appears consistent with the stellar PSF, so it is likely to be a real flare. The most powerful long-cadence–only flare candidate is at mission time 1039.049: W1906+40 nearly doubles in brightness, then decays over the next 2 hr back to the normal brightness. The pipeline, however, flags the initial increase as a possible cosmic ray strike, and if this is correct, the following "decay" simply represents the return of the CCD's sensitivity to normal. Flare rates should be viewed with caution for the lowest energy flares, since the contribution of a long-lived tail can be lost in the noise, and flares with low peaks will be missed altogether even in the short-cadence data. There is no apparent dependence of the flares on the rotation phase. It is also notable that four of the eight most powerful Quarter 14 flares might be related, the pair 1304.8597/1304.9353 and the pair 1336.1666/1336.1939.

In Figure 10, we plot the flare energy spectrum (Lacy et al. 1976) as the estimated flare energy as a function of the cumulative frequency ($\tilde{\nu }$). Additional monitoring would be needed to improve the frequency estimates and to check if the flare rate varies. Comparison to well studied dMe flare stars (Lacy et al. 1976; Gershberg 2005) indicates the flare energy spectrum is similar to that observed in flare stars but shifted downward for W1906+40: flares are ∼10–100 times less energetic at a given frequency, or alternatively flares of a given energy are less frequent on W1906+40 than on dMe flare stars. On the other hand, the white light flare rate on W1906+40 is comparable to that of the Sun (Neidig & Cliver 1983; Gershberg 2005), despite the L dwarf's much lower effective temperature and surface area. The observed rate for flares with energy >10³¹ erg is 10⁻³ to 10^−2.5 hr⁻¹, or 1–2 per month. For comparison, the Berger et al. (2010) limit on Hα-only L dwarf flares is a cumulative frequency of $\log \tilde{\nu } \,(\rm {hr}^{-1}) \lesssim -1.4$; those flares are too weak to be detected in the broad Kepler filter, because a change of 10 Å would change the photometry by only ∼0.0015%.

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