The χ Factor: Determining the Strength of Activity in Low‐Mass Dwarfs

Low‐mass stars are the most populous objects in our Galaxy, comprising roughly 70% of the stars in the solar neighborhood and almost half of the Galactic stellar mass (Henry et al. 1997). Until the last decade, the low luminosities of these stars prevented their study by large‐scale spectroscopic surveys. However, efforts such as the Palomar/Michigan State University Nearby‐Star Spectroscopic Survey (PMSU; Reid et al. 1995a, 2002; Hawley et al. 1996; Gizis et al. 2002), Two Micron All Sky Survey follow‐up (2MASS; Gizis et al. 2000a, 2000b), and New Luyten Two‐Tenths Catalog follow‐up (NLTT; Reid et al. 2004; Lépine et al. 2003) have characterized global properties of nearby M dwarfs, with samples numbering ≳1000 objects. Recently, the Sloan Digital Sky Survey (SDSS) has made it possible to study spectroscopic samples approaching 10,000 low‐mass stars (West et al. 2004).

In particular, these surveys have provided insight into the magnetic behavior of low‐mass stars. There are many diagnostics by which a star may be deemed to be magnetically active, including X‐ray and far‐ultraviolet (FUV) emission lines, but in optical wavelengths, Hα is the most readily available signature of activity. Although Hα in either emission or absorption may indicate the presence of a chromosphere, most large‐scale optical studies have used the more restrictive definition of Hα in emission. This practice is due to the difficulty in breaking the degeneracy between intermediate activity and truly weak chromospheres, both of which present as weak Hα absorption (see Giampapa et al. [1982] and Gizis et al. [2002] for a discussion of this phenomenon). In this paper, we use "active" to mean the presence of Hα in emission.

Increasingly large studies of active M dwarfs have built on one another to create our current picture of activity at the bottom of the main sequence. Originally, Joy & Abt (1974) showed a rise in the fraction of active M dwarfs toward later spectral types, with 100% of M5 stars showing activity. The PMSU investigation of Hawley et al. (1996) extended the trend with much better statistics and found that the fraction of active stars peaked at ∼60% at spectral type M5.5. Gizis et al. (2000b) observed a sample of later type stars from 2MASS and found that the incidence of activity rose to 100% at type M7, but declined at later types. Burgasser et al. (2000) added additional late M and L dwarfs to the sample and confirmed that the active fraction decreases past type ∼M7–M8. Most recently, West et al. (2004) used a much larger sample of low‐mass dwarfs from SDSS to further refine the trend. These data reveal a curiously elegant bell curve distribution for the incidence of magnetic activity in low‐mass stars. The fraction peaks at ∼73% of M8 stars being active.

Several of these studies consider the strength of activity, which is quantified as the ratio of the luminosity in Hα to the bolometric luminosity. The activity strength remains relatively constant (although with large scatter) at log (L_Hα/L_bol)∼ - 3.7 through the early‐ to mid‐M spectral types and appears to decline at types later than M6, while the active fraction is still rising. The increasing fraction of active stars may be an age effect (lower mass stars retain their activity longer) on the rising branch (M0–M8). The activity‐strength results indicate that some damping mechanism apparently sets in around type M6. At types later than M8, this damping mechanism (possibly related to the atmospheric temperature; see Mohanty et al. 2002) overwhelms the age effect and results in fewer active dwarfs at lower atmospheric temperature. The activity fraction and activity‐strength results, taken together, provide important constraints on models of magnetic dynamos and magnetic field production in M and L dwarfs, particularly as these objects span the mass range in which the stars become fully convective and develop degenerate cores.

Quantifying the strength of activity as the ratio between Hα luminosity and bolometric luminosity, while informative, is prone to systematic error. Determining luminosities depends on having well‐known distances, and as samples get larger (e.g., from SDSS), the stars are typically well beyond the immediate solar neighborhood (i.e., >25 pc from the Sun). Thus, precise trigonometric parallax measurements are generally not available, as in previous local surveys. In principle, one should be able to calculate the ratio L_Hα/L_bol and expect the distance dependencies to cancel out. However, in order to get L_bol, one must have determined M_bol, which relies on knowing M_V and having excellent color measurements and a reliable bolometric correction. The value of M_V generally comes from distance estimates based on spectroscopic and/or photometric parallax determinations, which are not always reliable or in agreement, primarily because of a paucity of calibration data at the bottom of the main sequence. Calculating L_Hα, on the other hand, relies on knowing the flux in the continuum near Hα. This is often obtained from the photometric colors (Reid et al. 1995b). Alternatively, it can be measured directly from spectrophotometrically calibrated data. Neither method is necessarily congruous with that used to obtain L_bol. Thus, in practice, the ratio usually contains two distances, typically determined in different ways, leading to possible systematic error.

An alternative method is to determine the ratio of the flux in the continuum near Hα to the bolometric flux. This ratio, which we call χ, can then be multiplied by the equivalent width in Hα for a given star to give the ratio L_Hα/L_bol. If one has excellent‐quality data, the ratio f_λ6560/f_bol can be calculated for each of the stars observed. In this paper, we use a well‐observed sample of nearby M dwarfs and early L dwarfs to determine χ as a function of several colors (I_C−K_s, V−I_C, r−i, and i−z) and of spectral type, in effect providing an improved "bolometric correction" for low‐mass dwarfs.

We drew our sample of calibrating stars for early spectral types (M0.5–M6.5) from the nearby 8 pc sample (Reid & Gizis 1997; Reid & Hawley 2000, Appendix A; Reid et al. 1995b; Hawley et al. 1996) and used a sample of cool standard stars from 2MASS for later spectral types (M4.5–L0; Gizis et al. 2000a; Kirkpatrick et al. 1999, 2000; Reid et al. 1999). The overlap of the two samples at midrange spectral types (M4.5–M6.5) ensured that there were no systematic differences in f_λ6560/f_bol for the two samples. The 8 pc sample has been well characterized photometrically (in the UBVR_CI_C Johnson‐Cousins filters, and also near‐IR J, H, and K filters) and spectroscopically. Spectra for the 8 pc sample were largely obtained using the spectrograph on the Palomar 60 inch (1.5 m) telescope, with a 600 lines mm⁻¹ grating, blazed at 6500 Å, giving a dispersion of 1.5 Å pixel⁻¹ and coverage from ∼6200 to 7400 Å. A 1^'' slit was used, giving a resolution element of ∼2.5 pixels. Most stars fainter than V = 16 were observed using the double spectrograph on the Palomar 200 inch (5.1 m) Hale telescope, using the 6800 dichroic as a beam splitter, with the blue camera set to cover 6200–6800 Å at a resolution of ∼0.55 Å pixel⁻¹ and the red camera covering 6900–7500 Å at ∼0.6 Å pixel⁻¹. A 1^'' slit gives a resolution of ∼3 pixels. Many of the 2MASS standards were observed using the low‐resolution imaging spectrograph (LRIS) on the Keck I 10 m telescope on Mauna Kea, Hawaii. The 1200 lines mm⁻¹ grating, centered at λ = 6800Å, yielded coverage from 6120 to 7500 Å at 0.5 Å pixel⁻¹. With a 1^'' slit, this gave a resolution of 1.5 Å. Visible photometry for the sample of late spectral types was not readily available, so I_C magnitudes were found by convolving the spectra with a Cousins I_C filter response curve. Infrared magnitudes for the late‐type objects were obtained from the 2MASS database. All spectra for the objects in our sample were spectrophotometrically calibrated in the region near Hα.

The method used to find χ is straightforward. Photometric data for the 8 pc sample were used with the bolometric corrections of Leggett et al. (1996) to find apparent bolometric magnitudes for the sample. The bolometric magnitudes m_bol were then transformed to apparent bolometric fluxes, f_bol. The values of the apparent continuum flux near Hα, f_λ6560 (defined as the mean between 6555 and 6560 Å), were measured directly from the calibrated spectra. For the late‐type stars, the spectra were convolved with a Cousins I_C filter response, and K_s (the short K filter used by 2MASS) was transformed to K_UKIRT (using relations from Carpenter [2001]) for use in the bolometric correction relations. Bolometric corrections were calculated using the Leggett et al. (2001) relation in I_C−K_UKIRT, which is more reliable for late spectral types. Bolometric fluxes were computed as for the earlier spectral types, and continuum fluxes at Hα were measured directly from the spectra. Combining the samples, we obtain χ = f_λ6560/f_bol over the entire spectral range. The mean values of χ by spectral type are shown in Table 1, while specific values for each object in the sample are given in Table 2. Figure 1 shows the data and fit to χ in I_C−K_s and V−I_C, and equations (1) and (2) provide the fits explicitly. Comparison in the overlap region of M4.5 V to M6.5 V shows no difference in the behavior of χ for the two samples.

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We are particularly interested in being able to use χ with our large sample of SDSS spectra. Unfortunately, transformations between the SDSS filter system and other common astronomical systems do not extend into the very red color range we require here. To address this problem, we computed fits between average colors by spectral type in traditional and SDSS filters. These fits were used to transform the colors for our sample onto the SDSS system. In § 3, Figure 2 shows the color transformations used, while equations (3) and (4) provide the relations. Figure 3 shows the χ results for the SDSS r−i and i−z colors, and the fits are given in equations (5) and (6).

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Equations (7) through (10) and Figures 4 and 5 provide fits to f_bol in terms of I_C, K_s, I_C−K_s, and i−z, for use by anyone wishing to obtain, for example, L_X/L_bol for a large sample of stars.

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After having obtained χ across the spectral range, we computed the best fit to the distribution as a function of color. Fits to log χ in V−I_C and I_C−K_s are presented below. Mean values for χ were computed for each spectral type, as shown in Table 1, enabling an estimate of an object's χ value in the event that color data are not readily available or cannot be synthesized.

The relationship between log χ and color can be expressed by the polynomials

for color ranges 1.89≤I_C−K_s≤5.12 and 1.91≤V−I_C≤4.40, respectively.

Obtaining fits in terms of SDSS filters was somewhat more complicated. We computed transformations between traditional and SDSS colors using average colors by spectral type. The fits between SDSS and Cousins‐2MASS colors, r−i to V−I_C and i−z to I_C−K_s, are shown in Figure 2. The fits obtained are

Using these relations, we transformed our sample onto the SDSS system and obtained fits to log χ in terms of r−i and i−z:

for color ranges 0.92≤r−i≤2.27 and 0.47≤i−z≤2.00, respectively.

In addition, we also provide fits to f_bol as a function of I_C, K_s, I_C−K_s, and i−z for our full sample:

In order to study the strength of magnetic activity as a function of spectral type, we require a distance‐independent way of determining the ratio L_Hα/L_bol. We used a sample of nearby stars and cool standards from 2MASS, spanning spectral types M0.5 to L0, with color ranges of 1.89≤I_C−K_s≤5.12 and 1.91≤V−I_C≤4.40. For the 8 pc sample stars, we obtained the bolometric flux from photometry combined with bolometric corrections. For the 2MASS standards, optical photometry was obtained by convolving the spectra with filter response, then bolometric corrections were applied. The flux in the continuum near Hα was obtained directly from the spectra. In both cases, we computed SDSS photometry for the sample using color transformations provided in § 3. We then found the ratio χ = f_λ6560/f_bol. In § 3, we provided fits to χ as a function of color (V−I_C, I_C−K_s, r−i, and i−z). Table 1 gives average χ values at each spectral type in the event that photometry is not available, while Table 2 provides χ values for all stars in our sample. The χ ratio can be multiplied by the measured equivalent width in Hα for a given star to yield the activity‐strength ratio L_Hα/L_bol. If the reader possesses excellent‐quality data, the method described here can also be used to calculate f_λ6560/f_bol for individual program stars.

This method has already yielded cleaner results for activity‐strength calculations in West et al. (2004), better illuminating trends in magnetic behavior at the end of the main sequence. Improved color transformations and a larger sample of stars with well‐calibrated photometry and spectra are required to reduce the scatter in these relations.