Near-infrared spectroscopy of M dwarfs. III. Carbon and oxygen abundances in late M dwarfs, including the dusty rapid rotator 2MASSI J1835379+325954^†

Abstract

Carbon and oxygen abundances of eight late M dwarfs are determined based on the near-infrared spectra of medium resolution (R = λ/Δ λ ≈ 20000). In late M dwarfs, dust forms in their photospheres for T_eff below about 2600 K, and this case applies to the M8.5 dwarf 2MASSI J1835379+325954 (hereafter 2MASS 1835+32) whose T_eff is 2275 K. The other seven objects with T_eff above 2600 K are analyzed with dust-free models. For the case of 2MASS 1835+32 analyzed with the dusty model, the surface temperature is higher by about 600 K due to the blanketing effect of the dust grains, mainly composed of iron grains, and the carbon and oxygen abundances are higher by 0.25 and 0.15 dex, respectively, compared to the analysis by the dust-free model. Once dust forms in the photosphere, the dust works as a kind of thermostat and the temperatures of the surface layers remain nearly the same as the condensation temperatures of the dust grains. For this reason, the temperatures of the surface layers of dusty dwarfs are relatively insensitive to the fundamental parameters, including T_eff. In addition, it appears that 2MASS 1835+32 is a rapid rotator, for which its equivalent widths (EWs) are thought to remain unchanged by the rotational broadening. This is, however, true only when the true continuum is well defined. Otherwise, the pseudo-continuum level depends on the rotational velocity and hence the EWs as well. For this reason, the derived abundances depend on the rotational velocity assumed: for the values of V_rotsin i = 37.6 and 44.0 km s⁻¹ available in the literature, the derived carbon and oxygen abundances differ by 0.23 and 0.14 dex, respectively, and we find that the higher value provides a better account of the observed spectrum. The resulting carbon and oxygen abundances in the eight late M dwarfs show no systematic difference from our results for the early and middle M dwarfs, and confirm the higher A_O/A_C ratio at the lower metallicity. In late M dwarfs, CO and H₂O remain as excellent abundance indicators of carbon and oxygen, respectively, except for additional uncertainty due to complexity associated with dust formation in the latest M dwarfs.

1 Introduction

The late M dwarfs are the last stars that shine by their own nuclear burning in their interiors and mark the end of the main sequence. It was shown that the lower limit of the stellar mass that maintains a high enough central temperature to ignite hydrogen burning is 0.08 M_⊙ (Hayashi & Nakano 1963; Kumar 1963). In recent years, more attention has been directed to the objects below the hydrogen-burning limit, referred to as brown dwarfs, rather than to the lowest mass hydrogen-burning objects, the latest M dwarfs. At the end of the 20th century, extensive searches for a brown dwarf have been attempted by several groups with different methods, and a genuine brown dwarf, Gliese 229B, was finally discovered by Nakajima et al. (1995) with immediate spectroscopic verification by Oppenheimer et al. (1995). Since then, many brown dwarfs were discovered by the large-scale surveys such as DENIS (Epchtein et al. 1999), 2MASS (Skrutskie et al. 2006), SDSS (York et al. 2000), UKIDSS (Lawrence et al. 2007), and WISE (Wright et al. 2010).

In observations, however, it is by no means easy to determine the stellar mass accurately, and hence it is not clear whether some of the latest M dwarfs are really stars or young brown dwarfs. Anyhow, some of our objects are situated at the boundary between stars and brown dwarfs, and we hope that a detailed spectroscopic study of such objects at relatively high resolution would be of some use in extending detailed spectroscopic analyses to very low mass objects. In fact, high resolution spectroscopy was already applied to a cool brown dwarf more than a decade ago by Smith et al. (2003), who observed the nearby T dwarf ϵ Indi Ba with a resolution as high as R ≈ 50000, and showed the advantages of high resolution spectroscopy over the other works mostly based on low resolutions. Thus, the very cool dwarfs including brown dwarfs are already within the capability of present-day high resolution infrared spectroscopy and we are even too late to work on M dwarfs.

Now, returning to M dwarfs, one problem in the latest M dwarfs is that dust may form in their photospheres. In fact, it was shown that the thermochemical conditions for the formation of dust grains such as corundum, iron, and enstatite are well met in the photospheres of M dwarfs with T_eff below about 2600 K, and actual model photospheres in thermal and convective equilibria incorporating dust formation have been generated (Tsuji et al. 1996a). The predicted spectra based on the dusty models appeared to be consistent with the known observation of the infrared spectrum of the latest M dwarf LHS 2924, while the dust-free models predicted H₂O bands to be too deep compared with the observed spectrum.

The evidence for dust has more clearly been shown in a peculiar cool dwarf GD 165B discovered by Becklin and Zuckerman (1988). This object showed a very red color but molecular bands were not as strong as expected for the possible low temperature suggested by the very red color. Such a characteristic could have been explained nicely by our dusty model (Tsuji et al. 1996b): the very red color is due to infrared excess caused by dust, which also weakens the molecular absorption bands at the same time. This object was later recognized as a brown dwarf and regarded as the prototype of L-type dwarfs (Kirkpatrick et al. 1999). Also, an analysis of optical spectra has shown additional spectroscopic evidence for dust in late M dwarfs (Jones & Tsuji 1997). Since then, the problem of dusty photospheres has been studied, mostly in connection with brown dwarfs, both observationally and theoretically (see, e.g., an extensive survey of such efforts by Helling et al. 2008a). More recently, modeling of cool dwarfs has made significant progress, extending to the cooler brown dwarfs and extra-solar planets (see reviews, e.g., Helling & Casewell 2014; Marley & Robinson 2015) on one hand, and also to self-consistent cloudy models of late M and L dwarfs (Witte et al. 2011) on the other.

In addition to the problem of dust, some of the late M dwarfs are known to be rapid rotators. One of our targets, 2MASS 1835+32, which was recognized as a nearby late-type M dwarf only recently (Reid et al. 2003), is also found to be a rapid rotator (Reiners & Basri 2010; Deshpande et al. 2012). In late M dwarfs in which molecular lines are already blended with each other, the rapid rotation further smears out the blended spectra. We have developed a method to analyze the spectrum without the true continuum by referring to the pseudo-continuum which is depressed appreciably by molecular bands (Tsuji & Nakajima 2014; Tsuji et al. 2015, hereafter Papers I and II, respectively). We find that our method can be extended quite well to the case of the pseudo-continuum smeared out further by the rapid rotation, and also can be applied to determine the rotational velocity more consistently.

In this paper, we first introduce our observed data (section 2) and then we discuss fundamental parameters such as effective temperatures of late M dwarfs (section 3). We then generate model photospheres, including the dusty case, and examine the effect of dust on the thermal structures and spectra with our simple dusty models (section 4). We first analyze seven non-rapid rotators, whose effective temperatures are found to be above 2600 K and hence can be assumed to be dust-free (section 5). Then, we examine the rotational broadening on the spectrum of 2MASS 1835+32 (section 6). We examine the effects of dust formation, rotational velocity, and fundamental parameters on our abundance analysis (section 7). Finally, we discuss our results on the carbon and oxygen abundances in M dwarfs, dust in late M dwarfs, and the coolest end of the main sequence (section 8).

2 Observations

We observed eight late M dwarfs between M4.5 and M8.5 as listed in table 1. Observations were carried out at the Subaru Telescope on 2014 August 31 (UT) using the echelle mode of the Infrared Camera and Spectrograph (IRCS) (Kobayashi et al. 2000) with adaptive optics. Six of the eight targets were bright enough in the visible range to be natural guide stars, while LP 412-31 and 2MASS 1835+32 were too faint and laser guide stars were used. The slit width of 0${}_{.}^{\prime \prime }$14 was sampled at 55 mas pixel⁻¹, and the resolution was about 20000 at K. The echelle setting was “K⁺,” which covered about half of the K window with the orders, wavelength segments, and pixel-wise dispersions given in table 2. The targets were nodded along the slit, and observations were taken in an ABBA sequence, where A and B stand for the first and second positions on the slit. The total exposure time ranged from 24 s (GJ 873) to 80 min (LP 412-31). The night was photometric at the beginning, but the condition gradually degraded and it was spectroscopic toward the end of the night. The visible seeing was better than 0${}_{.}^{\prime \prime }$5 throughout the night. Near midnight, a rapidly rotating B8 V star, HR 326, was observed as the calibrator of telluric transmission. Signal-to-noise ratios of reduced spectra are given for the 28th and 25th orders in table 1. H₂O and CO lines are analyzed in the 28th order (aperture 2), and H₂O lines are also analyzed in the 25th order (aperture 5).

Data reduction was carried out using the standard IRAF¹ routines in the imred and echelle packages. After extraction of one-dimensional spectra, wavelength calibrations were calculated using telluric absorption lines in the spectra of HR 326. After wavelength calibrations of one-dimensional spectra of the A and B positions, they were co-added to produce combined spectra. The combined spectra were normalized by the pseudo-continuum levels and then calibrated for telluric absorption using the spectra of HR 326.

3 Fundamental parameters

Angular diameter measurement by interferometry is not yet extended to late M dwarfs, and we apply the M_3.4–log T_eff relation [M_3.4 is the absolute magnitude at 3.4 μm based on the WISE data (Wright et al. 2010)] introduced in Paper I, with a slight modification for the latest M dwarfs.

3.1 Effective temperatures

For the M dwarfs with M_3.4 ≲ 9.0 (dM4.5–dM6) in our present sample, effective temperatures are estimated by the use of the M_3.4–log T_eff relation shown by the dashed line in figure 1 of Paper I (a part of which is reproduced in figure 1a by a dashed line). This method works well for these M dwarfs and the results for four objects, GJ 54.1, 873, 1002, and 1245B, are given in table 3.

For the later M dwarfs of M_3.4 > 9.0 (dM7–dM8.5), however, a dashed line (figure 1a) used to estimate T_eff was defined only by four objects, GJ 406, 644C, 752B, and 3849, whose effective temperatures were determined by the infrared flux method (Tsuji et al. 1996a). We estimate the physical parameters of these four objects in table 4 and examine the results in comparison with those of the evolutionary models by Baraffe et al. (1998). As shown in the lower right corner of the HR diagram in figure 17 of Paper I and reproduced in figure 1b, three of these four objects (GJ 406, 644C, and 3849) agree well with the theoretical HR diagram by Baraffe et al. (1998) shown by a solid line. However, one object (GJ 752B) deviates appreciably from the theoretical HR diagram. For this reason, we adopt these three objects (GJ 406, 644C, and 3849) as reliable calibration objects, and use these three objects to define a revised M_3.4–log T_eff relation shown by a solid line in figure 1a. The resulting values of T_eff based on the revised M_3.4–log T_eff relation are given in table 3 for GJ 752B, GAT 1370, LP 412-31, and 2MASS 1835+32.

3.2 Surface gravities

For estimating the surface gravity, radius and mass are needed. For this purpose, the effective temperature vs. radius relation given by equation (8) and the mass vs. radius relation given by equation (10) of Boyajian et al. (2012) are applied to the dM4.5–dM6 dwarfs and the results are given in table 3 together with the resulting values of log g.

For the later dM7–dM8.5 dwarfs, we try to extend the relations of Boyajian et al. (2012) by the use of our calibration stars in table 4. For this purpose, the mass is estimated with the use of the mass–luminosity (M_K) relation by Delfosse et al. (2000) extended to lower masses (M ≲ 0.1M_⊙) with the use of the theoretical mass–M_K relation by Baraffe et al. (1998), as shown in figure 2. This extension should be reasonable, since the empirical (dashed line) and theoretical (dotted line) mass–luminosity relations agree very well for M > 0.1M_⊙ in figure 2.

With the observed bolometric flux based on the photometry, the bolometric luminosity is derived with the known parallax, and with the T_eff based on the infrared flux method, the radius is estimated. The radius vs. effective temperature relation given by equation (8) of Boyajian et al. (2012) shown by a dashed line in figure 3 is extended to the lower temperatures with our calibration stars Nos. 1, 2, and 4 in table 4, as shown by a solid line in figure 3. It is to be noted that our empirical data for three calibration stars agree rather well with the theoretical relation based on the evolutionary models of the solar metallicity (Baraffe et al. 1998) shown by a dotted line in figure 3.

The radius vs. mass relation given by equation (10) of Boyajian et al. (2012) shown by a dashed line in figure 4 is extended to the lower masses (M ≲ 0.1M_⊙) with the use of the calibration stars Nos. 1, 2, and 4 in table 4, as shown by a solid line in figure 4. The dashed and solid lines are joined at M ≈ 0.15 M_⊙. For comparison, the relation predicted by the evolutionary models of the solar metallicity (Baraffe et al. 1998) is shown by a dotted line in figure 4. Again, the empirical relation based on our three calibration objects agrees rather well with the theoretical result of Baraffe et al. (1998).

Now, the radii and masses of our late M dwarfs are estimated with the use of the solid lines in figures 3 and 4, respectively, and the results for GJ 752B, GAT 1370, LP 412-31, and 2MASS 1835+32 are given in table 3 together with the resulting values of log g.

4 Model photospheres of late M dwarfs

We apply dust-free model photospheres for the M dwarfs with T_eff ≳ 2600 K (subsection 4.1). Since our sample now includes the latest M dwarf in which dust may form, we examine dusty models in some detail (subsection 4.2), and the effects of dust on the thermal structures (subsection 4.3) and spectra (subsection 4.4) are discussed.

4.1 Dust-free model photospheres of M dwarfs

The T_eff values of our seven program stars are higher than 2600 K (table 3), and we apply the dust-free models of case C to these seven objects. Case C includes Ca series based on the abundance case a (see table 1 of Tsuji 2002) adopting the classical solar C and O abundances (Anders & Grevesse 1989) and Cc series on the abundance case c, adopting the downward-revised C and O abundances (Allende Prieto et al. 2002). In our first iteration on CO spectra, we use the models of the Ca series from our UCM (Unified Cloudy Model) database.² For further iterations, we generate the specified model for each object based on T_eff and log g given in table 3, and the models are either Ca or Cc series depending on the carbon abundance determined by the first iteration on CO (section 5). These models are designated as Ca or Cc/T_eff/log g: for example, Ca3160c508 implies a model of Ca series, T_eff = 3160 K, and log g = 5.08. The micro-turbulent velocity is kept to 1 km s⁻¹ throughout.

4.2 Dusty model photospheres of the latest M dwarfs

Dust may form in the photosphere of M dwarfs with T_eff ≲ 2600 K (Tsuji 2002) and, since the T_eff value of our object 2MASS 1835+32 is found to be 2275 K (table 3), we consider the effect of dust in our analysis of this M8.5 dwarf. In late M dwarfs, however, dust formation has just started and only a small amount of dust grains should be formed. For this reason, we assume the simplest LTE (local thermodynamical equilibrium) model that dust forms everywhere so long as the temperature is lower than the condensation temperature, T_cond (T ≲ T_cond), and we refer to such a model as a fully dusty model of case B (Tsuji 2002). However, such a fully dusty model of case B is most difficult from the viewpoint of generating non-gray convective models, possibly because very large dust opacities compared with the gaseous opacities dominate throughout the upper layers. For this reason, we could not complete a grid for case B before, but computed only one sequence of log g = 5.0 (Tsuji 2002). We now need to generate some models of case B for the analysis of our target star and we consider this problem again in this section.

If dust forms, such an iterative procedure turns out to be unstable, possibly because very large dust opacities are highly sensitive to the changes of the physical conditions in general and also appear suddenly at T = T_cond. In such a case, we find that it is useful to temper the variation of F_rad(τ) somewhat. We prepare a small grid of the fully dusty models of case B to be applied to late M dwarfs; for T_eff between 2000 and 2600 K, log g = 5.0, 5.25, and 5.5, and abundances of case a and case c. The results are added to our UCM database.² We generally aim at a flux constancy within 1%, but flux errors as large as 3% remain in cooler models near T_eff = 2000 K in case B.³

4.3 Effect of dust on the thermal structures

In figure 5, the thermal structures of the fully dusty models (blue or black) are compared with those of the dust-free models (light sky or gray) for T_eff = 2600, 2300, and 2000 K (abundance case c and log g = 5.25). The condensation lines of corundum (Al₂O₃), iron (Fe), and enstatite (MgSiO₃) are shown by dashed lines. The dust-free models penetrate to the regime of T ≲ T_cond in the surface layers and the thermodynamical condition for dust formation is fulfilled in all the models shown in figure 5. Thus corundum first forms and temperatures are elevated somewhat by its blanketing effect. Then iron forms, and because of the larger abundance of iron, it has a dominant effect: the temperatures of the upper layers coincide with the condensation line of iron as shown in figure 5. This is a result that the photosphere is forced to be in radiative equilibrium, and it seems that iron works as a kind of thermostat; if a temperature T_f within the condensation line of iron in our model is increased by perturbation, then the iron grains will evaporate and the temperature will decrease because the blanketing effect of the iron grains disappears. But, if the temperature goes down below T_f, iron grains again form and the temperature will increase because of the blanketing effect of the iron grains. Such processes repeat and the temperature will eventually settle at the starting value T_f. As a result, the temperatures converge to the condensation line of iron and the thermal structures of the dusty models in the surface layers are not much different for late M dwarfs of different effective temperatures, in marked contrast to the dust-free models in which surface temperatures show large differences for different effective temperatures, as shown in figure 5. It is to be noted, however, that enstatite could not be formed since the photospheres are warmer than the condensation temperatures of enstatite because of the blanketing effect of iron and corundum.

4.4 Effect of dust on the spectra

The predicted spectra based on the dusty and dust-free models discussed in the previous subsection are shown in figure 6. In the model spectra of T_eff = 2600 K shown in the top of figure 6, the difference of the dusty (shown by blue or black) and dust-free (light sky or gray) cases is rather minor: the dusty model shows slightly weaker absorption only in the regions of strong absorption (e.g., at 2.5 μm) because of its higher surface temperature. Although the temperatures of the surface region are about 400 K higher in the dusty model than in the dust-free model (see figure 5), the matter density in the surface region is very low and this region has little effect on the emergent spectra except for the very strong absorption formed in the very surface. For this reason, we neglect the effect of dust for M dwarfs with T_eff ≈ 2600 K (e.g., GJ 752B, LP 412-31).

In the model spectra of T_eff = 2300 K shown in the middle of figure 6, the effect of dust is considerable, since the larger part of the surface region is now warmer (by about 600 K) in the dusty model than in the dust-free model (see figure 5). In particular, strong bands such as H₂O 1.9 and 2.7 μm bands as well as CO first overtone bands formed in the surface layers suffer an appreciable effect, while weak lines such as at the 2.2 μm region formed in the deeper layers show minor change. However, dust has a considerable effect even on the weak lines on the higher resolution spectra, as will be shown in section 7.

In the case of T_eff = 2000 K shown in the bottom of figure 6, the effect of dust is more drastic: the spectrum of the dusty model shows an overall excess compared with the dust-free model in the K band region and this is of course compensated for by the large extinction due to dust in the shorter wavelength region. The very strong absorption bands due to H₂O and CO in the dust-free model are much weaker in the dusty model because of the elevated temperatures in the larger part of the photosphere and also by the dust extinction. But T_eff = 2000 K is already in the regime of L dwarfs, and late M dwarfs are still free from such drastic effects of dust.

5 Seven M dwarfs of non-rapid rotators

Seven M dwarfs in our sample except for 2MASS 1835+32 are not rapid rotators, and also the effect of dust may not be important (T_eff > 2600 K). Then, CO and H₂O lines are analyzed by mini curves-of-growth (subsection 5.1) and by synthetic spectra (subsection 5.2), as in Papers I and II.

5.1 The mini–curve-of-growth analysis

5.1.1 First iteration on the CO blends

We analyze the CO blends of the 2–0 band listed in table 7 of Paper I with the solar carbon abundance of case a as an initial starting value for each object. Equivalent widths (EWs) are measured by referring to the pseudo-continuum, and the resulting values of log W/λ are given in table 5. We then apply the mini–curve-of-growth (CG) method from the beginning (i.e., not applying the conventional analysis as used in the first iteration on CO in Paper I).

The mini-CG method was developed through our Papers I to II. Briefly summarizing, we first generate synthetic spectra for an assumed initial abundance log A₀ and corrected abundances log A₀ ± δ. Then, equivalent widths, W's, are evaluated from these synthetic spectra in the same way as the observed EWs are measured, referring to the pseudo-continuum which is evaluated accurately by the use of the recent line-list of H₂O (Barber et al. 2006; Rothman et al. 2010). In the first iteration, we apply the models from the UCM grid and assume δ = 0.3. Then we measure equivalent widths, W's, on the predicted spectra calculated with the starting initial values of log A_C = −3.40⁴ and log A_O = −3.08 (case a for the initial model of the Ca series) and with log A_C = −3.40 ± δ and log A_O = −3.08 ± δ (δ = 0.3).⁵ These EWs are designated as log W(δ)/λ. Then, we generate a mini-CG plotting log W(δ)/λ against δ = −0.3, 0.0, and +0.3 for each blend. With this mini-CG, the observed EW is converted to the abundance correction to the assumed abundance of carbon. For more details about the mini CG method, see subsections 4.1 and 6.3 of Paper II.

This process is repeated blend by blend for all the EWs of CO measured in table 5. The resulting abundance corrections to the starting value of log $A_{\mathrm{C}}^{(0)}=-3.40$ for all the measured EWs given in table 5 are shown in figure 7 for our seven program stars. The mean abundance correction $\mathrm{\Delta }\mathrm{l}\mathrm{o}\mathrm{g}{A}_{\mathrm{C}}^{(1)}$ and the resulting carbon abundance log $A_{\mathrm{C}}^{(1)}=-3.40+\mathrm{\Delta }\mathrm{l}\mathrm{o}\mathrm{g}{A}_{\mathrm{C}}^{(1)}$ are given in the fifth and sixth columns of table 6, respectively, for each object.

5.1.2 First iteration on the H₂O blends in region B

Since H₂O lines are sufficiently strong in region B (defined in figure 3 and detailed in subsection 3.4 of Paper II) for late M dwarfs, we first analyze the H₂O blends in region B listed in table 4 of Paper II. EWs measured by referring to the pseudo-continuum are given in table 7.

Given that the carbon abundance is now known to be log $A_{\mathrm{C}}^{(1)}$ from the analysis of CO in the preceding subsection, we apply the specified model for each object noted in the second column of table 8. Assuming an initial starting value of log $A_{\mathrm{O}}^{(0)}=\mathrm{l}\mathrm{o}\mathrm{g}{A}_{\mathrm{C}}^{(1)}+0.30$⁠,⁶ the mini CG is generated from the synthetic spectra for log $A_{\mathrm{O}}^{(0)}$ and log $A_{\mathrm{O}}^{(0)}\pm \delta$ ⁷ for each object. Since this starting value of oxygen may be fairly good, as can be expected from our experience so far, we now apply the mini-CG method with δ = 0.1 instead of 0.3. The resulting oxygen abundance corrections $\mathrm{\Delta }\mathrm{l}\mathrm{o}\mathrm{g}{A}_{\mathrm{O}}^{(1)}$ are shown in figure 8. The mean abundance correction $\mathrm{\Delta }\mathrm{l}\mathrm{o}\mathrm{g}{A}_{\mathrm{O}}^{(1)}$ and the resulting oxygen abundance log $A_{\mathrm{O}}^{(1)}=(\mathrm{l}\mathrm{o}\mathrm{g}{A}_{\mathrm{C}}^{(1)}+0.30)+\mathrm{\Delta }\mathrm{l}\mathrm{o}\mathrm{g}{A}_{\mathrm{O}}^{(1)}$ are given in the third and fourth columns of table 8, respectively, for each object.

5.1.3 Second iteration on the CO blends

Starting from log $A_{\mathrm{C}}^{(1)}$ and log $A_{\mathrm{O}}^{(1)}$ in tables 6 and 8, respectively, we proceed to the second iteration on the carbon abundance, using the specified model for each object noted in the seventh column of table 6. The resulting second carbon abundance corrections $\mathrm{\Delta }\mathrm{l}\mathrm{o}\mathrm{g}{A}_{\mathrm{C}}^{(2)}$ show more or less similar patterns to those in figure 7. The second carbon abundance correction $\mathrm{\Delta }\mathrm{l}\mathrm{o}\mathrm{g}{A}_{\mathrm{C}}^{(2)}$⁠, which is generally smaller than $\mathrm{\Delta }\mathrm{l}\mathrm{o}\mathrm{g}{A}_{\mathrm{C}}^{(1)}$⁠, and the resulting carbon abundance log $A_{\mathrm{C}}^{(2)}=\mathrm{l}\mathrm{o}\mathrm{g}{A}_{\mathrm{C}}^{(1)}+\mathrm{\Delta }\mathrm{l}\mathrm{o}\mathrm{g}{A}_{\mathrm{C}}^{(2)}$⁠, are given in the eighth and ninth columns of table 6, respectively, for each object. Although we do not illustrate the results as in figure 7, the decrease of the probable errors in the resulting abundances (compare columns 6 and 9 in table 6) confirms that the accuracy of the analysis is improved by the use of the better starting values in the second iteration.

5.1.4 Second iteration on the H₂O blends in region B

Starting from log $A_{\mathrm{C}}^{(2)}$ and log $A_{\mathrm{O}}^{(1)}$ in tables 6 and 8, respectively, the second oxygen abundance corrections $\mathrm{\Delta }\mathrm{l}\mathrm{o}\mathrm{g}{A}_{\mathrm{O}}^{(2)}$ are obtained and the results also show more or less similar patterns to those of figure 8. The second oxygen abundance correction $\mathrm{\Delta }\mathrm{l}\mathrm{o}\mathrm{g}{A}_{\mathrm{O}}^{(2)}$ and the resulting oxygen abundance log $A_{\mathrm{O}}^{(2)}=\mathrm{l}\mathrm{o}\mathrm{g}{A}_{\mathrm{O}}^{(1)}+\mathrm{\Delta }\mathrm{l}\mathrm{o}\mathrm{g}{A}_{\mathrm{O}}^{(2)}$ are given in the fifth and sixth columns, respectively, of table 8 for each object.

5.1.5 Further iteration on the H₂O blends in region A

From our experience in Paper II, we first thought that it is sufficient to analyze the H₂O blends only in region B for late M dwarfs. However, it appears that the H₂O lines in region B are rather weak and noisy in the hottest object in our sample, GJ 873 (see figure 11a). Also, it appears that the spectrum of region B is disturbed by an unknown cause in the coolest object, LP 412-31, in our sample (see figure 11g). For these reasons, we decide to analyze the H₂O blends in region A (defined in figure 2 and detailed in subsection 3.3 of Paper II), where the H₂O lines are stronger. EWs are measured for the H₂O blends listed in table 2 of Paper II and the results are given in table 9.

We start from log $A_{\mathrm{C}}^{(2)}$ and log $A_{\mathrm{O}}^{(2)}$ in tables 6 and 8, respectively, and oxygen abundance corrections $\mathrm{\Delta }\mathrm{l}\mathrm{o}\mathrm{g}{A}_{\mathrm{O}}^{(3)}$ are obtained. The results are shown in figure 9. The oxygen abundance correction $\mathrm{\Delta }\mathrm{l}\mathrm{o}\mathrm{g}{A}_{\mathrm{O}}^{(3)}$ and the resulting oxygen abundance log $A_{\mathrm{O}}^{(3)}=\mathrm{l}\mathrm{o}\mathrm{g}{A}_{\mathrm{O}}^{(2)}+\mathrm{\Delta }\mathrm{l}\mathrm{o}\mathrm{g}{A}_{\mathrm{O}}^{(3)}$ are given in the third and fourth columns, respectively, of table 10 for each object. Note that the values of $\mathrm{\Delta }\mathrm{l}\mathrm{o}\mathrm{g}{A}_{\mathrm{O}}^{(3)}$ are generally less than the probable errors, and this fact confirms that the resulting oxygen abundances from region B are already well determined. However, the value of $\mathrm{\Delta }\mathrm{l}\mathrm{o}\mathrm{g}{A}_{\mathrm{O}}^{(3)}$ for GJ 873 is rather large compared with the probable error. This may be because the H₂O lines in region B are too weak in GJ 873 and hence their analysis cannot be accurate as noted above.

5.2 Synthetic spectra

5.2.1 CO 2–0 bandhead region

Based on log $A_{\mathrm{C}}^{(2)}$ and log $A_{\mathrm{O}}^{(2)}$ in tables 6 and 8, respectively, CO spectra near the 2–0 bandhead region are evaluated with the use of the specified models for our seven objects. The resulting predicted spectra, convolved with the slit function (Gaussian) of FWHM = 16 km s⁻¹, are compared with the observed ones in figure 10. Note that the observed and predicted spectra are both normalized by their pseudo-continua.

5.2.2 H₂O in region B

Based on log $A_{\mathrm{C}}^{(2)}$ and log $A_{\mathrm{O}}^{(2)}$ in tables 6 and 8, respectively, H₂O spectra in a part of region B (about one-fourth of region B) are evaluated with the use of the specified models given in the second column of table 8. The resulting predicted spectra are compared with the observed ones in figure 11. The resulting χ² values are given in the last column of table 8. We again find the correlation of the χ² values with the S/N ratios of the observed spectra (region B is in the same echelle order as CO bands). The χ² value of LP 412-31 is again very large and this is due to unknown disturbance⁸ on the observed spectrum, as seen in figure 11g (hatched region).

5.2.3 H₂O in region A

Based on log $A_{\mathrm{C}}^{(2)}$ and log $A_{\mathrm{O}}^{(3)}$ in tables 6 and 10, respectively, H₂O spectra in region A are computed with the use of the specified models for our seven objects. The resulting predicted spectra are compared with the observed ones in figure 12. The χ² values for the fit are given in the sixth column of table 10. The χ² values are large for GJ 752B, GJ 1002, and GAT 1370, and the S/N ratios for these objects are again high (table 1, fourth column). The χ² value for LP 412-31 is quite small, and this may reflect the very low S/N ratio on one hand and good overall fit without major defect in this region of the observed spectrum on the other.

6 Rotational broadening

In the later M dwarfs, it is known that the fraction of rapid rotators tends to be larger (e.g., Mohanty & Basri 2003; Jenkins et al. 2009). Our object, 2MASS 1835+32, is also found to be a fast rotator: the projected rotational velocity of 2MASS 1835+32 was measured to be 44.0 ± 4.0 km s⁻¹ (Reiners & Basri 2010) and 37.6 ± 5.0 km s⁻¹ (Deshpande et al. 2012). We adopt the mean value of 40.8 km s⁻¹ in the following analysis, and examine the effect of the uncertainty in the projected rotational velocity in subsection 7.4. So far, we have neglected rotational broadening in our analysis of M dwarfs, since it can be negligible compared to the rather large instrumental broadening of our medium resolution spectrograph (FWHM ≈ 16 km s⁻¹). In the case of 2MASS 1835+32, the rotational broadening is much larger than the FWHM of the slit function, and we neglect the width of the slit function for simplicity in the following analysis.

For comparison, the same high resolution spectrum is convolved with the slit function (Gaussian) of FWHM = 16 km s⁻¹, and the result is shown in figure 13b. Comparison of figures 13a and 13b reveals that most CO blends with reference numbers 1 to 10 are smeared out by the rotation, and the identities of these blends are lost. Only blends with reference numbers 11–15 conserve their identities and we will use the blends of reference nos. 12–15 in our analysis of CO. The blend of reference no. 11 is not used because the observed profile does not meet our criterion to accept a blend for our analysis (at least one wing reaches higher than half of the line depth). It is to be noted that the pseudo-continuum level of the rotationally broadened spectrum (figure 13a) is lower than that broadened by the slit function of the spectrograph (figure 13b). In figure 13c, the observed spectrum of 2MASS 1835+32 (dots) is compared with the rotationally broadened spectrum from figure 13a (thick line) after being renormalized by the pseudo-continuum.

Next, we examine the effect of rotation on the H₂O spectrum. For this purpose, we select the region between 22510 and 22660 Å, a part of region B noted in Paper II (see its table 4). The high resolution spectrum is evaluated with the use of the line-list by BT2-HIGHTEMP2010 (Barber et al. 2006; Rothman et al. 2010) and the result is shown by a thin line in figure 14a. The result convolved with the rotational profile of V_rotsin i = 40.8 km s⁻¹ is shown by a thick line. For comparison, the same high resolution spectrum convolved with the slit function of FWHM = 16 km s⁻¹ is shown in figure 14b on which the reference numbers introduced in Paper II are reproduced. Comparison of figures 14a and 14b reveals that the identities of some H₂O blends are conserved despite the rotational broadening, and some blends appear at slightly shifted positions, including different components. We select eight features near B01, 02, 03, 04 07, 09, 12, and 14 for our mini-CG analysis as indicated in figure 14c, where the observed spectrum of 2MASS 1835+32 (dots) is compared with the rotationally broadened spectrum of figure 14a (thick line).

However, we could not analyze the H₂O blends in region A noted in Paper II (see its table 2), since it is difficult to define the pseudo-continuum level for highly depressed strong H₂O bands which are further smeared out by rotation.

7 The M8.5 dwarf 2MASSI J1835379+325954

In the M 8.5 dwarf 2MASS 1832+35 in our sample, dust may form in its photosphere and this object is also known to be a rapid rotator. The analysis of the CO and H₂O spectra is done by the mini–curve-of-growth method (subsection 7.1) and by the spectral synthesis method (subsection 7.2), as for the seven earlier M dwarfs, but we apply the dusty models (subsection 4.2) and consider the rotational broadening (section 6). The effects of dust (subsection 7.3), rotational velocity (subsection 7.4), and fundamental parameters (subsection 7.5) are examined. Based on the examinations of the various factors affecting the spectrum of 2MASS 1835+32, a best possible solution on the abundances and projected rotational velocity in this dusty rapid rotator is suggested (subsection 7.6).

7.1 Mini–curve-of-growth analysis

We apply the fully dusty model of T_eff = 2280 K, Bc2280c526, assuming V_rotsin i = 40.8 km s⁻¹ in our analysis of the CO and H₂O blends in this subsection throughout.

7.1.1 First iteration on the CO blends

We measure the equivalent widths of the CO blends with reference numbers 12–15 on the observed spectrum by referring to the pseudo-continuum, and the results are given in the ninth column of table 5. We again apply the mini-CG method with the solar abundances of case c as initial starting values (log $A_{\mathrm{C}}^{(0)}=-3.61$ and log $A_{\mathrm{O}}^{(0)}=-3.31$⁠), and determine the abundance corrections needed to explain the observed log W/λ values in table 5. The resulting abundance corrections for the four lines are shown in figure 15a and the mean value Δlog $A_{\mathrm{C}}^{(1)}=-0.101$ is indicated by a dashed line. The resulting carbon abundance is log $A_{\mathrm{C}}^{(1)}=-3.711\pm 0.128$ (table 11, line 1).

7.1.2 First iteration on the H₂O blends

We measure the equivalent widths of the H₂O blends near B01, 02, 03, 04, 07, 09, 12, and 14 (see figure 14c) on the observed spectrum by referring to the pseudo-continuum, and the results are given in the ninth column of table 7. We again apply the mini-CG method to these blends. Since carbon abundance is now known to be log $A_{\mathrm{C}}^{(1)}=-3.711$⁠, we estimate log A_O to be larger by 0.30 dex as in the Sun (Allende Prieto et al. 2002). Thus we start from log $A_{\mathrm{O}}^{(0)}=-3.711+0.30=-3.411$⁠, and since this value may be a good starting value, we now apply the mini-CG method with δ = 0.1. The resulting abundance corrections from the eight blends are shown in figure 16a and the mean value Δ log $A_{\mathrm{O}}^{(1)}=+0.008$ is indicated by a dashed line. The result is log $A_{\mathrm{O}}^{(1)}=-3.403\pm 0.024$⁠, as given in table 11 (line 2).

7.1.3 Confirmation by second iteration

Given that we have now determined log $A_{\mathrm{C}}^{(1)}$ and log $A_{\mathrm{O}}^{(1)}$⁠, we repeat the mini-CG analysis with these values as starting values. Since the CO blends are contaminated with the weak H₂O lines, revised oxygen abundance may have some effect on the determination of carbon abundance from the CO blends. The resulting mean abundance correction is only −0.025 (table 11, line 3), confirming that the first iteration is already close to the solution, and this may be because the oxygen abundance assumed in the first iteration is already close to the final result (as will be confirmed in table 11). With the revised carbon abundance, the H₂O blends are also examined and the resulting mean correction is only −0.012 (table 11, line 4).

We confirm that the first iteration is already near the convergence value. For this reason, we will skip the second iteration in the analysis to be given hereafter (subsections 7.3, 7.4, and 7.5), since these analyses are for comparison purposes to examine the effect of different assumptions. For consistency in these analyses, we will use the result of the first iteration given in table 11 (lines 1 and 2) as the reference.

7.2 Synthetic spectra

Based on the carbon and oxygen abundances given in table 11 (lines 1 and 2), synthetic spectra of CO and H₂O are computed and compared with the observed spectra in figures 17a and 18a, respectively. Although only four CO blends are used to obtain the carbon abundance from the CO spectrum, the whole CO spectrum including the bandhead region is reasonably well reproduced with the abundances based on the mini-CG analysis. The χ² values for CO and H₂O are 5.803 and 2.035 (table 12, line 1), respectively.¹⁰

7.3 Effect of dust on abundance determination

We apply our dusty model in our abundance analysis in subsections 7.1 and 7.2. We now examine the case of a dust-free model to assess the effect of dust in our analysis. For this purpose, we use the dust-free model of the same values of T_eff = 2280 K, log g = 5.26, and V_rot sin i = 40.8 km s⁻¹ as the dusty model we applied in subsection 7.1. We assume the same initial abundances of log $A_{\mathrm{C}}^{(0)}=-3.61$ and log $A_{\mathrm{O}}^{(0)}=-3.31$⁠, and apply the mini-CG method to the CO blends as for the dusty model. The resulting abundance corrections are shown in figure 15b. The mean value of Δlog $A_{\mathrm{C}}^{(1)}=-0.358\pm 0.078$ is 0.257 dex smaller compared with the result of Δlog $A_{\mathrm{C}}^{(1)}=-0.101\pm 0.128$ using the dusty model.

Then, we assume the revised carbon abundance log $A_{\mathrm{C}}^{(1)}=-3.61-0.358=-3.968$ and log $A_{\mathrm{O}}^{(0)}=-3.968+0.3=-3.668$ as starting values, and analyze the H₂O blends. The resulting abundance corrections are shown in figure 16b and the mean value of Δlog $A_{\mathrm{O}}^{(1)}=0.144\pm 0.034$ is compared with the result using the dusty model, Δlog $A_{\mathrm{O}}^{(1)}=+0.008\pm 0.024$⁠. However, the starting values are different from those of the dusty case this time, and what is to be compared is the resulting log $A_{\mathrm{O}}^{(1)}=-3.668+0.144=-3.524$ of the dust-free case with log $A_{\mathrm{O}}^{(1)}=-3.411+0.008=-3.403$ of the dusty case. Summarizing, the carbon and oxygen abundances based on the dust-free model are log $A_{\mathrm{C}}^{(1)}=-3.968\pm 0.078$ and log $A_{\mathrm{O}}^{(1)}=-3.524\pm 0.034$ (table 11, lines 5 and 6), or smaller by 0.257 and 0.121 dex, respectively, compared with the case by the dusty model (table 11, lines 1 and 2).

We also compute the synthetic spectra of CO and H₂O, and the results are shown in figures 17b and 18b, respectively. We also evaluate the χ² values, which turn out to be χ²(CO) = 10.369 and χ²(H₂O) = 2.879 by the dust-free model (table 12, line 2), compared with χ²(CO) = 5.803 and χ²(H₂O) = 2.035 by the dusty model (table 12, line 1). Thus, the abundances based on the dusty model provide a better account of the observed spectra both of CO and H₂O. Also, the resulting carbon and oxygen abundances based on the dust-free model appear to be somewhat too small for an M dwarf of the disk population. For these reasons, we may conclude that the dusty model better represents the photospheric structure of 2MASS 1832+35 and that it is more appropriate for abundance analysis.

7.4 Effect of the rotational velocity

So far, we have applied the projected rotational velocity of V_rotsin i = 40.8 km s⁻¹, which is the mean value of the two determinations by different groups (section 6). During our analysis, we noticed that the projected rotational velocity has an appreciable effect on the abundance determination. This is because the EWs are evaluated from the synthetic spectrum by referring to the pseudo-continuum level which depends on the rotational broadening, in our mini-CG analysis. To examine the effect of the rotational velocity, we apply the mini-CG method to the CO blends by assuming the lower value of V_rotsin i = 37.6 km s⁻¹ (Deshpande et al. 2012) and the higher value of V_rotsin i = 44.0 km s⁻¹ (Reiners & Basri 2010), both using the dusty model Bc2280c526 with the initial starting abundances of log $A_{\mathrm{C}}^{(0)}=-3.61$ and log $A_{\mathrm{O}}^{(0)}=-3.31$⁠. The resulting abundance corrections are shown in figures 15c and 15d for the cases of low and high rotational velocities, respectively. The resulting carbon abundances are log $A_{\mathrm{C}}^{(1)}=-3.794\pm 0.094$ and −3.566 ± 0.199 (table 12, lines 7 and 9) for the low and high rotational velocities, respectively. Thus the difference of the resulting carbon abundances is 0.228 dex for a difference in V_rot sin i values by 6.4 km s⁻¹.

We also repeat the mini-CG method for the H₂O blends and the results are shown in figures 16c and 16d for the cases of low and high rotational velocities, respectively. The resulting oxygen abundances are log $A_{\mathrm{O}}^{(1)}=-3.461\pm 0.036$ and −3.318 ± 0.029 (table 12, lines 8 and 10) for the cases of low and high rotational velocities, respectively, and the difference is 0.143 dex for a difference in V_rotsin i values by 6.4 km s⁻¹.

The synthetic spectra of CO are shown in figures 17c and 17d for the cases of low and high rotational velocities, respectively. The observed spectrum remains unchanged in figures 17c and 17d, but the pseudo-continuum level of the synthetic spectrum is lower¹¹ in the case of the higher rotational velocity than in the case of the lower rotational velocity for the reason noted at the beginning of this subsection. For this reason, EWs of the CO blends measured from the synthetic spectrum of the higher rotational velocity are smaller than those measured from the spectrum of the lower rotational velocity, as can directly be seen in figures 17c and 17d. Then, a higher abundance is required to explain the observed spectrum by the predicted spectrum based on the higher rotational velocity. The fitting appears to be better for the case of the higher rotational velocity for which the χ² value is 3.126, while it is 9.406 for the case of the lower rotational velocity (table 12, lines 3 and 4).

Similar analyses on the H₂O blends are shown in figures 16c and 16d for the cases of low and high rotational velocities, respectively. The fits appear to be somewhat better for the case of the higher rotational velocity, since the χ² values are 2.568 and 1.820 for the low and high rotational velocities, respectively (table 12, lines 3 and 4). By the way, χ²(CO) = 5.803 and χ²(H₂O) = 2.035 for the case of V_rotsin i= 40.8 km s⁻¹ discussed in subsection 7.2. In conclusion, the highest rotational velocity of V_rotsin i= 44.0 km s⁻¹ provides the best account of the observed spectra (compare lines 1, 3, and 4 of table 12), and possibly the best carbon and oxygen abundances.

7.5 Effect of the fundamental parameters

Finally, we examine the effect of the uncertainty in the fundamental parameters, especially of T_eff by ±100 K. For this purpose, we generate new dusty models of T_eff = 2180 and 2380 K. The gravities are estimated from the T_eff–radius–mass relations discussed in section 3. We apply our dusty models with the low and high values of T_eff, Bc2180c528 and Bc2380c525, respectively, with the initial starting abundances of log $A_{\mathrm{C}}^{(0)}=-3.61$ and log $A_{\mathrm{O}}^{(0)}=-3.31$⁠. We also assume V_rotsin i = 40.8 km s⁻¹. The resulting carbon abundance corrections by the mini-CG analysis for the low and high T_eff models are shown in figures 15e and 15f, respectively. It appears that the resulting carbon abundance corrections are Δlog $A_{\mathrm{C}}^{(1)}=+0.0003$±0.136 and −0.094 ± 0.142 (table 11, lines 11 and 13) for the low and high T_eff, respectively. For comparison, Δlog $A_{\mathrm{C}}^{(1)}=-0.101$ ±0.128 for our adopted case of T_eff = 2280 K (table 11, line 1). Thus the changes of T_eff by ± 100 K result in rather minor changes to the resulting carbon abundances. This should certainly be due to special circumstance noted in subsection 4.3 that the temperatures of the surface layers converge to the condensation lines of iron.

We repeat the same analysis on H₂O blends and the resulting abundance corrections are shown in figures 16e and 16f for the cases of low and high T_eff, respectively. The resulting oxygen abundances are log $A_{\mathrm{O}}^{(1)}=-3.324\pm 0.023$ and −3.388 ± 0.023 (table 11, lines 12 and 14) for T_eff = 2180 and 2380 K, respectively. For comparison, log $A_{\mathrm{O}}^{(1)}=-3.403\pm 0.024$ for T_eff = 2280 K (table 11, line 2). Again, the oxygen abundance depends little on T_eff. This is a favorable result, since the effective temperature is difficult to determine in very cool M dwarfs since no angular diameter measurement is available yet.

We then compute the synthetic spectra of CO for the abundances determined for the cases of low and high T_eff, and the results are shown in figures 17e and 17f, respectively. We also compute the synthetic spectra of H₂O for the abundances determined for the cases of low and high T_eff and the results are shown in figures 18e and 18f, respectively. The results are again not much different compared with the reference case, and the χ² values are nearly the same for three cases (compare lines 1, 5, and 6 of table 12).

7.6 Carbon and oxygen abundances in 2MASS 1835+32

In the latest M dwarf 2MASS 1835+32 in our sample, the dust formation and rapid rotation introduce additional uncertainty in the abundance determinations. The effect of the dust formation is appreciable (subsection 7.3), but this is based on the simplest assumption on dust formation based on the thermodynamics alone. The limitation of such a simple approach will be discussed in subsection 8.2, but we think that even our simple treatment of dust provides a better account of the observed spectrum of 2MASS 1835+32 compared with the case based on the dust-free model, and we decide to apply our dusty models in our abundance analysis of this object.

The effect of the rotational velocity turns out to be unexpectedly large. We first apply the mean value (V_rotsin i = 40.8 km s⁻¹) of the two available values in the literature (section 6), but we find that the higher value (V_rotsin i = 44.0 km s⁻¹) of the two literature values gives the better account of the observed spectrum (subsection 7.4). The empirical determinations of the rotational velocity are anyhow difficult in late M dwarfs since the true continuum level of the observed spectrum cannot be well defined in general and, further, abundances and physical parameters (or model photosphere applied) must be properly assigned. We believe that our analysis can be a way to clear these requirements. Since our analysis suggests the higher value of the known V_rotsin i values, we examine a still higher value of V_rotsin i = 48.0 km s⁻¹ and repeat the analysis outlined in subsection 7.4. The results are log A_C = −3.451 ± 0.298 and log A_O = −3.245 ± 0.044. However, we can use only four lines of CO, and we notice that the internal consistency (p.e.) of our CO analysis turns out to be worse for the higher V_rotsin i value. Also, we find that χ² = 3.476 for the comparison of the observed and predicted spectra of CO, and V_rotsin i = 48.0 km s⁻¹ does not provide any improvement over the result based on 44.0 km s⁻¹ for which χ² = 3.126 (table 12, line 4).

The abundances of log A_C = −3.711 and log A_O =−3.403 based on V_rotsin i = 40.8 km s⁻¹ (table 11, lines 1 and 2) are somewhat too low for a possibly young object like 2MASS 1835+32 which is not spun down yet. On the other hand, the abundances of log A_C = −3.566 and log A_O = −3.318 based on V_rotsin i = 44.0 km s⁻¹ (table 11, lines 9 and 10) appear to be acceptable for 2MASS 1835+32, which may be a relatively young M dwarf as noted above. Also, inspection of table 12 reveals that this case provides the best account of the observed spectrum among the six possible cases we have examined. For these reasons, we adopt log A_C = −3.57 ± 0.20 and log A_O = −3.32 ± 0.03 together with V_rotsin i = 44.0 km s⁻¹ for 2MASS 1835+32 (table 13).

8 Discussion

8.1 Carbon and oxygen abundances in M dwarfs

We take the weighted mean (with the numbers of lines used as weights) of the oxygen abundances resulting from the analyses of regions A (table 10) and B (table 8) for the seven M dwarfs for which the two regions have been analyzed. The results and that from region B alone (2MASS 1835+32) are summarized in table 13, together with the carbon abundances from tables 6 and 11 (lines 9 and 10). Also, the classical and more recent solar carbon and oxygen abundances by Anders and Grevesse (1989) and Asplund et al. (2009), respectively, are included in table 13 for comparison.

The resulting values of log A_O/A_C (given in the fourth column of table 13) are plotted against the values of log A_C in figure 19 by red (or black) filled circles. The results for 38 M dwarfs analyzed in Papers I and II are reproduced from figure 15 of Paper II, as shown by green (or gray) filled circles. The two representative solar values are shown by marks. Inspection of figure 19 reveals that the late M dwarfs show no systematic difference from the early and middle M dwarfs, and our conclusion outlined in Paper II that the A_O/A_C ratios are larger at lower metallicities and gradually decrease for higher metallicities is strengthened with the additional data on the eight late M dwarfs.

Although the large production of oxygen in the metal-poor era is explained as due to Type II supernovae/hypernovae in the early Galaxy (Nomoto et al. 2013), the large differential effect in the production of carbon and oxygen is by no means well understood (Gustafsson et al. 1999), and accurate determinations of the carbon and oxygen abundances will still be needed. In particular, our analysis is limited to the M dwarfs in the disk, and to extend our analysis to the halo M dwarfs (M subdwarfs) is the next major step. However, besides the observational difficulty of observing faint M subdwarfs, the K band region of the metal deficient M dwarfs is badly depressed by the strong collision-induced absorption (CIA) due to H₂–H₂ and H₂–He pairs (Borysow et al. 1997). In fact, the CO and H₂O bands we have analyzed in our present work are almost unseen on the observed spectra of a few late M subdwarfs, at least by low resolutions (Burgasser & Kirkpatrick 2006). For this reason, abundance analysis of M subdwarfs will be quite challenging even if higher resolution spectra of M subdwarfs can be obtained by future large telescopes.

By the way, the effect of H₂ CIA on the abundance analysis is by no means well evaluated so far as we are aware, and we examine this problem in the case of a relatively cool M dwarf GJ 752B as an example. In figure 20, we plot the predicted spectrum of CO based on our model of GJ 752B (Cc2640c526) but disregarding the H₂ CIA due to H₂–H₂ and H₂–He pairs (blue/black solid line), and compare it with that based on the same model but including the H₂ CIA (light sky/gray solid line) as we have done so far (i.e., the same as figure 10f). The resulting CO spectrum computed disregarding the H₂ CIA is stronger than that including the H₂ CIA, as expected. Then, we apply the mini-CG analysis based on our model of GJ 752B but disregarding the H₂ CIA to the CO lines, and determine the abundance correction to the log A_C based on the model including the H₂ CIA (log A_C = −3.550 ± 0.066, see table 6). The result is Δlog A_C = −0.196 ± 0.081, as shown in figure 21, and the effect of neglecting the H₂ CIA is to reduce the logarithmic carbon abundance by 0.196 dex. Thus, the effect of the H₂ CIA is appreciable even in the solar metallicity case, even if it is not so drastic as in the metal-poor cases (Borysow et al. 1997).

8.2 Dust in the photospheres of M dwarfs

In subsection 4.2, we assumed that dust forms everywhere in the photospheres of M dwarfs so long as the thermochemical condition for condensation is fulfilled. However, it is now known that dust does not fill the whole photosphere where the thermochemical condition of condensation is fulfilled, at least in L and T dwarfs. In fact, if dust forms this way, more dust should be formed in the cooler dwarfs, and hence the cooler dwarfs should increasingly be red. However, observations never follow such an expectation. For example, the J − K color first shows reddening from late M to L dwarfs, but it turns to blue after late L dwarfs (e.g., Knapp et al. 2004), suggesting that dust in the photospheres should decrease towards late T dwarfs.

We have developed a simple model referred to as the unified cloudy model, UCM (Tsuji 2002). Dust grains first form at T_cond, the condensation temperature, but dust grains grow larger at cooler temperatures and will precipitate at a certain temperature, which we referred to as the critical temperature, T_cr. Then, dust grains exist only in the limited region of T_cr ≲ T ≲ T_cond, or dust grains appear in a form of cloud. In an L dwarf, T_eff (approximately the temperature where the optical depth is about unity) is sufficiently higher than T_cond (see figure 5) as well as T_cr (1700, 1800, 1900 K, and T_cond). For this reason, the dust cloud appears in the optically thin region and the L dwarf appears dusty. On the other hand, T_eff of a T dwarf is generally lower than T_cond as well as T_cr, and the dust cloud is situated in the optically thick region. For this reason, the dust cloud will have little observable effect and the T dwarf appears observationally as if it is dust-free. Then, observed characteristics of the spectra of L and T dwarfs could naturally be accounted for as a single sequence of T_eff in our UCMs.

However, further progress in the photometry of L and T dwarfs revealed that the infrared colors such as J − K (Knapp et al. 2004) show a large variation at a given effective temperature (Vrba et al. 2004). This fact suggests that the infrared colors are not determined by T_eff alone, and we proposed that the critical temperature, T_cr, introduced in our UCMs, should be changing at a given T_eff (Tsuji 2005). Since the value of T_cr essentially defines the thickness of the dust cloud,¹² we concluded that the thickness of the dust cloud should be changing at a fixed T_eff. Further, the change of the spectra from L6.5 to T3.5, for example, could be explained as a result of increasing T_cr from 1700 K to T_cond while T_eff remains nearly constant at T_eff ≈ 1300 K (see figure 10 of Tsuji 2005). This fact implies that the spectra as well as infrared colors of L and T dwarfs are not determined primarily by T_eff but should be determined by T_cr or the thickness of the dust cloud. Given that dust has such a large and definitive effect on the observed spectra and colors, the values of T_cr can be inferred from observations in L and T dwarfs.

Although our cloudy models could be applied to L and T dwarfs, we are not sure if the same cloudy models can be applied to late M dwarfs. In fact, we have no means by which to estimate the value of T_cr in the case of M dwarfs, since the effect of dust on the observed spectra is rather subtle compared with that in L and T dwarfs. For this reason, we do not apply the cloudy models included in our UCMs to M dwarfs, but apply the simple fully dusty models of case B, also included as a subset in our UCM grid. If dust grains form a thin cloud in M dwarfs, our treatment provides a maximum estimate of the effect of dust.

Such a limitation of our simple approach basically based on the thermodynamics alone has been overcome by the recent development of the self-consistent treatment of dust formation coupled with the photospheric structure (Helling et al. 2008b, 2008c). These authors followed the kinematic processes of nucleation, dust growth, and evaporation, resulting in gravitational setting in the atmosphere, and the resulting dust cloud structure was incorporated into the photospheric model structure. They have applied their method to the substellar atmospheres for a wide range of metallicity extending to [M/H] as low as −6.0, and showed that dust clouds form even in the most metal-poor substellar objects (Witte et al. 2009). They have also applied their method to evaluate the synthetic spectra of cool dusty dwarfs, including late M dwarfs, and showed that the observed infrared spectra of late M and L dwarfs could be fitted well with their model predictions (Witte et al. 2011). Such a success is quite encouraging for the spectral analysis of dusty dwarfs, and we hope that their method will be extended to the abundance analysis on high resolution spectra of the dusty M dwarfs and subdwarfs in the near future.

Inspection of figure 22 reveals that oxygen atoms are mostly depleted by CO, H₂O, and SiO. We consider only three dust species (corundum, iron, and enstatite), and the dust grain that works as a sink of oxygen is only corundum (Al₂O₃), since enstatite (MgSiO₃) does not form in our dusty model yet (see figure 5). In the abundance analysis based on H₂O, the fraction of oxygen that is depleted by corundum must be corrected for, but we skip this correction since the corundum consumes only about 1% of the oxygen and hence its effect on the oxygen abundance is about 1%. If enstatite forms, its effect on the oxygen abundance should be larger, since Si and Mg consume more than 10% of the oxygen because of their higher abundance (about 10% of the oxygen abundance). For this reason, we are considering enstatite as a sink of oxygen in our chemical equilibrium, but this effect is not shown in figure 22, since enstatite does not form in our model of the late M dwarf 2MASS 1835+32 (see figure 5). Then, the uncertainty in our oxygen abundance due to dust formation may be about 1% within the framework of our present LTE analysis. Certainly, the dust species considered by us are limited (only three) and the effect of element depletion should be different if more dust species are considered (Helling et al. 2008c). But the amount of the oxygen depletion, for example, is limited by the abundances of non-volatile elements (Al, Mg, Si, etc.) that combine with oxygen and we hope that an approximate effect of dust formation on abundance analysis can be represented by the species that consumes the larger amount of oxygen atoms (e.g., corundum in our case discussed above under the absence of enstatite). However, it is certainly necessary to consider more dust species if higher accuracy in abundance determination is required.

8.3 The end of the main sequence

Probably, our present sample includes some objects near the end of the main sequence. The latest ones, such as GAT 1370, LP 412-31, and 2MASS 1835+32 are even referred to as brown dwarfs in SIMBAD, but the reason for these assignments is by no means clear to us. Unfortunately, we do not know how to exactly discriminate a brown dwarf from a star for a particular object in question. This is because there is no definite spectral criterion by which to identify the substellar nature. The Li test (Robolo et al. 1992) applied to field objects is effective in confirming the brown dwarf status of objects less massive than 0.06 M_⊙, while the three objects mentioned above are likely to be more massive than 0.06 M_⊙ (table 3). This method requires reasonably high spectral resolution and may be difficult to be used as a spectroscopic criterion in spectral classification level. Spectral classification of late M dwarfs, however, is also hampered by the presence of dust in their photospheres. Unlike atomic and molecular lines that show definitive signatures of their origins, dust spectra have no such signature. For this reason, it is not yet known how the effect of dust can be taken into account in the spectral classification of late M dwarfs. Anyhow, it is difficult to define the end of the main sequence observationally and we simply follow the spectral classification that has classified our objects as M dwarfs in the present work.

On the other hand, the observed HR diagram at the end of the main sequence is well explained by the evolutionary model by Baraffe et al. (1998) (subsection 6.4 of Paper I) and consistent with their theoretical HR diagram (see figure 17 in Paper I). Also, some physical parameters predicted by their model are well consistent with the observed values (figures 2–4). For this reason, our estimations of some physical parameters are helped by their evolutionary model (section 3). Also, chemical abundances are shown to be rather normal down to the end of the main sequence, even though our analysis has been limited to the carbon and oxygen abundances so far, and our result appears to be consistent with the present idea of the Galactic chemical evolution (e.g., figure 19).

9 Concluding remarks

Our spectroscopic analysis of M dwarfs is greatly helped by the recent progress in observations, including the angular diameter measurements by interferometry (e.g., Boyajian et al. 2012), infrared spectroscopy using new detectors (e.g., Kobayashi et al. 2000), and high precision astrometry (e.g., van Leeuwen 2007) and photometry (e.g., Wright et al. 2010) from space. A remaining problem in the present paper is that the effective temperatures of the latest M dwarfs cannot be determined by directly measured angular diameters, and we hope that the angular diameter measurements can be extended to late M dwarfs in the near future.

We have determined the carbon and oxygen abundances in 38 + 8 = 46 M dwarfs from the CO and H₂O spectra through our Papers I, II, and III, and the carbon abundances in an additional four early M dwarfs in Paper I. So far, M dwarfs might not be expected to be proper objects for abundance determinations in general, but we now believe that such a prejudice should be reconsidered. In fact, we think that the stellar carbon and oxygen abundances can best be determined from the numerous CO and H₂O lines, respectively, observed in the spectra of M dwarfs, especially because stable CO and H₂O molecules are the major species of carbon and oxygen, respectively, in M dwarfs, and hence their abundances are almost unaffected by the uncertainty in the photospheric structures (Paper II). For this reason, the resulting carbon and oxygen abundances are insensitive to the imperfect model photospheres used in the abundance analysis. This fact shows a marked contrast to the well-known difficulties in determining carbon and oxygen abundances from other molecular or atomic lines usually pretty model-sensitive in solar-type stars. An example of such an advantage is that the atypical nature of the recently revised solar carbon and oxygen abundances (Asplund et al. 2009) for its metallicity, compared with the nearby unevolved stars, has been demonstrated clearly by our analysis (see figure 15 of Paper I and figure 16 of Paper II for C and O, respectively), because of the higher internal consistency of our analysis based on M dwarfs compared with the other works based on the solar-type stars.

Also, even though the true continuum cannot be seen in the spectra of M dwarfs, the pseudo-continuum defined by numerous lines of H₂O can be well defined observationally and, at the same time, it can theoretically be evaluated fairly accurately thanks to the recent high precision line list of H₂O (e.g., Barber et al. 2006; Rothman et al. 2010). Then, the basic principle of the quantitative analysis of the stellar spectra can be applied essentially in the same way whether the true or the pseudo-continuum is referred to, and the difficulty of the continuum in cool stars is resolved, at least in the near-infrared spectra of M dwarfs (Paper I). Also, our analysis is not necessarily limited to the well-defined unblended lines but can be extended to any blends by the use of a flexible method such as the mini–curve-of-growth analysis.

In late M dwarfs, dust formation and rotation introduce additional problems in the spectroscopic analysis. In particular, the effect of dust on abundance analysis involves complicated problems such as cloud formation, element depletion by dust grains, dust properties, and so on, and impressive progress is being achieved in this field (subsection 8.2). In the present paper, however, we have restricted our analysis to a simple treatment based on the thermodynamical argument, and we hope that the effect of dust formation on abundance determination can be estimated approximately. Except for the dusty M dwarfs, CO and H₂O remain excellent abundance indicators of carbon and oxygen, respectively, even in late M dwarfs (Paper III).

What we have done in our present work, however, is limited to the carbon and oxygen abundances determined from the CO and H₂O spectra, respectively, and it is true that this is due to some favorable conditions in these particular cases. It is still to be investigated if similar analysis can be extended to other molecular or atomic spectra, but it is by no means trivial if such an extension can be done easily. However, given the recent progress in observations and in basic data such as molecular data, we believe that stellar spectroscopy will provide great possibilities in uncovering abundant information coded in high resolution stellar spectra.

We thank the anonymous referee for careful reading of our text and for helpful suggestions, especially on cloud formation in dusty M dwarfs.

We thank the staff of the Subaru Telescope and S. Sorahana for the help with observations.

This research makes use of data products from the Wide-field Infrared Survey Explorer which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by NASA.

This research has made use of the VizieR catalog access tool and the SIMBAD database, both operated at CDS, Strasbourg, France, and of the RECONS database at www.recons.org.

Computations were carried out on the common use data analysis computer system at the Astronomy Data Center, ADC, of the National Astronomical Observatory of Japan.

Appendix. Additional spectroscopic data for CO lines (2–0 band)

In addition to the CO blends referred to as Nos. 1–14 in table 7 of Paper I, we use an additional blend referred to as No. 15 in this paper; the spectroscopic data for this blend composed of two CO lines are given in table 14.

References