THE QUADRUPLE PRE-MAIN-SEQUENCE SYSTEM LkCa 3: IMPLICATIONS FOR STELLAR EVOLUTION MODELS

The visible-light spectroscopy of LkCa 3 was collected at the Harvard-Smithsonian Center for Astrophysics (CfA) over a period of a little more than nine years beginning in 1985 November. This is largely the same material used by Mathieu (1994) to obtain the preliminary single-lined orbit of the system, but the details are reported here for the first time. Observations were acquired with nearly identical echelle spectrographs (Digital Speedometers; Latham 1985, 1992) attached to the 1.5 m Tillinghast reflector at the F. L. Whipple Observatory on Mount Hopkins (AZ), and the 4.5 m equivalent MMT also on Mount Hopkins, prior to its conversion to a monolithic 6.5 m telescope. A single echelle order 45 Å wide was recorded using intensified photon-counting Reticon detectors at a central wavelength of about 5190 Å, which includes the lines of the Mg i b triplet. The resolving power of these instruments was λ/Δλ ≈ 35,000, and the signal-to-noise ratios of the observations range from 9 to 21 per resolution element of 8.5 km s⁻¹. A total of 58 exposures were obtained, although 4 were later discarded because of moonlight contamination or other problems.

The standard, one-dimensional cross-correlation functions for these spectra clearly show a peak corresponding to the star reported by Mathieu (1994), which has a variable radial velocity (RV) with a period of 12.94 days. Occasionally, however, a second peak of similar height is also visible (see Figure 1). Shortly after the Mathieu (1994) publication, the spectra of LkCa 3 were subjected to an analysis with TODCOR, a two-dimensional cross-correlation technique that had just been introduced by Zucker & Mazeh (1994). TODCOR is designed to measure radial velocities from double-lined spectra, and uses two templates, one for each set of lines. It was discovered that the velocities from the second set of lines were not changing with the same period as the first set, but varied instead with a shorter period of 4.06 days. LkCa 3 is a near-equal brightness visual binary with an angular separation (∼05) that is smaller than the width of the spectrograph slit (1''), which immediately suggested the quadruple nature of the system, with each visual component being in turn a spectroscopic binary. We refer to the 12.94 day binary as system A with components Aa and Ab, and to the stars in the other system as Ba and Bb, although from these observations alone we cannot identify which visual component is A and which is B.

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This discovery languished for a time, and only more recently did we return to the system in an attempt to detect the secondary star of the 12.94 day system. For this, we re-analyzed the spectra using TRICOR, an extension of TODCOR to three dimensions (Zucker et al. 1995). The effort was successful, and showed the faint secondary to be only half as massive as the primary. A similar exercise with TRICOR resulted in the detection of the secondary of the 4.06 day binary, which also has about half the mass of its primary.

The TRICOR velocities obtained from these two analyses are likely biased to some extent due to line blending from the fourth star, whose presence is ignored in each case. With proof that the spectra are in fact quadruple-lined, we proceeded to re-analyze them one more time with QUADCOR, which is an extension of TODCOR to four dimensions (Torres et al. 2007). In previous studies of solar-type stars with similar spectroscopic material, we have typically used synthetic spectra as templates for the cross-correlations, based on model atmospheres by R. L. Kurucz. Here, however, all components are M stars, and calculated spectra become less realistic at these low temperatures, particularly at high resolution. Instead, we used strong exposures of M stars obtained with the same instrumentation, covering a range of spectral types. We applied rotational broadening following the prescription of Gray (2005). A large number of template combinations were tried, guided initially by the results described below from the NIR spectra. In those observations the visual components are spatially resolved, so the spectra are only double-lined, making template selection easier. For the rotational broadening we were also guided by the results from Nguyen et al. (2012), described later in Section 3, who resolved some of the components spectroscopically with observations of much higher signal-to-noise ratio than ours.

For the 12.94 day binary, the templates that provided the best match to the observations, as measured by the cross-correlation value averaged over all exposures, are GJ 49 (M1.5; adopted RV = −5.48 km s⁻¹) and GJ 699 (M4; RV = −108.77 km s⁻¹), rotationally broadened to 12 km s⁻¹ and 4 km s⁻¹, respectively. For the 4.06 day binary, we used GJ 846 (M0; RV = +18.61 km s⁻¹) and GJ 15 A (M2; RV = +12.72 km s⁻¹), broadened to 16 km s⁻¹ and 14 km s⁻¹, respectively. Our final QUADCOR velocities are presented in Table 1, along with the individual uncertainties. These are typically 1.7 and 5.7 km s⁻¹ for stars Aa and Ab, and 2.6 and 11 km s⁻¹ for stars Ba and Bb. To monitor the zero point of our velocity system, we obtained exposures of the dusk and dawn sky, and applied small run-to-run corrections as described by Latham (1992). These corrections are included in the measurements listed in Table 1.

In addition to the radial velocities, we determined the light ratios between the stars at the mean wavelength of our spectra (5190 Å; see Torres et al. 2007). The values, averaged over all exposures, are ℓ_Ab/ℓ_Aa = 0.161 ± 0.012, ℓ_Ba/ℓ_Aa = 0.938 ± 0.042, and ℓ_Bb/ℓ_Aa = 0.223 ± 0.018. The fractional light contributions are thus 0.43 (Aa), 0.07 (Ab), 0.40 (Ba), and 0.10 (Bb). As a result, the combined light of A is the same as that of B, at least at these wavelengths.

NIR spectra were obtained at the Keck II 10 m telescope (Mauna Kea, HI) using the facility's cross-dispersed cryogenic spectrograph NIRSPEC (McLean et al. 1998, 2000). NIRSPEC employs a 1024 × 1024 InSb array detector, which can accommodate several high-resolution spectral orders simultaneously. We observed in the H band centered at ∼1.555 μm, corresponding to NIRSPEC order 49. The advantage of this spectral region is that while it lacks significant contamination from telluric absorption lines, thus avoiding the need to divide by the spectrum of an early-type star, numerous OH night sky emission lines are distributed across the spectral range (Rousselot et al. 2000), supplying an excellent means of inherent wavelength and zero-point calibration. It is also rich in deep atomic and molecular lines found in G- through M-type stars. Spectra were recorded in sets of four exposures in an "ABBA" dither pattern to allow for background subtraction and elimination of bad pixels. A two-pixel slit width yielded a spectral resolution of λ/Δλ ≈ 30, 000. In total, five observations were obtained in a period between 2003 and 2010 (Table 2).

On UT 2006 December 14 and 2010 December 12, we used NIRSPEC in high angular resolution mode behind the Keck II adaptive optics (AO) system. This facilitated separation of the ∼05 pair, LkCa 3 A and B. Natural seeing conditions on these nights were ∼1'' and 05, respectively. Because of multiple reflections in the complex AO light path, integration times per exposure were 300 s, even for these relatively bright stars. The NIRSPEC slit is re-imaged behind the AO system by a factor of ∼11, making it prohibitively small for collecting light from OH night sky emission lines. Comparison lamp lines were thus recorded to accomplish the dispersion solution and set the zero point.

On the other three nights (UT 2003 November 5, 2004 December 25, and 2010 November 22), we did not use AO. Although the seeing was relatively good (04, 03, and 06, respectively) the point-spread functions (PSFs) of the spectral traces overlapped in the two-dimensional spectra. With the higher throughput, shorter integration times of 60–120 s per exposure were adequate.

Spectra were extracted with the REDSPEC IDL package,⁶ available on the Keck Web site and specifically designed for the analysis of NIRSPEC data. These procedures correct for spatial and spectral distortions in the two-dimensional spectra. Reduction of the AO data was trivial; the same procedure was followed as in, e.g., Rosero et al. (2011). However, to extract individual spectra from the overlapping PSFs in the non-AO data, it was necessary to use REDSPEC to extract the rectified two-dimensional spectra and then apply further operations to separate LkCa 3 A and B. Customized IDL procedures were employed to determine the best Gaussian parameters for the A and B PSFs. The stellar signal for each of the 1024 columns in the cross-dispersion direction of the rectified two-dimensional spectrum was then fit with two overlapping model Gaussians, varying the heights to identify the flux (the Gaussian maximum) of each component individually. The reduced and barycenter-corrected spectra for order 49 are shown in Figures 2 and 3.

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Since our procedures separate LkCa 3 A and B, the NIR spectra are only double lined, and the velocity analysis is considerably simpler than in the optical. Furthermore, with previous knowledge of the orbits from our optical spectra, the NIR observations allow us to unambiguously identify star A (P = 12.94 days) as the western component and star B (P = 4.06 days) as the eastern one.

Radial velocities were measured using the TODCOR algorithm of Zucker & Mazeh (1994). As with our optical spectra, we correlated each of our observations against a grid of observed template spectra of high signal-to-noise ratio (Prato et al. 2002b) seeking the highest correlation coefficient averaged over all exposures. These IR template spectra were rotationally broadened by convolving them with the line-broadening function of Gray (2005). For LkCa 3 A, the best fit was produced by combining the M2 star GJ 752 A (adopted RV = +34.2 km s⁻¹) rotationally broadened to 12 km s⁻¹ with the M4 star GJ 402 (RV = −3.1 km s⁻¹) spun up to 6 km s⁻¹ as the primary and secondary templates, respectively. For LkCa 3 B, the best templates were 61 Cyg B (K7; RV = −65.4 km s⁻¹) and GJ 436 (M2.5; RV = +7.8 km s⁻¹), rotated to 18 and 14 km s⁻¹, respectively. The nominal spectral types of these templates are within one subtype of the templates adopted in the optical for the same components, and the rotational velocities are also quite similar (within 2 km s⁻¹). Average values for the latter are reported later in Section 4.

The flux ratios in the H band were found to be 0.47 ± 0.06 for LkCa 3 A (ratio Ab/Aa) and 0.52 ± 0.08 for LkCa 3 B (Bb/Ba), which are averages over the five observations in each case. The NIR radial velocities in the heliocentric frame are presented in Table 2, along with their estimated uncertainties.

We observed LkCa 3 on UT 2008 January 17 with the AO system at the W. M. Keck Observatory, using the near-infrared camera NIRC2 (Wizinowich et al. 2000). A series of 10 dithered images were collected in the narrowband H and K continuum regions, using a five-point dither pattern with an offset of 2''. Each image consisted of 10 co-added 0.5 s exposures. The images were flat-fielded using dark-subtracted, medianed dome flats. Pairs of dithered images were subtracted to remove the sky background.

We used the brighter, western component (LkCa 3 A) as a PSF reference to measure the relative separation and flux ratio of the eastern component relative to the western one. We corrected the positions for geometric distortions in the detector, used a plate scale of 9.952 ± 0.001 mas pixel⁻¹, and subtracted 0252 ± 0009 from the raw position angles to correct for errors in the orientation of the camera relative to true north (Yelda et al. 2010). Taking the average of the H- and K-band images, we measured an angular separation of 483.73 ± 0.28 mas and a position angle of 69068 ± 0035 east of north at Julian date 2,454,482.872. We also determined the flux ratio of the eastern relative to the western component to be 0.942 ± 0.011 at H and 0.9404 ± 0.0074 at K. Uncertainties were derived by analyzing multiple images individually and computing the standard deviation.

While the angular separation between the visual components of LkCa 3 has remained the same since its discovery by Leinert et al. (1993), the position angle has decreased significantly by almost 10°. This could be a sign of orbital motion, but it may also be the result of a small difference in proper motion between two unbound members of the star-forming region. Our spectroscopic orbital solutions described in the next section indicate that the center-of-mass velocities of LkCa 3 A and B are very close to each other (within 0.4 km s⁻¹), but this, too, is consistent with either scenario. Though the current observations are insufficient to confirm the physical association, an estimate of the likelihood of a chance alignment in Taurus strongly favors the binary hypothesis. Based on the stellar densities near LkCa 3 reported by Gómez et al. (1993), we compute the probability of finding two unbound stars within only 05 of each other to be ∼10⁻⁷, and for the remainder of the paper we will assume that LkCa 3 A and B are physically bound. At the distance to the object of 133 pc estimated in Section 5.1, the projected linear separation of the visual binary is approximately 64 AU, and the orbital period roughly 400 yr.

The orbital periods of LkCa 3 A (12.94 days) and LkCa 3 B (4.06 days) are on either side of the observed tidal circularization period for PMS stars (considered as a more or less coeval population of young objects), which represents the transition between circular and eccentric orbits (Mathieu et al. 1992; Mazeh 2008). This transition period is currently believed to be near 7.6 days, and is marked by the binary system RX J1603.9−3938, which has the longest-period circular orbit among bona fide PMS stars (Melo et al. 2001; Covino et al. 2001). Our finding that the orbit of LkCa 3 B is essentially circular is consistent with this picture. Two other effects of tidal forces, the alignment of the spin axes with the orbital axis and the synchronization of the stellar rotations with the mean orbital motion, are predicted to occur much earlier than circularization (e.g., Hut 1981), so we may reasonably assume that these conditions also apply to the two components of LkCa 3 B. In other words, i_rot = i_orb and P_rot = P_orb. If that is the case, then our measurement of the projected rotational velocities (which are strictly vsin i_rot) provides a way to infer the mean stellar densities as described by Torres et al. (2002) if we assume solid-body rotation. From the orbital elements and vsin i, the density ρ of star Ba in solar units may be expressed as

$$\begin{eqnarray} \log \rho _{\rm Ba} &=& -1.8724 - 2\log P_{\rm orb} + 2 \log (K_{\rm Ba}+K_{\rm Bb}) \nonumber\\ &&+\log K_{\rm Bb} - 3 \log (v_{\rm Ba}\sin i), \end{eqnarray} \tag{ 1 }$$

where P_orb is in days, the velocity semi-amplitudes K_Ba and K_Bb are in km s⁻¹, and the numerical constant is an update of that of Torres et al. (2002). A similar expression holds for log ρ_Bb.

The non-circular orbit we find for LkCa 3 A (with e = 0.1735 ± 0.0090) is consistent with the fact that its period is longer than the transition period for PMS stars. Rotational synchronization is therefore not guaranteed, so we cannot assume that P_rot = P_orb. We note, however, that long-term photometric monitoring of LkCa 3 has shown a clear periodic signal with an amplitude of one-tenth or two-tenths of a magnitude attributed to rotational modulation due to spots, although the multiplicity of the system has previously made it difficult to attribute the signal to any particular component. The period reported by Bouvier et al. (1995) from a 45 day observing campaign is 7.2 ± 0.6 days, and a similar result of 7.38 days was obtained by Norton et al. (2007) over a two-month period. More extensive observations by Grankin et al. (2008) between 1992 and 1998 yielded rotation periods in close agreement for each of five separate seasons that had sufficient observations, with an average of 7.35 days. More recently, Xiao et al. (2012) reported a period of 3.689 days from observations carried out over two months. This is very nearly half of the previous values, suggesting that it may be a harmonic of the true period.

The rotation signal in LkCa 3 is likely due to either star Aa or Ba, simply on the basis that they are much brighter than the secondaries. We proceed under this assumption, and return to it below. Furthermore, if our earlier hypothesis of synchronized rotation for Ba and Bb is correct, then the signal would be associated with star Aa, which is also marginally the brighter star. Evidence supporting this comes from the intensive photometric observations by Norton et al. (2007), totaling about 1350 measurements over a two-month period in a broad passband similar to V+R. Their power spectrum of the observations shows a hint of a second peak near a period of 4 days. Our re-analysis of those data is illustrated in Figure 7. The top panel presents a Lomb–Scargle periodogram of the original observations with the 7.38 day signal clearly visible. The photometric amplitude of this signal is approximately 0.10 mag, and we estimate the uncertainty in the period to be 0.40 days. The peaks at periods under two days are all aliases with the 1 day cycle of the observations. This is seen more clearly in the second panel, where we have subjected the same data to the CLEAN algorithm of Roberts et al. (1987), which reduces the impact of the window function. We then subtracted a sine curve with a period of 7.38 days from the original measurements, and computed a Lomb–Scargle periodogram of the residuals (Figure 7(c)). A prominent peak is revealed at a period of 4.06 ± 0.13 days, with a false alarm probability smaller than 10⁻³, determined from Monte Carlo simulations. This signal has a smaller photometric amplitude of about 0.05 mag. Once again the peaks at much shorter periods are aliases, as shown by the CLEANed power spectrum in the bottom panel of the figure. The close similarity with the orbital period of LkCa 3 B (P_orb = 4.0676115; Table 3) strongly suggests that this second signal corresponds to the rotational signature of star Ba, tidally locked to the mean orbital motion, as we had assumed. We conclude that the 7.38 day signal corresponds indeed to star Aa, which is then spinning more rapidly than the expected pseudo-synchronous rate (P_pseudo = 11.0 days; Hut 1981), and also faster than the periastron rate (9.0 days).

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The evidence that the above signals originate on the primaries of LkCa 3 A and B rather than the secondaries relies on statistics from variability studies of non-accreting periodic T Tauri stars. If star Ab were the source of the 7.38 day signal, then the photometric amplitude corrected for the light contribution of the other components of the quadruple system (Table 4) would be in excess of 1.5 mag. This is a very rare occurrence among WTTSs (typical amplitudes are about 0.1–0.2 mag; see, e.g., Stassun et al. 1999; Grankin et al. 2008). On the other hand, if attributed to star Aa, the intrinsic amplitude would be only 0.24 mag, which is fairly common. Similarly for the 4.06 day signal, the intrinsic amplitudes we would infer for stars Ba and Bb are 0.5 mag and 0.12 mag, respectively. Though not unprecedented, the larger value is seen much less frequently than the smaller one in photometric studies of young stars. In any event, assignment of the observed signal to the primary or secondary in this case is less important, as rotational synchronization is likely to occur in both components given the short period and circular orbit, as discussed earlier.

With our measurement of the projected rotational velocity of Aa from Table 4 and knowledge of its rotation period, we may compute its mean density in a similar way as was done for LkCa 3 B with a slight modification of Equation (1):

$$\begin{eqnarray} \log \rho _{\rm Aa} &=& -1.8724 + \log P_{\rm orb} -3\log P_{\rm rot} \nonumber \\ &&+2 \log (K_{\rm Aa}+K_{\rm Ab}) + \log K_{\rm Ab}\nonumber \\ &&-3 \log (v_{\rm Aa}\sin i) + 1.5\log (1-e^2). \end{eqnarray} \tag{ 2 }$$

However, this still requires the assumption that the spin axis of star Aa is aligned with the axis of the orbit (i_rot = i_orb), which may or may not be true given the longer orbital period of the binary and its young age.

The top panel of Figure 8 shows that the mean densities of stars Ba and Bb computed in this way, and also that of Aa, seem to agree quite well with predictions from our best-fit 1.6 Myr isochrone, even though they were not used in that fit. It is important to note that in our earlier model comparison there is a certain amount of degeneracy between distance and age because evolutionary tracks are mostly vertical in a diagram of magnitude versus color (or temperature) such as Figure 6; a similarly good agreement with the models can be obtained for a smaller distance if one chooses an older age. On the other hand, the mean stellar densities from Equations (1) and (2) are completely independent of distance, but strongly constrain the age, and are therefore complementary to the brightness measurements. Since the top panel of Figure 8 suggests that a somewhat better fit may be had with a slightly younger age, we repeated the comparison incorporating the densities of stars Ba and Bb as additional constraints, though not that of star Aa because of lingering doubts regarding its spin-orbit alignment. The new fit gives an age of 1.4 ± 0.2 Myr, a distance of $D = 133.5_{-6.5}^{+6.0}$ pc, and inclination angles for the binaries of $i_{\rm A} = 70.5_{-3.5}^{+4.5}$ deg and i_B = 57.0 ± 2.5 deg, which are not very different from our previous results. This new best-fit isochrone is shown with the density measurements in the bottom panel of Figure 8. The agreement with the flux measurements is not significantly altered. The uncertainties are now smaller, especially for the distance and age, because use of the densities suppresses the age–distance correlation to a significant degree. This is illustrated graphically in Figure 9. The masses of the stars do not change appreciably, but the radii are now somewhat larger because of the slightly younger age (approximately 1.63 R_☉ for stars Aa and Ba, and 1.27 R_☉ for Ab and Bb).

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Given the multiple constraints on the models afforded by the observations for LkCa 3, we investigated the possibility that the data might enable us to determine the distance to each binary independently, and perhaps disprove our assumption that LkCa 3 A and B are physically associated. Because some of these constraints including the flux ratios and combined magnitudes involve properties of both binaries, it is not possible to carry out completely independent solutions, as the individual colors and magnitudes of the binary components are strongly correlated. Instead, we performed a joint solution involving all four stars as before, but we allowed the distance as well as the age and inclination angle of each binary to vary independently, for a total of six free parameters. However, since the mean stellar density constraint is applied only for stars Ba and Bb, the parameters for LkCa 3 A are more poorly determined, and the distances turn out to be indistinguishable within their formal errors: $142_{-21}^{+22}$ pc for LkCa 3 A and $135.5_{-9.6}^{+7.5}$ pc for LkCa 3 B. The ages are $0.9_{-0.3}^{+0.8}$ Myr and $1.6_{-0.2}^{+0.4}$ Myr, and the inclination angles $i_{\rm A} = 76.6_{-6.7}^{+17.4}$ deg and $i_{\rm B} = 55.5_{-2.0}^{+2.6}$ deg, respectively. The parameters for LkCa 3 B are close to those reported previously, as expected from the strength of the density constraint.

The poor performance of the Lyon models of Baraffe et al. (1998) in the case of LkCa 3 may seem surprising given that they have been employed in numerous studies since their appearance, and have even been used to help define the temperature scale for late-type PMS stars (see below). On relatively few occasions have difficulties with these models been documented in the young-star literature (for examples see Luhman & Reike 1998; Hillenbrand & White 2004). This may have to do with the nature and large number of observational constraints for LkCa 3, in contrast with the typical situation for young stars. A somewhat similar case to LkCa 3 in terms of the strength of the observational constraints is that of GG Tau, which is another hierarchical quadruple PMS system comprised of two visual binaries in the Taurus–Auriga star-forming region. This system is often regarded as a benchmark for testing evolutionary calculations for young stars, and the observations for GG Tau are generally considered to support the validity of the Lyon models, in apparent contradiction with our findings for LkCa 3.

For GG Tau, the individual spectral types (K7+M0 and M5+M7) and luminosities (for an assumed distance of 140 pc) are known for all four components from the work of White et al. (1999). Additionally, the total mass of one of the binaries (GG Tau A) has been estimated from measurements of the orbital velocities of the circumbinary disk (M_{Aa + Ab} = 1.28 ± 0.07 M_☉; Dutrey et al. 1994; Guilloteau et al. 1999). As in LkCa 3, the four stars in GG Tau form a "mini-cluster" of sorts (on the assumption that they are coeval), which means they provide an unusually strong constraint on stellar evolution theory because a successful model must be able to reproduce the properties of all four components at a single age. This was exploited by White et al. (1999) to test the Lyon models, among others, and in the process to establish a conversion between spectral types and effective temperatures for late-type PMS stars, which has been a persistent source of uncertainty in the field.

The authors showed that the solar-metallicity Lyon models with α_ML = 1.0, which are the same ones we used for LkCa 3, provide a good match to the effective temperatures and luminosities of all four stars in GG Tau (see their Figure 6), but yield a predicted total mass of 2.00 ± 0.17 M_☉ for GG Tau A, which is considerably larger than observed. Thus, these models fail, as is the case with LkCa 3. The temperatures in this analysis were converted from spectral types, but were allowed some freedom for the cooler stars Ba and Bb in order to produce the best fit, which resulted in an implied temperature scale for the SpT/T_eff conversion that is intermediate between dwarfs and giants. The two scales converge for the hotter stars Aa and Ab, which were not adjusted separately. White et al. (1999) also tested Lyon models computed with a higher mixing length parameter of α_ML = 1.9, appropriate for the Sun. These were found to produce a similarly good agreement with the temperatures (readjusted for Ba and Bb) and luminosities, and a lower mass of 1.46 ± 0.10 M_☉ for GG Tau A, which is in better accord with the dynamically measured value. Very similar conclusions were reached by Luhman (1999) for the α_ML = 1.9 models from Lyon, apparently lending support to their accuracy.

We point out, however, that the α_ML = 1.9 models employed by the above authors are actually "hybrids," since published tabulations from the Lyon group with this mixing length parameter only reach down to 0.6 M_☉. Below this mass, the authors reverted to the Baraffe et al. (1998) models with α_ML = 1.0 on the assumption that variations in α_ML are inconsequential under 0.6 M_☉ (Chabrier & Baraffe 1997; Baraffe et al. 1998). As it turns out, 1 Myr models for the two values of α_ML have about the same luminosity at this mass, but temperatures that differ significantly by some 230 K, with the α_ML = 1.9 calculations being hotter (see also Weinberger et al. 2013). Both White et al. (1999) and Luhman (1999) were then forced to artificially connect the α_ML = 1.0 isochrones below 0.6 M_☉ in some smooth fashion with the α_ML = 1.9 isochrones at some mass above 0.6 M_☉ in order to bridge the discontinuity, which results in a kink at 0.6 M_☉ and an unsmooth and likely unphysical change in stellar properties across the boundary.

This situation is less than ideal because the α_ML = 1.9 models were computed with a helium abundance Y appropriate for the Sun that is not the same as that assumed for the α_ML = 1.0 models. A larger Y for these young ages (as in the α_ML = 1.9 models) typically leads to a lower luminosity at a given temperature, with some dependence on mass, so in a strict sense joining the isochrones is inconsistent. Furthermore, the assumption that the models become insensitive to the mixing length parameter below about 0.6 M_☉ is not necessarily valid at early ages for stellar models with low surface gravities and cool temperatures (Baraffe et al. 2002), and may only hold true for much lower masses (see also Burrows et al. 1989; D'Antona & Mazzitelli 1994; Luhman & Reike 1998). Indeed, a comparison of published Lyon models with α_ML = 1.9 and α_ML = 1.0 shows that the difference in effective temperature at a fixed mass of 0.6 M_☉ rises considerably toward younger ages, from about 60 K at 10 Myr to 120 K at 3 Myr and 230 K at 1 Myr, which is the age regime relevant for GG Tau.

This undermines the conclusion that the Lyon models for "α_ML = 1.9" used by White et al. (1999) and Luhman (1999) agree with the measurements for GG Tau, as the comparison may not be very meaningful.

Given that the Dartmouth models seem to work well for LkCa 3, it is of interest to see how they fare when tested against the observed properties of GG Tau. This comparison is shown in Figure 12 for an age of 1 Myr. The spectral types and luminosities adopted for stars Aa and Ab are those reported by White et al. (1999), and for stars Ba and Bb we used the revised determinations by Luhman (1999). Spectral types were converted to effective temperatures with the intermediate scale of Luhman (1999) for Ba and Bb and the dwarf scale for Aa and Ab since the difference with the giant scale is insignificant at these earlier spectral types.⁸ The temperatures were assigned a conservative uncertainty of 150 K, corresponding approximately to a spectral type error of one subtype.

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In Figure 12, star Bb is outside the range of masses covered by the Dartmouth models, but there is excellent agreement for the other three components of GG Tau. The dynamical mass estimate for GG Tau A is also well reproduced by these models: the sum of the inferred masses of stars Aa and Ab is 1.19 ± 0.06 M_☉, to be compared with the measured value of 1.28 ± 0.07 M_☉. The individual masses we derive from the Dartmouth models are M_Aa = 0.63 ± 0.05 M_☉, M_Ab = 0.56 ± 0.04 M_☉, and M_Ba = 0.15 ± 0.02 M_☉.

We note that the SpT/T_eff conversion of Luhman (1999) used in this comparison was established largely using GG Tau itself (specifically, the Ba and Bb components) along with the hybrid "α_ML = 1.9" models from Lyon. Using this to test the Dartmouth models may therefore seem to be inconsistent. However, as seen in Figure 12, the Lyon and Dartmouth isochrones converge near star Ba (slightly under 0.2 M_☉), so the fact that the Dartmouth model is able to match the temperature of this star is not surprising. Whether it would also be able to reproduce the temperature of the Bb component using the same SpT/T_eff conversion if the calculations were extended to lower masses remains to be seen.

We conclude from the above that, as in the case of LkCa 3, the available constraints for the quadruple system GG Tau favor the PMS Dartmouth models over those from Lyon with α_ML = 1.0, at least at these young ages. It is quite possible, however, that Lyon models with the higher value of α_ML = 1.9, if they were available for masses below 0.6 M_☉, might also perform well, although this cannot be verified at the moment. A second conclusion we may draw is that the Dartmouth models do not appear to be inconsistent with the SpT/T_eff prescription advocated by Luhman (1999) in which a dwarf-like temperature scale is adopted for young stars earlier than M0 and an intermediate scale for later-type stars, although further observational constraints between M1 and M5 are required to confirm this.

Based on the examples of GG Tau and LkCa 3, we would expect that in other cases the use of the Dartmouth models instead of the α_ML = 1.0 Lyon models for PMS stars would result in generally smaller inferred masses and slightly younger ages, although this will depend to some extent on the constraints available.