On the Age of the TRAPPIST-1 System

The locations of TRAPPIST-1 and other late M-type spectral standards and field stars on the M_Ks versus ($V-{K}_{s}$) CMD are shown in Figure 1. The comparison data were drawn from Dieterich et al. (2014) and Winters et al. (2015) for nearby M dwarfs with trigonometric parallaxes. A polynomial fit to these data between $3.5\lt (V-{K}_{s})\lt 11.8$ yields

$$\begin{eqnarray}&&{M}_{{Ks}}\,=\,26.987-20.6315(V-{K}_{s})\\ &&\,+\,6.88044{(V-{K}_{s})}^{2}-1.01665{(V-{K}_{s})}^{3}\\ &&\,+\,0.0707374{(V-{K}_{s})}^{4}-0.00188517{(V-{K}_{s})}^{5}.\end{eqnarray} \tag{ 1 }$$

The root mean square scatter is 0.48 mag. Recent spectroscopic surveys of field M dwarfs found median metallicities ranging from [Fe/H] ≃ +0.04 (Newton et al. 2014) to [Fe/H]  ≃ −0.03 (Mann et al. 2015), with 1σ dispersions of ∼0.2 dex. Hence, this CMD sequence is largely representative of a solar-metallicity population.

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Also shown on the CMD are recent isochrones of low-mass stars of solar composition from Baraffe et al. (2015) for ages of 0.2, 0.3, 0.5, 3, and 8 Gyr; and masses of 0.07, 0.08, and $0.09\,{M}_{\odot }$, spanning current mass estimates for TRAPPIST-1. These isochrones reproduce the empirical locus for late M dwarfs to an accuracy of ∼0.1–0.2 mag in M_Ks or ∼0.2 mag in $(V-{K}_{s})$ color, but not for the greater spread, which is likely due to metallicity scatter (López-Morales 2007). Binaries are relatively rare among late M dwarfs (∼10%–20%; Allen 2007; Kraus & Hillenbrand 2012) and most field stars will be older than the pre-main-sequence contraction timescale of hundreds of Myr.

TRAPPIST-1 lies 0.15 ± 0.04 mag above (brighter than) the empirical locus. This offset could be interpreted as TRAPPIST-1 being a 0.2–0.3 Gyr old, 0.07 ${M}_{\odot }$ brown dwarf. However, this interpretation would predict a density of 38 ${\rho }_{\odot }$ based on the Baraffe et al. (2015) models, which is much lower than the measured density from transit observations (see below). We can also rule out contamination from a stellar or substellar companion based on previous adaptive optics surveys (Bouy et al. 2003; Gizis et al. 2003; Siegler et al. 2003; Janson et al. 2012) and multi-epoch radial velocity studies (Tanner et al. 2012; Barnes et al. 2014).

Instead, the magnitude offset of TRAPPIST-1 is likely due to a slightly supersolar metallicity. The offset is similar to that of the M8 V standard VB 10 (aka GJ 752B), and both it and its M3 V companion, GJ 752A, are slightly metal-rich ([Fe/H] = 0.09 ± 0.12 and 0.10 ± 0.08, respectively; Newton et al. 2014; Mann et al. 2015), which is consistent with the spectroscopic value for TRAPPIST-1 (Gillon et al. 2016). Alternately, attributing the 0.48 mag scatter in the absolute magnitudes of late M dwarfs entirely to metallicity scatter (∼0.2 dex) would imply a metallicity-absolute magnitude gradient of Δ[Fe/H]/ΔM_Ks ≃ −0.4 dex mag⁻¹, with metal-rich stars being brighter. Under this interpretation, the −0.15 mag absolute magnitude offset for TRAPPIST-1 would be consistent with [Fe/H] = +0.06 dex, which is again consistent with the spectroscopic metallicity,

Unfortunately, current evolutionary models for the very lowest-mass stars and brown dwarfs are defined only for solar metallicities, preventing us from disentangling age and metallicity effects. Our CMD analysis therefore reinforces a supersolar metallicity for TRAPPIST-1, but cannot constrain its age.

Average stellar density, an observable from transit lightcurve analysis (Seager & Mallén-Ornelas 2003), provides an independent check on the evolutionary state of TRAPPIST-1. Figure 1 shows the predicted average densities for stars and brown dwarfs from the solar-metallicity evolutionary models of Burrows et al. (1997, 2001) and Baraffe et al. (2015) as a function of age, constrained to have the observed luminosity of TRAPPIST-1. At this luminosity, young, contracting substellar objects have densities that increase up to an age of approximately 500 Myr; beyond 1 Gyr (corresponding to hydrogen-burning low-mass stars), densities plateau. The Burrows et al. models predict densities 17% higher than the Baraffe et al. models, corresponding to radii that are 6% smaller. The average density of TRAPPIST-1 is on the rising portion of this trend, and intersects with the Burrows et al. and Baraffe et al. models at ages of 0.3 and 0.4 Gyr, respectively, corresponding to masses at the hydrogen-burning mass limit ($0.071\,{M}_{\odot }$ and $0.079\,{M}_{\odot }$).

These comparisons again suggest that TRAPPIST-1 could be relatively young. However, two factors confound this interpretation. First, there are the previously noted metallicity effects. Among −0.04 < [Fe/H] < 0.12 low-mass stars, López-Morales (2007) found that evolutionary models underpredict stellar radii by 10%–20%, which is a correction factor sufficient to bring the density plateaus of the Baraffe et al. and Burrows et al. models in line with the observed density of TRAPPIST-1. Second, magnetic activity can also modulate the radii of low-mass stars and brown dwarfs (Chabrier et al. 2007; López-Morales 2007; Reiners et al. 2007; MacDonald & Mullan 2009; Mohanty et al. 2010). Empirical relations from Stassun et al. (2012) predict a modest effect: 0%–4% based on X-ray emission, and 4%–6% based on Hα emission. However, theoretical models by Chabrier et al. (2007) show radii can range over $0.10\mbox{--}0.14\,{R}_{\odot }$ at $M=0.08\,{M}_{\odot }$ for (black) spot surface coverage of up to 50%, which is more than sufficient to cover the difference between TRAPPIST-1's observed stellar density and models. Kepler data have confirmed the presence of cool, stable magnetic spots on TRAPPIST-1 (Luger et al. 2017; Vida et al. 2017), so these may play a role in radius inflation.

Given that both metallicity and magnetic activity likely play a role in setting the radius and average stellar density of TRAPPIST-1, current evolutionary models do not allow us to extract meaningful age constraints from this physical parameter.

Multiple studies have reported the absence of 6708 Å Li i absorption in the optical spectrum of TRAPPIST-1, indicating depletion of this element in its fully convective interior (Reiners & Basri 2009; Burgasser et al. 2015). As noted above, theoretical models of solar-metallicity stars and brown dwarfs more massive than $0.06\,{M}_{\odot }$ show full depletion within ∼200 Myr (Bildsten et al. 1997; Burke et al. 2004). Similarly, the Burrows et al. (1997, 2001) and Baraffe et al. (2015) evolutionary models predict a minimum age of 190 Myr for a $0.06\,{M}_{\odot }$ brown dwarf with the observed luminosity of TRAPPIST-1. These estimates provide a lower limit to TRAPPIST-1's age that is already incorporated into the age range proposed by Filippazzo et al. (2015).

Low-resolution near-infrared spectra presented in Gillon et al. (2016) are generally consistent with the M8 dwarf spectral standard VB 10. However, there are some notable peculiarities, including weaker FeH absorption and a more triangular H-band peak (Figure 2). Application of the Allers & Liu (2013) surface gravity indices yield a gravity classification of INT-G for this source, suggesting a low surface gravity and young age (∼100–300 Myr). Moreover, comparison of its near-infrared spectrum to the entirety of the SpeX Prism Library (Burgasser 2014) uncovers an excellent match to the M7 dwarf 2MASS J2352050−110043 (Cruz et al. 2007; hereafter 2MASS J2352−1100), which is a source identified by Gagné et al. (2015a) and Aller et al. (2016) as a possible kinematic member of the ∼110 Myr AB Doradus association (Luhman et al. 2005; Barenfeld et al. 2013). These lines of evidence again suggest TRAPPIST-1 could be a young brown dwarf.

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However, 2MASS J2352−1100 lacks Li i absorption, expected for a 110 Myr M8 dwarf; and it kinematic association with AB Doradus is based on proper motion and estimated distance alone, and may be spurious. TRAPPIST-1's kinematics firmly rule out membership in AB Doradus or any nearby young moving group (Malo et al. 2013; Gagné et al. 2014); and neither TRAPPIST-1 nor 2MASS J2352−1100 exhibit the enhanced VO absorption generally seen in the spectra of low-gravity M and L dwarfs (Kirkpatrick et al. 2008; Cruz et al. 2009; Allers & Liu 2013). The near-infrared spectrum of TRAPPIST-1 is also a good match to those of the M8 dwarfs LP 938-71 and 2MASS J2341286−113335 (Figure 2), neither of which are reported to be unusually young or active (Cruz et al. 2007; Schmidt et al. 2007).

We conclude that the INT-G gravity classifications for both TRAPPIST-1 and 2MASS J2352−1100 are unrelated to youth, and may arise from other physical factors, as previously reported for the high velocity M6.5 2MASS J02530084+1652532 (aka Teegarden's star, V_tan = 93 km s⁻¹; Teegarden et al. 2003; Henry et al. 2006; Gagné et al. 2015b) and the d/sdM7 metal-poor companion GJ 660.1B (Aganze et al. 2016). Given the apparent interplay between surface gravity and metallicity effects in index-based gravity diagnostics for late M dwarfs, we discount the INT-G classification as evidence of youth for TRAPPIST-1.

While age–metallicity correlations are generally weak among stellar populations, the dispersion of stellar metallicities increases for populations older than a few Gyr, and median metallicity decreases for populations older than ≈10 Gyr (Edvardsson et al. 1993; Haywood et al. 2013; Bergemann et al. 2014). Hence, comparison of TRAPPIST-1's metallicity to population distributions can provide a statistical constraint on its age.

To quantify this diagnostic, we examined the age and metallicity distributions of stars drawn from the Spectroscopic Properties of Cool Stars (SPOCS; Valenti & Fischer 2005) and an updated analysis of the Geneva-Copenhagen Survey (GCS; Casagrande et al. 2011). In both samples, ages are inferred by comparison of CMDs to model isochrones (for GCS, we used the ages inferred from the Padova isochrones; Bertelli et al. 2008), while metallicities are determined from spectroscopic measurements in SPOCS and Strömgren photometry in GCS. For both samples, we selected $M\leqslant 1\,{M}_{\odot }$ stars with −0.04 < [Fe/H] < +0.12 and parallactic distances within 30 pc, and constructed an age distribution by assuming a uniform likelihood for each star between the minimum and maximum ages (SPOCS) or 16% and 84% (±1σ for a normal distribution) Padova isochronal ages (GCS). These distributions are shown in Figure 3. In both samples, stars younger than 1–2 Gyr and older than 11–12 Gyr are relatively rare. The SPOCS age distribution peaks at young ages, which is enhanced by the metallicity constraint. The GCS age distribution is flat between 2 and 10 Gyr, with a slight preference toward younger ages with the metallicity constraint. While the maximum likelihood ages are quite different between these samples, their distributions overlap, and we infer ages of ${3.2}_{-1.0}^{+6.0}$ Gyr for SPOCS and ${5.5}_{-2.3}^{+3.7}$ Gyr for GCS. As anticipated, the uncertainties are considerable and do little to reduce the overall uncertainty in the system's age.

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Reiners & Basri (2009) and Burgasser et al. (2015) previously reported statistically equivalent UVW kinematics for TRAPPIST-1 based on radial velocity and proper motion measurements, the latter study concluding that the star is a borderline thin/thick disk star based on the criteria of Bensby et al. (2003). We update this analysis using the more precise radial velocity reported in Barnes et al. (2014) and astrometry reported in Weinberger et al. (2016). The corresponding heliocentric UVW velocities are given in Table 1. Adopting the Local Standard of Rest (LSR) correction of Dehnen & Binney (1998) used by Bensby et al. (2003), ${{UVW}}_{\odot }\,=[+10.00,+5.25,+7.17]$, we find probabilities of kinematic association P(thin) = 81%, P(thick) = 19%, and P(halo) < 0.1%. With P(thick)/P(thin) = 0.23, this star remains a borderline thin/thick disk star by the Bensby et al. (2003) criteria.

To derive a quantitative estimate of TRAPPIST-1's kinematic age, we took advantage of the fact that the V-velocity asymmetric drift of stellar populations increases over time as the velocity scatter increases (${V}_{a}\propto {\sigma }_{U}^{2};$ Strömberg 1924). We again used the GCS sample and examined the age distribution of stars with negative V velocities like TRAPPIST-1. Figure 4 shows the same mass- and distance-constrained distribution as Figure 3, but with the additional constraint that V < V_T1 = −66 km s⁻¹. In this case we see a tilt toward older ages, with a maximum likelihood value of 9.8 Gyr, albeit with a wide uncertainty range (3.9–10.5 Gyr). We find a similar distribution (albeit with a very small sample) when a metallicity constraint was also applied to the GCS sample.

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As a second approach, we applied a Bayesian method to solve for the probability distribution function of a star's age given its UVW velocities:

$$\begin{eqnarray}&&P(\mathrm{age}| \mathrm{UVW})\propto P(\mathrm{UVW}| \mathrm{age})P(\mathrm{age}).\end{eqnarray} \tag{ 2 }$$

Here, P(age) is the a priori distribution of stellar ages while P($\mathrm{UVW}| \mathrm{age}$) is the distribution of stellar UVW velocities as they evolve over time. We considered three different age priors in our analysis: a constant age distribution (the constant star formation rate) up to 12 Gyr; the age distribution of GCS stars with $M\leqslant 1\,{M}_{\odot }$ and $d\leqslant 30$ pc without constraints on metallicity or V-velocity; and a constant age distribution with an additional "heating" term modeled after Aumer & Binney (2009) that deweights older populations that spend less time in the immediate solar neighborhood,

$$\begin{eqnarray}&&\mathrm{ln}P\propto -\displaystyle \frac{{\rm{\Delta }}{\rm{\Phi }}(Z)}{{\sigma }_{W}{(t)}^{2}}.\end{eqnarray} \tag{ 3 }$$

Here, ${\rm{\Delta }}{\rm{\Phi }}(Z)$ is the vertical gravitational potential difference from the mid-plane to a Galactic height Z, taken to be 50 pc; and ${\sigma }_{W}(t)$ is the vertical velocity dispersion over time t in Gyr, calculated as

$$\begin{eqnarray}&&{\sigma }_{W}(t)=8.388{(t+0.01)}^{0.445}\,\mathrm{km}\,{{\rm{s}}}^{-1}\end{eqnarray} \tag{ 4 }$$

(Aumer & Binney 2009). For the Galactic potential, we used the four-component disk, halo, and bulge model of Barros et al. (2016) at a Galactic radius of 8 kpc. UVW velocities as a function of age were drawn using the age-dispersion relations of Aumer & Binney (2009), including an asymmetric drift term ${V}_{a}=-\tfrac{{\sigma }_{U}^{2}}{74\,\mathrm{km}\,{{\rm{s}}}^{-1}}$. We drew 10⁷ stars at each simulated time step and computed the fraction of draws as a function of age whose UVW velocities were within 5σ of those of TRAPPIST-1.

Figure 4 shows the resulting distribution of stellar ages for our three age priors. All are strongly skewed toward older ages, with maximum likelihood values ranging from 10.2 Gyr (GCS) to 12 Gyr (constant). The GCS and heating model priors produce similar distributions within 11 Gyr, the former dropping off rapidly beyond this in accord with the underlying sample age distribution. Using the median values as best estimates, we infer ages of ${9.2}_{-2.7}^{+1.1}$ Gyr and ${8.7}_{-2.9}^{+2.3}$ Gyr for the GCS and heating model priors, which are on the high end but consistent with the other age diagnostics.

While the timescales for rotation spin down and activity decline for ultracool dwarfs become exceedingly long compared to more massive stars (West et al. 2008; Irwin et al. 2011), there is evidence that both properties do evolve in a measurable way (e.g., Burgasser et al. 2015). Early analysis of TRAPPIST-1 has suggested that its 3.295 day rotation period is average for late-type M dwarfs, suggesting a "middle-age" star (3–8 Gyr from Luger et al. 2017).

To quantify this, we compared the rotation period of TRAPPIST-1 to those of mid- and late M dwarfs observed through the MEarth program (Nutzman & Charbonneau 2008) as reported in Newton et al. (2016). Selecting subsamples of stars with significant periodic variability ($A/{\sigma }_{A}$ > 2) in $0.02\,{M}_{\odot }$ bins over masses of $0.07\mbox{--}0.18\,{M}_{\odot }$, we computed the fraction of each subsample that had periods longer than TRAPPIST-1. Figure 5 shows the resulting trend, illustrating that roughly 60% of the stars observed in this study were slower rotators, which suggests that they are older. If we assume (simplistically) that rotation declines monotonically over time, and use the GCS sample as an age prior, this analysis would suggest an age of ∼4–5 Gyr for TRAPPIST-1, depending on the assumed age of the Milky Way.

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However, the rotation periods of very low-mass stars at late ages are highly sensitive to initial conditions and the mechanism for angular momentum loss. Figure 5 shows the evolution of rotation period for a $0.08\,{M}_{\odot }$ star following the angular momentum loss rate prescription of Chaboyer et al. (1995) and Krishnamurthi et al. (1997) as previously applied to low-mass stars (e.g., Bouvier et al. 1997; Reiners & Basri 2008; Irwin et al. 2011). We assumed a critical (or saturation) rotation rate ${\omega }_{\mathrm{crit}}={\omega }_{\mathrm{crit},\odot }\tfrac{{\tau }_{\odot }}{\tau }=1.86{\omega }_{\odot }$, where ${\omega }_{\mathrm{crit},\odot }=10{\omega }_{\odot }$ and τ is the convective overturn time assumed proportional to ${M}^{-2/3}$ (Reiners & Basri 2008). We used the time-dependent radii for an $M=0.08\,{M}_{\odot }$ from the models of Baraffe et al. (2003). We considered both "slow" and "fast" prescriptions for spin down⁴ from Irwin et al. (2011), as well as a range of initial rotation velocities at ∼20–50 Myr of $20\mbox{--}100\,{\omega }_{\odot }$, based on measurements for $M\lt 0.35\,{M}_{\odot }$ stars in NGC 2547 reported in the same study. As shown in Figure 5, different angular momentum loss prescriptions produce widely dispersed rotation periods beyond 300 Myr, spanning nearly three orders of magnitude by 10 Gyr. This range is well-matched to the range of observed rotation periods for significantly variable $0.06\mbox{--}0.10\,{M}_{\odot }$ stars in the sample of Newton et al. (2016), which span 0.11–364 days. TRAPPIST-1's rotation period resides between the slow and fast evolutionary tracks. Since neither the specific mechanism of spin down nor the initial rotation rate are known for this source, at best we can conclude that TRAPPIST-1 is likely older than 300 Myr, the age at which the "fast" track periods exceed 3.3 days.

Low-mass stars show clear age-activity correlations related to the spin down of stars and reduction of rotationally driven magnetic dynamos (Skumanich 1972; Feigelson & Lawson 2004; Covey et al. 2008). As spin-down timescales increase for the lowest-mass stars, saturated magnetic emission can persist for even slowly rotating stars ($P\lt 86$ day) with little correlation between the incidence of emission and rotation period (West et al. 2008, 2015). However, there is evidence for a correlation between the strength of Hα emission and "stratigraphic" age (the distance from the Galactic plane) that continues through the end of the M dwarf sequence.

Persistent Hα emission from TRAPPIST-1 is weaker than emission in over half of the active late M dwarfs near the Sun (Figure 6), suggesting an age in the upper half of this sample. This comparison stands in contrast to claims that TRAPPIST-1 is "highly active," which suggests youth (e.g., Bourrier et al. 2017; Vida et al. 2017). The perception that TRAPPIST-1 is highly active is also related to its flaring emission, specifically the detection of a supersolar flare in its K2 lightcurve, with an integrated energy E = 10³³ erg (Vida et al. 2017). Again, context is essential. M dwarfs typically have higher optical and infrared flare rates and flare energies than solar-type stars (Davenport et al. 2012), and TRAPPIST-1's flare duty cycle during the K2 monitoring period, ≈0.1%⁵ as reported by Vida et al. (2017), is more than an order of magnitude below the 3 ± 1% inferred for M7–M9 dwarfs by Hilton et al. (2010). The cumulative flare frequency distribution for TRAPPIST-1 as a function of energy, also reported in Vida et al. (2017), is similarly depressed by a factor of $\approx 4$ compared to other M6-M8 dwarfs (Hilton 2011; Gizis et al. 2017a). Finally, the K2 flare, while dramatic, is not unique; Gizis et al. (2017b) report an E > 4 × 10³³ erg flare from an L0 dwarf observed with K2, and Schmidt et al. (2014, 2016) have reported E > 10³⁴ erg flares from M8 and L1 dwarfs detected in the All-Sky Automated Survey for Supernovae survey (Shappee et al. 2014).

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Taken together, these activity metrics suggest TRAPPIST-1 is older than the typical late M dwarf in the solar neighborhood, but remains an active star. Given the lack of empirical calibrations for age/activity among the latest M dwarfs, we are unable to more specifically quantify TRAPPIST-1's age from these measures.