On the Age of the TRAPPIST-1 System

TL;DR

This paper estimates the age of the TRAPPIST-1 system at approximately 7.6 billion years using multiple empirical methods, providing insights into its planetary stability and habitability potential.

Contribution

It offers a comprehensive, multi-faceted age assessment of TRAPPIST-1, integrating various observational constraints to refine its age estimate.

Findings

TRAPPIST-1 is approximately 7.6 billion years old.

The star is slightly metal-rich with [Fe/H] ≈ +0.06.

The star's radius exceeds model predictions by 8-14%.

Abstract

The nearby (d = 12 pc) M8 dwarf star TRAPPIST-1 (2MASS J23062928-0502285) hosts a compact system of at least seven exoplanets with sizes similar to Earth. Given its importance for testing planet formation and evolution theories, and for assessing the prospects for habitability among Earth-size exoplanets orbiting the most common type of star in the Galaxy, we present a comprehensive assessment of the age of this system. We collate empirical age constraints based on the color-absolute magnitude diagram, average density, lithium absorption, surface gravity features, metallicity, kinematics, rotation, and magnetic activity; and conclude that TRAPPIST-1 is a transitional thin/thick disk star with an age of 7.6$\pm$2.2 Gyr. The star's color-magnitude position is consistent with it being slightly metal-rich ([Fe/H] $\simeq$ +0.06), in line with its previously reported near-infrared spectroscopic metallicity; and it has a radius (R = 0.121$\pm$0.003 R$_{\odot}$) that is larger by 8-14% compared to solar-metallicity evolutionary models. We discuss some implications of the old age of this system with regard to the stability and habitability of its planets.

Tables

Table 1. Table 1: Stellar Parameters for TRAPPIST-1

Parameter	Value	Units	Ref.
Physical Parameters
${\log }_{10}L/L_{\mathrm{bol}}$	$-$3.28$\pm$0.03	dex	1
Teff	2560 $\pm$ 50	K	2
[Fe/H]	+0.04 $\pm$ 0.08	dex	2
$P_{\mathrm{rot}}$	3.295$\pm$0.003	day	3
$v\sin i$	6	km s-1	4
${\log }_{10}$(LHα/Lbol)	$-$4.85 to $-$4.60	dex	5–8
${\log }_{10}$(LX/Lbol)	$-$3.52$\pm$0.17	dex	9
Radius	0.117$\pm$0.004	R⊙	2
$\cdots$	0.121$\pm$0.003	R⊙	10
Mass	0.080$\pm$0.007	M⊙	2
Density	50.7${}_{-2.2}{}^{}{}_{}{}^{+1.2}$	${\rho }_{\odot }$	2
Age	7.6$\pm$2.2	Gyr	10
Astrometric/Kinematic Parameters
$\alpha$	346.6250957	deg	11
$\delta$	-5.0428081	deg	11
${\mu }_{\alpha }$	922.0 $\pm$ 0.6	mas yr-1	12
${\mu }_{\delta }$	$-$471.9 $\pm$ 0.9	mas yr-1	12
$\pi$	80.09 $\pm$ 1.17	mas	12
Distance	12.49 $\pm$ 0.18	pc	12
$v_r$	-51.688 $\pm$ 0.014	km s-1	4
$U$	-43.8 $\pm$ 0.7	km s-1	10
$V$	-66.3 $\pm$ 0.5	km s-1	10
$W$	11.0 $\pm$ 0.3	km s-1	10
$S$	80.0 $\pm$ 0.7	km s-1	10
Photometric Parameters
$V$	18.75 $\pm$ 0.03	mag	13
$R_C$	16.401 $\pm$ 0.004	mag	14
$I_C$	13.966 $\pm$ 0.002	mag	14
$J$	11.35 $\pm$ 0.02	mag	15
$H$	10.72 $\pm$ 0.02	mag	15
$K_s$	10.30 $\pm$ 0.02	mag	15
$W1$	10.07 $\pm$ 0.02	mag	11
$W2$	9.81 $\pm$ 0.02	mag	11
$W3$	9.51 $\pm$ 0.04	mag	11
$V-K_s$	8.45 $\pm$ 0.04	mag	13,15
$M_V$	18.27 $\pm$ 0.04	mag	10
$M_J$	10.87 $\pm$ 0.04	mag	10
$M_{K_s}$	9.81 $\pm$ 0.04	mag	10

Equations

$$M_{Ks}=26.987-20.6315(V-K_s)+6.88044(V-K_s{)}^2-1.01665(V-K_s{)}^3$$

$$+0.0707374(V-K_s{)}^4-0.00188517(V-K_s{)}^5$$

$$P(age|UVW)\propto P(UVW|age)P(\mathrm{age})$$

$$\ln P\propto -\frac{\Delta \Phi (Z)}{{\sigma }_W(t{)}^2}$$

$${\sigma }_W(t)=8.388(t+0.01{)}^{0.445}$$

Full text

stars: activity — stars: atmospheres — stars: low-mass — stars: individual (2MASS J23062928-0502285, TRAPPIST-1)

††journal: ApJ††software: astropy (Astropy Collaboration et al., 2013), SPLAT (Burgasser, 2014)

&

1 Introduction

TRAPPIST-1 (2MASS J23062928$-$0502285; Gizis et al. 2000) is as an ultracool M8 dwarf 12 pc from the Sun which was recently identified to host at least seven Earth-sized planets, three orbiting within the star’s habitable zone (Gillon et al., 2016, 2017; Luger et al., 2017). The planets were identified by both ground-based and space-based transit observations, and span orbit periods of 1.5–19 days and orbital semi-major axes of 0.011-0.062 AU (21-114 stellar radii). These observations and others have provided important constraints on the physical parameters of the star itself, summarized in Table 1.

One crucial parameter of TRAPPIST-1 that is poorly constrained by observations to date is its age, due to the weak empirical age diagnostics currently available for ultracool M dwarfs. While rotation (gryochronology), activity diagnostics, and lithium depletion are standard age-dating tools for solar-type stars (Soderblom, 2010), the physical properties of ultracool dwarfs restrict the application of these tools at the bottom of the main sequence. The low ionization of ultracool dwarf photospheres reduces their coupling with magnetic winds, resulting in spin-down timescales that can exceed the age of the Milky Way Galaxy (West et al., 2008). Depletion of lithium, which provide approximate ages for solar-type stars up to 1–2 Gyr (Sestito & Randich, 2005), is complete for fully-convective low-mass stars by $\sim$200 Myr, and so is useful only for young ultracool dwarfs (Stauffer et al., 1998). Spectral age diagnostics, such as surface gravity sensitive features, are also limited to ages older than $\sim$300 Myr (Allers & Liu, 2013). While the kinematics of TRAPPIST-1 suggest it to be an “old disk” star (Leggett, 1992; Burgasser et al., 2015), such labels are insufficient to firmly constain an age. Filippazzo et al. (2015) report an fairly unconstrained age constraint of 0.5–10 Gyr; while Luger et al. (2017) adopt a more constrained, but still broad, range of 3–8 Gyr. In contrast, several studies in the literature have argued that TRAPPIST-1 must be young based on the strength of its nonthermal magnetic emission (Bourrier et al., 2017; O’Malley-James & Kaltenegger, 2017).

Age is necessary for understanding the formation, orbital evolution, stability, and surface evolution (including habitability) of the planets orbiting TRAPPIST-1. With this in mind, we present an analysis of various age-related empirical diagnostics for this source that allow us to more precisely quantify its age. In Section 2 we analyze in turn the color-absolute magnitude diagram, average density, lithium abundance, surface gravity features, metallicity, kinematics, rotation, and magnetic activity as age diagnostics for the star. In Section 3 we summarize these into a concordance age of 7.6$\pm$2.2 Gyr, and discuss implications on the stability and habitability of the planetary system.

\startlongtable

2 Analysis

2.1 Age Constraints from the HR Diagram

The locations of TRAPPIST-1 and other late M-type spectral standards and field stars on the $M_{Ks}$ versus ($V-K_{s}$) color-absolute magnitude diagram (CMD) are shown in Figure 1. The comparison data are 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 $<$ ($V-K_{s}$) $<$ 11.8 yields:

[TABLE]

The rms scatter among field M dwarfs is 0.48 mag. Recent spectroscopic surveys of field M dwarfs find median metallicities ranging from [Fe/H]  $\simeq$ +0.04 (Newton et al., 2014) to [Fe/H]  $\simeq$ -0.03 (Mann et al., 2015), with 1$\sigma$ dispersions of $\sim$0.2 dex. Hence, this CMD sequence is largely representative of a solar metallicity population.

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*⊙, spanning the mass estimates for TRAPPIST-1. These isochrones reproduce the empirical locus for late-M dwarfs to an accuracy of $\sim$0.1-0.2 mag in MKs* or $\sim$0.2 mag in ($V-K_{s}$) color; but not the greater spread, which is likely due to metallicity scatter (López-Morales, 2007). Binaries are relatively rare among late M dwarfs ($\sim$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$\pm$0.04 mag above (brighter than) the empirical locus. This offset could be interpretted as TRAPPIST-1 being a 0.2–0.3 Gyr-old, 0.07 M*⊙*  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 (Siegler et al., 2003; Bouy et al., 2003; Gizis et al., 2003; Janson et al., 2012) and multi-epoch radial velocity surveys (Tanner et al., 2012; Barnes et al., 2014).

Instead, the magnitude offset of TRAPPIST-1 is likely due to a slightly super-solar metallicity. The offset is similar to that of the M8 V standard VB 10 (aka GJ 752B), and it and its M3 V companion GJ 752A both have [Fe/H]  $\simeq$ 0.1 (Newton et al., 2014; Mann et al., 2015), consistent with the spectroscopic value for TRAPPIST-1 (Gillon et al., 2016). Alternately, attributing the 0.48 mag scatter in absolute magnitude for the CMD empirical locus entirely to metallicity scatter ($\sim$0.2 dex) would imply a metallicity-absolute magnitude gradient of $\Delta$[Fe/H]/$\Delta$MKs $\simeq$ $-$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, again consistent with the spectroscopic metallicity.

Unfortunately, currently-available 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.

2.2 Age Constraints from Stellar Density

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 stellar 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 Gyr and 0.4 Gyr, respectively, corresponding to masses at the hydrogen-burning mass limit (0.071 M*⊙* and 0.079 M*⊙*).

These comparisons again suggest that TRAPPIST-1 is relatively young. However, two factors must be considered. First, there are the previously-noted metallicity effects. Among $-$0.04 $<$ [Fe/H] $<$ 0.12 low-mass stars, López-Morales (2007) found evolutionary models underpredict stellar radii by 10–20%, 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 (López-Morales, 2007; Reiners et al., 2007; Chabrier 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, 4–6% based on H$\alpha$ emission. However, theoretical models by Chabrier et al. (2007) show radii can range over 0.10–0.14 R*⊙* at M = 0.08 M*⊙* for (black) spot coverage of up to 50%, more than sufficient to cover the offset in TRAPPIST-1’s stellar density. Kepler data have confirm 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 average stellar density of TRAPPIST-1, the current unavailability of appropriate evolutionary models encapsulating these effects again prevents us from extracting meaningful age constraints from this physical parameter.

2.3 Age Constraints from Lithium Depletion

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*⊙* show full depletion within $\sim$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*⊙* 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).

2.4 Age Constraints from Surface Gravity Features

Low-resolution near-infrared spectra presented in Gillon et al. (2016) are generally consistent with the M8 dwarf spectral standard VB 10. Howeer, 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 yields a gravity classification of INT-G for this source, suggesting a low surface gravity and young ($\sim$100-300 Myr) age. Moreover, comparison of its near-infrared spectrum to the entirety of the SpeX Prism Library (Burgasser, 2014) uncovers an excellent match to the M7 dwarf 2MASS J2352050$-$110043 (Cruz et al. 2007; hereafter 2MASS J2352$-$1100), a source identified by Gagné et al. (2015a) and Aller et al. (2016) as a possible kinematic member of the $\sim$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.

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 sources 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 metrics for late M dwarfs, we discount the INT-G classification as evidence of youth for TRAPPIST-1.

2.5 Age Constraints from Metallicity

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 $\approx$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 $\leq$ 1 M*⊙* 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 maxium ages (SPOCS) or 16% and 84% 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 2–10 Gyr with a slight preferance 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${}^{+6.0}_{-1.0}$ Gyr for SPOCS and 5.5${}^{+3.7}_{-2.3}$ Gyr for GCS. As anticipated, the uncertainties are considerable and do little to reduce the overall uncertainty in the system’s age.

2.6 Age Constraints from Kinematics

Reiners & Basri (2009) and Burgasser et al. (2015) previously reported 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.

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:

[TABLE]

Here, $P$(age) is the apriori distribution of stellar ages while $P$(UVW$|$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 (constant star formation rate) up to 12 Gyr; the age distribution of GCS stars with M $\leq$ 1 M*⊙* and $d\leq$ 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,

[TABLE]

Here, $\Delta\Phi(Z)$ 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 in km s*-1* over time $t$ in Gyr, calculated as

[TABLE]

(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 $V_{a}=-\frac{\sigma_{U}^{2}}{74~{}{\rm km~{}s^{-1}}}$. 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 5$\sigma$ 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 accordance with the underlying sample age distribution. Using the median values as best estimates, we infer ages of 9.2${}^{+1.1}_{-2.7}$ Gyr and 8.7${}^{+2.3}_{-2.9}$ Gyr for the GCS and heating model priors, which are on the high end but consistent with the other age diagnostics.

2.7 Age Constraints from Rotation

While the timescales for rotation spindown 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 measureable 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*⊙* bins over masses of 0.07–0.18 M*⊙*, 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, suggesting 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 $\sim$4–5 Gyr for TRAPPIST-1, depending on the assumed age of the Milky Way.

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*⊙* 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 $\omega_{crit}$ = $\omega_{crit,\odot}\frac{\tau_{\odot}}{\tau}$ = 1.86$\omega_{\odot}$, where $\omega_{crit,\odot}$ = 10$\omega_{\odot}$ and $\tau$ 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*⊙* from the models of Baraffe et al. (2003). We considered both “slow” and “fast” prescriptions for spindown111In the notation of Chaboyer et al. (1995) these are $K_{slow}$ = 1.20$\times$1045 g cm2 s = 1.25$\times$10*-10* M*⊙* R${}_{\odot}^{2}$ s and $K_{fast}$ = 1.12$\times$1047 g cm2 s = 1.16$\times$10*-8* M*⊙* R${}_{\odot}^{2}$ s, respectively. from Irwin et al. (2011), as well as a range of initial rotation velocities at $\sim$20-50 Myr of 20–100 $\omega_{\odot}$, based on measurements for M $<$ 0.35 M*⊙* 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 three orders of magnitude by 10 Gyr. This range is well-matched to the range of observed rotation periods for significantly variable 0.06–0.10 M*⊙* 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.

2.8 Age Constraints from Activity

Low-mass stars show clear age-activity correlations related to the spindown of stars and reduction of rotationally-driven magnetic dynamos (Skumanich, 1972; Feigelson & Lawson, 2004; Covey et al., 2008). As spindown timescales increase for the lowest-mass stars, saturated magnetic emission can persist for even slowly rotating stars ($P<86$ 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 H$\alpha$ emission and “stratigraphic” age (distance from the Galactic plane) that continues through the end of the M dwarf sequence.

Persistent H$\alpha$ 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”, suggesting 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, and specifically the detection of a super-solar 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, $\approx$0.1%222Based on a typical flare time scale of 1-2 min = 0.02–0.03 hr and the median time between flares 28.1 hr. as reported by Vida et al. (2017), is more than an order of magnitude below the 3$\pm$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$\times$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).

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.

3 Discussion

Combining our age probability distribution functions from metallicity and kinematics, and lower limits from the absence of lithium absorption and measured rotation period, we deduce a concordance age of 7.6$\pm$2.2 Gyr for TRAPPIST-1 (Figure 7). This is inconsistent with some of the qualitative estimates reported in the literature (e.g., “relatively young”; Bourrier et al. 2017; “young”, O’Malley-James & Kaltenegger 2017), and is on the high end of the 3–8 Gyr age adopted by Luger et al. (2017). This older age has important implications on both the stability and habitability of its orbiting planets.

In terms of stability, N-body simulations presented in Gillon et al. (2017) showed the planetary system to be consistently unstable on timescales $<$0.5 Myr, with only an 8% change of surviving 1 Gyr. This is refuted by the much older age we infer for the TRAPPIST-1 star. However, recent simulations show that the resonant configuration of these planets is in fact highly stable through disk migration on timescales of 50 Myr (1010 orbits), with or without eccentricity dampening. That this system appears to have persisted for over 5 Gyr, despite dynamial interactions that are readily detectable through transit timing variations (Gillon et al., 2017; Wang et al., 2017), suggests that the resonant configuration is indeed inherently stable.

In terms of habitability, despite TRAPPIST-1’s modest emission as compared to other late-M dwarfs, the radiation and particle environment is still extreme as compared to the Earth (Bolmont et al., 2017; Wheatley et al., 2017; Garraffo et al., 2017). Based on current estimates of XUV-driven mass loss, the high energy emission of TRAPPIST-1 is likely sufficient to have evaporated an Earth’s ocean of water mass from each of the TRAPPIST-1 planets except $g$ and $h$ over the system’s lifetime (Bolmont et al., 2017; Bourrier et al., 2017). Moreover, the stripping of atmospheres and oceans may be enhanced by direct interaction between stellar and planetary magnetic field lines, which could funnel stellar wind particles directly to the planets’ surfaces (Garraffo et al., 2016, 2017). On the other hand, current estimates of the planets’ densities are generally below Earth’s average density (Gillon et al., 2017; Wang et al., 2017), suggesting volatile-rich worlds that may have ample reservoirs; while ocean evaporation and hydrogen loss could result in an oxygen- and ozone-rich atmosphere that could shield the surface from high UV fluxes (Luger & Barnes, 2015; O’Malley-James & Kaltenegger, 2017). Transit spectroscopy measurements of the atmospheres of these planets are currently insufficient to detect the signatures of all but the lightest elements (de Wit et al., 2016), but the James Webb Space Telescope should have the sensitivity to detect Earth-like atmospheres around these plants, if they exist (Barstow & Irwin, 2016).

Finally, we note that agreement between the observed luminosity, average stellar density and evolutionary models can be achieved if the star’s radius is modestly inflated relative to model predictions. Our analysis indicates that a radius of 0.121$\pm$0.003 R*⊙* is needed to bring both Burrows et al. (1997, 2001) and Baraffe et al. (2015) evolutionary models in line with the observed properties of TRAPPIST-1. This radius is formally consistent with the value adopted in Gillon et al. (2016), and represents a modest 3% increase in planetary radii and 11% decrease in inferred planetary densities, less than current uncertainties (Gillon et al., 2017; Wang et al., 2017).

The authors acknolwedge discussions with Eric Agol, Vincent Bourrier, Amaury Triaud, and Valerie van Grootel that aided in the preparation of the manuscript; and thank the Hon. John Culberson of Texas’s 7th congressional district, US House of Representatives, for asking about the age of TRAPPIST-1 during his visit to JPL in February 2017, which spurred the writing of this paper. EEM acknowledges the NASA NExSS program for support. AJB acknowledges funding support from the National Science Foundation under award No. AST-1517177. Part of this research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. This document does not contain export controlled information (URS266250). This material is based upon work supported by the National Aeronautics and Space Administration under Grant No. NNX16AF47G issued through the Astrophysics Data Analysis Program This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France; NASA’s Astrophysics Data System Bibliographic Services; the M, L, T, and Y dwarf compendium housed at http://DwarfArchives.org; and the Spex Prism Libraries at http://www.browndwarfs.org/spexprism.