ON THE AGE AND BINARITY OF FOMALHAUT

An isochronal age for Fomalhaut can be estimated through comparing its T_eff and luminosity to modern evolutionary tracks. The stellar parameters for Fomalhaut are fairly well determined due to its brightness and proximity, which has enabled the star to have its diameter measured interferometrically. Here I estimate refined H-R diagram parameters for Fomalhaut and estimate an isochronal age.

Basic stellar parameters for Fomalhaut are listed in Table 2. Davis et al. (2005) estimated the bolometric flux of Fomalhaut to be 8.96 ± 0.25 × 10⁻⁹ W m⁻², which I adopt. Combining this with the revised Hipparcos parallax from van Leeuwen (2007) of ϖ = 129.81 ± 0.47 mas (d = 7.704 ± 0.028 pc), this results in a bolometric luminosity for Fomalhaut of 16.63 ± 0.48 L_☉ or log(L/L_☉) = 1.221 ± 0.013 dex.²Absil et al. (2009) resolved a small amount of K band excess due to circumstellar dust and reported a revised limb-darkened diameter taking into account all the available VLTI data: θ_LD = 2.223 ± 0.022 mas. Using the relation from van Belle et al. (1999) (T_eff = (2341 K) × (f_bol/θ²_LD)^(1/4), where f_bol is in units of 10⁻⁸ erg cm⁻² s⁻¹ and θ_LD is in mas) with the bolometric flux from Davis et al. (2005) and the limb-darkened diameter from Absil et al. (2009), I derive a new T_eff of 8590 ± 73 K and radius 1.842 ± 0.019 R_☉. The T_eff is only slightly lower than recent estimates (e.g., Davis et al. 2005).

To calculate an isochronal age, I overlay the new H-R diagram point for Fomalhaut on the evolutionary tracks of Bertelli et al. (2008) (Figure 1, top). I assume that Fomalhaut has a chemical composition similar to the proto-Sun, with an asteroseismically motivated (and diffusion corrected) composition of Y = 0.27 and Z = 0.017 (see Serenelli & Basu 2010 and references therein).³ I generate H-R diagram positions by Monte Carlo sampling the bolometric flux, parallax, and limb-darkened radius values from their quoted values and uncertainties (assuming a normal distribution), and interpolating within the Bertelli et al. (2008) tracks. The T_eff and luminosity of Fomalhaut are consistent with a mass of 1.95 ± 0.02 M_☉ and age of 433 ± 36 Myr. The uncertainties only take into account observational errors and not systematic uncertainties in chemical composition and input physics. To estimate systematic uncertainties due to different assumed solar composition and input physics, I estimate calculate ages and masses for three more sets of tracks (see Table 3). The expectation ages for the four sets of tracks are very similar,⁴ and the mean of the ages from the four tracks is 450 Myr with a 22 Myr (5%) rms scatter (which is a reasonable estimate of the systematic error considering slightly different assumed protosolar abundances and input physics). Considering the typical observational error in age (±33 Myr; 7%), this suggests a total isochronal age uncertainty of ±40 Myr (9%). For the four sets of tracks, the average mass is 1.923 M_☉, with ±0.014 M_☉ rms (systematic error component) and ±0.016 M_☉ scatter due to the observational errors. Note that this new mass (1.92 ± 0.02 M_☉) is similar to previous estimates (e.g., Kalas et al. 2008), but 16% lower than the 2.3 M_☉ quoted by Chiang et al. (2009). The new estimated mass is in line with observed trends in T_eff and log(L/L_☉) versus mass for main-sequence (MS) stars in eclipsing binaries (Malkov 2007), which empirically predict ∼1.9–2.0 M_☉.

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Although overlooked in most recent literature on Fomalhaut, the star has a likely stellar companion: TW PsA (GJ 879, HIP 113283). That TW PsA and Fomalhaut appear to share proper motion and parallax appears to have been first noticed by Luyten (1938). TW PsA is an active K4Ve star (Keenan & McNeil 1989), at a projected separation of 196 (∼7100''), and has been listed as among the widest (∼50,000 AU) candidate binaries known (Gliese & Jahreiß 1991). The proximity of TW PsA and its approximate co-motion with Fomalhaut led to Barrado y Navascues et al. (1997) using TW PsA to age-date Fomalhaut. Shaya & Olling (2011) included Fomalhaut and TW PsA as a wide binary in their Bayesian search for multiple systems in the Hipparcos catalog, and the system was one of only two binaries with separations >0.25 pc identified within 10 pc.

Given the utility of TW PsA to age-dating Fomalhaut, we should test the physicality of the purported binary system using the best available astrometry. The best available astrometric and radial velocity data for Fomalhaut and TW PsA are listed in Table 2, and their degree of similarity is striking. I adopt the literature mean v_R for Fomalhaut from Gontcharov (2006; 6.5  ± 0.5 km s⁻¹). This should reflect center-of-mass motion, as the revised Hipparcos astrometric analysis was able to statistically fit an unperturbed single-star solution for Fomalhaut's astrometry (van Leeuwen 2007). The astrometric acceleration from the original Hipparcos reduction reported by Chiang et al. (2009) is statistically insignificant (1.7σ) and likely spurious. For TW PsA, I adopt the v_R from Nordström et al. (2004), who reported a mean v_R for TW PsA of 6.6 km s⁻¹ over seven epochs over 3794 days. The star apparently showed remarkable stability, with a quoted rms of ±0.1 km s⁻¹.

I calculate the Galactic velocity vectors U, V, and W (in the direction of Galactic center, rotation, and North Galactic pole, respectively) from the astrometry and radial velocities of Fomalhaut and TW PsA. The three-dimensional (3D) velocities are listed in Table 2. The degree of similarity is embarrassingly good, as their velocities agree within ±0.1 km s⁻¹ in all three directions. The barycentric speeds of the star differ by only 0.1 ± 0.5 km s⁻¹. Just how remarkable is the agreement in velocities? I generated a catalog of 3D velocities for 34,817 stars with updated Hipparcos astrometry from van Leeuwen (2007; those with positive parallaxes) and literature mean radial velocities from Gontcharov (2006). The only stars with 3D velocities within 1 km s⁻¹ of Fomalhaut are TW PsA and HIP 3800 (0.99 km s⁻¹ different). This suggests that <10⁻⁴ of field stars have velocities within 1 km s⁻¹ of Fomalhaut's, and that the similarity in velocities between Fomalhaut and TW PsA (separated by only ∼0.3 pc) is more than just a coincidence.

What is the 3D separation of Fomalhaut and TW PsA? Taking the parallax values and uncertainties from van Leeuwen (2007), I generate 10⁴ Monte Carlo realizations of the distances to Fomalhaut and TW PsA (assuming d = 1/parallax). Fomalhaut has a parallax distance of 7.704 ± 0.028 pc, while TW PsA is at 7.609 ± 0.036 pc. The 3D separations in the simulations have a median separation of 0.280^+0.019_{− 0.012} pc (57.4^+3.9_{− 2.5} kAU; 68%CL range quoted). Many plausible binary systems are known with larger separations (e.g., Shaya & Olling 2011). Assuming that the census of the nearest 100 star systems is complete,⁵ the local density of star systems is 0.085 pc⁻³. The chances of having an unrelated star (system) within 0.28 pc of a nearby star is approximately 1 in 130.

So the proximity in space and velocity between Fomalhaut and TW PsA appear to be more than coincidental, but are they bound? The escape velocity from Fomalhaut (1.92 M_☉) at the separation of TW PsA is 0.21 km s⁻¹, suggesting that the observed difference in velocities (0.1 ± 0.5 km s⁻¹) is statistically consistent with the hypothesis of TW PsA and Fomalhaut constituting a bound pair. If one sets the semimajor axis of TW PsA's orbit to be equal to the observed 3D separation, and adopt a TW PsA mass of 0.73 M_☉ (Casagrande et al. 2011), then one estimates an orbital period of ∼8 Myr. The predicted amplitude of the orbital velocities would be 0.06 km s⁻¹ for Fomalhaut and 0.15 km s⁻¹ for TW PsA.

Rotation rates among late-F through early-M dwarfs appear to spin-down as they age through magnetically braking, approximately as rotation period ∝ age^1/2 (Skumanich 1972). Using the rotation period from Busko & Torres (1978; 10.3 days) and the gyrochronology curves from Mamajek & Hillenbrand (2008), and assuming ±1.1 days rms fit to the gyro relation, I estimate a gyrochronology age of 410 ± 80 Myr.

Attempts to derive an isochronal age for TW PsA were discussed by Barrado y Navascues et al. (1997). Here I use the T_eff and luminosity from Table 2 to derive new estimates of isochronal ages from evolutionary tracks. Using the pre-MS evolutionary tracks of D'Antona & Mazzitelli (1997), TW PsA is consistent with having a mass of 0.71 M_☉ and an age of 52 Myr. Using the Baraffe et al. (1998) pre-MS tracks, TW PsA is consistent with having a mass of 0.72 M_☉ and 66 Myr. As discussed by Barrado y Navascues et al. (1997) in their review of TW PsA's other youth diagnostics (Li, activity), it is unlikely that the star is <100 Myr. The spectroscopic surface gravity appears to be log(g) ≃ 4.5–4.7 (e.g., Dall et al. 2005), again consistent with an MS star. In light of those findings, I consider the star to be a young MS star, rather than pre-MS. Hence, I consider the pre-MS isochronal age estimates to represent strict lower limits to TW PsA's age (i.e., >50 Myr), rather than useful age estimates themselves.

TW PsA is also a coronal X-ray source, with L_X = 10^28.33 erg s⁻¹, and fractional X-ray luminosity of log(L_X/L_bol) = −4.57 (Wright et al. 2011). Although the star's color is slightly redder (B − V = 1.1) than the range probed by the calibrations in Mamajek & Hillenbrand (2008), using the X-ray age relation from Equation (A3) of that paper, this log(L_X/L_bol) value would be consistent with an age of ∼380 Myr. Given the scatter in X-ray luminosities among stars in clusters of similar mass (∼0.4 dex; Mamajek & Hillenbrand 2008), the age uncertainty is approximately ⁺⁴⁷⁰_{− 220} Myr.

As pointed out by Barrado y Navascues et al. (1997), TW PsA shows detectable Li (EW(Li i λ6707) = 33 ± 2 mÅ) consistent with an abundance of log N(Li) = 0.6. The recent photometric T_eff from Casagrande et al. (2011) that I adopt (T_eff = 4594 ± 80 K) is not far from the T_eff originally adopted by Barrado y Navascues et al. (1997) (4500 K). Barrado y Navascues et al. (1997) plotted the Li i abundances for members of four clusters of different ages (their Figure 2; Pleiades, M34, UMa, Hyades), and given its completeness, there is little reason to repeat the plot here. What has changed in the past 15 years is the age scale for these benchmark clusters. Barrado y Navascues et al. (1997) adopted the following age scale: Pleiades, 85 Myr; M34, 200 Myr; UMa, 300 Myr; and Hyades, 700 Myr. More recent evolutionary tracks are leading to slightly older ages among the younger clusters. I adopt the following age scale: Pleiades, 130 Myr (Barrado y Navascués et al. 2004), 220 Myr (Meibom et al. 2011); UMa, 500 Myr (King et al. 2003), and Hyades, 625 Myr (Perryman et al. 1998). The Li abundance for TW PsA appears to be intermediate between the Hyades and Pleiades, and M34 and UMa. It is more Li-poor than the Pleiades and M34 stars (hence >220 Myr), but more Li-rich than the UMa stars and Hyades (hence <500 Myr). Based on the Li abundances alone, I adopt an estimate of 360 ± 140 Myr.

The three independent age estimates for TW PsA listed in Table 1 are consistent with a weighted mean age of 400 ± 70 Myr. This age estimate for TW PsA is independent of any genetic association with Fomalhaut.