THE SOLAR NEIGHBORHOOD. XXX. FOMALHAUT C

Adopted optical and infrared magnitudes from 0.4 μm (B band) to 22 μm (W4 band) are compiled in Table 2. V-band magnitudes of 12.618 ± 0.012 (Reid et al. 2003) and 12.62 ± 0.02 (UCAC4; Zacharias et al. 2013; Henden et al. 2012) have been previously reported for LP 876-10. Additionally, we measure Johnson V = 12.59 ± 0.03 based on three photometric nights of imaging with the Small and Moderate Aperture Research Telescope System's (SMARTS) 0.9 m telescope taken during 2004–2006, using 14'' diameter aperture and the standards of Landolt (1992, 2007). This measurement is consistent with preliminary values previously reported from this program (Bartlett 2007; Bartlett et al. 2007). Jao et al. (2005) and Winters et al. (2011) describe the REsearch Consortium On Nearby Stars (RECONS)¹² photometry program. Pojmanski (1997) presents time series V-band ASAS photometry for this star (349 observations between UT 2000 November 22 and UT 2009 October 7 with quality flag A) with mean V = 12.62 and rms scatter of 0.06 mag. The photometric errors of the individual measurements are typically ∼0.03 mag, so approximately ∼0.05 mag of the scatter in the reported V magnitudes appear to be due to intrinsic stellar variability. All of these previously mentioned V magnitudes are calibrated to the Johnson system, either through Landolt standards (Reid et al. 2003; APASS/UCAC4, RECONS) or Hipparcos (ASAS). The SuperWASP project took many photometric measurements of LP 876-10 in a V band calibrated to the Tycho-2 V_T photometric system (Pollacco et al. 2006; Butters et al. 2010; Høg et al. 2000). This photometry is later discussed in Section 2.8 for the purposes of measuring the rotation period, but was not included in our assessment of the mean Johnson V magnitude. Based on photometry measured independently by Reid et al. (2003), Henden et al. (2012), Pojmanski (1997), and the RECONS observations with the SMARTS 0.9 m telescope, we adopt a mean V magnitude of 12.62 ± 0.01 mag.

The parallax and proper motion of LP 876-10 have been measured during the long-term astrometry program carried out by RECONS at the SMARTS 0.9 m telescope. Jao et al. (2005) describes the astrometry program; however, we briefly summarize the program here. A filter is selected from the Johnson–Kron–Cousins VR_KCI_KC filter set that provides a well-exposed reference field that, ideally, encircles the target star. Throughout the course of the observations, the same pointing (to within a few pixels) and filter are used. Centroids for the reference field and parallax star are extracted using SExtractor (Bertin & Arnouts 1996) and corrected for differential color refraction using VR_KCI_KC photometry of the reference and science target stars (see Section 2.1). Relative parallax and proper motion of the target star are solved for using the Gaussfit program.¹³ Correction from relative to absolute parallax is done by estimating the mean distance to the reference field stars, again, using VR_KCI_KC photometry and the photometric distance relations of Henry et al. (2004).

LP 876-10 was included in the RECONS astrometric survey due to its close predicted photometric distance (7.2 ± 0.8 pc; Reid et al. 2003), which is consistent with preliminary parallax solutions from this program (Bartlett 2007; Bartlett et al. 2007). Based on 25 astrometric nights from 2004 to 2012, we derive an absolute trigonometric parallax of 132.07 ± 1.19 mas and a proper motion of 378.1 ± 0.4 mas yr⁻¹ at position angle (P.A.) 1180 ± 01 east of north. When the proper and parallactic motions are removed from the star's position, the residuals show no hint of curvature or any pattern that would suggest the existence of an unseen companion (see Section 2.4). At distance d = 7.57 ± 0.07 pc, the three-dimensional (3D) separation of LP 876-10 from Fomalhaut is only 0.77 ± 0.01 pc (158$^{+2}_{-1}$ kAU), and from TW PsA it lies only 0.987$^{+0.006}_{-0.005}$ pc (203 ± 1 kAU) away (Fomalhaut B; Mamajek 2012). Figure 2 summarizes the positions, separations, and proper motion vectors for Fomalhaut, TW PsA, and LP 876-10.

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Blinking images suggest that the neighboring high proper motion star LP 876-11 could be a proper motion companion to LP 876-10; LP 876-11 is located 18 away from LP 876-10 at 42° east of north. However, we determine a proper motion for LP 876-11 of 321.3 ± 0.7 mas yr⁻¹ at P.A. 1430 ± 02 east of north, which is inconsistent with the measured motion for LP 876-10. Using 12 color–magnitude relations from Henry et al. (2004), we estimate a photometric distance to LP 876-11 of 730 ± 120 pc. We measure a trigonometric parallax of LP 876-11 ¹⁴ of 1 ± 2 mas, consistent with the photometric distance. We conclude that LP 876-11 is not physically associated with LP 876-10.

A spectrum of LP 876-10 was taken with the CRIRES spectrograph on the 8.4 m Very Large Telescope UT1 (Antu) telescope on UT date 2009 June 16 as part of a near-infrared radial velocity survey of nearby late-type M dwarfs (Bean et al. 2010). The CRIRES spectrum has wavelength coverage 2.292–2.349 μm over the effective 4096 × 512 focal plane detector, a mosaic of four Aladdin III InSb arrays (Kaeufl et al. 2004). The slit width was 02, yielding a resolving power of R ≃ 100,000 (resolution is 3 km s⁻¹ at 2 pixel sampling). The signal-to-noise ratio (S/N) in the continuum of the spectrum was ∼170–220. By fitting a broadened and shifted PHOENIX model spectrum from the Gaia Version 2.0 library (Hauschildt et al. 1999; Brott & Hauschildt 2005) to the spectrum of LP 876-10, we determine a sizeable projected rotation velocity of vsin i = 22 ± 2 km s⁻¹; a heliocentric radial velocity of +6.5 ± 0.5 km s⁻¹ was also measured. Slit viewer images of LP 876-10 appear pointlike, and there is no sign of duplicity in the CRIRES spectrum. A more detailed spectroscopic analysis of LP 876-10 will be presented in a forthcoming paper (A. Seifahrt et al., in preparation).

While neither the astrometry nor the spectroscopy data are consistent with LP 876-10 being a binary, it is listed as a double star in the Washington Double Star catalog (WDS; Mason et al. 2001)¹⁵ as WDS 22481-2422 and with the discovery identifier "WSI 138."¹⁶ A single observation is reported for epoch 2010, with a reported companion at separation 05 at P.A. = 144°, with magnitudes of 12.80 and 14.80 (presumably V-band, as the combined magnitude (12.64) is similar to the adopted V magnitude in Table 2). We are unable to confirm the existence of the companion reported in the WDS. In the 118 frames taken during 25 nights, with FWHMs in the range 12 to 28, LP 876-10 appeared to be a point source, with no evidence of elongation. With only a single observation, the possibility remains that the reported WDS companion may be a chance alignment between this high proper motion star and a background star (B. D. Mason 2013, private communication). However, we believe that a background star is unlikely to explain this discrepancy. Based on the UCAC4 position of LP 876-10 for epoch 2000.0 (Zacharias et al. 2013), the proper motion calculated in this paper, and the separation/P.A. value listed in the WDS, we estimate that the WSI 138 companion reported in the WDS had an approximate ICRS position 22:48:04.76–24 :22:09.1 (epoch 2010). The only object listed in any Vizier-queryable catalog within 2'' of this position is the WISE detection of LP 876-10 itself (05 away) during 2010. No plausible optical–IR counterpart within 2'' of this position exists in the USNO-B1.0, SuperCOSMOS, GSC, and Two Micron All Sky Survey (2MASS) catalogs. It seems very unlikely that a bright (V = 14.8) background star can explain the faint companion to LP 876-10 reported in the WDS. If the companion was real and physically associated with LP 876-10, then its absolute magnitude (M_V = 15.40) would correspond to a 0.11 M_☉ star on the calibration of Delfosse et al. (2000). Given the projected separation (05 = 3.8 AU), these values would predict an orbital period of ∼13.5 yr. Assuming zero eccentricity and face-on projection, one would predict an orbital motion of ∼27 deg yr⁻¹ and a photocentric amplitude of ∼110 mas.

The predicted photocentric amplitude would be about half (50 mas over 8 yr) of the full amplitude (110 mas over ∼13.5 yr) during the observations to date. As seen in Figure 3, the astrometric solution using only parallax and proper motion is quite good, and any gravitational perturbations on LP 876-10 must be at the <10 mas level over ∼8 yr, which easily rules out the predicted signal for the companion reported in the WDS. Table 3 shows that the differences between the long-term proper motions (e.g., SuperCOSMOS, USNO-B1.0, PPMX, and UCAC4) are largely within ∼5–10 mas yr⁻¹ (rms) of the 8 yr baseline proper motion calculated in this survey, further suggesting that it would be difficult to hide a ∼50 mas yr⁻¹ perturbation of the photocentric motion. As the purported WDS companion should have a period only somewhat longer than the duration of our RECONS astrometric data set, and with a predicted photocentric amplitude similar in size to the observed parallax, we conclude that it is unlikely that the companion reported in the WDS catalog is real.

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We estimated T_eff for LP 876-10 by fitting the photometry in Table 2 to the BT-Settl grid of synthetic stellar spectra, which vary by effective temperature, metallicity, and surface gravity (Allard et al. 2012). Twenty-two colors consisting of combinations of the bands V, R_KC, I_KC, J, H, K_s, W1, W2, and W3 were compared to grid interpolations based on models, and the best fit yielded an interpolated temperature of T_eff = 3132 K and solar metallicity. We estimated the uncertainty in T_eff due to metallicity and surface gravity by individually varying these parameters by one increment (0.2 dex) and measuring the effect on the resultant T_eff. The uncertainty in the T_eff breaks down approximately as follows: ±33 K from the dispersion in color-based T_eff estimates for the best fit, ±50 K due to metallicity uncertainty, and ±25 K due to uncertainty in log(g). Together this yields an overall T_eff uncertainty of ±65 K. The systematic error due to the validity of the BT-Settl models is unknown; however, our derived T_eff should be comparable to M dwarf T_eff values derived using the same models (indeed Rajpurohit et al. 2013; similarly derives T_eff ≃ 3100–3200 K for M4 dwarfs like LP 876-10 using BT-Settl models). The best-fitting BT-Settl synthetic spectrum had T_eff = 3100 K, [Fe/H] = 0.0, and log(g) = 5.0. From considerations of the star's color–magnitude diagram position (Section 2.6), we predict that LP 876-10 has a slightly subsolar metallicity, and lies near the zero-age main sequence for ∼0.2 M_☉ stars (log(g) ≃ 5.06; Baraffe et al. 1998). The best-fitting BT-Settl synthetic spectrum is then adjusted via an iterative process to produce a match to the observed photometry. The process determines a small λ-dependent polynomial correction factor that is applied to the synthetic spectrum to cause small modifications in order to produce the best fit to the photometry (details of the technique are described in S. Dieterich et al. 2013, in preparation).

By directly integrating the spectral energy distribution made by fitting the photometry in Table 2 with solar composition BT-Settl models, we estimate m_bol = 9.994 ± 0.020, luminosity = (1.763 ± 0.042) × 10³¹ erg s⁻¹, log(L/L_☉) = −2.337 ± 0.010, absolute bolometric magnitude M_bol = 10.597 ± 0.026, and bolometric correction BC_V = m_bol − V = −2.62 ± 0.02 (adopting solar parameters from Mamajek 2012; Pecaut & Mamajek 2013). Combining this luminosity with our previous T_eff estimate, we estimate a radius of 0.23 ± 0.01 R_☉. Combined with our estimate of the projected rotation velocity vsin i (22 ± 2 km s⁻¹), this places an upper limit on the rotation period of LP 876-10 of 0.55 ± 0.05 day (see Section 2.8).

Using our new parallax and the photometry in Table 2, we estimate absolute magnitudes of M_V = 13.21 ± 0.02 and $M_{\rm K_s}$ = 7.81 ± 0.03. From Table 2, we calculate a (V − K_s) color of 5.40 ± 0.02 mag. Using the (V − K_s) versus M_V relations from Henry et al. (2004) and Johnson & Apps (2009), we predict photometric distances of 7.9 ± 1.5 pc and 7.7 ± 1.4 pc, respectively, in excellent agreement with our trigonometric parallax distance. The agreement between the trigonometric parallax distance and the available photometric distances (Reid et al. 2003; this section) is also indicative that LP 876-10 is unlikely to have an unresolved companion of similar mass, and it is more likely to be a main-sequence, rather than pre-main-sequence, star.

We can constrain the metallicity and age using the star's color–magnitude data. In Figure 4, we plot the star's (V − K_s) color versus absolute magnitude M_V and use the metallicity color–magnitude calibration of Johnson & Apps (2009) to predict a metallicity of [Fe/H] = −0.07 dex (estimated accuracy ±0.06 dex). The calibration of Schlaufman & Laughlin (2010) predicts a metallicity of [Fe/H] = −0.15 dex. These are in reasonable agreement with the high S/N estimate for TW PsA (Fomalhaut B) from Barrado y Navascues et al. (1997) ([Fe/H] = −0.11 ± 0.02). Other published [Fe/H] estimates for TW PsA are −0.01 ± 0.09 (Santos et al. 2004) and −0.20 (Morell 1994). Hence, both the photometric metallicity estimate for LP 876-10 and the spectroscopic metallicity estimates for TW PsA are self-consistent, and consistent with being very slightly subsolar ([Fe/H] ≃ −0.1 dex).

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Using the Delfosse et al. (2000) M_V versus mass calibration for field M dwarfs (i.e., mixed metallicities and ages), the approximate mass of LP 876-10 is ∼0.20 M_☉. Interpolating within the Baraffe et al. (1998) tracks, one finds that solar composition stars with masses of greater than 0.163 M_☉ are not ever predicted to be as faint as M_V = 13.21 mag (see Figure 4). As the tracks are first and foremost tracing luminosity evolution as a function of mass and age, we also examine the constraints that the luminosity of LP 876-10 can provide. Through fitting BT-Settl models to the photometry, we estimate the luminosity to be log(L/L_☉) = −2.337 ± 0.010 dex. We find that the Baraffe et al. (1998) and Dotter et al. (2008) solar composition tracks give essentially identical predictions that no stars with masses greater than 0.197 M_☉ are predicted to have luminosities this low. Using those tracks, we estimate that it takes a 0.2 M_☉ star ∼300 Myr to reach within ∼0.01 mag of the zero-age main sequence (the actual minimum in luminosity and radius occurs around ∼400–500 Myr). The appearance of LP 876-10 on the zero-age main sequence for [M/H] ≃ −0.1 is commensurate with the adopted age for Fomalhaut A and B (440 Myr; Mamajek 2012).

As seen in Figure 4, the Baraffe et al. (1998) isochrones do not accurately reproduce the empirical main sequence from Johnson & Apps (2009) in this color regime, so our lower bound on the age of LP 876-10 is only approximate. Naively interpolating the mass and age of Fomalhaut C from the evolutionary tracks and isochrones would yield a mass of ∼0.11 M_☉ and an age of ∼60 Myr. However, as can be seen in Figure 4, a 125 Myr isochrone (log(age/yr) = 8.1) from the same tracks fails to replicate the intrinsic color–magnitude sequence for the ∼125 Myr old Pleiades (Barrado y Navascués et al. 2004). For the V–Ks color (5.4) of LP 876-10, the combination of the Pleiades color–magnitude sequence from Stauffer et al. (2007) and mean Pleiades distance from Soderblom et al. (2005) yields a Pleiades absolute magnitude of M_V = 12.24. The Baraffe et al. (1998) isochrones for age 125 Myr (log(age/yr) = 8.1) predict absolute magnitude M_V = 13.57 for V–K_s = 5.4,¹⁷ i.e., 1.33 mag too faint! As summarized by Bell et al. (2012), "for all optical colors, no pre-main-sequence models follows the observed Pleiades sequence for temperatures cooler than 4000 K." Estimating isochronal ages using pre-main-sequence evolutionary tracks is quite problematic, with large systematic differences between tracks (see review by Soderblom 2010). For all of these reasons, we do not adopt the pre-main-sequence mass and isochronal age interpolated from the evolutionary tracks and isochrones in Figure 4, and instead constrain the age based on its proximity to the main sequence and infer the mass based on main-sequence absolute magnitude versus mass considerations. Given the empirical and theoretical constraints previously discussed, we adopt a mass of 0.18 ± 0.02 M_☉ for Fomalhaut C.

Photometric data from the online SuperWASP archive¹⁸ (Butters et al. 2010) consisting of 14,991 measurements for LP 876-10 were extracted for two observing seasons (2007–2008). To search for a rotation period, we selected SuperWASP photometry from a single well-sampled season (2008) with V_SuperWASP magnitudes between 12.46 and 12.70, with magnitude and photometric errors of less than 0.2 mag, and with a good TAMFLUX2 flag extraction. SuperWASP photometry is calibrated to the Tycho-2 V_T system (Pollacco et al. 2006; Høg et al. 2000). There were 3162 points for subsequent analysis. To remove 1 day aliasing effects, all points during a single observing night were adjusted so that their average equaled the average seasonal magnitude of LP 876-10. A Lomb–Scargle (LS) periodogram with associated False Alarm Probabilities (FAPs) was calculated following Press et al. (1992), and the resultant periodogram is plotted in Figure 5. There is significant power (FAP < 0.001) seen in the LS periodogram of LP 876-10 at periods of 0.195, 0.242, 0.318, and 0.466 days. A period of P = 0.466 day would correspond to an equatorial velocity of 26 km s⁻¹, which is only slightly larger than the observed vsin i (22 km s⁻¹, corresponding to a maximum period of 0.55 day). For the star's mass and radius, we estimate a breakup velocity and period (following Townsend et al. 2004) of 386 km s⁻¹ and P_breakup = 0.03 day, respectively. Hence, any of the periods between ∼0.03 and ∼0.55 day are possible, given the breakup and vsin i constraints, respectively. The fastest rotation period among 41 nearby field M dwarfs in the MEarth survey of Irwin et al. (2011) is 0.28 day. We test the robustness of the detection by injecting artificial sinusoidal (P = 0.466 day) signals into a Gaussian distributed photometric data set with the same time cadence as the LP 876-10 data set. These tests indicate that the 0.195, 0.242, and 0.318 day peaks are aliasing effects due to the irregular time sampling of the light curve. We conclude that the P = 0.466 day peak is most likely due to the rotation of the star.

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Unfortunately, a rotation period of ∼0.5 day for a ∼0.2 M_☉ star places negligible constraint on its age. Mid-M stars with rotation periods faster than 1 day are a nearly ubiquitous feature of stellar samples between ages of ∼2 Myr and ∼10 Gyr (see Figure 12 of Irwin et al. 2011). Figure 11 of Irwin et al. (2011) plots the masses of field M dwarfs versus their rotation periods measured by the MEarth survey. For stars of ∼0.2 M_☉, a rotation period of ∼0.466 day is fast, however not unprecedented. Indeed, Irwin et al. (2011) finds that mid-M dwarfs like LP 876-10 can have periods of less than 1 day whether they are thin-disk or thick-disk stars. The survey of Irwin et al. (2011) had little difficulty finding kinematically old (>7 Gyr) thick-disk M dwarfs with sub-day rotation periods. We conclude that attempts to age-date LP 876-10 via gyrochronology/rotation constraints appear fruitless.

Not only is LP 876-10 fast rotating, but, unsurprisingly, it appears to be a coronally active star as well. Voges et al. (1999) ranked LP 876-10¹⁹ as the most likely optical counterpart of the ROSAT All-Sky Survey (RASS) Bright Source Catalog (BSC) X-ray source 1RXS J224803.5-242240. The X-ray counterpart is 35'' away from LP 876-10. However, the RASS BSC position error is 15'', and LP 876-10 is the brightest optical source within 40'', indicating that it is the likely X-ray source (Neuhaeuser et al. 1995). 1RXS J224803.5-242240 appears to be the brightest RASS X-ray source within a degree of LP 876-10. The fact that the position of the brightest RASS X-ray source within a degree of LP 876-10 lies within 40'' of the rapidly rotating, nearby M dwarf suggests to us that LP 876-10 is almost certainly the optical counterpart of 1RXS J224803.5-242240. The RASS BSC (Voges et al. 1999) reports a soft X-ray flux of 0.142 counts s⁻¹ (28% uncertainty) with HR1 hardness ratio of −0.23 ± 0.21, detected over a short exposure time of 176 s. Using the energy conversion factor relation from Fleming et al. (1995), this translates to a coronal X-ray flux in the soft X-ray band (0.2–2.4 keV) of roughly 1.01 × 10⁻¹² erg s⁻¹ cm⁻². At d = 7.57 pc, this corresponds to an X-ray luminosity of L_X ≃ 10^27.84 erg s⁻¹. This implies log(L_X/L_bol) ≃ −3.41, i.e., a very active star close to X-ray saturation. This corroborates the very high projected rotational velocity measured spectroscopically (vsin i = 22 km s⁻¹), which should induce strong magnetic activity.

With our best measurements of the proper motion, radial velocity, and parallax, we calculate the 3D Galactic velocity of LP 876-10 to be (U, V, W) = −5.3 ± 0.2, −7.6 ± 0.3, −11.9 ± 0.4 km s⁻¹. Comparing these values to those for Fomalhaut and Fomalhaut B (TW PsA), we find that LP 876-10's velocity only differs from that of Fomalhaut by 1.1 ± 0.7 km s⁻¹, and that of Fomalhaut B by 1.1 ± 0.5 km s⁻¹. Using the LSR velocity ellipsoid for both dM and dMe dwarfs estimated by Reid et al. (2002, their unweighted solution), and adopting the solar peculiar velocity with respect to the LSR from Schönrich et al. (2010), we naively only expect roughly 1 in ∼55,000 field M dwarfs to have UVW velocities within 1.1 km s⁻¹ of Fomalhaut, and roughly 1 in ∼12,000 field M dwarfs to have a velocity within 2 km s⁻¹.

Henry et al. (2006) report 239 M dwarfs within 10 pc that have accurate trigonometric parallaxes. These numbers are updated at recons.org, with a count as of 2012 January 1 of 248, which corresponds to a number density of 0.059 pc⁻³. This space density implies that within a sphere of radius 1 pc surrounding Fomalhaut, we would expect to find 0.25 M dwarfs. Hence, we estimate the probability that a random M dwarf could appear within 1 pc of Fomalhaut, and sharing its velocity within less than 2 km s⁻¹, as approximately 1 in ∼10^4.7 (and sharing its velocity within less than 1.1 km s⁻¹ as roughly 1 in ∼10^5.3). For comparison, one would expect to have to encircle a sphere ∼36 pc in radius in the local Galactic disk in order to find another M dwarf whose motion randomly agreed with that of Fomalhaut within less than 2 km s⁻¹. Our probability estimates do not take into account the similarity in the spectroscopic metallicity of TW PsA and the photometric metallicity of LP 876-10, which provides further agreement. We conclude that LP 876-10 appears to be related to Fomalhaut and TW PsA beyond a reasonable doubt.

Fomalhaut was listed by Barrado y Navascues (1998) as a potential member of the Castor Moving Group (CMG). The co-motion of LP 876-10 with Fomalhaut may be less significant if Fomalhaut is immersed in a swarm of co-moving stars like the purported CMG. The origin and nature of moving groups like the CMG is an active field of study (e.g., Famaey et al. 2005; Murgas et al. 2013). That the CMG represents a kinematic group of stars of common age and birthsite is unlikely.

Calculating revised space motions for the 14 CMG "members" ("Y" or "Y?" members) from Barrado y Navascues (1998), using revised Hipparcos astrometry (van Leeuwen 2007) and the best available radial velocities (Barbier-Brossat & Figon 2000; Gontcharov 2006), we find that the CMG stars have a median velocity of (U, V, W) = −11.1 ± 1.9, −8.6 ± 0.8, −9.7 ± 1.0 km s⁻¹, with standard deviations of 6.1, 3.6, and 4.2 km s⁻¹. The scatters are much larger than the typical velocity errors, and larger than the one-dimensional velocity dispersions of nearby clusters and associations (<1.5 km s⁻¹; Madsen et al. 2002; Mamajek 2010). The velocity for Fomalhaut differs from the CMG median velocity by 5.6 ± 2.3 km s⁻¹. The list of "final" members in Barrado y Navascues (1998) comprises ∼27 M_☉ of stars spread out over a volume of ∼55,000 pc³, implying that the density of CMG members in the solar neighborhood is roughly ∼0.004× the local disk density (0.12 M_☉ pc⁻³; van Leeuwen 2007). The stellar systems in the CMG have negligible interaction with one another, and so their motions are completely dominated by the local Galactic potential.

The velocity differences between Fomalhaut and individual CMG members are illuminating, and we discuss the famous CMG members Vega, LP 944-20, and Castor itself, in more detail. Vega is a proposed fellow CMG star of either similar age (455 ± 13 Myr; Yoon et al. 2010) or somewhat older age (700$^{+150}_{-75}$ Myr; Monnier et al. 2012) than that of Fomalhaut (440 ± 40 Myr; Mamajek 2012). Could Vega and Fomalhaut be related? Using the revised Hipparcos astrometry for Vega and its mean radial velocity reported by Parthasarathy & Lambert (1987), we estimate for Vega a velocity of (U, V, W) = −15.9 ± 0.7, −6.2 ± 0.5, −7.7 ± 0.3 km s⁻¹. Vega's velocity differs from that of Fomalhaut by 10.9 ± 1.0 km s⁻¹, and only 10 Myr ago their separations differed by ∼110 ±10 pc. Another nearby famous CMG "member" is the nearby candidate brown dwarf LP 944-20²⁰ (Ribas 2003). Adopting the astrometry from Tinney (1996) and a mean radial velocity of +9.0 ± 0.5 km s⁻¹ (based on measurements from Martín et al. 2006), we calculate a velocity for LP 944-20 of (U, V, W) = −12.2 ± 0.4, −5.6 ± 0.3, −2.8 ± 0.3 km s⁻¹. LP 944-20 is currently situated 6.6 pc away from Fomalhaut, and its velocity differs from that of Fomalhaut by 10.9 ± 0.8 km s⁻¹. Only 10 Myr ago, LP 944-20 and Fomalhaut were separated by ∼100 ± 8 pc, and were only more widely separated in the past. We also investigated whether there was any association between Fomalhaut and Castor itself. For the Castor sextuplet system, we adopt the recent parallax estimate from Torres & Ribas (2002), the long-term system proper motion from PPMX (Röser et al. 2008), and the center-of-mass radial velocity estimate from Heintz (1988; we adopt a radial velocity uncertainty of 1 km s⁻¹ in our calculations). These values are consistent with the Castor system having a velocity of (U, V, W) = −7.5 ± 0.7, −3.7 ± 0.6, −11.5 ± 0.4 km s⁻¹. Fomalhaut is currently ∼21 pc away from the Castor system, differing in velocity by a significant margin (4.9 ± 1.1 km s⁻¹), and only 10 Myr ago Fomalhaut and Castor were separated by ∼50 ± 5 pc and were even more distant in the past (more than 700 pc 100 Myr ago).

Despite these stars (the Fomalhaut system, Vega, LP 944-20, and the Castor system) being young and having somewhat similar velocities, their velocities are well-constrained enough and different enough that it is clear that they were not in the vicinity of one another even in the recent past, let alone a couple of Galactic orbits ago. We conclude that the CMG is comprised of stars from different birthsites rather than a coeval system, and hence "membership" to the CMG does not provide useful age constraints for the Fomalhaut system (or Vega, LP 944-20, Castor, or other CMG members).

One predicts that stellar companions in multiple systems can exist with separations up to their tidal (Jacobi) radius with respect to the Galactic potential. Jiang & Tremaine (2010) parameterize the tidal radius r_t as:

$$\begin{equation} r_{\rm t}\,= \left\lbrace \frac{G(M_1 + M_2)}{4\Omega A} \right\rbrace ^{1/3}, \end{equation} \tag{ 1 }$$

where G is the Newtonian gravitational constant, M₁ and M₂ are the masses of the stars, Ω is the Galactic angular circular speed (orbital velocity divided by Galactocentric radius), and A is the Oort parameter. Adopting modern estimates of the relevant Galactic parameters, and rewriting the expression from Jiang & Tremaine (2010), we estimate the tidal radius to be:

$$\begin{equation} r_{\rm t} = 1.35\,{\rm pc}\,\left\lbrace \frac{M_{\rm total}}{M_{\odot }} \right\rbrace ^{1/3}. \end{equation} \tag{ 2 }$$

Summing the masses of the Fomalhaut system components (2.83 M_☉), one predicts a tidal radius of ∼1.9 pc. The tidal radius for Fomalhaut A alone is ∼1.7 pc. Hence, the separation of ∼0.8 pc between Fomalhaut C and A is not dynamically implausible for a bound system. Recent systematic surveys for wide-separation pairs using modern astrometric databases have started to yield many previously unrecognized parsec-scale common proper motion pairs (e.g., Caballero 2010; Shaya & Olling 2011), making Fomalhaut C less unusual than it may have appeared even a decade ago. Stable orbits for timescales longer than a Gyr are also possible for separations larger than the tidal radius if the distant companion orbits retrograde to the Galactic rotation (e.g., Makarov 2012). Indeed, higher-precision radial velocities for the components of the Fomalhaut system and taking into account the sub-kilometer second⁻¹ effects of convective blueshift and gravitational redshift may lend themselves to providing a test as to whether Fomalhaut C is orbiting Fomalhaut AB either retrograde or prograde to the Galactic rotation (V. Makarov 2013, private communication).

Fomalhaut A and B are separated by Δ_AB = 57.4$^{+3.9}_{-2.5}$ kAU, and Fomalhaut C is separated by Δ_AC = 158.2$^{+2.3}_{-1.2}$ kAU from A and by Δ_BC = 203.4$^{+1.0}_{-0.8}$ kAU from B. We calculate the position of the barycenter (center of mass) for the system using the Galactic (X, Y, Z) positions and masses in Table 1: (X, Y, Z)_com = 3.08, 1.13, −6.93 pc. Converting this position to the equatorial ICRS coordinate system yields (α, δ) = 344179, −29792, at a distance of 7.67 pc. We can make a rough estimate of the orbital period of C around the AB pair. C is currently located ∼0.77 pc from the system's center of mass. If C is currently near apastron (not an unreasonable assumption given that binary stars will spend most of their time near apastron), and if C's periastron must almost certainly be larger than B's current separation from the system barycenter (0.24 pc), then a reasonable first estimate of C's orbit is a ∼ 0.5 pc and e ∼ 0.5. For the total mass of the Fomalhaut system (2.83 M_☉), this translates to an approximate orbital period of ∼20 Myr, or ∼5% of the system's age. The predicted orbital velocity of LP 876-10 around the Fomalhaut system barycenter would be ∼0.15 km s⁻¹. Given the masses and configuration of the AB pair, the escape velocity of C is ∼0.2 km s⁻¹.

How stable is Fomalhaut C's orbit with respect to A and B? Obviously, the orbit of AB and AB-C are not well constrained. We only have fairly accurate estimates of the relevant mass ratios and current separations, while the semimajor axes and eccentricities are unknown. The mass of C is very small compared to that for the AB pair (μ = M_C/(M_A + M_B) ≃ 0.07), and its current separation from the center of mass for the system is approximately 159 kAU. Based on simulations of test particles in the vicinity of binary systems of varying semimajor axis, mass ratio, and eccentricity, Holman & Wiegert (1999) provided estimates of the widest stable orbit around a member of a binary system (S-type orbits), and the closest orbit around both members of a binary system (P-type orbits). If the current A–B separation is equivalent to its semimajor axis (assume e = 0; a = 57.4 kAU), then the minimum stable semimajor axis for C is predicted to be ∼135 kAU. Tokovinin (1998) estimates that the mean eccentricity for wide binary pairs is 〈e〉 ≃ 2/3. If Fomalhaut B is currently near apastron with e ∼ 2/3, then a∼ 34 kAU and the minimum stable semimajor axis for Fomalhaut C is ∼140 kAU. There are plausible ranges of orbital parameters for Fomalhaut B and C that would be dynamically stable over many orbits.

Could LP 876-10 be genetically related to Fomalhaut AB, but we are "catching it in the act" of being an unbound escapee of the Fomalhaut system? We argue that this is very unlikely. LP 876-10 has a velocity statistically consistent with that of Fomalhaut A and B (ΔS = 1.1 ± 0.7 km s⁻¹). If the star actually had a velocity difference of >0.2 km s⁻¹ (i.e., above escape velocity), with respect to the Fomalhaut AB barycenter, then it would not spend much time in the vicinity of Fomalhaut or near its tidal radius. The approximate timescale that LP 876-10 would spend within Fomalhaut's tidal radius is approximately t ≃ r_t/ΔS, where r_t ≃ 1.9 pc is the tidal radius of Fomalhaut AB, and we posit that v must be larger than the escape velocity (0.2 km s⁻¹ = 0.2 pc Myr⁻¹). Hence,

$$\begin{equation} t_{{\rm Myr}}\, \simeq \, \frac{r_t}{\Delta S} < 9.5\, {\rm Myr}. \end{equation} \tag{ 3 }$$

For ΔS ∼ v_esc ∼ 0.2 km s⁻¹, LP 876-10 could spend of the order of ∼10 Myr within the tidal radius of Fomalhaut. For a velocity difference of ∼1 km s⁻¹, LP 876-10 would spend only ∼2 Myr. Velocity differences between LP 876-10 and Fomalhaut of ΔS greater than 2.5 km s⁻¹ are ruled out at 95% confidence, so timescales for LP 876-10 being unbound and within the tidal radius of Fomalhaut shorter than ∼0.8 Myr are ruled out. Hence, t would have to be of the order of ∼1–10 Myr if LP 876-10 is unbound to Fomalhaut AB. For a main-sequence lifetime of Fomalhaut A of ∼0.9 Gyr, this suggests that for LP 876-10 to be an unbound member of the Fomalhaut system in a state of disintegration, then we would have to be witnesses to an unusual dynamical state predicted to occur over ∼0.01%–0.1% over the lifetime of Fomalhaut A. This seems rather unlikely, and the simplest explanation for the agreement in velocities at the kilometer-per-second level between LP 876-10 and Fomalhaut A and B, and its position within the tidal radius of Fomalhaut AB, is that LP 876-10 is a third bound component of the Fomalhaut system.