We present a massluminosity relation MLR for red dwarfs spanning a range of masses from 0.62  M SUB SUB to the end of the stellar main sequence at 0.08  M SUB SUB. The relation is based on 47 stars for which dynamical masses have been determined primarily using astrometric data from Fine Guidance Sensors FGS 3 and 1r whitelight interferometers on the Hubble Space Telescope HST and radial velocity data from McDonald Observatory. For our HSTFGS sample of 15 binaries component mass errors range from 0.4 to 4.0 with a median error of 1.8. With these and masses from other sources we construct a Vband MLR for the lower main sequence with 47 stars and a Kband MLR with 45 stars with fit residuals half of those of the V band. We use GJ 831 AB as an example obtaining an absolute trigonometric parallax  SUBabsSUB  125.3  0.3 mas with orbital elements yielding  M SUBASUB0.270 0.004  M SUB SUB and  M SUBBSUB0.145 0.002  M SUB SUB. The mass precision rivals that derived for eclipsing binaries. A remaining major task is the interpretation of the intrinsic cosmic scatter in the observed MLR for lowmass stars in terms of physical effects. In the meantime useful mass values can be estimated from the MLR for the ubiquitous red dwarfs that account for 75 of all stars with applications ranging from the characterization of exoplanet host stars to the contribution of red dwarfs to the mass of the universe. P Based on observations made with the NASAESA Hubble Space Telescope obtained at the Space Telescope Science Institute which is operated by the Association of Universities for Research in Astronomy Inc. under NASA contract NAS526555.
['astrometry', 'binaries: close', 'stars: distances', 'stars: late-type', 'techniques: interferometric', 'techniques: radial velocities', 'Astrophysics - Solar and Stellar Astrophysics']
The Solar Neighborhood. XXXVII The MassLuminosity Relation for Mainsequence M Dwarfs
{'The Solar Neighborhood XXXVII: The Mass-Luminosity Relation for Main Sequence M Dwarfs 1': 'G. F. Benedict 2 , T. J. Henry 3 , 11 , O. G. Franz 4 , B. E. McArthur 2 , L. H. Wasserman 4 , Wei-Chun Jao 5 , 11 , P. A. Cargile 6 , S. B. Dieterich 7 , 11 , A. J. Bradley 8 , E. P. Nelan 9 , and A. L. Whipple 10', 'ABSTRACT': 'We present a Mass-Luminosity Relation (MLR) for red dwarfs spanning a range of masses from 0.62 M glyph[circledot] to the end of the stellar main sequence at 0.08 M glyph[circledot] . The relation is based on 47 stars for which dynamical masses have been determined, primarily using astrometric data from Fine Guidance Sensors (FGS) 3 and 1r, white-light interferometers on the Hubble Space Telescope (HST) , and radial velocity data from McDonald Observatory. For our HST /FGS sample of 15 binaries, component mass errors range from 0.4% to 4.0% with a median error of 1.8%. With these and masses from other sources, we construct a V -band MLR for the lower main sequence with 47 stars, and a K -band MLR with 45 stars with fit residuals half of those of the V -band.  \nWe use GJ831 AB as an example, obtaining an absolute trigonometric parallax, π abs = 125 . 3 ± 0 . 3 milliseconds of arc, with orbital elements yielding M A = 0 . 270 ± 0 . 004 M glyph[circledot] and M B = 0 . 145 ± 0 . 002 M glyph[circledot] . The mass precision rivals that derived for eclipsing binaries.  \nA remaining major task is the interpretation of the intrinsic cosmic scatter in the observed MLR for low mass stars in terms of physical effects. In the meantime, useful mass values can be estimated from the MLR for the ubiquitous red dwarfs that account for 75% of all stars, with applications ranging from the characterization of exoplanet host stars to the contribution of red dwarfs to the mass of the Universe.  \nSubject headings: astrometry - interferometry - stars: binary - stars: radial velocities - stars: late-type - stars: distances - stars: masses', '1. Introduction': "With the exception of the Hertzsprung-Russell (HR) diagram, the main sequence MassLuminosity Relation (MLR) is perhaps the single most important 'map' of stellar astronomy because the entire evolution of an individual star depends on its mass. The MLR allows astronomers to convert a relatively easily observed quantity, luminosity, to a more revealing characteristic, mass, providing a better understanding of the object's nature. When considering populations of stars, an accurate MLR permits luminosity functions to be converted into mass functions, and drives estimates of the stellar contribution to the mass of the Galaxy.  \nBasic observations of stars include measurements of apparent magnitudes, colors, surface temperatures ( T eff ), and metallicities. From these attributes we can derive luminosities and radii, given combinations of distance measurements and/or directly measured angular sizes from long-baseline interferometry. Yet, the crucial masses remain elusive, typically only measured dynamically in binary systems. Thus, when stellar modelers compare their theoretical results with real stars to test the accuracy of their efforts, they often assume masses for their test stars. For example, at a mass of 0.3 M glyph[circledot] , models from Allard et al. (2000), Baraffe et al. (2015), and Dotter (2016) predict M V values of 11.42, 11.61, and 11.25 respectively. An accurate MLR allows choices to be made between various modeling approaches, such as the treatment of stellar ages, convection, turbulent mixing, rotation, magnetic activity, and mass loss (Andersen 1991). Furthermore, the need for accurate mass estimates for the smallest stars has come to the forefront with the discovery of exoplanets associated with M dwarf stars (see Lurie et al. 2014 for a recent list of the nearest M dwarfs  \nwith planets). More accurate estimates of host star masses translate directly into more reliable companion planet masses, and consequently, more accurate bulk compositions for planets that transit.  \nThe first robust MLR for M dwarf stars was reported in Henry & McCarthy (1993), who provided relations at V JHK for 37 stars, of which 26 had masses less than 0 . 6 M glyph[circledot] . Early results from the project described here can be found in Henry et al. (1999), where an improved MLR at V was pivotal. A modest update was presented in Delfosse et al. (2000), which included many of the same stars. Torres et al. (2010) summarized the entire stellar MLR using eclipsing systems as of a few years ago, outlining how few M dwarf stars have accurate masses because these tiny stars rarely eclipse.  \nHere we present astrometry from the Hubble Space Telescope (HST) and other sources, in combination with ground-based radial velocities (RVs), to improve the low-mass region of the MLR that is complicated by age, metallicity, and magnetic effects. This is the realm of the red dwarfs, typically of spectral type M, that slowly descend to the Main Sequence along a nearly vertical Hyashi track in the HR diagram (Palla 2012). This results in an M dwarf population that in the solar neighborhood spans a wide range of ages (Riedel et al. 2014). In addition, M dwarf luminosities are affected by composition (metallicity), which is a function of the mixing of the interstellar medium (with sources internal and external to the Galaxy), star formation rate, and birth date within the Galaxy (e.g., Sneden, Lawler, & Cowan 2002). To complicate matters even further, a more poorly understood factor affecting Mdwarf luminosity is magnetism, which induces short and long-term photometric variability (Hosey et al. 2015).  \nIn this paper we model HST Fine Guidance Sensor (FGS) astrometric, and, if available, radial velocity (RV) measurements simultaneously to obtain parallaxes, proper motions, orbits, and component masses for 15 systems containing 30 M dwarfs (Table 1). Our resulting masses have a median precision of 1.8%, which constitutes a significant advance from previous efforts. The derived parallaxes, which take into account the orbital motions of the binaries, are usually marked improvements - typically by factors of 5-10 over available parallaxes in the Yale Parallax Catalog (van Altena, Lee & Hoffleit 1995, hereafter YPC) and Hipparcos (van Leeuwen et al. 2007) - and are key in deriving the high-accuracy masses. We derive the first masses for stars in six systems, improve masses over previous efforts by others for seven systems, and improve upon our own HST efforts for two systems: GJ 748 AB in (Benedict et al. 2001) and GJ 791.2 AB in (Benedict et al. 2000). Our reductions and analyses for the latter two systems include a few new HST /FGS measurements and have improved radial velocity measurements, yielding more precise masses.  \nIn Sections 2 and 3 we describe the various sources of astrometric and RV data that  \nwere used in our modeling. In Section 4 we discuss in detail the data and modeling process for GJ 831 AB, as an example of our technique to obtain masses. Section 5 summarizes modeling notes and resulting masses for the other 14 systems. In Section 6 we describe the measurements for the M V and M K values used to create the MLRs. Section 7 describes additional systems in which red dwarfs with high-quality masses are available to be used in the MLRs. In Section 8 we construct a V -band MLR from the set of 47 stars and a K -band MLR using 45 stars. We discuss our results in Section 9, and conclude with a summary in Section 10.", '2.1. HST Fine Guidance Sensors': "An HST Fine Guidance Sensor (FGS) provides two types of astrometric data. Like all interferometers, an FGS produces a fringe that we interrogate either by tracking the fringe position (POS mode) or by scanning to build up a fringe image (TRANS mode). TRANS mode yields the structure of the fringe from which it is possible to derive the separation and position angle of a secondary relative to a primary in a binary system, essential in establishing the relative orbit. POS mode permits the measurement of the position of the primary star (brighter component) relative to a local frame of reference stars, essential for determining proper motion, parallax, and mass fraction. Franz et al. (1998) and Benedict et al. (2011) contain details of HST FGS's TRANS and POS mode reduction and analysis, respectively. Detectability and measurement precisions for a given binary star's components critically depend on star brightnesses, component brightness differences, and the binary angular separation. All HST /FGS astrometry also depends on the Optical Field Angle Distortion (OFAD) calibration (McArthur et al. 2002). This calibration reduces as-built HST telescope and FGS distortions with magnitude ∼ 1 '' to below 2 milliseconds of arc (mas) over much of the FGS field of regard. The present work benefits from a POS mode OFAD revision more recent (2012) than reported in McArthur et al. (2006). The 15 red dwarf systems targeted in this program and discussed in the following sections rely on HST TRANS and/or POS mode measurements, and are listed in Tables 1 and 2.  \nFor binary stars the FGS transforms images of the two components into two overlapping fringes, sometimes called the 'S-curves'. For perfect fringes and component separations greater than the resolution of HST at λ =580 nm, about 40 mas, the presence of a companion should have little effect on the component A position obtained from the fringe-zero crossing. For separations smaller than 40 mas the measurement of the position of the brighter component of a blended image will be biased toward the fainter component (van de Kamp 1967),  \npossibly requiring a photocenter correction. Because an FGS has two orthogonal axes, the separation along each interferometer axis, not the total separation on the sky, determines the photocenter correction. For any arbitrary HST orientation, components A and B can have separations along the FGS axes from 0.0 mas to the actual full separation.  \nDuring this project we observed with two FGS units: FGS 3 from 1992-2000, and FGS 1r from 2000-2009. FGS1r replaced the original FGS 1 during HST Servicing Mission 3A in late 1999. Our original analysis of GJ 748 AB (Benedict et al. 2001) incorporated POS mode photocenter corrections, as does the present re-analysis. These corrections, ranging from -3.2 to +2.3 mas, are for the FGS 3 X-axis a complicated function of separation because FGS3 delivered far from perfect fringes with substantial instrumental structure (Franz et al. 1991). FGS1r produces higher quality fringes (Nelan 2012), and corrections are similar along both axes and similar in size to the Y-axis correction seen in Benedict et al. (2001) for GJ 748 AB. These corrections (typically a maximum of 2 mas) are negligible for systems with component ∆ V ≥ 2 . 3 (∆ V values listed in Table 10). The systems with ∆ V ≤ 2 . 3 and typically smaller separations are GJ 469 AB, GJ 748 AB, GJ 831 AB, and GJ 1081 AB, and G 250-029 AB. Generally, only a few measurements require photocenter corrections.", '2.2.1. Traditional Ground-Based Methods': 'We included visual, photographic, and CCD observations of separations and position angles from Geyer et al. (1988) for our analysis of GJ 65 AB (Section 5.13). These (relative to HST ) low-precision measurements are primarily from the USNO 61in reflector and stretch to nearly twice the full period of GJ 65 AB, proving extremely useful for this long orbital period system.', '2.2.2. Adaptive Optics': 'We include a single observation of G 193-027 AB from Beuzit et al. (2004), who used the Adaptive Optics Bonnette system on the Canada-France-Hawaii Telescope. For GJ 65 AB we include five VLT/NACO measures of position angle and separation (Kervella et al. 2016).', '2.2.3. Aperture Masking Interferometry': 'For our analysis of GJ 623 AB (Section 5.4) we included astrometric observations (Martinache et al. 2007) performed with the PHARO instrument on the Palomar 200in (5m) telescope and with the NIRC2 instrument on the Keck II telescope. Separations have typical errors of 2 mas. Position angle errors average 0 . · 5.', '2.2.4. Speckle Interferometry': 'Measurements are included for GJ 22 AC from McCarthy et al. (1991) and for GJ 473 AB from Henry et al. (1992) and Torres et al. (1999), who used a two-dimensional infrared speckle camera containing a 58 × 62 pixel InSb array on the Steward Observatory 90in telescope. We also include infrared speckle observations by Woitas et al. (2003), who obtained fourteen separation and position angle measurements for GJ 22 AC with the near-infrared cameras MAGIC and OMEGA Cass at the 3.5m telescope on Calar Alto. These instruments are capable of taking fast sequences of short time exposures (t exp ∼ 0 . 1 sec) that enable speckle interferometry observations at the infrared wavelengths where red dwarfs emit significant flux, and where atmospheric turbulence is reduced compared to visual wavelengths. We also include a few speckle observations at optical wavelengths from the Special Astrophysical Observatory 6m BTA and 1m Zeiss (Balega et al. 1994), from the CFHT (Blazit et al. 1987), and from the DSSI on the WIYN 3.5 m (Horch et al. 2012).', '2.2.5. Other HST Astrometry': 'Where available, we use astrometric observations from HST instruments other than the FGSs, including the Faint Object Camera (FOC) (Barbieri et al. 1996), the Faint Object Spectrograph (FOS) (Schultz et al. 1998), the Near-Infrared Camera and Multi-Object Spectrometer (NICMOS) (Golimowski et al. 2004), and the Wide-Field Planetary Camera 2 (WFPC2) (Schroeder et al. 2000; Dieterich et al. 2012).', '3. Radial Velocities': 'Adding radial velocities (RVs) to the astrometry allows us to completely sample the motion of a binary system; in tandem, the two data sets improve the accuracies of the final component masses. RVs have particular value for the longer-period systems lacking  \nconsistent astrometric coverage of the entire orbit (GJ 22 AC, GJ 234 AB, GJ 469 AB, and GJ623AB). We have RVs for 11 system components out of the 30 for which we present new masses. The value of the RV data sets depends the brightnesses and magnitude differences of components in each binary, given in Table 2 and Table 10. Our RV measurements, listed in Table 3, are from two sources, described next. RV orbit semi-amplitudes and systemic velocities for the 11 stars in 7 systems are given in Table 4.', '3.1. McDonald Cassegrain Echelle Spectrograph': 'We obtained most RV data with the McDonald 2.1m Struve telescope and Sandiford Cassegrain Echelle spectrograph (McCarthy et al. 1993), hereafter CE. The CE delivers a dispersion equivalent to 2.5 km s -1 /pix ( R = λ/ ∆ λ =60,000) with a wavelength range of 5500 ≤ λ ≤ 6700 ˚ A spread across 26 orders (apertures). The McDonald data were collected during thirty-three observing runs from 1995 to 2009 and reduced using the standard IRAF (Tody 1993) echelle package tools, including the cross-correlation tool fxcorr . We visually inspected the resulting cross correlation functions (CCF) to select the better of the 26 apertures, typically using half for each binary. Four systems (GJ 469 AB, GJ 748 AB, GJ791.2AB, GJ831AB) had detectable double peaks in their CCF; these have ∆ V = 1.59 to 3.27. For these we used an IRAF script to average the lower-noise CCF. We then used a Gaussian multi-peak fitting routine in the GUI-based commercial package IGOR 2 to derive component ∆RVs. We obtained the RVs for the three single peak CCF systems (GJ 22 AC, GJ234AB, and GJ623AB) through a weighted average of the fxcorr output velocities. For all systems, we averaged the velocities derived from observations taken during the span of a few days (our typical observing run lasted four nights) to form average absolute heliocentric velocities at the epochs plotted in Sections 4.6 and 5.', '3.2. HET and Other Sources': 'Some GJ623AB velocities came from the Hobby-Eberly Telescope using the Tull Spectrograph, as did our high-resolution, high signal-to-noise spectrum of GJ 623 AB used as a template for all cross correlations. We identify a few other sources of radial velocities in the individual object modeling notes below (Section 5). The HET became fully functional about halfway through our CE campaign, so we decided to remain with the McDonald 2.1m CE system for the entire datasets for the six other systems with RVs.', '4. An Analysis of GJ 831 AB': "As a guideline for all 15 systems for which we determine masses, the following sections provide details of the analysis path leading to the determination of component masses for GJ831AB. Section 5 provides summary information for the other 14 systems, indicating where the mass derivations differ from GJ 831 AB.  \nS'egransan et al. (2000) summarizes previous knowledge of the orbit and mass of this system.", '4.1. The GJ831AB Astrometric Reference Frame': 'Figure 1 shows the distribution in FGS coordinates of the 10 sets of five reference star (numbers 2, 3, 4, 5, and 10) POS mode measurements for the GJ 831 AB reference frame, where 1 indicates GJ 831 AB. The elongated pattern is impressed by the pickle-like shape of the FGS field of regard and the requirement that HST roll to keep its solar panels fully illuminated throughout the year. Not all reference stars were measured at each epoch, nor were all stars measured by the same FGS. We acquired the first six POS data sets with FGS 3 and the last four sets with FGS 1r. We obtained a total of 71 reference star observations and 29 observations of GJ 831 AB. We note in Table 6 that we have more position angle and separation TRANS mode measurements (18) over 6.61 yr than we have for component A POS mode measurements (10) over 4.07 yr.', '4.2. GJ831AB Radial Velocities': 'We have a single source of RV data for this system, including 13 measurement epochs spanning 11.9 years (Table 3) obtained with the McDonald 2.1m telescope and Sandiford Cassegrain Echelle spectrograph (McCarthy et al. 1993) (hereafter, CE). Treating GJ831AB as a double lined spectroscopic binary, we obtained velocities for both components at orbital phases 0 . 3 ≤ φ ≤ 0 . 98 (Figure 5, right). Unfortunately, our monitoring missed a critical phase, periastron.', '4.3.1. Reference Stars': 'As for of our previous parallax projects, e.g., Benedict et al. (2007, 2009, 2011); McArthur et al. (2011) we include as much prior information as possible for our modeling. Quantities that inform the modeling include estimates of reference star absolute parallaxes, various color indices for the reference stars, proper motions from the PPMXL (Roeser et al. 2010), and FGS lateral color calibrations (e.g., Benedict et al. 1999, section 3.4). In contrast to much of our previous parallax work (e.g., Harrison et al. 1999), for this project we estimate our reference star absolute parallaxes using only color and proper motion information.  \nBecause the parallaxes determined for the binary systems will be measured with respect to reference frame stars that have their own parallaxes, we must either apply a statistically derived correction from relative to absolute parallax (van Altena, Lee, & Hoffleit 1995) or estimate the absolute parallaxes of the reference frame stars. We choose the latter methodology for the FGS reference fields discussed here. The colors and luminosity class of a star can be used to estimate the absolute magnitude, M V , and V -band absorption, A V . The absolute parallax is then simply,  \nπ abs = 10 -( V -M V +5 -A V ) 5 (1)  \nGiven the galactic latitude of GJ 831 AB ( glyph[lscript] II = -40 · ) and external estimates of low reddening (Schlafly & Finkbeiner 2011), a J -K vs. V -K color-color diagram (Figure 2) supports an initial assignment of spectral type and dwarf luminosity class to all but reference star ref-5 (with an initial estimate of K2III).  \nTo confirm the reference star spectral type and luminosity classes estimated from all available photometry we employ the technique of reduced proper motions (Stromberg 1939; Gould & Morgan 2003; Gould 2004). We obtain proper motions from PPMXL and J and K photometry from 2MASS for sources in a 2 · × 2 · field centered on GJ 831 AB. Figure 3 shows a reduced proper motion diagram (RPM), H K = K + 5log( µ ) plotted against J -K color index for that field, where µ is the proper motion vector absolute value in arcsec yr -1 . If all stars had the same transverse velocities, Figure 3 would be equivalent to an HR diagram. GJ831AB and associated reference stars are plotted with ID numbers from Table 5. Comparing with our past reduced proper motion diagrams (Benedict et al. 2011, 2014; McArthur et al. 2011), all but ref-5 and ref-10 lie on or near the main sequence.  \nIn a quasi-Bayesian approach we input all priors as observations with associated errors, not as hardwired quantities known to infinite precision. Input proper motion values have  \ntypical errors of 4-6 mas yr -1 for each coordinate. The lateral color and cross-filter calibrations and the B -V color indices are also treated as observations with error. Where there is tension between the color-color and RPM diagrams, in this case for ref-5 and -10, we run the models with two sets of inputs and adopt the classification that produces the smaller χ 2 . The best model results included ref-5 as an M dwarf and ref-10 as a G subgiant. Table 5 contains V magnitudes, colors, estimated spectral types, M V , input prior parallaxes (estimated from photometry and proper motion) and final parallaxes (as final astrometric modeling results) for the five reference stars used in the GJ 831 AB field.', '4.3.2. The M Dwarf Binaries': 'Parallax was not the primary goal of the FGS effort. Consequently, for the more reference star-poor fields and those systems with very poor sampling of the parallactic ellipse we introduce previously determined science target parallaxes as priors. While not included as a prior in the GJ 831 AB modeling, we will flag parallax prior inclusion in the modeling notes for other systems in Section 5. Ultimately, the HST /FGS do provide significantly improved parallaxes in most cases.  \nThe derived orbital solutions benefit from a relationship between the two astrometric modes (POS and TRANS) and the RV measurements, which together are enforced by the constraint (Pourbaix & Jorissen 2000)  \nα A sin i π abs = PK A √ (1 -e 2 ) 2 π × 4 . 7405 (2)  \nQuantities derived only from astrometry (parallax, π abs , primary perturbation orbit size, α A , and inclination, i ) are on the left, and quantities derivable from both radial velocities and astrometry (the period, P in years and eccentricity, e ), or radial velocities only (the RV amplitude for the primary, K A in km s -1 ), are on the right. An object traveling 4.7405 km s -1 will move 1 AU in 1 year. When radial velocities for both components exist, we employ an additional constraint, substituting K B for K A and the component B orbit size, α B = a -α A , for α A , where a is the orbital semimajor axis. Finally, given the simple orbital mechanics of these binary systems, we constrain the longitudes of periastron passage, ω A and ω B to differ by 180 · .', '4.4. The Astrometric Model': "From the astrometric data we determine the scale, rotation, and offset 'plate constants' relative to an arbitrarily adopted constraint epoch (the so-called 'master plate') for each observation set. The GJ831AB reference frame contains five stars, but only four were observed at each epoch. Hence, we constrain the scales along X and Y to equality and the two axes to orthogonality. The consequences of this choice are minimal. For example, imposing these constraints on the Barnard's Star astrometry discussed in Benedict et al. (1999) results in an unchanged parallax and increases the error by 0.1 mas, compared to a full 6 parameter model (substituting D for -B and E for A in Equation 6, below).  \nOur reference frame model becomes, in terms of standard coordinates  \nx ' = x + lc x ( B -V ) (3)  \ny ' = y + lc y ( B -V ) (4)  \nξ = Ax ' + By ' + C -µ α ∆ t -P α π α -ORBIT α (5)  \nη = -Bx ' + Ay ' + F -µ δ ∆ t -P δ π δ -ORBIT δ (6)  \nwhere x and y are the measured coordinates from HST , lc x and lc y are lateral color corrections (see section 3.4 of Benedict et al. 1999), and B -V represents the color of each star, either from SIMBAD or estimated from the spectral types suggested by Figure 2. A and B are scale and rotation plate constants, C and F are offsets, µ α and µ δ are proper motions, ∆ t is the epoch difference from the mean epoch, P α and P δ are parallax factors, and π α and π δ are the parallaxes in RA and DEC. ξ and η are relative positions that (once scale, rotation, parallax, the proper motions and the ORBIT are determined) should not change with time. All Equation 5 and 6 subscripts are in RA and DEC because the master constraint plate was rolled into the RA DEC coordinate system before the analysis. We obtain the parallax factors from a JPL Earth orbit predictor (Standish 1990), upgraded to version DE405. Orientation to the sky for the master plate is obtained from ground-based astrometry from the PPMXL (Roeser et al. 2010) with uncertainties in the field orientation of ± 0 . · 1. This orientation also enters the modeling as an observation with error. Note that because we switched FGS units on HST during the sequence of GJ 831 AB observations, two different OFAD and lateral color calibrations entered the modeling. Finally, ORBIT is an offset term that is a function of the traditional astrometric and RV orbital elements listed in Table 9.  \nFor the astrometric solution we solve simultaneously for a position within our reference frame, a parallax, and a proper motion for the target GJ 831 AB and all reference stars. In addition, for GJ 831 AB the orbital period (P), the epoch of passage through periastron in  \nmodified Julian days (T 0 ), the eccentricity (e), and the position angle of the line of nodes (Ω), are constrained to be equal in the RV and two modes of astrometry. We also constrain the angle ( ω ) in the plane of the true orbit between the line of nodes and the major axis to differ by 180 · for the component A and B orbits. Only RV provides information with which to determine the velocity half-amplitudes ( K A , K B ) and γ , the systemic velocity. For systems with both radial velocities and astrometry, the constraint described in Section 4.3.2 ties POS, TRANS, and radial velocities together, yielding a single self-consistent description of the binary system.", '4.5. Assessing Reference Frame Residuals': "From histograms of the astrometric residuals for 71 reference star and 29 GJ 831 AB position measurements (Figure 4), we conclude that we have obtained corrections at the ∼ 1 mas level in the region available at all HST roll angles (an inscribed circle centered on the pickle-shaped FGS field of regard). The resulting reference frame 'catalog' in ξ and η standard coordinates (Equations 5 and 6) was determined with median absolute errors of σ ξ = 0 . 6 and σ η = 0 . 9 mas in X and Y, respectively.  \nTo determine if there might remain unmodeled - but possibly correctable - systematic effects at the 1 mas level, we plotted the GJ 831 AB reference frame X and Y residuals against a number of spacecraft, instrumental, and astronomical parameters. These included X, Y positions within the pickle, radial distances from the pickle center, reference star V magnitudes and B -V colors, HST spacecraft roll angles, and epochs of observation. We saw no obvious trends, other than an expected increase in positional uncertainty with reference star magnitude.", '4.6. Results of GJ 831 AB Simultaneous Modeling': 'The results of this simultaneous solution are as follows. Average absolute value residuals for the RV and POS/TRANS observations of GJ 831 AB are given in Tables 4 and 6, respectively. The absolute parallax and the proper motion are presented in Table 7 (errors are 1 σ ), where previous parallaxes are also listed. Compared to Hipparcos , our precision has improved knowledge of the parallax by a factor of 20. Residuals to the orbit fits for each astrometric measurement are individually listed in Table 8, while Table 9 contains the orbital parameters with formal (1 σ ) uncertainties.  \nFigure 5, left, illustrates component A and B astrometric orbits. The POS mode com-  \nponent A residuals and the TRANS residuals (component A-B separations) are all smaller than the dots in Figure 5. Figure 5, right, shows all RV measurements, RV residuals, and the predicted velocity curve from the simultaneous solution, phased to the derived orbital period.', '4.7. GJ831AB Component Masses': "Our orbit solution and derived absolute parallax (Equation 2) provide an orbital semimajor axis, a in AU, from which we can determine the system mass through Kepler's Third Law. Given P and a , we solve the expression  \na 3 /P 2 = ( M A + M B ) = M tot (7)  \nto find (in solar units) M tot = 0 . 414 ± 0 . 006 M glyph[circledot] . At each instant in the orbits of the two components around the common center of mass,  \nM A / M B = α B /α A (8)  \na relationship that contains only one observable, α A , the perturbation orbit size. Instead, we calculate the mass fraction  \nf = M B / ( M A + M B ) = α A / ( α A + α B ) = α A /a, (9)  \nwhere α B = a -α A . This parameter, f (also given in Table 9), ratios the two quantities directly obtained from the observations - the perturbation orbit size ( α A from POS mode) and the relative orbit size ( a from TRANS mode), both shown in Figure 5 and listed in Table 9. From these we derive a mass fraction of 0.3489 ± 0.0031. Equations 7, 8, and 9 yield M A = 0 . 270 ± 0 . 004 M glyph[circledot] and M B = 0 . 145 ± 0 . 002 M glyph[circledot] , indicating that both components are low mass red dwarfs with mass errors of ∼ 1.5%, a considerable improvement over the S'egransan et al. (2000) 4% determinations. We collect these component masses and those for the other systems discussed below in Table 10, which also includes V , K , the magnitude differences ∆ V , ∆ K , and the M V , and M K values used to create the MLRs.", '5. Modeling Notes and Masses for Other Systems': 'The presentation order in this section for the other 14 systems observed with HST /FGS3 and FGS1r is dictated by dwindling observational resources. We first discuss component mass results for the six other systems with astrometric and RV measurements, then provide notes on the derivation of component masses for the remaining eight systems with astrometric measurements only. The final two systems discussed analyze only position angle and separation measurements (TRANS and/or ground-based). Among all 15 systems, we provide the first mass determinations for six binary pairs: GJ 54 AB, GJ 469 AB, GJ 831 AB (above), GJ1081AB, G 193-027AB, and G 250-029AB. For historical context, we provide comparisons with previous mass determinations.  \nIn parallel with our detailed discussion of GJ 831 AB, RV measurements are listed in Table 3 for the first six systems, with RV results (component semi-amplitudes, system center of mass velocities, number of observations, and residuals to the orbital fits) in Table 4. Astrometric results for all 15 systems are collected in Table 6 (reference star information, study durations, and average absolute value residuals to the orbital fits for POS, TRANS), Table 7 (previous parallaxes and derived HST parallaxes and proper motions), and Table 8 (observation dates, individual position angles, separations, residuals to the orbital fits, and measurement sources). All analyses yield the orbital elements in Table 9 and component masses in Table 10. All of these Tables list the systems in Right Ascension order.', '5.1. GJ22AC': "The GJ22AC system is a triple, with the two close components known as A and C. Our modeling of GJ 22 AC included the four sources of position angle and separation measurements noted in Table 8. HST TRANS astrometric measurements came from FGS3 and FGS1r, while modeling included POS from only FGS 1r in a reference frame of four stars. We also utilized infrared speckle interferometry measurements from McCarthy et al. (1991) and Woitas et al. (2003). All but one of the POS mode epochs were secured at one year intervals, very poorly sampling the parallactic ellipse. In this situation, simultaneous modeling including RVs and the Equation 2 constraint could push the parallax to unrealistic values. Therefore we modeled radial velocities (only for GJ 22 A due to the large ∆ V = 3 . 08 value) in combination with the astrometry, with a parallax from Hipparcos (van Leeuwen et al. 2007) used as a prior.  \nOur derived parallax has a larger error (0.6 mas) than typical for an HST parallax (0.20.3 mas) due to the less than ideal sampling. Still, even though all epochs sampled nearly  \nthe same part of the parallactic ellipse, they serve to scale the size of that ellipse well enough to provide a useful parallax. We obtain a parallax similar to Hipparcos (van Leeuwen et al. 2007), but the precision has improved by a factor of four. The average absolute value of the residuals to the orbit fit (Table 6) for both the TRANS and speckle observations is 〈| res |〉 = 10.1 mas. For TRANS only, the average absolute value residuals are 〈| res |〉 = 3.3 mas, and for speckle-only, 〈| res |〉 = 13.4 mas.  \nTable 9 contains the GJ 22 AC orbital parameters with formal (1 σ ) uncertainties. The orbital semimajor axis is now extremely well-determined, with an error approaching 0.1%. The orbital period is known to 0.3%, while the parallax error (0.6%) dominates the derived mass errors. The left panel of Figure 6 provides component A and B orbits and shows the observations that entered into the modeling. Component A-B separation residuals from TRANS mode and speckle observations are illustrated by the offsets (+ and × next to o). Regarding the RV measurements (Figure 6, right), we note that GJ 22 AC has a companion, GJ 22B, separated from GJ22AC by 4 '' , with an orbital period on the order of 320 yr (Hershey 1973). Figure 7 shows the RV orbit as a function of time, not phase. We may (at a low level of significance) be detecting the AC-B orbital motion as a slope in the RV residuals. We obtain component masses of M A = 0 . 405 ± 0 . 008 M glyph[circledot] (2.0% error) and M C = 0 . 157 ± 0 . 003 M glyph[circledot] (1.9%). These masses support the excellent early work on this system by Hershey (1973), who found M A = 0 . 40 M glyph[circledot] and M C = 0 . 13 M glyph[circledot] , followed by McCarthy et al. (1991) who found M A = 0 . 362 ± 0 . 053 M glyph[circledot] (15%) and M C = 0 . 123 ± 0 . 018 M glyph[circledot] (15%).", '5.2. GJ234AB': 'Our modeling of GJ 234 AB included astrometry measurements from both FGS 3 and FGS1r. Treating GJ234AB as a single-lined spectroscopic binary (due to the large ∆ V = 3.08 value), we obtained velocities from the CE for component A at most orbital phases, as seen in Figure 8, right. Modeling included RVs for the A component and astrometry with five reference stars and only four plate parameters. Most of the POS mode observations were secured at one year intervals, very poorly sampling the parallactic ellipse, with only one POS measurement at the other extreme of the parallactic ellipse, so we used a Hipparcos parallax as a prior. We obtained 11 TRANS observations of the separation and position angle for AB and 7 POS observations of component A and the reference frame.  \nOur observations have reduced the parallax error by a factor of 7-10 compared to YPC and Hipparcos . Table 9 contains the GJ 234 AB orbital parameters with formal (1 σ ) uncertainties, with both a and P now known to better than 0.2%. The left panel of Figure 8 illustrates component A and B orbits with observed positions indicated. The POS mode  \ncomponent A residuals and the TRANS residuals are all smaller than the dots and circles. We obtain masses of M A = 0 . 223 ± 0 . 002 M glyph[circledot] (0.9% error) and M B = 0 . 109 ± 0 . 001 M glyph[circledot] (0.9%). These are among the best mass measurements known for red dwarfs, rivaling accuracies for components in eclipsing systems. These masses are larger than those reported by Probst (1977), M A = 0 . 13 ± 0 . 04 M glyph[circledot] (31%) and M B = 0 . 07 ± 0 . 02 M glyph[circledot] (29%), who summarized much of the early work on this important binary. Our new masses are larger but consistent with those of Coppenbarger et al. (1994), who found M A = 0 . 179 ± 0 . 047 M glyph[circledot] (26%) and M B = 0 . 083 ± 0 . 023 M glyph[circledot] (28%).', '5.3. GJ469AB': "GJ469AB astrometric measurements came from FGS3 and FGS1r and radial velocities came from the CE. Modeling was identical to that used for GJ 831 AB, although the astrometric reference frame included only three stars. Hence, we included a Hipparcos parallax (van Leeuwen et al. 2007) as a prior.  \nThe FGS measurements have improved our knowledge of GJ 469 AB's absolute parallax of by a factor of 8-13 compared to YPC and Hipparcos . Table 9 contains the system's orbital parameters and formal (1 σ ) uncertainties, with the largest error, 0.3%, in the semimajor axis. Figure 9, left, illustrates component A and B orbits, with effectively complete coverage of the orbit for the TRANS observations. The POS mode component A residuals and the TRANS residuals (component A-B separations) are all smaller than the dots. Figure 9, right, shows the RV measurements, residuals, and velocity curves predicted from the simultaneous solution. We obtain component masses of M A = 0 . 332 ± 0 . 007 M glyph[circledot] (2.4% error) and M B = 0 . 188 ± 0 . 004 M glyph[circledot] (2.7%) listed in Table 10. These are the first mass determinations for this system.", '5.4. GJ623AB': 'Our modeling of the challenging GJ 623 AB system included three sources of astrometry and two sources of RVs. With ∆ V = 5 . 3, GJ 623 AB is an extremely difficult system for HST TRANS mode observations; our campaign yielded only two usable measurements. Fortunately, in addition to the HST POS and TRANS astrometric observations, there are six aperture masking observations from Martinache et al. (2007) and a single HST /FOS observation (Barbieri et al. 1996) available that were included in our modeling. Figure 10, left, illustrates the orbital phase coverage contributions of the three astrometric sources.  \nThe inclusion of ground-based relative orbit observations greatly increased the number of measurements of position angle and separation and the time span over which to establish orbital elements. Treating GJ 623 AB as a single lined spectroscopic binary, we obtained from the CE and from the HET (Endl et al. 2006) velocities for component A at most orbital phases, as shown in the right panel of Figure 10. In addition to including velocities only for component A, we modeled GJ623AB (and 5 reference stars, all observed at each of the 13 POS mode epochs of observation) with six plate coefficients, replacing -B with D and A with E in Equation 6, thereby allowing unequal scales along each axis. The addition of two additional coefficients reduced the χ 2 to degrees of freedom ratio by 47%, to near unity.  \nWe have improved our knowledge of the parallax of GJ 623 AB by a factor of four compared to Hipparcos , and by over a factor of ten compared to the YPC. Table 9 contains the GJ623AB orbital parameters with formal (1 σ ) uncertainties, with the semimajor axis now known to 0.6%. We obtain component masses of M A = 0 . 379 ± 0 . 007 M glyph[circledot] (1.8% error) and M B = 0 . 114 ± 0 . 002 M glyph[circledot] (1.8%). We have improved upon the previous results of Martinache et al. (2007), who found M A = 0 . 371 ± 0 . 015 M glyph[circledot] (4.0%) and M B = 0 . 115 ± 0 . 002 M glyph[circledot] (1.7%), and the first masses measured by McCarthy and Henry (1986), M A = 0 . 51 ± 0 . 16 M glyph[circledot] (31%) and M B = 0 . 11 ± 0 . 029 M glyph[circledot] (26%), which provided a breakthrough in infrared speckle imaging sensitivity at the time.', '5.5. GJ748AB': 'The binary GJ748AB was the first system with a relative orbit determined using the HST FGSs Franz et al. (1998). The system was subsequently analyzed for mass determinations in Benedict et al. (2001), and we revisit GJ 748 AB here for several reasons. We have an additional TRANS observation obtained with FGS 1r to add to the previous observations secured with FGS 3, extending our coverage by 12 years. This reduces the uncertainty in the orbital period modestly, from 0.5 to 0.3 days. The OFAD has improved from that applied in 2000, and we now use all reference star POS measurements; the previous study discarded any reference star not observed at each epoch, while the present analysis averages four reference stars per epoch. The primary RV source remains Benedict et al. (2001), although we add a single new epoch. Finally, we now apply our quasi-Bayesian modeling technique to these data.  \nWe have improved our knowledge of the parallax by a factor of 8-10 compared to YPC and Hipparcos . We note that the revised proper motion now agrees more closely with Hipparcos . Table 9 contains the GJ 748 AB orbital parameters with formal (1 σ ) uncertainties, and we find that the parameters are consistent with the orbit in Benedict et al.  \n(2001), as expected. Figure 11, left, provides the component A and B orbits, in which the POS mode component A residuals and the TRANS residuals (component A-B separations) are all smaller than the dots. Figure 11, right, contains all RV measurements, RV residuals, and velocity curves predicted from the simultaneous solution. We obtain component masses of M A = 0 . 369 ± 0 . 005 M glyph[circledot] (1.3% error) and M B = 0 . 190 ± 0 . 003 M glyph[circledot] (1.6%). The previous result (Benedict et al. 2001) had slightly higher masses and similar errors: M A = 0 . 379 ± 0 . 005 M glyph[circledot] and M B = 0 . 192 ± 0 . 003 M glyph[circledot] .', '5.6. GJ791.2AB': 'We also re-analyze GJ 791.2 AB, previously investigated in Benedict et al. (2000). Our motivations, similar to those for GJ 748 AB, now include a set of H α radial velocities for both components. With system total magnitude V = 13 . 06 and ∆ V = 3 . 27, detecting both components of GJ 791.2 AB in both the RV data and with HST /FGS is difficult. In addition, GJ791.2AB is a rapid rotator (Delfosse et al. 1998), further hindering RV efforts. However, both components exhibit strong H α emission, permitting RV determination for each component, although with less precision (0.7 km s -1 ) than for our other systems with RVs (typically 0.3 km s -1 ). We also now have two additional TRANS observations and an additional epoch of POS measurements. All but the last set of POS astrometric measurements came from FGS3; the last is from FGS 1r. Modeling is exactly as for GJ 831 AB.  \nThis re-analysis has improved our knowledge of the parallax of GJ 791.2 AB by 30% compared to our previous effort, and by a factor of ten versus the YPC. The new analysis, incorporating priors rather than relying on a correction to absolute derived from a Galactic model, has increased the accuracy of the parallax. Table 9 contains the GJ 791.2 AB orbital parameters with formal (1 σ ) uncertainties with an error of 0.5% in the semimajor axis, which is 5 mas smaller than in Benedict et al. (2000). The derived orbital period is unchanged. Figure 12, left, provides the component A and B orbits, in which the POS mode component A residuals and the TRANS residuals (component A-B separations) are all smaller than the dots. Figure 12, right, contains all H α RV measurements, RV residuals, and velocity curves predicted from the simultaneous solution. We obtain component masses of M A = 0 . 237 ± 0 . 004 M glyph[circledot] (1.7% error) and M B = 0 . 114 ± 0 . 002 M glyph[circledot] (1.8%). The previous analysis (Benedict et al. 2000) yielded significantly higher masses; M A = 0 . 286 ± 0 . 006 M glyph[circledot] and M B = 0 . 126 ± 0 . 003 M glyph[circledot] , which placed both components far from other stars in the MLR. The new masses, due to a smaller orbital semimajor axis, bring the components of GJ 791.2 closer to the MLR defined by the ensemble of system components (Section 8, below).', '5.7. GJ1005AB': 'We have chosen to reanalyze these data, originally collected and analyzed by Hershey & Taff (1998), with the hope that our Bayesian approach might offer some small improvement. Hershey & Taff (1998) obtained component masses of M A = 0 . 179 ± 0 . 003 M glyph[circledot] and M B = 0 . 112 ± 0 . 002 M glyph[circledot] , precision difficult to improve upon.  \nWe have no RVs for this system, nor for any of the remaining systems discussed below. All HST astrometric observations, POS and TRANS, were made using FGS 3. The parallactic ellipse is well-sampled, as is the component A-B separation with 17 TRANS observations. The component A orbit is somewhat less-well sampled, but better than, for example, GJ 22 AC. The major difference in modeling, compared to GJ 831 AB, involves the number of plate coefficients. The reference frame consist of only two stars. Consequently, we replace the coefficients A and B in Equations 5 and 6 with cos ( A ) and sin ( A ), where A is a rotation angle. In essence we assume the scale given by our improved OFAD, just as done by Hershey & Taff (1998) with an older OFAD. The average absolute value residuals for the POS and TRANS observations of GJ1005AB given in Table 6 are ∼ 1 mas, indicating adequate astrometry even with only three coefficients.  \nOur parallax agrees almost perfectly with the Hershey & Taff (1998) value, but our approach has improved the precision by a factor of four, and represents a vast improvement by factors of ∼ 20-30 over the YPC and Hipparcos values. Table 9 contains the GJ 1005 AB orbital parameters with formal (1 σ ) uncertainties, with both a and P now known to 0.2%. Figure 13 provides component A and B orbits with observed positions indicated. The POS mode component A residuals and the TRANS residuals (component A-B separations) are all smaller than the dots and circles in Figure 13. We obtain masses of M A = 0 . 179 ± 0 . 002 M glyph[circledot] (1.1% error) and M B = 0 . 112 ± 0 . 001 M glyph[circledot] (0.9%), values that are virtually identical to and with errors slightly smaller than reported in Hershey & Taff (1998).', '5.8. GJ1245AC': 'The GJ1245 system is a triple, consisting of components A, B, and C, with three components near the end of the stellar main sequence. Our modeling of GJ 1245 AC included six HST POS and 10 TRANS astrometric measurements from FGS3 and FGS1r, and two observations from WFPC2 (Schultz et al. 1998). The POS mode epochs of observations included seven reference stars and were secured at one year intervals, again, very poorly sampling the parallactic ellipse, and sampling less than half the component A orbit (Figure 14). Consequently, we included a YPC parallax as a prior. Note that the parallax  \nerror (0.5 mas, Table 7) is larger than for a typical HST parallax due to the less than ideal sampling. Even though all epochs sampled almost exactly the same part of the parallactic ellipse, they serve to scale the size of that ellipse well enough to provide a parallax whose error does not adversely affect the determination of the final masses. In our final solution, we obtain a parallax error that is half as large at the YPC parallax used as a prior.  \nThe 10 TRANS mode data points sample three-quarters of the relative orbit and nicely map the component A-C separations and position angles over time, as shown in Figure 14. The POS mode component A residuals and the TRANS residuals (component A-C separations) are all smaller than the dots and circles. We obtain masses of M A = 0 . 111 ± 0 . 001 M glyph[circledot] (0.9% error) and M C = 0 . 076 ± 0 . 001 M glyph[circledot] (1.3%). These masses are consistent with the first detailed study of the system by McCarthy et al. (1988), who found M A = 0 . 14 ± 0 . 03 M glyph[circledot] (21%) and M C = 0 . 10 ± 0 . 02 M glyph[circledot] (20%).  \nThese are among the best red dwarf masses yet determined, which is important because component C has the lowest mass of any object in this study. Its mass corresponds to 79 . 8 ± 1 . 0 M Jup , placing it at the generally accepted main sequence hydrogen burning demarkation boundary of ∼ 80 M Jup (Dieterich et al. 2014 and references therein). In addition, Kepler has collected extensive photometric data. Astrometry carried out with these data (Lurie et al. 2015) are consistent with our GJ 1245 AC orbit and may have detected AC-B motion.', '5.9. G 250-029 AB': 'All astrometric measurements came from HST /FGS3 and FGS1r. The eight POS measurements contain only three reference stars, but sample the parallactic ellipse better than for GJ 1245 AC, although not well-enough to preclude the introduction of a lowerprecision parallax prior from Hipparcos as a prior. Fourteen TRANS observations of the component A-B separations sample nearly the entire orbit (Figure 15).  \nOur parallax agrees with the Hipparcos value, but has reduced the error by a factor of eight, and represents a factor of 16 improvement over the value in YPC. Figure 15 provides component A and B orbits with observed positions indicated. Again, the POS mode component A residuals and the TRANS residuals (component A-B separations) are all smaller than the dots and circles in the Figure. We obtain masses of M A = 0 . 350 ± 0 . 005 M glyph[circledot] (1.7% error) and M B = 0 . 187 ± 0 . 004 M glyph[circledot] (2.2%). These are the first mass determinations for this system.', '5.10. GJ1081AB': 'Our modeling of GJ 1081 AB included only HST POS and TRANS astrometric observations made using FGS 1r. The five epochs of POS mode observations had six reference stars and were secured at one year intervals, very poorly sampling the parallactic ellipse. Hence, we included a parallax from the YPC as a prior. The POS observations sparsely sample the perturbation of the primary, while the 11 TRANS observations nicely map the entire relative orbit (Figure 16).  \nOur parallax agrees with the YPC value, but our precision has reduced the parallax error of GJ 1081 AB by a factor of four. Figure 16 provides component A and B orbits with observed positions indicated, where the POS and TRANS residuals are smaller than the dots and circles in the Figure. Due to the high orbital inclination and paucity of POS observations, we obtain relatively poor mass precision for this system. We find component masses of M A = 0 . 325 ± 0 . 010 M glyph[circledot] (3.2% error) and M B = 0 . 205 ± 0 . 007 M glyph[circledot] (3.4%). These are the first mass determinations for this system.', '5.11. GJ54AB': 'GJ54AB has the shortest orbital period in our sample, with a period of 1.15 yr. Our modeling of GJ 54 AB included HST POS (with three reference stars) and TRANS from FGS1r, and one NICMOS astrometric measurement (Golimowski et al. 2004). Again, the timing of the six epochs of POS mode observations sparsely sampled the parallactic ellipse, so we included a Hipparcos parallax prior. The eight TRANS observations sample the entire relative orbit (Figure 17), and the NICMOS measurement falls squarely between two TRANS observations on the orbit, although occurs nearly two years before the first FGS 1r observation.  \nOur derived parallax is larger than the Hipparcos value and has an error six times smaller. Hipparcos likely struggled to measure an accurate value because the orbital period is so close to the 1.00 yr periodicity of the tracing of the parallax ellipse - without knowledge of the resolved positions of the two stars in the binary, an accurate parallax determination is a challenge. Figure 17 provides the component A and B orbits with observed positions indicated. The model yields component masses of M A = 0 . 432 ± 0 . 008 M glyph[circledot] (2.0% error) and M B = 0 . 301 ± 0 . 006 M glyph[circledot] (2.0%). These are the first mass determinations for this system.', '5.12. G 193-027 AB': 'Our modeling of G 193-027 AB included HST POS and TRANS from FGS1r, one AO observation (Beuzit et al. 2004), and one NICMOS astrometric measurement (Golimowski et al. 2004). The POS mode observations included five reference stars and most of the six epochs were secured at one year intervals, very poorly sampling the parallactic ellipse. Hence, we initially included a parallax from Khrutskaya et al. (2010) as a prior, although that parallax had no correction from relative to absolute parallax. Ultimately, the χ 2 markedly decreased when we removed that low-precision and perhaps inaccurate prior.  \nBecause of the typical large component separation (100-170 mas), and small ∆ V = 0.30, the POS observations half the time locked on component B instead of component A. We modified our model to deal with this complication by constraining the parallax of component A to equal the parallax of B and by solving for two POS mode orbits. In Figure 18 we plot observation set numbers on the component A orbit that should have been established for the six sets of POS measurements; the POS observations sparsely sample the perturbation of the primary because they locked on component B for sets 2, 3, and 4. The relatively large parallax error of 1.4 mas (Table 7) reflects the paucity of POS mode observations.  \nTable 9 contains the G 193-027 AB orbital parameters from a model not including the Khrutskaya et al. (2010) parallax prior. Figure 18 provides component A and B orbits with observed positions indicated. Because we had relatively few component A POS observations, we obtain somewhat poorer mass precision for this system, finding M A = 0 . 126 ± 0 . 005 M glyph[circledot] (4.0% error) and M B = 0 . 124 ± 0 . 005 M glyph[circledot] (4.0%), with the errors driven almost entirely by the parallax error. The two components are of nearly equal mass, consistent with their low ∆ V . These are the first mass determinations for this system.', '5.13. GJ65AB': 'The 10 HST TRANS-only observations were made using FGS 3, covering only 18 degrees of orbital position angle during 1.1 yr. Seven POS mode observations exist for component A (shown as dots in Figure 19), but the reference frame consisted of a single star at each epoch, and not always the same single star. Consequently, we model only the relative orbit. We included a large sample of valuable visual, photographic, and CCD observations of separation and position angle (Geyer et al. 1988) for our analysis of GJ 65 AB. These low-precision (relative to HST ) measurements came primarily from the USNO 61in reflector and span 48 years. Even though of lower precision, they are extremely useful for this system, which has an orbital period of 26.5 yr, the longest among the 15 systems we observed with HST /FGS.  \nAlso useful for extending the orbit sampling; five VLT/NACO (Kervella et al. 2016) and one HST /WFPC2 (Dieterich et al. 2012) measure of position angle and separation.  \nModeling only relative position angle and separation measurements yields a , P , T 0 , e , i , Ω, and ω B . Figure 19 provides the component AB relative orbit with observed positions indicated. The USNO residuals are obviously not smaller than the circles in the Figure. The average absolute value residual for the HST TRANS observations is 〈| res |〉 = 3.5 mas, considerably smaller than the corresponding 81 mas value for the USNO and 9.6 mas for the VLT/NACO measurements. By adopting a mass fraction ( f = 0 . 494 ± 0 . 004, Geyer 1988) and absolute parallax ( π abs = 373 . 7 ± 2 . 7 mas) from the YPC, we obtain component masses of M A = 0 . 120 ± 0 . 003 M glyph[circledot] (2.5% error) and M B = 0 . 117 ± 0 . 003 M glyph[circledot] (2.5%), agreeing with the recent Kervella et al. (2016) values of M A = 0 . 123 ± 0 . 004 M glyph[circledot] (3.5% error) and M B = 0 . 120 ± 0 . 004 M glyph[circledot] (3.6%)', '5.14. GJ473AB': 'The sky within the FGS FOV near GJ 473 AB contains no usable POS mode reference stars. Hence, as with GJ 65 AB we again model only the relative orbit with astrometry, using several sources: HST (TRANS-only) from FGS3 and FGS1r, a single HST FOS measurement (Schultz et al. 1998), and optical (Balega et al. 1994; Blazit et al. 1987; Horch et al. 2012) and near-infrared speckle (Henry et al. (1992) and Torres et al. 1999) observations. These lower-precision (relative to HST ) measurements are extremely useful for this long period (15.8 yr) system.  \nFigure 20 provides the component AB relative orbit with observed positions indicated. The speckle residuals are generally not smaller than the circles in the Figure. The mean absolute value residual for the HST TRANS observations is 2.5 mas, considerably smaller than the corresponding 17 mas value for the speckle measurements. By adopting a mass fraction of f = 0 . 477 ± 0 . 008 from Torres et al. (1999) and absolute parallax of π abs = 235 . 5 ± 2 . 9 mas from the RECONS astrometry program at the CTIO/SMARTS 0.9m (Henry et al. 2006), we obtain component masses of M A = 0 . 124 ± 0 . 005 M glyph[circledot] (4.0% error) and M B = 0 . 113 ± 0 . 005 M glyph[circledot] (4.1%). The relatively large mass errors are driven almost entirely by the parallax error. These masses are somewhat lower than those reported by Torres et al. (1999), M A = 0 . 143 ± 0 . 011 M glyph[circledot] (7.7% error) and M B = 0 . 131 ± 0 . 010 M glyph[circledot] (7.6%), who used 56 astrometry observations taken between 1938 and 1998; however, only four highquality FGS observations were available then, compared to the 11 FGS observations used here.', '6. Absolute Magnitudes': 'The orbital elements summarized for the 15 systems in Table 9 are used to derive the component masses given in Table 10. To place these stars on an empirical MLR, we also require the component absolute magnitudes. For the MLR at optical wavelengths, we use V photometry from Weis (1984, 1996) and the RECONS program at the CTIO/SMARTS 0.9m (see Winters et al. 2015). For ∆ V , we use values in our earlier work (Henry et al. 1999) as well as results from the FGS observations presented here to derive component magnitudes. The ∆ m measurements from the FGS TRANS observations using F583W have been transformed to ∆ V , as described in Henry et al. (1999).  \nAt infrared wavelengths, we use 2MASS K s photometry (Skrutskie et al. 2006) for the combined light of all 15 systems in Table 10. The ∆ K values have been adopted from infrared speckle measurements previously reported in Henry & McCarthy (1993), HST /NICMOS measurements reported in Dieterich et al. (2012), and new Gemini-N observations using the Near InfraRed Imager (NIRI) with the ALTAIR facility level adaptive optics system Dieterich (2015). We have not made any transformations between the infrared K band filters given that the ∆ K measurement errors tend to be larger than the slight adjustments due to differences in the various K filters.  \nUsing our parallaxes in Table 7 and the ∆ V and ∆ K values, we find the component absolute magnitudes presented in Table 10. For these very nearby systems we have assumed no absorption ( A V = 0). Parallaxes of this precision do not require correction for LutzKelker-Hanson bias (Lutz & Kelker 1973; Hanson 1979) in the derived absolute magnitudes.', '7. Additional Low Mass Binary Systems': 'To boost the number of stars used in the MLRs at both optical and infrared wavelengths, we augment our sample of 15 binaries with red dwarfs in 9 additional systems having highquality mass, M V , and M K measurements. In Table 11, we list their parallaxes, V and K -band absolute magnitudes, and masses computed by other investigators. Ground-based astrometry has been used for all systems, except the eclipsing binaries GJ 278 CD (YY Gem), GJ 630.1 AB (CM Dra), GJ 2069 AC (CU Cnc), and GU Boo. A few systems are particularly worthy of note:  \nGJ166C is a tertiary in the 40 Eri system, in a ∼ 250 yr orbit orbit with the white dwarf GJ 166B. Hence, the mass error of 0.029 M glyph[circledot] is the largest for any star in the sample.  \nGJ2005ABC is an important triple with components near the end of the stellar main sequence. Only a small portion of the A-BC orbit has been observed, so the resulting mass determinations for A are not of sufficiently high quality to be included in this paper; here we concentrate on the BC pair. The three stars have combined V = 15.28 ± 0.02 from our CTIO/SMARTS 0.9m photometry, and individual ABC magnitudes of V = 15.35 ± 0.04, 18.68 ± 0.06, and 19.07 ± 0.07 from Leinert et al. (2000). In the infrared, K = 8.24 ± 0.03 for ABC from 2MASS. Using ∆ K AB = 1.20 ± 0.03 and ∆ K AC = 1.55 ± 0.09 from Leinert et al. (1994), the components have individual magnitudes of K A = 8.73 ± 0.03, K B = 9.93 ± 0.04 and K C = 10.28 ± 0.04. The weighted mean parallax of 128.49 ± 1.50 mas is from the RECONS astrometry program at the CTIO/SMARTS 1.5m (129.47 ± 2.48 mas, Costa et al. 2005) and a new value from the CTIO/SMARTS 0.9m program (127.93 ± 1.88 mas). We adopt the BC orbit and fractional mass (0.48 ± 0.01) from Kohler et al. (2012), with the new parallax to obtain masses of M B = 0.079 ± 0.003 M glyph[circledot] and M c = 0.073 ± 0.003 M glyph[circledot] .  \nGJ2069AC is an eclipsing binary that is part of the quintuple red dwarf system known as CU Cnc (components A, C, and E) and CV Cnc (components B and D). There was no parallax in YPC and the various parallaxes derived using Hipparcos data are of poor quality, with values of 78.05 ± 5.69 (original reduction), 85.16 ± 6.42 mas (Pourbaix et al. 2003), and 90.37 ± 8.22 mas (van Leeuwen et al. 2007). The first parallax value was used by Delfosse et al. (1999), which led them to conclude that the eclipsing components were subluminous. Here we provide a much better value of 64.80 ± 1.43 mas from the RECONS astrometry program at the CTIO/SMARTS 0.9m that utilizes 11 years of astrometry data. This represents the weighted mean of values determined for ACE (63.59 ± 1.92 mas) and BD (66.32 ± 2.15 mas) - the system is observed as two sources at the resolution of the 0.9m. With this parallax, the components of the eclipsing pair AC now fall closer to other stars in the MLR, particularly in the K -band.', '8. The MLR': 'We plot all 47 stars with their masses, M , and absolute magnitudes, M V , from Tables 10 and 11 on the V - and K -band MLR shown in Figures 21 and 22. For the MLR at K only the GJ 54 AB system is missing, lacking individual M K values. The object identifications in bold denote new and/or improved absolute magnitudes and masses derived for this paper using the HST /FGS and RV data presented here. The primaries are plotted in blue; the secondaries in red.  \nWith the caveat that we are mixing stars of various ages, metallicities, and magnetic properties, we fit the V and K mass-absolute magnitude distribution with a double exponent  \n(with offset) to provide an empirical MLR for the full range of M dwarf masses from 0.6 M glyph[circledot] to the end of the stellar main sequence at 0.08 M glyph[circledot] . The form of the fitted equation is:  \nM V = y 0 + A 1 exp { -( Mx 0 ) τ 1 } + A 2 exp { -( Mx 0 ) τ 2 } (10)  \nwhere the coefficients are in Table 12.  \nThis function serves as a smoothing tool. We chose the double exponent because it yielded a lower χ 2 and smaller RMS residuals than, for example, a single exponential with offset ( χ 2 ten times larger, RMS residuals 4 times larger). Because both absolute magnitudes and masses have errors, we fit Equation 10 to the points using GaussFit (Jefferys et al. 1988), a modeling tool that fairly assesses errors in both variable sets. The top panels of Figure 21 and Figure 22 display residuals to the fit for both absolute magnitude ( M V and M K ) and mass. The RMS values are 0.19 mag in M V and 0.023 M glyph[circledot] in mass. The RMS values to the fit are 0.09 mag in M K and 0.014 M glyph[circledot] in mass for the MLR at K . The coefficients for the MLR at K are also given in Table 12, along with the RMS values for both MLRs.  \nRather than knowing the mass of an object and needing to derive its luminosity, astronomers more typically want to estimate a mass from an absolute magnitude. Therefore, in the two panels of Figure 23 we plot mass against M V and M K , and provide a method to estimate a mass given an absolute magnitude in a fundamental band at either optical or near-infrared wavelengths. Fit with fifth-order polynomials with magnitude offsets, x 0 , we obtain expressions that can be used to estimate masses, M , as functions of M V or M K (for M V ≤ 19 , M K ≤ 10):  \nM = C 0 + C 1 ( M V -x 0 ) + C 2 ( M V -x 0 ) 2 + C 3 ( M V -x 0 ) 3 + C 4 ( M V -x 0 ) 4 (11)  \nwhere V can be replaced with K . Table 13 contains coefficient and offset values for the M V and M K relations. As examples, a measured M V = 14 would yield M = 0 . 15 ± 0 . 03 M glyph[circledot] , while M K = 9 would yield M = 0 . 12 ± 0 . 02 M glyph[circledot] , where the errors are simply the RMS scatter about the polynomial fits.  \nIn Figure 24 we compare the V and K -band MLRs with recent models from Baraffe et al. (2015) and Dotter (2016). Note that the models agree with the K -band MLR better than with the V -band MLR. Thus, the K -band fluxes are better modeled better than the V -band fluxes, which show twice the residual offsets to the Equation 10 fits. These results confirm and extend the results of Henry & McCarthy (1993) and Delfosse et al. (2000). We shall return to the larger V -band scatter in § 9. These MLRs are the most robust ever  \nestablished for M dwarfs, yet could be improved with additional mass measurements around ∼ 0.5 M glyph[circledot] , ∼ 0.3 M glyph[circledot] , and below ∼ 0.1 M glyph[circledot] .', '9. Discussion': "Whyis the scatter in the V -band MLR twice that of the corresponding K -band MLR? To reiterate, age, chemical composition, and magnetic properties can all affect the luminosities and temperatures of stars, as well as rotation, mixing length and other properties. Here we address the first three properties.  \nAge attributes are particularly complicated because the MLR at low masses is a mapping that is complicated by both relative age and absolute age. Regarding relative age, once formed, M stars descend slowly to the main sequence along a nearly vertical Hayashi track (Palla 2012), so young stars have higher luminosities than stars on the main sequence. In fact, because in the age of the Universe, none of the low mass stars investigated here have left the main sequence through evolution, the 24 systems in this sample are a heterogeneous mix of stars with ages potentially spanning ∼ 10 Gyr, although for a roughly uniform star formation rate, only a few percent might lie above the main sequence. Determining the age of an M dwarf is notoriously difficult, and usually based on circumstantial evidence such as x-ray emission, flare rates, etc., unless it is associated with a more massive star of known age (e.g., Mamajek et al. 2013) or a particular cluster or moving group (e.g., Riedel et al. 2011). As for absolute age, an M star's luminosity is affected by its composition (metallicity), which is a function of birth date within the Galaxy. More metal-rich M dwarfs, generally formed later in the history of the Galaxy, can appear fainter due to line blocking. Absolute ages might be established through gyrochonology, e.g., Cargile et al. (2014); Meibom et al. (2015); a more rapidly rotating M dwarf would presumably be younger than one rotating more slowly, although a reliable relation for M dwarfs remains elusive.  \nChemical composition (Metallicity) is, in principle, a more directly measurable quantity than age. In the past, accurate M dwarf metallicities were difficult to quantify. Relative metallicites could be inferred from single project efforts (Bean 2007; Jenkins et al. 2009) despite their lack of agreement, and such comparisons clearly indicated that some stars possess fewer metals than others. For M dwarfs observed at optical wavelengths, the CaH bands are typically used, although these features are confounded by the interplay of effects due to both metallicity and gravity (Jao et al. 2008). There has been considerable work on M dwarf metallicities in recent years (Rojas-Ayala et al. 2012; Onehag et al. 2012; Terrien et al. 2012; Neves et al. 2014; Mann et al. 2014, 2015; Newton et al. 2015; Lindgren & Heiter 2015). While there are subtle effects that remain challenging (including the effect of low log  \ng for young M dwarfs that have not yet contracted to their main sequence radii), the broad consistency between most of this work suggests that accurate metallicities (perhaps at least as accurate as ± 0.1 dex) are no longer out of reach.  \nMagnetic properties are generally explored via measurements of chromospheric activity (flaring, x-ray emission, spots) and are correlated with rapid rotation (Wright et al. 2011; Stelzer et al. 2013), attributes that are all typically seen in young M dwarfs. Photometric studies of M dwarfs at various timescales have shown that these small stars exhibit wide ranges in variability. For periods of minutes to hours, some stars flare often (Kowalski et al. 2013), while others are quiescent for long periods of time (Jao et al. 2011). Spots on the surfaces of M dwarfs reveal the presence of magnetic fields, and can be used to measure rotation periods of days to months (Benedict et al. 1998; Irwin et al. 2011). Recent work has shown that like the Sun, M dwarfs also undergo long-period photometric changes lasting several years, presumably connected to magnetic cycles in the stars (Hosey et al. 2015). Variations in magnitude could change a derived absolute magnitude, adding scatter to the MLR.  \nThe formal errors in Table 10 effectively eliminate mass and absolute magnitude errors as explanations for the offsets of most of the stars in the MLRs, although errors in luminosity may be invoked for GU Boo A and B, as well errors in mass for GJ 166 C and GJ 860 A. Some combination of the three factors outlined above can explain the remaining scatter we see in Figures 21 and 22. Table 14 collects the presently available metallicity, rotation velocity, radius, and x-ray emission information for the systems considered in this paper. Here we attempt to explain a few of the significant departures in the V and K -band MLR using these data.  \nThe components of GJ1005 are elevated above the MLR at V . The [Fe/H] = -0.47 value for this system is among the lowest in the sample, as are the rotation velocity and x-ray flux. This system is likely older than most others in the sample, but absorption that affects the V band is muted in K , so the components fall on the infrared MLR.  \nSimilar to GJ 1005, both components of G 193-027 are elevated above the MLR at V . The [Fe/H] = -0.46 value is virtually identical to GJ 1005, although G 193-027's rotation velocity and x-ray luminosity fall in the middle of the sample distribution. We suspect that this is mildly metal-poor system, again causing it to be elevated in the MLR at V , but not at K .  \nBoth components of GJ 791.2 lie below the MLRs. The stars exhibit rapid rotation and relatively large x-ray emission compared to the other stars in the sample, which suggests relative youth. This, in turn, suggests that they are metal-rich and consequently have lower  \nthan expected luminosities.  \nBoth components of GJ 831 lie below the MLR at V , yet fall squarely on the MLR at K . This is the most metal-rich [Fe/H] = +0.3 measurement available for the 24 systems, which may reduce the flux at V but not at K and lead to the seemingly discordant locations on the two MLRs.  \nThe GJ1245ABC triple is in the original Kepler FOV, and both the AC and B components have been found to rotate rapidly and exhibit significant x-ray emission (Lurie et al. 2015). X-ray emission is an age proxy. Youth can be correlated with metallicity which can effect absolute magnitude. In contrast to GJ 791.2, the puzzle is that neither A nor C strays far from the MLR at V or K . We note that at the very lowest masses in the MLRs presented, there are very few points with empirically determined masses to constrain the fits.  \nTwo remaining systems worthy of note are GJ 747 and G 250-029. All four components in these systems are elevated in the MLR at V . There is nothing unusual about the available data for GJ 747, while G 250-029 lacks (Table 14) any additional knowledge to which we might appeal for explanation.  \nThe eclipsing system GJ 2069 yielded (Ribas 2003) an extremely high mass precision, but the absolute magnitudes would place these components about 0.5 magnitude below the MLR smoothing functions in M V . The system has one of the highest metallicities in Table 14, perhaps partially explaining this deviation. Given that the Figure 21 smoothing fits depend on the measured mass and luminosity errors, we chose to exclude this system from the V -band fitting.  \nTurning the question about scatter in the MLRs around, one might ask why both components of the GJ 65 system, which have high rotational v sin i, perhaps indicative of youth (and presumably higher metallicity), are not further from other objects with similar masses on any MLR? Clearly, there remains work to be done to understand the complex interplay of the effects that set locations on the MLRs.  \nUltimately, the predictive utility of an MLR depends on the scatter about any smoothing function. In the absence of physically-based corrections to absolute magnitude, even the lower-scatter K -band MLR (Figure 22) exhibits deviations about the smoothing function that at some mass levels exceeds 15%. For example, at M K =7.8 the scatter is ± 0.035 M glyph[circledot] , or 18% at the corresponding mass location of M = 0.2 M glyph[circledot] . This particular mass region, exhibiting the highest scatter, may signal the onset of full convection (c.f. Houdebine & Mullan, 2015).  \nFinally, Figure 25 presents an M K vs. ( V -K ) HR diagram for all systems with ∆ K measurements (Table 10) and masses determined using FGS data. We derive absolute magnitudes using our relatively precise HST parallaxes (Table 7). Also plotted are stellar models for 0.1 and 10 Gy from Baraffe et al. (2015) and the single M dwarfs Proxima Centauri and Barnard's Star, using HST parallaxes from Benedict et al. (1999) to derive absolute magnitudes. As expected, the presumably co-eval components in systems usually lie close to the same age track. We note that the components of GJ 791.2 lie close to the Baraffe et al. (2015) 0.1 Gyr model, supporting a young age. Note also that GJ 1245 A and C lie close to the 10 Gyr model trace, even though their rapid rotation and x-ray flux (Table 14) suggest youth. Late-type stars like GJ 1245AC can remain active on a timescale of ∼ 8 Gyr (see West et al. 2008), so it is perhaps not surprising that the stars might be quite old but still active. Apparently, this particular bandpass HR diagram is not an infallible indicator of youth, nor the adopted markers for youth foolproof.", '10. Conclusions': "- 1. With HST Fine Guidance Sensors (FGS) we obtained fringe tracking (POS) observations with either FGS 3 and/or FGS 1r of 13 low-mass binary systems, each with 6-16 epochs of observations typically spanning 5 years. These yielded absolute parallaxes with an external error better than 1% and proper motions with average errors 0.3 mas y -1 in RA and Dec.\n- 2. Fringe tracking (POS) observations of the primary, including photocenter corrections where required, provide a perturbation orbit.\n- 3. Fringe scans (TRANS) combined with POS observations provide a mass fraction relative to an astrometric reference frame. TRANS observations also provide ∆ V measurements (Henry et al. 1999). For two systems we acquired only TRANS measurements, requiring external determinations of the mass fractions.\n- 4. Radial velocities, primarily from the McDonald Observatory 2.1m telescope and Cassegrain Echelle spectrograph, were combined with astrometry for seven systems, increasing the accuracy of their final mass results.\n- 5. We derive 30 component masses in 15 binary red dwarf systems with a median precision 1.8% and an average precision of 2.1%. These are the first mass determinations for 12 stars in 6 systems. These masses are augmented with additional red dwarfs with high-quality masses in 9 additional systems.  \n- 6. We provide system magnitudes at UBVRIJHK and component magnitude differences, ∆ V and ∆ K , many reported for the first time here. New trigonometric parallaxes from both HST and the RECONS astrometry program at CTIO, as well as literature values, are used with the photometry to obtain accurate component M V and M K values.\n- 7. This study provides high quality data for 47 red dwarfs with masses between 0.616 M glyph[circledot] and 0.076 M glyph[circledot] , M V = 8.68-19.63, and M K = 5.29-10.27. Many of the secondaries lie below 0.2 M glyph[circledot] , the crucial region where age begins to play a significant role in the luminosities of stars. With these data, we establish Mass-Luminosity Relations (MLRs) in both the optical V and near-infrared K bands.\n- 8. The V and K MLRs are fit with double exponentials to provide empirical conversions of masses to absolute magnitudes, and vice versa. We find that the K -band MLR exhibits a scatter in mass half that of the corresponding V -band relation.\n- 9. The predictive capability of even the smaller-scatter K -band MLR appears to be limited by age, composition, and magnetic differences between low mass stars. At M K =7.8 the scatter is 0.035 M glyph[circledot] , or 18% at the corresponding mass location of M = 0.2 M glyph[circledot] .\n- 10. Future work will disentangle the effects of age, metallicity, and magnetic properties on luminosity, and enable us to construct MLRs that consider additional factors that affect low mass stars.  \nSupport for this work was provided by NASA through grants GTO NAG5-1603, GO6036, 6047, 6566, 6764, 6882, 6883, 6884, 7491, 7493, 7894, 8292, 8728, 8729, 8774, 9234, 9408, 9972, 10104, 10613, 10773, 10929, 11299, and 12629 from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. The RECONS program has been supported by the National Science Foundation through grants AST 05-07711, AST 09-08402, and AST 14-12026.  \nThis publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by NASA and the NSF. This research has made use of the SIMBAD and Vizier databases and Aladin, operated at CDS, Strasbourg, France, the NASA/IPAC Extragalactic Database (NED) which is operated by JPL, California Institute of Technology, under contract with the NASA, and NASA's Astrophysics Data System Abstract Service.  \nWe thank Linda Abramowicz-Reed for her unflagging and expert FGS instrumental support over the last 25 years. Cassegrain Echelle (CE) Spectrograph observing and data reduction assistants included J. Crawford, Aubra Anthony, Iskra Strateva, Tim Talley, Amber Armstrong, Robert Hollingsworth, and Jacob Bean. We thank Dave Doss, John Booth, and many other support personnel at McDonald Observatory for their cheerful assistance over many years, as well as the staff at CTIO for their continued support of the CTIO/SMARTS 0.9m, where the RECONS astrometry/photometry program is carried out. Finally, thanks to Dr. Michael Endl for the use of additional unpublished HET radial velocities for GJ 623 A, and to R. Andrew Sevrinsky for the improved parallax for GJ 473 AB.  \nLastly, we thank an anonymous referee for their prompt attention to this paper, and for the many suggestions that materially improved the final version.", 'REFERENCES': "Allard F., Hauschildt P.H., & Schweitzer A., 2000. ApJ, 539, 366  \nAndersen J., 1991. A&A Rev., 3, 91  \nBalega I.I., Balega Y.Y., Belkin I.N., et al., 1994. A&AS, 105  \nBaraffe I., Homeier D., Allard F., et al., 2015. A&A, 577, A42  \nBarbieri C., De Marchi G., Nota A., et al., 1996. A&A, 315, 418  \nBean J.L., 2007. M dwarf metallicities and exoplanets . 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Radial Velocities a  \nTable 3-Continued  \nTable 3-Continued  \nTable 3-Continued  \nb mJD=JD-2400000.0  \nTable 4. Results from Radial Velocities a  \na All velocity units in km s -1 . Errors for RV amplitudes K A , K B , and system velocity, γ , are just below each entry. #A, #B are numbers of unique epochs of RV measurements. A 〈| res |〉 and B 〈| res |〉 are average absolute value RV residuals in km s -1 .  \nTable 5. GJ831AB Reference Star Characteristics and Final Parallaxes  \nTable 6. Astrometry Data Spans and Residuals a  \na #ref = number of astrometric reference stars; 〈 V 〉 ref = average V magnitude of the reference stars; POS span = length of POS campaign; #POS = number of epochs of POS observation; POS 〈| res |〉 = average absolute value of POS astrometric residuals; TRANS span = length of TRANS campaign; #TRANS = number of epochs of TRANS observation; TRANS 〈| res |〉 = average absolute value of TRANS astrometric residuals, unless noted with other observations.  \nb For some orbit plots the number of POS plotted appears less than listed here because of overplotting and plot resolution.  \nc includes measurements in addition to HST TRANS  \nTable 7. Parallaxes and Proper Motions  \nTable 8. Component Position Angles, Separations, Residuals and Sources a  \nTable 8-Continued  \nTable 8-Continued  \nTable 8-Continued  \nTable 8-Continued  \nTable 8-Continued  \nTable 8-Continued  \nTable 8-Continued  \nTable 8-Continued  \nTable 8-Continued  \na Non-FGS measurements: B04=Beuzit et al. (2004), adaptive optics; B87=Blazit et al. (1987), optical speckle; B94=Balega et al. (1994), optical speckle; FOC=Barbieri et al. (1996), HST /FOC; FOS=Schultz et al. (1998), HST /FOS; H12=Horch et al. (2012), optical speckle; H92=Henry et al. (1992), infrared speckle; M07=Martinache et al. (2007), aperture masking; M91=McCarthy et al. (1991), infrared speckle; NIC=Golimowski et al. (2004), HST /NICMOS; VIS/CCD= Geyer et al. (1988); visual, photographic, and CCD measurements; W03=Woitas et al. (2003), infrared speckle; WFPC2=Schroeder et al. (2000), Dieterich et al. (2012) HST /WFPC2; NACO=Kervella et al. (2016) VLT/NACO", 'T able 9. Orbital Elemen ts a': "<!-- image -->  \nT able 10. Masses, M V , and M K for Red Dw arfs observ ed with HST /F GS a  \n<!-- image -->  \nsec.  \n=  \nSecondary  \npri,  \n=  \ny  \nmar  \ni  \nPr  \nare  \nts  \nonen  \nComp  \n).  \nglyph[circledot]  \nM  \n(  \nmasses  \nsolar  \nin  \n)  \nM  \n(  \nMasses  \na  \n2.  \nable  \nT  \nin  \nreferences  \nfrom  \nalues  \nv  \nV  \nb  \nc References f or ∆ V and ∆ K : * = this pap er, B96 = Barbieri et al. (1996), B00 = Benedict et al. (2000), H93 = Henry & McCarth y (1993), H99 = Henry et al. (1999)  \n2MASS  \nfrom  \nd  \nTable 11. Other Mass Determinations  \na Sources: * = this paper, C05 = Costa et al. (2005), D00 = Delfosse et al. (2000), D15 = Dieterich (2015), F99 = Forveille et al. (1999), H12 = Holberg et al. (2012), H99 = Henry et al. (1999), L00 = Leinert et al. (2000), L05 = L'opez-Morales & Ribas (2005), M09 = Morales et al. (2009), P97 = Perryman et al. (1997), R03 = Ribas (2003), S00 = S'egransan et al. (2000), S08 = Seifahrt et al. (2008), T02 = Torres & Ribas (2002), T06 = Tamazian et al. (2006), YPC = van Altena et al. (1995)  \nTable 12. MLR Fits and Residuals a  \na Parameters in the Equation 10 double exponential.  \nb RMS of absolute magnitude, M , and mass, M , residuals  \nc DOF = degrees of freedom for each fit  \nTable 13. Polynomial Fits for Mass as a Function of Absolute Magnitude a  \na Parameters in the Equation 11 polynomial.  \nTable 14. Factors Potentially Influencing Deviations from MLR  \na F13 = Feiden & Chaboyer (2013), G05 = Glebocki & Gnacinski (2005), I11 = Irwin et al. (2011), J09 = Jenkins et al. (2009), NB14 = Neves et al. (2014), NC15 = Newton et al. (2015), P01 = Pasinetti Fracassini et al. (2001), R12 = Rojas-Ayala et al. (2012), E13 = Eiroa et al. (2013), S13 = Stelzer et al. (2013), W11 = Wright et al. (2011), T15 = Terrien et al. (2015), G14 = Gaidos et al. (2014), K12 = Koleva & Vazdekis (2012), K16 = Kervella et al. (2016)  \nb Assumes same [Fe/H] as GJ 570 A, a K5V star.  \nFig. 1.- Positions within FGS 3 of GJ 831 AB and reference stars (ID numbers from Table 5) for all ten observation sets. The '1' indicates the location of the GJ 831 AB photocenter. Reference stars 6-9, identified in the original HST Phase 2 submission, were never observed, either due to faintness or placement within the FGS FOV.  \n<!-- image -->  \nFig. 2.J -K vs. V -K color-color diagram for GJ 831 AB and reference stars described in Table 5. The dashed line is the locus of dwarf (luminosity class V) stars of various spectral types; the dot-dashed line is for giants (luminosity class III) from Cox (2000). The reddening vector indicates A V =1.0 for the plotted color systems. For this field at Galactic latitude glyph[lscript] II = -40 · we estimate 〈 A V 〉 = 0.00 ± 0.06 magnitude with a maximum of 0.11 (Schlafly & Finkbeiner 2011).  \n<!-- image -->  \nFig. 3.- Reduced proper motion diagram for 2378 stars taken from a 2 · × 2 · field centered on GJ831AB. Star identifications are shown for astrometric reference stars 2,3,4,5, and 10 in Table 5. Star '1' is GJ 831 AB. H K for all numbered stars is calculated using proper motions from the PPMXL and K magnitude from 2MASS. See Section 4.3 for a discussion of ref-5 and ref-10.  \n<!-- image -->  \nFig. 4.- Histograms of GJ 831 AB and reference star residuals resulting from the application of the model (Equations 5 and 6) incorporating both astrometry and radial velocities.  \n<!-- image -->  \nFig. 5.- Left: GJ831A (dots, POS orbit predicted positions) and component B (open circles, TRANS orbit predicted positions). All observations, POS and TRANS and A and B component radial velocities, were used to derive the orbital elements listed in Table 9. POS and TRANS astrometric residuals (average absolute value for which listed in Table 6) are smaller than the symbols. The POS mode points either side of p.a=90 · each represent two temporally close epochs. The arrow indicates the direction of orbital motion. Right: RV measurements (Table 3) using CE on the McDonald 2.1m, phased to the orbital period determined from a combined solution including astrometry and RVs. The error bars are 1σ . The dashed lines are component velocities predicted from the orbital parameters derived in the combined solution. Middle and top panels illustrate component B and A RV residuals (average absolute value for which listed in Table 4) from the combined solution.  \n<!-- image -->  \n<!-- image -->  \nFig. 6.- Left: GJ 22 A (dots, POS orbit predicted positions) and component C (open circles) TRANS and speckle (McCarthy et al. 1991; Woitas et al. 2003) orbit predicted positions. All observations, POS, TRANS (+), and speckle ( × ), and component A radial velocities were used to derive the orbital elements listed in Table 9. POS astrometric residuals (average absolute value for which listed in Table 6) are smaller than the dot symbols. Right: component A RV measurements from the present study using CE on the McDonald 2.1m, phased to the orbital period determined from a combined solution including astrometry and RVs. The error bars are 1σ . The dashed line is the velocity predicted from the orbital parameters derived in the combined solution. Top panel: Component A RV residuals (average absolute values, listed in Table 4) from the combined solution.  \n<!-- image -->  \nFig. 7.- GJ 22 A radial velocities used to derive the orbital elements listed in Table 9 plotted against mJD rather than orbital phase. GJ 22 AC has a companion, GJ 22 B, separated by 4', with a period, P glyph[similarequal] 320 yr. We may be detecting that motion as a slope in the RV residuals (top panel).  \n<!-- image -->  \nFig. 8.- Left: GJ234A (dots, POS orbit predicted positions) and component B (open circles) TRANS orbit predicted positions. All observations, POS, TRANS, and component A radial velocities were used to derive the orbital elements listed in Table 9. POS, TRANS astrometric residuals are smaller than the symbols. Right: Component A RV measurements from the present study using CE on the McDonald 2.1m, phased to the orbital period determined from a combined solution including astrometry and RV. The error bars are 1σ . The dashed line is the velocity predicted from the orbital parameters derived in the combined solution. Top panel: Component A RV residuals (average absolute values, listed in Table 4) from the combined solution.  \n<!-- image -->  \narc seconds  \nFig. 9.- Left: GJ469A (dots, POS orbit predicted positions) and component B (open circles, TRANS orbit predicted positions). All observations, POS and TRANS and A and B component radial velocities, were used to derive the orbital elements listed in Table 9. POS and TRANS astrometric residuals (average absolute values listed in Table 6) are smaller than the symbols. Right: RV measurements from the present study using CE on the McDonald 2.1m, phased to the orbital period determined from a combined solution including astrometry and RVs. The error bars are 1σ . The dashed lines are velocities predicted from the orbital parameters derived in the combined solution. Middle and top panels: Component B and A RV residuals (average absolute values listed in Table 4) from the combined solution.  \n<!-- image -->  \n0.6  \n0.4  \n0.2  \nPhase (P= 4209.48  \nd  \n)  \n]  \n-1  \ns  \n[km  \nds  \ni  \ns  \nre  \nV  \nR  \n]  \n-1  \ns  \n[km  \nV  \nR  \n0.4  \n0.0  \n-0.4  \n1.0  \n0.5  \n0.0  \n-0.5  \n-1.0  \n0  \n-2  \n-4  \n-6  \n-8  \n-10  \n-12  \n-14  \n0.0  \n0.8  \nA  \nB  \n1.0  \n<!-- image -->  \nFig. 10.- Left: GJ623A (dots, POS orbit predicted positions) and component B (+) TRANS, FOC (*) (Barbieri et al. 1996), and ( × ) Aperture Masking (Martinache et al. 2007) orbit predicted positions. All observations, POS, TRANS, FOC, Aperture Masking, and component A radial velocities, were used to derive the orbital elements listed in Table 9. POS, TRANS, FOC, and Aperture Masking astrometric residuals (RMS for which listed in Table 6) are smaller than the symbols. Right: Component A RV measurements from the present study using CE on the McDonald 2.1m and the Tull Spectrograph on HET, phased to the orbital period determined from a combined solution including astrometry and RVs. The dashed lines are velocities predicted from the orbital parameters derived in the combined solution. Top panel: Component A RV residuals (average absolute values, CE and HET, listed in Table 4) from the combined solution. We plot 1σ error bars in this case on the residuals. They are smaller than the points in the lower panel.)  \n<!-- image -->  \n<!-- image -->  \nFig. 11.- Left: GJ748A (dots, POS orbit predicted positions) and component B (open circles, TRANS orbit predicted positions). All observations, POS and TRANS and A and B component radial velocities, were used to derive the orbital elements listed in Table 9. POS and TRANS astrometric residuals (average absolute value for which listed in Table 6) are smaller than the symbols. Right: RV measurements from the present study using CE on the McDonald 2.1m, phased to the orbital period determined from a combined solution including astrometry and RVs. The error bars are 1σ . The dashed lines are velocities predicted from the orbital parameters derived in the combined solution. Middle and top panels: Component B and A RV residuals (average absolute value for which listed in Table 4) from the combined solution. The four component B velocities with errors larger than 0.5 km s -1 come from Marcy & Benitz (1989).  \n<!-- image -->  \nFig. 12.- Left: GJ 791.2 A (dots, POS orbit predicted positions) and component B (open circles, TRANS orbit predicted positions). All observations, POS and TRANS and A and B component radial velocities (from H α emission lines only), were used to derive the orbital elements listed in Table 9. POS and TRANS astrometric residuals (average absolute value for which listed in Table 6) are smaller than the symbols. Right: RV measurements of the H α emission lines from the present study using CE on the McDonald 2.1m, phased to the orbital period determined from a combined solution including astrometry and RVs. The error bars are 1σ . The dashed lines are velocities predicted from the orbital parameters derived in the combined solution. Middle and top panels: Component B and A RV residuals (average absolute value for which listed in Table 4) from the combined solution. Note that the less massive component B has stronger H α emission, resulting in slightly smaller RV residuals for B (0.66 km s -1 ) than for component A (0.70 km s -1 ).  \n<!-- image -->  \nFig. 13.- GJ 1005 A (dots, POS orbit predicted positions) and component B (open circles) TRANS orbit predicted positions. All observations were used to derive the orbital elements listed in Table 9. Component B TRANS and component A POS astrometric residuals (average absolute value for which listed in Table 6) are smaller than their symbols.  \n<!-- image -->  \nFig. 14.- GJ1245A (dots, POS orbit predicted positions); WFPC2 (*) and component C (open circles) TRANS orbit predicted positions. All observations were used to derive the orbital elements listed in Table 9. Component C TRANS and component A POS astrometric residuals (average absolute value for which listed in Table 6) are smaller than their symbols.  \n<!-- image -->  \nFig. 15.- G 250-029 A (dots, POS orbit predicted positions) and component B (open circles) TRANS orbit predicted positions. All observations were used to derive the orbital elements listed in Table 9. Component B TRANS and component A POS astrometric residuals (average absolute value for which listed in Table 6) are smaller than their symbols.  \n<!-- image -->  \nFig. 16.- GJ 1081 A (dots, POS orbit predicted positions) and component B (open circles) TRANS orbit predicted positions. All observations were used to derive the orbital elements listed in Table 9. Component B TRANS and component A POS astrometric residuals (average absolute value for which listed in Table 6) are smaller than their symbols.  \n<!-- image -->  \nFig. 17.- GJ 54 A (dots, POS orbit predicted positions) and component B (+) TRANS orbit predicted positions. The solution included one HST NICMOS observation ( × ). All observations were used to derive the orbital elements listed in Table 9. Component B (TRANS and NICMOS) and component A POS astrometric residuals (average absolute value for which listed in Table 6) are smaller than their symbols.  \n<!-- image -->  \nFig. 18.- G193-027A (numbers, POS orbit predicted positions), component B TRANS (+), AO ( × ), and NICMOS (*) orbit predicted positions. POS, which normally provides astrometry for component A, locked onto component B for sets 2, 3, and 4. All observations were used to derive the orbital elements listed in Table 9. Component B (TRANS, AO, and NICMOS) and component A and B POS astrometric residuals (average absolute value for which listed in Table 6) are smaller than their symbols.  \n<!-- image -->  \narc seconds  \nFig. 19.- GJ65 component B orbit predicted (open circles) and measured positions (identified in the legend) with sizes proportional to average measurement error. All A-B position angle and separation observations (sources identified in Table 8) were used to derive the orbital elements listed in Table 9. No POS mode observations entered into the modeling due to a paucity of reference stars.  \n<!-- image -->  \narc seconds  \nFig. 20.- GJ473AB relative orbit. Calculated positions are plotted on the orbit as open circles. Measurements from TRANS (+), speckle ( × ), and HST FOS (*) illustrate the residuals. All observations were used to derive the orbital elements listed in Table 9. No POS mode observations entered into the modeling.  \n<!-- image -->  \narc seconds  \nFig. 21.V -band MLR . The dashed line is a double exponential function fit (Equation 10) to masses and M V . Primaries are in blue; secondaries in red. M V and M residuals show differences between observed values and a fit to Equation 10 using GaussFit. We list fit coefficients and M V , M residual RMS in Table 12. Errors for M V and mass are plotted on their respective residual points. Points are GJ numbers without GJ prefix, except for GU Boo (GUB), G250-029 (G250), and G193-027 (G193). Boldface denotes new and/or improved masses and absolute magnitudes from this work.  \n<!-- image -->  \nΟ  \nFig. 22.K -band MLR . The dashed line is a double exponential function fit (Equation 10) to masses and M K . Primaries are in blue; secondaries in red. M K and M residuals show differences between observed values and a fit to Equation 10 using GaussFit. We list fit coefficients and M K , M residual RMS in Table 12. Errors for M V and mass are plotted on their respective residual points. Note the marked decrease in residual size. Same point labeling as Figure 21. GJ 54, lacking K -band photometry, is not included.  \n<!-- image -->  \nΟ  \nFig. 23.V and K -band Luminosity-Mass Relations. The dashed lines are 5th order polynomial function fits (Equation 11) to masses and absolute magnitudes. Primaries are in blue; secondaries in red. Residuals are tagged with mass errors from Tables 10 and 11. We list fit coefficients in Table 13. The relations lose predictive utility for M V > 19 and M K > 10.  \n<!-- image -->  \n<!-- image -->  \nFig. 24.V and K -band Mass-Luminosity Relations from Baraffe et al. (2015) and Dotter (2016) along with actual measured absolute magnitudes and masses. Primaries are in blue; secondaries in red. Models agree in some mass ranges better than in others with K -band models having overall best agreement.  \n<!-- image -->  \nFig. 25.- M K , (V-K) 0 Hertzsprung-Russell diagram for all systems with K magnitudes (Tables 10, 11). We derive absolute magnitudes from HST parallaxes (Table 7). Also plotted are stellar models for 0.1 and 10 Gy from Baraffe et al. (2015) and the single M dwarfs Proxima Cen and Barnard's Star. The box contains a clump consisting of GJ 234 A, G250-029B, GJ748B, GJ469B, GJ1081B, GJ791.2A, and GJ1005A. An age spread can explain some of the scatter. As expected most components lie close to the same age track.  \n<!-- image -->"}