THE SOLAR NEIGHBORHOOD. XXXVII. THE MASS–LUMINOSITY RELATION FOR MAIN-SEQUENCE M DWARFS^*

An HST 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 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- or POS-mode measurements and are listed in Tables 1 and 2.

For 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), possibly 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.

During this project we observed with two FGS units: FGS 3 from 1992 to 2000, and FGS 1r from 2000 to 2009. FGS 1r 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 reanalysis. These corrections, ranging from −3.2 to +2.3 mas, are for the FGS 3 X axis a complicated function of separation because FGS 3 delivered far from perfect fringes with substantial instrumental structure (Franz et al. 1991). FGS 1r 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 ${\rm{\Delta }}V\geqslant 2.3$ (ΔV values listed in Table 10). The systems with ΔV ≤ 2.3 and typically smaller separations are GJ 469 AB, GJ 748 AB, GJ 831 AB, GJ 1081 AB, and G 250-029 AB. Generally, only a few measurements require photocenter corrections.

2.2.1. Traditional Ground-based Methods

2.2.2. Adaptive Optics

2.2.3. Aperture Masking Interferometry

2.2.4. Speckle Interferometry

2.2.5. Other HST Astrometry

We obtained most RV data with the McDonald 2.1 m Struve telescope and the 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/Δλ = 60,000) with a wavelength range of 5500 ≤ λ ≤ 6700 Å spread across 26 orders (apertures). The McDonald data were collected during 33 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, GJ 791.2 AB, GJ 831 AB) had detectable double peaks in their CCF; these have ΔV = 1.59–3.27. For these we used an IRAF script to average the lower-noise CCF. We then used a Gaussian multipeak fitting routine in the GUI-based commercial package IGOR¹¹ to derive component ΔRVs. We obtained the RVs for the three single-peak CCF systems (GJ 22 AC, GJ 234 AB, and GJ 623 AB) 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.

Some GJ 623 AB velocities came from the Hobby–Eberly Telescope (HET) 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 RVs 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 2.1 m CE system for the entire data sets for the six other systems with RVs.

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 affected 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 years than we have for component-A POS-mode measurements (10) over 4.07 years.

Zoom In Zoom Out Reset image size

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 2.1 m telescope and the Sandiford Cassegrain Echelle spectrograph (McCarthy et al. 1993). Treating GJ 831 AB 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, for example Benedict et al. (2007, 2009, 2011) and 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, and 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.

Because 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 et al. 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

$$\begin{eqnarray}&&{\pi }_{\mathrm{abs}}={10}^{\tfrac{-(V-{M}_{V}+5-{A}_{V})}{5}}.\end{eqnarray} \tag{ 1 }$$

Given the galactic latitude of GJ 831 AB (ℓ^II = −40°) and external estimates of low reddening (Schlafly & Finkbeiner 2011), a J−K versus 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).

Zoom In Zoom Out Reset image size

To 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 the 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. GJ 831 AB  and associated reference stars are plotted with ID numbers from Table 5. Comparing with our past RPM diagrams (Benedict et al. 2011, 2014; McArthur et al. 2011), all but ref-5 and ref-10 lie on or near the main sequence.

Zoom In Zoom Out Reset image size

Table 5.  GJ 831 AB Reference-star Characteristics and Final Parallaxes

ID	V	V − K	J − K	Sp.T	MV	πest (mas)	πabs (mas)
ref-2	13.41	1.38 ± 0.10	0.34 ± 0.03	G0V	4.4	1.6 ± 0.5	1.7 ± 0.2
ref-3	13.65	1.73 0.11	0.47 0.04	G7V	5.4	2.2 0.7	2.2 0.2
ref-4	13.02	1.62 0.10	0.42 0.04	G7V	5.4	3.0 1.0	3.0 0.3
ref-5	14.35	3.01 0.10	0.87 0.05	M1V	9.2	9.0 3.0	8.4 0.9
ref-10	14.67	1.58 0.11	0.45 0.04	G5IV	2.5	0.4 0.2	1.4 0.2

Download table as:  ASCIITypeset image

In 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 typical 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 ref-10, we run the models with two sets of inputs and adopt the classification that produces the smaller χ². 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 does provide significantly improved parallaxes in most cases.

The 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)

$$\begin{eqnarray}&&\displaystyle \frac{{\alpha }_{{\rm{A}}}\,\sin \,i}{{\pi }_{\mathrm{abs}}}=\displaystyle \frac{{{PK}}_{{\rm{A}}}\sqrt{(1-{e}^{2})}}{2\pi \times 4.7405}.\end{eqnarray} \tag{ 2 }$$

Quantities 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 RVs and astrometry (the period P in years and eccentricity, e) or RVs 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 RVs 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°.

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 GJ 831 AB 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 six-parameter model (substituting D for −B and E for A in Equation (6), below).

Our reference-frame model becomes, in terms of standard coordinates

$$\begin{eqnarray}&&x^{\prime} =x+{{lc}}_{x}(B-V)\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&y^{\prime} =y+{{lc}}_{y}(B-V)\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&\xi ={Ax}^{\prime} +{By}^{\prime} +C-{\mu }_{\alpha }{\rm{\Delta }}t-{P}_{\alpha }{\pi }_{\alpha }-{\mathrm{ORBIT}}_{\alpha }\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&\eta =-{Bx}^{\prime} +{Ay}^{\prime} +F-{\mu }_{\delta }{\rm{\Delta }}t-{P}_{\delta }{\pi }_{\delta }-{\mathrm{ORBIT}}_{\delta }\end{eqnarray} \tag{ 6 }$$

where 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 the SIMBAD database or estimated from the spectral types suggested by Figure 2. Here, A and B are scale and rotation plate constants, C and F are offsets, μ_α and μ_δ are proper motions, Δt 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 R.A. and decl. The parameters 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 R.A. and decl. because the master constraint plate was rolled into the R.A. decl. 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 ±01. 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.

For 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 modified Julian days (T₀), 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 RVs and astrometry, the constraint described in Section 4.3.2 ties POS, TRANS, and RVs together, yielding a single self-consistent description of the binary system.

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.

Zoom In Zoom Out Reset image size

To 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 and 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.

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 1σ), 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 (1σ) uncertainties.

Table 6.  Astrometry Data Spans and Residuals^a

ID	#ref	$\langle V\rangle$ ref	POS	POSb	POS	TRANS	TRANS	TRANS
			span (year)	#obs	$\langle | \mathrm{res}| \rangle$ (mas)	span (year)	#obs	$\langle | \mathrm{res}| \rangle$ (mas)
GJ 1005 AB	2	14.5	4.36	12	1.2	4.36	17	1.4
GJ 22 AC	4	14.4	6.06	6	0.8	9.22	9	3.3
GJ 22 AC	⋯	⋯	⋯	⋯	⋯	15.04	23c	10.1
GJ 54 AB	3	13.22	4.56	6	0.9	9.15	9c	2.2
GJ 65 AB	⋯	⋯	⋯	⋯	⋯	1.47	13	3.5
GJ 65 AB	⋯	⋯	⋯	⋯	⋯	63.0	51d	81.4
GJ 1081 AB	6	14.05	4.00	5	0.7	10.47	11	2.1
GJ 234 AB	5	14.21	5.5	7	0.8	13.90	11	2.4
G 250-029 AB	3	15.27	7.65	7	2.6	12.08	14	1.9
G 193-027 AB	5	11.76	3.02	6	0.9	8.95	10c	1.3
GJ 469 AB	3	12.73	6.53	7	0.9	10.33	12	1.8
GJ 473 AB	⋯	⋯	⋯	⋯	⋯	10.46	11	2.5
GJ 473 AB	⋯	⋯	⋯	⋯	⋯	23.00	25c	11.3
GJ 623 AB	5	14.04	3.14	13	1.0	12.18	9c	4.5
GJ 748 AB	4	12.93	1.83	15	1.3	14.78	18	1.4
GJ 1245 AC	7	11.81	5.00	6	0.9	9.45	12c	3.7
GJ 791.2 AB	3	13.70	5.04	16	0.8	12.84	5	2.8
GJ 831 AB	5	13.85	4.07	10	1.4	6.61	17	1.8

Notes.

^a#ref—number of astrometric reference stars; $\langle V\rangle$ref—average V magnitude of the reference stars; POS span—length of POS campaign; #POS—number of epochs of POS observation; POS $\langle | \mathrm{res}| \rangle$—average absolute value of POS astrometric residuals; TRANS span—length of TRANS campaign; #TRANS—number of epochs of TRANS observation; TRANS $\langle | \mathrm{res}| \rangle$—average absolute value of TRANS astrometric residuals, unless noted with other observations. ^bFor some orbit plots, the number of POS plotted appears less than listed here because of overplotting and plot resolution. ^cIncludes measurements in addition to HST TRANS. ^dGround-based astrometry only.

Download table as:  ASCIITypeset image

Table 9.  Orbital Elements^a

ID	a ('')	αA ('')	inc (°)	Ω (°)	P (days)	T0	ecc	ωA (°)	ωB (°)	f
GJ 1005 AB	0.3037	0.1169	146.1	62.8	1666.1	49850.4	0.364	166.6	−13.4	0.385
	0.0005	0.0004	0.2	0.4	2.5	0.8	0.001	0.5	0.5	0.002
GJ 22 AC	0.5106	0.1425	43.7	178.3	5694.2	51817.2	0.163	284.5	104.5	0.279
	0.0007	0.0011	0.2	0.2	14.9	5	0.002	0.5	0.5	0.002
GJ 54 AB	0.1264	0.0517	126.6	92.2	418.5	53944.6	0.174	46.8	226.8	0.410
	0.0007	0.0003	0.4	0.5	0.2	1.6	0.004	1.4	1.4	0.003
GJ 65 AB	2.0458	⋯	128.3	147.6	9612	41358.4	0.617	⋯	104.6	0.494
	0.0066	⋯	0.3	0.5	12	12	0.004	⋯	0.5	0.003
GJ 1081 AB	0.2712	0.104	97.1	231.5	4066.1	48919.1	0.848	51	231	0.386
	0.0027	0.0013	0.2	0.2	27.5	28.9	0.004	0.7	0.7	0.005
GJ 234 AB	1.0932	0.3537	53.2	210.6	6070.1	51277.1	0.382	220.4	40.4	0.324
	0.0007	0.0015	0.1	0.1	10.2	1.3	0.001	0.1	0.1	0.001
G 250-029 AB	0.4417	0.1539	109.7	107.8	4946.3	54696.1	0.482	238.9	58.9	0.348
	0.0009	0.0023	0.1	0.1	2.2	1.7	0.002	0.2	0.2	0.005
G 193-027 AB	0.1564	0.0773	130.8	173.6	1195.5	51340.5	0.236	72.9	252.9	0.495
	0.0008	0.0008	0.5	0.6	1.4	3.3	0.004	0.9	0.9	0.006
GJ 469 AB	0.3139	0.1131	107.6	190.4	4223	54548.2	0.302	−90.8	89.2	0.360
	0.0008	0.0009	0.1	0.2	2.9	3.2	0.003	0.2	0.2	0.003
GJ 473 AB	0.9167	⋯	103.3	143.5	5772.3	54554	0.301	⋯	350.7	0.477
	0.0017	⋯	0.7	0.1	9.4	8.7	0.002	⋯	0.5	0.008
GJ 623 AB	0.2397	0.0556	152.5	98.3	1367.4	45838.7	0.629	65.4	245.4	0.231
	0.0014	0.0003	0.2	0.5	0.6	2.8	0.004	0.5	0.5	0.002
GJ 748 AB	0.148	0.0503	131.6	−0.4	901.7	50033.2	0.45	27.3	207.3	0.340
	0.0004	0.0003	0.3	0.2	0.3	0.8	0.002	0.4	0.4	0.002
GJ 1245 AC	0.8267	0.336	135.7	261.2	6147	51506.8	0.334	216.1	36.1	0.406
	0.0008	0.0027	0.1	0.2	17	2.1	0.002	0.2	0.2	0.003
GJ 791.2 AB	0.1037	0.0336	145.4	102.2	538.6	51022.7	0.558	12.2	192.2	0.324
	0.0005	0.0003	0.6	1.1	0.1	0.9	0.005	1.5	1.5	0.003
GJ 831 AB	0.1448	0.0505	49.8	326.1	704.9	51164	0.39	190.6	10.6	0.349
	0.0005	0.0004	0.4	0.4	0.5	1.1	0.002	0.9	0.9	0.003

Note.

^aa—semimajor axis of relative orbit; α—semimajor axis of perturbation of component A; ${\omega }_{{\rm{A}}}$—longitude of periastron passage of component A; f—mass fraction—${{ \mathcal M }}_{\sec }$/(${{ \mathcal M }}_{\mathrm{pri}}$+${{ \mathcal M }}_{\sec }$).

Download table as:  ASCIITypeset image

Figure 5, left, illustrates component A and B astrometric orbits. The POS-mode component-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.

Zoom In Zoom Out Reset image size

Our orbit solution and derived absolute parallax (Equation (2)) provide an orbital semimajor axis, a in astronomical units, from which we can determine the system mass through  Kepler's third law. Given P and a, we solve the expression

$$\begin{eqnarray}&&{a}^{3}/{P}^{2}=({{ \mathcal M }}_{{\rm{A}}}+{{ \mathcal M }}_{{\rm{B}}})={{ \mathcal M }}_{\mathrm{tot}}\end{eqnarray} \tag{ 7 }$$

to find (in solar units) ${{ \mathcal M }}_{\mathrm{tot}}=0.414\pm 0.006\,{{ \mathcal M }}_{\odot }$. At each instant in the orbits of the two components around the common center of mass,

$$\begin{eqnarray}&&{{ \mathcal M }}_{{\rm{A}}}/{{ \mathcal M }}_{{\rm{B}}}={\alpha }_{{\rm{B}}}/{\alpha }_{{\rm{A}}},\end{eqnarray} \tag{ 8 }$$

, a relationship that contains only one observable, α_A, the perturbation orbit size. Instead, we calculate the mass fraction

$$\begin{eqnarray}&&f={{ \mathcal M }}_{{\rm{B}}}/({{ \mathcal M }}_{{\rm{A}}}+{{ \mathcal M }}_{{\rm{B}}})={\alpha }_{{\rm{A}}}/({\alpha }_{{\rm{A}}}+{\alpha }_{{\rm{B}}})={\alpha }_{{\rm{A}}}/a,\end{eqnarray} \tag{ 9 }$$

where α_B = $a-{\alpha }_{{\rm{A}}}.$ 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)–(9) yield ${{ \mathcal M }}_{{\rm{A}}}=0.270\pm 0.004\,{{ \mathcal M }}_{\odot }$ and ${{ \mathcal M }}_{{\rm{B}}}=0.145\pm 0.002\,{{ \mathcal M }}_{\odot }$, indicating that both components are low-mass red dwarfs with mass errors of ∼1.5%, a considerable improvement over the Ségransan 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 ΔV and ΔK, and the M_V and M_K values used to create the MLRs.

Table 10.  Masses, M_V, and M_K for Red Dwarfs Observed with HST/FGS^a

ID	${{ \mathcal M }}_{\mathrm{pri}}$	${{ \mathcal M }}_{\sec }$	Vb	ΔV	refc	MVpri	MVsec	Kd	ΔK	refc	MKpri	MKsec
GJ 1005 AB	0.179 ± 0.002	0.112 ± 0.001	11.48 ± 0.01	2.39 ± 0.05	*	12.70 ± 0.01	15.12 ± 0.09	6.39 ± 0.02	1.23 ± 0.02	*	7.80 ± 0.02	9.03 ± 0.03
GJ 22 AC	0.405 0.008	0.157 0.003	10.28 0.03	3.08 0.10	H99	10.32 0.03	13.40 0.10	6.04 0.02	1.93 0.03	*	6.19 0.02	8.12 0.04
GJ 54 AB	0.432 0.008	0.301 0.006	9.82 0.02	1.04 0.05	*	10.70 0.03	11.70 0.04	5.13 0.02	...	...	...	...
GJ 65 AB	0.120 0.003	0.117 0.003	12.08 0.02	0.45 0.08	H99	15.49 0.04	15.94 0.05	5.33 0.02	0.16 0.02	*	8.87 0.03	9.03 0.03
GJ 1081 AB	0.325 0.010	0.205 0.007	12.21 0.03	1.67 0.10	H99	11.49 0.04	13.16 0.09	7.34 0.04	0.96 0.01	*	6.79 0.04	7.75 0.04
GJ 234 AB	0.223 0.002	0.109 0.001	11.14 0.03	3.08 0.05	H99	13.11 0.03	16.19 0.06	5.49 0.02	1.58 0.02	*	7.63 0.02	9.21 0.03
G 250-029 AB	0.350 0.005	0.187 0.004	10.95 0.03	1.61 0.08	H99	11.07 0.03	12.68 0.07	6.35 0.02	1.03 0.07	*	6.61 0.03	7.64 0.05
G 193-027 AB	0.126 0.005	0.124 0.005	13.29 0.03	0.30 0.05	*	14.16 0.05	14.46 0.05	7.78 0.02	0.10 0.02	*	8.74 0.03	8.84 0.04
GJ 469 AB	0.332 0.007	0.188 0.004	12.05 0.02	1.59 0.05	H99	11.69 0.03	13.28 0.05	6.96 0.04	1.01 0.02	*	6.74 0.04	7.75 0.04
GJ 473 AB	0.124 0.005	0.113 0.005	12.47 0.04	−0.01 0.05	H99	15.09 0.05	15.08 0.05	6.04 0.02	0.00 0.01	*	8.65 0.03	8.65 0.03
GJ 623 AB	0.379 0.007	0.114 0.002	10.28 0.03	5.28 0.10	B96	10.77 0.03	16.05 0.10	5.92 0.02	2.87 0.14	H93	6.48 0.02	9.35 0.13
GJ 748 AB	0.369 0.005	0.190 0.003	11.10 0.04	1.81 0.04	H99	11.25 0.04	13.06 0.05	6.29 0.02	1.09 0.07	*	6.59 0.03	7.68 0.06
GJ 1245 AC	0.111 0.001	0.076 0.001	13.41 0.03	3.29 0.05	H99	15.17 0.03	18.46 0.06	6.85 0.02	1.05 0.01	*	8.91 0.02	9.96 0.02
GJ 791.2 AB	0.237 0.004	0.114 0.002	13.13 0.04	3.27 0.10	B00	13.46 0.04	16.73 0.10	7.31 0.02	1.28 0.10	*	7.87 0.03	9.15 0.08
GJ 831 AB	0.270 0.004	0.145 0.002	12.02 0.02	2.10 0.06	H99	12.66 0.02	14.76 0.06	6.38 0.02	1.20 0.01	*	7.18 0.02	8.38 0.02

Notes.

^aMasses (${ \mathcal M }$) in solar masses (${{ \mathcal M }}_{\odot }$). Components are Primary—pri, Secondary—sec. ^bV values from references in Table 2. ^cReferences for ΔV and ΔK: *—this paper, B96—Barbieri et al. (1996), B00—Benedict et al. (2000), H93—Henry & McCarthy (1993), H99—Henry et al. (1999). ^dFrom 2MASS.

Download table as:  ASCIITypeset image

The GJ 22 AC 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 FGS 3 and FGS 1r, 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 RVs (only for GJ 22 A due to the large ${\rm{\Delta }}V=3.08$ value) in combination with the astrometry, with a parallax from Hipparcos (van Leeuwen et al. 2007) used as a prior.

Our derived parallax has a larger error (0.6 mas) than typical for an HST parallax (0.2–0.3 mas), due to the less-than-ideal sampling. Still, even though all epochs sampled nearly the 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 $\langle | \mathrm{res}| \rangle =10.1$ mas. For TRANS only, the average absolute-value residuals are $\langle | \mathrm{res}| \rangle =3.3$ mas, and for speckle only, $\langle | \mathrm{res}| \rangle =13.4$ mas.

Table 9 contains the GJ 22 AC orbital parameters with formal (1σ) 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 22 B, separated from GJ 22 AC by 4'', with an orbital period on the order of 320 years (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 ${{ \mathcal M }}_{{\rm{A}}}=0.405\pm 0.008\,{{ \mathcal M }}_{\odot }$ (2.0% error) and ${{ \mathcal M }}_{{\rm{C}}}\ =0.157\pm 0.003\,{{ \mathcal M }}_{\odot }$ (1.9%). These masses support the excellent early work on this system by Hershey (1973), who found ${{ \mathcal M }}_{{\rm{A}}}=0.40\,{{ \mathcal M }}_{\odot }$ and ${{ \mathcal M }}_{{\rm{C}}}=0.13\,{{ \mathcal M }}_{\odot }$, followed by McCarthy et al. (1991), who found ${{ \mathcal M }}_{{\rm{A}}}=0.362\pm 0.053\,{{ \mathcal M }}_{\odot }$ (15%) and ${{ \mathcal M }}_{{\rm{C}}}=0.123\pm 0.018\,{{ \mathcal M }}_{\odot }$ (15%).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Our modeling of GJ 234 AB included astrometry measurements from both FGS 3 and FGS 1r. Treating GJ 234 AB as a single-lined spectroscopic binary (due to the large ΔV = 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 seven POS observations of component A and the reference frame.

Zoom In Zoom Out Reset image size

Our 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 (1σ) 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 the observed positions indicated. The POS-mode component-A residuals and the TRANS residuals are all smaller than the dots and circles. We obtain masses of ${{ \mathcal M }}_{{\rm{A}}}=0.223\pm 0.002\,{{ \mathcal M }}_{\odot }$ (0.9% error) and ${{ \mathcal M }}_{{\rm{B}}}=0.109\pm 0.001\,{{ \mathcal M }}_{\odot }$ (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), ${{ \mathcal M }}_{{\rm{A}}}=0.13\pm 0.04\,{{ \mathcal M }}_{\odot }$ (31%) and ${{ \mathcal M }}_{{\rm{B}}}=0.07\pm 0.02\,{{ \mathcal M }}_{\odot }$ (29%), who summarized much of the early work on this important binary. Our new masses are larger than but consistent with those of Coppenbarger et al. (1994), who found ${{ \mathcal M }}_{{\rm{A}}}=0.179\pm 0.047\,{{ \mathcal M }}_{\odot }$ (26%) and ${{ \mathcal M }}_{{\rm{B}}}\ =0.083\pm 0.023\,{{ \mathcal M }}_{\odot }$ (28%).

GJ 469 AB astrometric measurements came from FGS 3 and FGS 1r, and RVs 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.

The FGS measurements have improved our knowledge of GJ 469 AB's absolute parallax by a factor of 8–13 compared to YPC and Hipparcos. Table 9 contains the system's orbital parameters and formal (1σ) 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 ${{ \mathcal M }}_{{\rm{A}}}=0.332\pm 0.007{{ \mathcal M }}_{\odot }$ (2.4% error) and ${{ \mathcal M }}_{{\rm{B}}}=0.188\pm 0.004\,{{ \mathcal M }}_{\odot }$ (2.7%) listed in Table 10. These are the first mass determinations for this system.

Zoom In Zoom Out Reset image size

Our modeling of the challenging GJ 623 AB system included three sources of astrometry and two sources of RVs. With ΔV = 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. The 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 GJ 623 AB (and five 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 ratio of χ² to degrees of freedom by 47%, to near unity.

Zoom In Zoom Out Reset image size

We 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 10 compared to the YPC. Table 9 contains the GJ 623 AB orbital parameters with formal (1σ) uncertainties, with the semimajor axis now known to 0.6%. We obtain component masses of ${{ \mathcal M }}_{{\rm{A}}}=0.379\pm 0.007\,{{ \mathcal M }}_{\odot }$ (1.8% error) and ${{ \mathcal M }}_{{\rm{B}}}=0.114\pm 0.002\,{{ \mathcal M }}_{\odot }$ (1.8%). We have improved upon the previous results of Martinache et al. (2007), who found ${{ \mathcal M }}_{{\rm{A}}}=0.371\pm 0.015\,{{ \mathcal M }}_{\odot }$ (4.0%) and ${{ \mathcal M }}_{{\rm{B}}}=0.115\ \pm 0.002\,{{ \mathcal M }}_{\odot }$ (1.7%), and the first masses measured by McCarthy & Henry (1987), ${{ \mathcal M }}_{{\rm{A}}}=0.51\pm 0.16{{ \mathcal M }}_{\odot }$ (31%) and ${{ \mathcal M }}_{{\rm{B}}}=0.11\pm 0.029{{ \mathcal M }}_{\odot }$ (26%), which provided a breakthrough in infrared speckle imaging sensitivity at the time.

The binary GJ 748 AB 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 day. 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.

We 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 (1σ) uncertainties, and we find that the parameters are consistent with the orbit in Benedict et al. (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 ${{ \mathcal M }}_{{\rm{A}}}=0.369\pm 0.005\,{{ \mathcal M }}_{\odot }$ (1.3% error) and ${{ \mathcal M }}_{{\rm{B}}}=0.190\pm 0.003\,{{ \mathcal M }}_{\odot }$ (1.6%). The previous result (Benedict et al. 2001) had slightly higher masses and similar errors: ${{ \mathcal M }}_{{\rm{A}}}=0.379\pm 0.005\,{{ \mathcal M }}_{\odot }$ and ${{ \mathcal M }}_{{\rm{B}}}=0.192\pm 0.003\,{{ \mathcal M }}_{\odot }$.

Zoom In Zoom Out Reset image size

We also reanalyze 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 Hα RVs for both components. With system total magnitude V = 13.06 and ΔV = 3.27, detecting both components of GJ 791.2 AB in both the RV data and with HST/FGS is difficult. In addition, GJ 791.2 AB is a rapid rotator (Delfosse et al. 1998), further hindering RV efforts. However, both components exhibit strong Hα 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 FGS 3; the last is from FGS 1r. Modeling is exactly as for GJ 831 AB.

This reanalysis has improved our knowledge of the parallax of GJ 791.2 AB by 30% compared to our previous effort, and by a factor of 10 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 (1σ) 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 Hα RV measurements, RV residuals, and velocity curves predicted from the simultaneous solution. We obtain component masses of ${{ \mathcal M }}_{{\rm{A}}}=0.237\pm 0.004\,{{ \mathcal M }}_{\odot }$ (1.7% error) and ${{ \mathcal M }}_{{\rm{B}}}\ =0.114\pm 0.002\,{{ \mathcal M }}_{\odot }$ (1.8%). The previous analysis (Benedict et al. 2000) yielded significantly higher masses, ${{ \mathcal M }}_{{\rm{A}}}=0.286\ \pm 0.006\,{{ \mathcal M }}_{\odot }$ and ${{ \mathcal M }}_{{\rm{B}}}=0.126\pm 0.003\,{{ \mathcal M }}_{\odot }$, 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).

Zoom In Zoom Out Reset image size

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 ${{ \mathcal M }}_{{\rm{A}}}=0.179\pm 0.003\,{{ \mathcal M }}_{\odot }$ and ${{ \mathcal M }}_{{\rm{B}}}=0.112\pm 0.002\,{{ \mathcal M }}_{\odot }$, a precision difficult to improve upon.

We 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 consists 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 GJ 1005 AB given in Table 6 are ∼1 mas, indicating adequate astrometry even with only three coefficients.

Our 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 (1σ) uncertainties, with both a and P now known to 0.2%. Figure 13 provides component A and B orbits with the 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 ${{ \mathcal M }}_{{\rm{A}}}=0.179\pm 0.002\,{{ \mathcal M }}_{\odot }$ (1.1% error) and ${{ \mathcal M }}_{{\rm{B}}}=0.112\pm 0.001\,{{ \mathcal M }}_{\odot }$ (0.9%), values that are virtually identical to and with errors slightly smaller than reported in Hershey & Taff (1998).

Zoom In Zoom Out Reset image size

The GJ 1245 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 FGS 3 and FGS 1r 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 error (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 as the YPC parallax used as a prior.

Zoom In Zoom Out Reset image size

The 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 ${{ \mathcal M }}_{{\rm{A}}}=0.111\pm 0.001\,{{ \mathcal M }}_{\odot }$ (0.9% error) and ${{ \mathcal M }}_{{\rm{C}}}=0.076\pm 0.001\,{{ \mathcal M }}_{\odot }$ (1.3%). These masses are consistent with the first detailed study of the system by McCarthy et al. (1988), who found ${{ \mathcal M }}_{{\rm{A}}}=0.14\pm 0.03\,{{ \mathcal M }}_{\odot }$ (21%) and ${{ \mathcal M }}_{{\rm{C}}}=0.10\pm 0.02\,{{ \mathcal M }}_{\odot }$ (20%).

These 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 $79.8\ \pm 1.0\,{{ \mathcal M }}_{\mathrm{Jup}}$, placing it at the generally accepted main-sequence hydrogen-burning demarcation boundary of $\sim 80\,{{ \mathcal M }}_{\mathrm{Jup}}$ (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) is consistent with our GJ 1245 AC orbit and may have detected AC–B motion.

All astrometric measurements came from HST/FGS 3 and FGS 1r. 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 lower-precision parallax prior from Hipparcos as a prior. Fourteen TRANS observations of the component A–B separations sample nearly the entire orbit (Figure 15).

Zoom In Zoom Out Reset image size

Our parallax agrees with the Hipparcos value but has reduced the error by a factor of eight, and it represents a factor of 16 improvement over the value in YPC. Figure 15 provides component A and B orbits with the 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 ${{ \mathcal M }}_{{\rm{A}}}=0.350\pm 0.005\,{{ \mathcal M }}_{\odot }$ (1.7% error) and ${{ \mathcal M }}_{{\rm{B}}}=0.187\pm 0.004\,{{ \mathcal M }}_{\odot }$ (2.2%). These are the first mass determinations for this system.

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).

Zoom In Zoom Out Reset image size

Our 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 the observed positions indicated, where the POS and TRANS residuals are smaller than the dots and circles in the figure. Because of the high orbital inclination and paucity of POS observations, we obtain relatively poor mass precision for this system. We find component masses of ${{ \mathcal M }}_{{\rm{A}}}=0.325\pm 0.010\,{{ \mathcal M }}_{\odot }$ (3.2% error) and ${{ \mathcal M }}_{{\rm{B}}}=0.205\pm 0.007\,{{ \mathcal M }}_{\odot }$ (3.4%). These are the first mass determinations for this system.

GJ 54 AB has the shortest orbital period in our sample, with a period of 1.15 year. Our modeling of GJ 54 AB included HST POS (with three reference stars) and TRANS from FGS 1r 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 the two TRANS observations on the orbit, although it occurs nearly two years before the first FGS 1r observation.

Zoom In Zoom Out Reset image size

Our 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 year 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 the observed positions indicated. The model yields component masses of ${{ \mathcal M }}_{{\rm{A}}}=0.432\pm 0.008\,{{ \mathcal M }}_{\odot }$ (2.0% error) and ${{ \mathcal M }}_{{\rm{B}}}=0.301\pm 0.006\,{{ \mathcal M }}_{\odot }$ (2.0%). These are the first mass determinations for this system.

Our modeling of G 193-027 AB included HST POS and TRANS from FGS 1r, 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 χ² markedly decreased when we removed that low-precision and perhaps inaccurate prior.

Because of the typical large component separation (100–170 mas) and small ΔV = 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.

Zoom In Zoom Out Reset image size

Table 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 the observed positions indicated. Because we had relatively few component-A POS observations, we obtain somewhat poorer mass precision for this system, finding ${{ \mathcal M }}_{{\rm{A}}}=0.126\pm 0.005\,{{ \mathcal M }}_{\odot }$ (4.0% error) and ${{ \mathcal M }}_{{\rm{B}}}=0.124\pm 0.005\,{{ \mathcal M }}_{\odot }$ (4.0%), with the errors driven almost entirely by the parallax error. The two components are of nearly equal mass, consistent with their low ${\rm{\Delta }}V$. These are the first mass determinations for this system.

The 10 HST TRANS-only observations were made using FGS 3, covering only 18° of orbital position angle during 1.1 year. 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 61 in 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 years, the longest among the 15 systems we observed with HST/FGS. Also useful for extending the orbit sampling are five VLT/NACO (Kervella et al. 2016) measures and one HST/WFPC2 (Dieterich et al. 2012) measure of position angle and separation.

Zoom In Zoom Out Reset image size

Modeling only relative position angle and separation measurements yields a, P, T₀, e, i, Ω, and ω_B. Figure 19 provides the component-AB relative orbit with the 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 $\langle | \mathrm{res}| \rangle =3.5$ 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 et al. 1988) and absolute parallax (π_abs = 373.7 ± 2.7 mas) from the YPC, we obtain component masses of ${{ \mathcal M }}_{{\rm{A}}}=0.120\pm 0.003\,{{ \mathcal M }}_{\odot }$ (2.5% error) and ${{ \mathcal M }}_{{\rm{B}}}=0.117\pm 0.003\,{{ \mathcal M }}_{\odot }$ (2.5%), agreeing with the recent Kervella et al. (2016) values of ${{ \mathcal M }}_{{\rm{A}}}=0.123\pm 0.004\,{{ \mathcal M }}_{\odot }$ (3.5% error) and ${{ \mathcal M }}_{{\rm{B}}}=0.120\pm 0.004\,{{ \mathcal M }}_{\odot }$ (3.6%).

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 FGS 3 and FGS 1r, a single HST FOS measurement (Schultz et al. 1998), and optical (Blazit et al. 1987; Balega et al. 1994; 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 year) system.

Figure 20 provides the component-AB relative orbit with the 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 an absolute parallax of π_abs = 235.5 ± 2.9 mas from the RECONS astrometry program at the CTIO/SMARTS 0.9 m (Henry et al. 2006), we obtain component masses of ${{ \mathcal M }}_{{\rm{A}}}=0.124\pm 0.005{{ \mathcal M }}_{\odot }$ (4.0% error) and ${{ \mathcal M }}_{{\rm{B}}}=0.113\pm 0.005\,{{ \mathcal M }}_{\odot }$ (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), ${{ \mathcal M }}_{{\rm{A}}}=0.143\pm 0.011\,{{ \mathcal M }}_{\odot }$ (7.7% error) and ${{ \mathcal M }}_{{\rm{B}}}=0.131\pm 0.010\,{{ \mathcal M }}_{\odot }$ (7.6%), who used 56 astrometry observations taken between 1938 and 1998; however, only four high-quality FGS observations were available then, compared to the 11 FGS observations used here.

Zoom In Zoom Out Reset image size