# Inner and Outer Orbits in 13 Resolved Hierarchical Stellar Systems

A. Tokovinin  Cerro Tololo Inter-American Observatory — NSF’s NOIRLab, Casilla
603, La Serena, Chile<EMAIL_ADDRESS>
###### Abstract

Orbits of inner and outer subsystems in 13 triple or higher-order stellar
systems are computed or updated using position measurements and, in three
cases, radial velocities. The goal is to determine mutual orbital
inclinations, period ratios, and masses to complement the statistics of
hierarchical systems. Effect of the subsystems on the motion in the outer
orbits (wobble) is explicitly modeled to determine inner mass ratios. Stars
studied here (HD 5408, 8036, 9770, 15089, 29310, 286955, 29316, 140538,
144362, 154621, 156034, 185655, and 213235) are bright and nearby (from 15 pc
to 150 pc). Their inner periods range from 1.7 yr to 49 yr, and the outer
periods from 83 to 2400 yr. Some long-period outer orbits are poorly
constrained. Four astrometric inner orbits and one outer orbit are computed
for the first time.

###### Subject headings:

binaries:visual; binaries:general

##  1\. Introduction

Architecture of hierarchical stellar systems is interesting for several
reasons. Period ratios, mutual orbit orientation, and distribution of masses
are related to the processes of formation and early dynamical evolution of
these systems, and therefore inform us on the star formation in general. Most
stars are born in groups, hence properties of binary and higher-order systems
are very relevant. Life and death of stars is also affected by their
companions in various ways. Close pairs can form within hierarchical systems
by their dynamical evolution on a long (compared to the formation) time scale.
Late mergers or collisions can create unusual objects. Hamers (2020) explored
outcomes of dynamical evolution using data on real hierarchical systems
collected in the Multiple Star Catalog (MSC, Tokovinin, 2018a).

This study is aimed at calculation or improvement of visual or astrometric
orbits of the inner and outer subystems belonging to the same hierarchy. There
are only a few dozen systems where this is possible; the list of 54 such
objects with outer periods less than 1000 yr is given in Tokovinin (2017b). On
the one hand, most outer systems with their typically long periods lack
adequate coverage needed for orbit calculation. On the other hand, resolution
of inner subsystems is possible when they are relatively close to the Sun and
their periods are not too short. Combination of both constraints results in a
sample of hierarchical systems with comparable periods and separations and,
typically, with similar masses. Obviously, the observed sample is not
representative of hierarchical systems in the statistical sense. However,
hierarchies with comparable periods are most interesting from the dynamical
perspective because interactions between their subsystems might be important.

The observed motions in a hierarchical system are modeled here by two
Keplerian orbits. In reality dynamical interactions cause evolution of those
orbits over time. These effects are too small to be detected in our systems.
Given the orbital elements and the estimated masses, the long-term dynamical
evolution can be explored numerically (e.g. Xia & Fu, 2015; Hamers, 2020), but
this is outside the scope of the present study.

Table 1List of multiple systems  WDS | HD | HIP | $V$ |  $\varpi$aaParallax codes: G — Gaia (Gaia collaboration, 2018), D — dynamical. | OuterbbSystem types: V — visual orbit, A — astrometric orbit, S1 — single-lined spectroscopic orbit, * — first orbit.  | $P_{\rm out}$ | InnerbbSystem types: V — visual orbit, A — astrometric orbit, S1 — single-lined spectroscopic orbit, * — first orbit.  | $P_{\rm in}$ | Masses | $N_{\rm comp}$
---|---|---|---|---|---|---|---|---|---|---
(J2000) |  |  | (mag) | (mas) |  | (yr) |  | (yr) | (${\cal M}_{\odot}$ ) |
00568$+$6022 | 5408 | 4440 | 5.55 | 5.6 D | V | 84.1 | V,A,S1 | 4.9 | (3.4+4.1)+3.4 | 4
01198$-$0031 | 8036 | 6226 | 5.87 | 7.76 G | V | 1000: | V,A | 27.1 | 2.5+(2.1+1.8) | 3
01350$-$2955 | 9770 | 7372 | 7.08 | 45.83 G | V | 121 | V,A | 4.56 | (0.9+1.5)+0.6 | 4
02291$+$6724 | 15089 | 11569 | 4.52 | 22.22 G | V | 2400: | V,A | 48.7 | (2.1+0.9)+1.3 | 5
04375$+$1509 | 29310 | 21543 | 7.54 | 20.69 G | V,S1 | 125 | S1,A* | 2.0 | (1.1+0.3)+0.7 | 3
04397$+$0952 | 286955 | 21710 | 9.19 | 34.04 G | V,S1 | 285 | S1,A* | 1.7 | (0.75+0.15)+0.5 | 4
04400$+$5328 | 29316 | 21730 | 5.35 | 13.0 D | V | 660: | V,A* | 26.3 | (1.9+1.5)+1.5 | 3
15440$+$0231 | 140538 | 77052 | 5.87 | 67.71 G | V | 900: | V,A | 6.6 | 1.0+(0.3+0.3) | 3
16057$-$0617 | 144362 | 78849 | 6.36 | 12.03 G | V | 300: | V,A | 5.0 | (1.6+1.6)+1.1 | 4
17066$+$0039 | 154621 | 83716 | 8.28 | 17.90 G | V | 900: | V,A | 6.3 | 1.0+(0.7+0.7) | 3
17157$-$0949 | 156034 | 84430 | 6.98 | 7.9 D | V | 137 | V | 5.3 | 1.7+(1.4+1.3) | 3
19453$-$6823 | 185655 | 97196 | 10.06 | 21.0 G | V* | 84 | V,A | 4.4 | 0.8+(0.5+0.4) | 3
22300$+$0426 | 213235 | 111062 | 5.51 | 18.84 G | V | 124 | V,A* | 2.1 | 1.7+(1.2+1.1) | 3

Table 1 presents the 13 hierarchies studied here. Its first three columns
identify the systems by the Washington Double Star (WDS, Mason et al., 2001)
codes based on J2000 positions, HD, and HIP numbers. The combined visual
magnitudes come from Simbad, the parallaxes are either measured by Gaia (Gaia
collaboration, 2018) or estimated from the masses and orbits as explained
below. Types of the orbits (visual, astrometric, or spectroscopic) and periods
of the outer and inner subsystems are given. Astrometric or visual orbits
computed for the first time here are marked by asterisks. Some of these
hierarchies have more than three components, but they are effectively treated
here as triples. Components’ masses of these triples and the total number of
components are given in the last two columns (inner subsystem in brackets).

Most of these bright and nearby hierarchies had previous determinations of
their inner and outer visual orbits or even a joint analysis (e.g. Heintz,
1996), so why a new study is needed? First, outer and inner orbits are fitted
here jointly, including the wobble caused by the subsystem. Second, recent
speckle data are used and appropriate weights are applied; radial velocities
(RVs) are included when available to improve the accuracy. Masses of the
components are estimated using new Gaia parallaxes and absolute magnitudes (or
isochrones for evolved stars). Consistency between parallaxes, orbits, and
estimated mass sums is achieved; elements of poorly constrained long-period
orbits are chosen to match the mass sums. Finally, accuracy of the orbital
elements is evaluated, as well as the reliability of mutual inclination and
period ratios computed from these orbits. Two astrometric inner orbits of
spectroscopic subsystems, two visual inner orbits, and one outer orbit are
first-time solutions. This study continues previous work on the orbits of
hierarchical systems (Tokovinin & Latham, 2017, 2020; Tokovinin, 2018b).

Components of multiple systems are designated by sequences of letters and
numbers, as in the WDS. Subsystems are denoted as two components joined by a
comma. Thus, A,B means a subsystem with components A and B, while AB stands
for the composite component (center of mass) in a wider pair AB,C. I try to
avoid using the outdated discoverer codes given in the WDS, but mention them
for completeness in the notes on individual systems.

Section 2 presents the methods briefly. The main part of the paper is Section
3, where each hierarchy is discussed individually, and masses and other
parameters are estimated. A short summary in Section 4 closes the paper.

##  2\. Data and methods

The orbit of a visual binary is defined by the seven Campbell elements
$P,T,e,a,\Omega,\omega,i$ (in standard notation). Fitting these elements to
position measurements $\theta(t)$ and $\rho(t)$ is a classical problem with
many solutions given in the literature. I use the least-squares fit with
weights inversely proportional to the square of measurement errors
$\sigma^{2}_{i}$. However, the actual errors are not Gaussian and are not well
known, leaving room for subjective choices; different orbits can be derived
from the same data, depending on the method and weights.

The observed motion in a resolved triple system (e.g. Aa,Ab and A,B) is
modeled here by two Keplerian orbits and the wobble factor $f$, hence there
are 15 free parameters; more parameters (e.g. the inner and outer RV
amplitudes $K_{1}$, $K_{3}$, and the system velocity $\gamma$) are fitted when
the RV measurements are available. The angles $\Omega$ and $\omega$ are chosen
to fit the RV of the primary component, otherwise both can be changed
simultaneously by 180°. The wobble factor is the ratio of the wave amplitude
in the outer orbit caused by the subsystem to the inner semimajor axis. When
the outer positions of Aa,B refer to the resolved inner subsystem, $f=q/(1+q)$
is directly related to the inner mass ratio $q_{\rm Aa,Ab}$; it is also called
the fractional mass (Heintz, 1996; Tokovinin & Latham, 2017). When the
positions of the outer component B are measured relative to the photo-center
of the unresolved inner subsystem, e.g. A,B, then the wobble factor
$f^{*}=f-r/(1+r)$ depends on the light ratio $r=10^{-0.4\Delta m}$. For
example, if the inner subsystem consists of two identical stars, $f=0.5$ and
$f^{*}=0$. When the inner subsystem is not resolved, I set its semimajor axis
to the value calculated from the period and mass and fit the wobble factor to
determine the axis of the astrometric orbit.

I use the IDL code orbit3.pro111Codebase:
http://dx.doi.org/10.5281/zenodo.321854 to fit both orbits simultaneously
(Tokovinin & Latham, 2017; Tokovinin, 2018b; Tokovinin & Latham, 2020). Its
modification orbit4.pro distinguishes between outer positions measured from a
resolved and unresolved inner system; the factor $f$ is fitted, while the
ratio $f^{*}/f$ is fixed to a value deduced from a model, e.g. zero for equal-
mass subsystems. Often accurate speckle measurements refer to the resolved
subsystem Aa,B, while the remaining unresolved micrometer measurements are so
crude that the photo-center wobble is lost in their noise, making the
distinction between $f$ and $f^{*}$ irrelevant.

The position measurements were retrieved from the WDS database (Mason et al.,
2001) on my request and, where possible, complemented by the recent speckle
interferometry at the Southern Astrophysical Research (SOAR) 4.1 m telescope
(Tokovinin et al., 2020, and references therein). The latest SOAR observations
are made in 2020.8. The SOAR measurements and the data from telescopes with
apertures of 6 m and larger are assigned errors of 1–2 mas, speckle
interferometry at other 4 m class telescopes is assumed to have errors of 5
mas. For smaller apertures, the errors are correspondingly larger. The visual
micrometer measurements are assumed to have errors from 0$\farcs$05 to
0$\farcs$25, depending on the separation. These errors are poorly known, while
the old data vary in quality, depending on the observer. I liberally deleted
the largest historical outliers and down-weighted some other micrometer
measurements. The system of weights used here privileges the speckle data and
is different from the weights adopted by the orbit catalog (Hartkopf et al.,
2001). The existing outer orbits of triple systems were typically computed by
simple Keplerian fits, neglecting the wobble; its explicit modeling here
improves the accuracy.

Some long-period outer orbits considered here have only a short arc covered by
the observations, resulting in the loosely constrained elements. Previously
computed orbits of these pairs have grade 5 (preliminary) in the catalog
(Hartkopf et al., 2001). While the standard approach uses only the position
measurements, I consider here additional information on the mass sum,
parallax, and, in some cases, the RVs. Large errors of some elements (e.g.
period or inclination) in the initial, unconstrained, fit indicate that a wide
range of orbits is compatible with the data. Poorly defined elements usually
strongly correlate with other elements. I fix some poorly defined elements to
satisfy the additional constraints on the mass sum and fit the remaining
elements; the formal errors of the fitted elements in such orbits are in fact
only lower limits. Even uncertain long-period orbits of hierarchical systems
are useful because they accurately represent the observations, serve as a
reference for measuring the wobble, and allow calculation of the mutual
inclination $\Phi$. In most cases, the true ascending nodes of the inner
and/or outer orbits are not known, and two alternative values of $\Phi$ are
possible (Tokovinin, 2017b).

Masses of main-sequence stars are estimated from their absolute visual
magnitudes using standard relations (Pecaut & Mamajek, 2013). The parallaxes
are measured by Gaia (Gaia collaboration, 2018). However, its current data
release does not account for the non-linear motion of binaries. Gaia
parallaxes and proper motions (PMs) of binary stars may have large errors and
biases, and the relation between these effects and the orbital parameters is
complex (Penoyre et al., 2020). The short-term PM of binaries measured by Gaia
often differs from their long-term PM deduced from the Hipparcos and Gaia
positions. This PM anomaly (Brandt, 2018) is a manifestation of astrometric
acceleration; sometimes it helps to constrain or verify the orbits. I use the
more accurate Gaia parallaxes of other (non-binary) components of multiple
systems when they are available. Otherwise, dynamical parallaxes computed from
the orbits and the estimated masses are currently the best distance
measurements.

All orbit plots in the next section (e.g. Figure 1) are similar. They show the
outer orbit, including the wobble, by the wavy line. Measurements of the outer
pair (asterisks if resolved, squares and crosses otherwise) are connected to
the expected positions on the orbit by short dotted lines. The scale is in
arcseconds, the orientation is standard (north up, east left), and the primary
component (large asterisk) is located at the coordinate origin. The orbit of
the inner subsystem is plotted at the center, on the same scale, by the
magenta dash ellipse and triangles. Even if the inner subsystem belongs to the
secondary component, its orbit is plotted around the center; in this case, the
wobble factor is negative. Each plot also contains a graph representing the
hierarchical structure of the multiple system (the mobile diagram). Masses of
stars and orbital periods in italics accompany these diagrams.

Table 2Orbital Elements  WDS | System | $P$ | $T$ | $e$ | $a$ | $\Omega$ | $\omega$ | $i$ | $f$ | $K$
---|---|---|---|---|---|---|---|---|---|---
|  | (yr) | (yr) |  | (′′) | (deg) | (deg) | (deg) |  | (km s-1 )
00568$+$6022 | Aab,Ac | 4.899 | 2003.565 | 0.260 | 0.0309 | 146.8 | 279.2 | 47.3 | 0.346 | 11.18
|  |  $\pm$0.009 |  $\pm$0.038 |  $\pm$0.017 |  $\pm$0.0012 |  $\pm$2.5 |  $\pm$2.9 |  $\pm$3.2 |  $\pm$0.065 |  $\pm$0.24
00568$+$6022 | A,B | 84.10 | 1953.59 | 0.225 | 0.237 | 171.5 | 343.5 | 53.6 | … | 4.30
|  |  $\pm$0.84 |  $\pm$0.83 |  $\pm$0.010 |  $\pm$0.006 |  $\pm$0.9 |  $\pm$3.6 |  $\pm$0.9 | … | fixed
01198$-$0031 | B,C | 27.13 | 1994.92 | 0.205 | 0.1108 | 41.2 | 343.2 | 18.5 |  $-$0.466 | …
|  |  $\pm$0.08 |  $\pm$0.19 |  $\pm$0.010 |  $\pm$0.0014 |  $\pm$11.4 |  $\pm$13.0 |  $\pm$3.5 |  $\pm$0.015 | …
01198$-$0031 | A,BC | 1000 | 1686.7 | 0.540 | 1.450 | 123.1 | 102.8 | 37.9 | … | …
|  | fixed |  $\pm$17.4 | fixed |  $\pm$0.021 |  $\pm$9.8 |  $\pm$5.5 |  $\pm$1.1 | … | …
01350$-$2955 | A,B | 4.5595 | 2010.127 | 0.3142 | 0.1669 | 168.6 | 298.9 | 11.9 | 0.640 | 2.92
|  |  $\pm$0.0014 |  $\pm$0.006 |  $\pm$0.0022 |  $\pm$0.0006 |  $\pm$6.1 |  $\pm$6.0 |  $\pm$1.7 |  $\pm$0.012 |  $\pm$0.69
01350$-$2955 | AB,C | 120.88 | 1955.24 | 0.230 | 1.571 | 132.5 | 54.3 | 36.5 | … | …
|  |  $\pm$0.83 |  $\pm$0.95 |  $\pm$0.011 |  $\pm$0.018 |  $\pm$3.0 |  $\pm$4.0 |  $\pm$1.2 | … | …
02291$+$6724 | Aa,Ab | 48.72 | 1993.21 | 0.637 | 0.423 | 176.6 | 328.2 | 148.2 | 0.322 | …
|  |  $\pm$0.45 |  $\pm$0.05 |  $\pm$0.004 |  $\pm$0.004 |  $\pm$1.8 |  $\pm$1.9 |  $\pm$1.3 |  $\pm$0.013 | …
02291$+$6724 | A,B | 2400 | 940 | 0.40 | 6.50 | 188.0 | 113.3 | 102.9 | … | …
|  | fixed |  $\pm$47 | fixed | fixed |  $\pm$0.9 |  $\pm$3.4 |  $\pm$0.3 | … | …
04375$+$1509 | Aa,Ab | 2.0097 | 1999.818 | 0.405 | 0.037 | 319.3 | 339.2 | 144.0 | 0.223 | 3.85
|  |  $\pm$0.0009 |  $\pm$0.011 |  $\pm$0.014 | fixed |  $\pm$11.1 |  $\pm$3.0 | fixed |  $\pm$0.043 |  $\pm$0.06
04375$+$1509 | A,B | 124.9 | 2049.6 | 0.20 | 0.667 | 324.0 | 260.5 | 73.3 | … | 2.23
|  |  $\pm$22.6 |  $\pm$12.5 | fixed |  $\pm$0.080 |  $\pm$2.2 |  $\pm$44.3 |  $\pm$1,8 | … |  $\pm$0.23
04397$+$0952 | Aa1,Aa2 | 1.6732 | 2013.958 | 0.426 | 0.046 | 286.5 | 267.7 | 62.7 | 0.170 | 4.47
|  |  $\pm$0.0005 |  $\pm$0.013 |  $\pm$0.018 | fixed |  $\pm$10.2 |  $\pm$3.6 |  $\pm$14.3 |  $\pm$0.030 |  $\pm$0.10
04397$+$0952 | Aa,Ab | 285 | 1999.8 | 0.19 | 1.637 | 335.7 | 253.0 | 78.6 | … | 1.91
|  | fixed |  $\pm$76 |  $\pm$0.09 |  $\pm$0.136 |  $\pm$4.9 |  $\pm$145 |  $\pm$1.5 | … |  $\pm$0.57
04400$+$5328 | A,B | 26.34 | 1988.98 | 0.846 | 0.1727 | 12.6 | 42.9 | 138.4 | 0.446 | …
|  |  $\pm$0.05 |  $\pm$0.03 |  $\pm$0.005 |  $\pm$0.0023 |  $\pm$2.5 |  $\pm$2.6 |  $\pm$1.2 |  $\pm$0.009 | …
04400$+$5328 | AB,C | 660 | 2011.7 | 0.405 | 1.666 | 286.2 | 105.1 | 132.5 | … | …
|  | fixed |  $\pm$2.7 |  $\pm$0.015 |  $\pm$0.019 |  $\pm$1.8 |  $\pm$5.4 |  $\pm$1.9 | … | …
15440$+$0231 | Ba,Bb | 6.57 | 2020.09 | 0.357 | 0.189 | 21.4 | 230.0 | 70.4 |  $-$0.500 | …
|  |  $\pm$0.29 |  $\pm$0.08 |  $\pm$0.038 |  $\pm$0.008 |  $\pm$2.7 |  $\pm$9.3 | fixed | fixed | …
15440$+$0231 | A,B | 900 | 1936.1 | 0.435 | 7.20 | 54.9 | 330.9 | 138.1 | … | …
|  | fixed |  $\pm$9.9 |  $\pm$0.030 | fixed |  $\pm$7.4 |  $\pm$3.4 |  $\pm$1.5 | … | …
16057$-$0617 | Aa,Ab | 4.999 | 2013.16 | 0.370 | 0.0521 | 117.3 | 8.7 | 127.0 | 0.485 | …
|  |  $\pm$0.005 |  $\pm$0.06 |  $\pm$0.009 |  $\pm$0.0011 |  $\pm$2.5 |  $\pm$5.7 |  $\pm$1.7 |  $\pm$0.053 | …
16057$-$0617 | A,B | 300 | 2048.5 | 0.611 | 0.877 | 107.6 | 141.0 | 155.0 | … | …
|  | fixed |  $\pm$4.1 |  $\pm$0.033 |  $\pm$0.009 |  $\pm$24.0 |  $\pm$26.3 | fixed | … | …
17066$+$0039 | Ba,Bb | 6.311 | 2013.04 | 0.400 | 0.0700 | 2.4 | 195.2 | 29.9 |  $-$0.521 | …
|  |  $\pm$0.031 |  $\pm$0.03 |  $\pm$0.010 |  $\pm$0.0014 |  $\pm$6.2 |  $\pm$7.0 |  $\pm$2.9 |  $\pm$0.031 | …
17066$+$0039 | A,B | 900 | 1963.04 | 0.596 | 2.264 | 97.7 | 356.9 | 40.0 | … | …
|  | fixed |  $\pm$1.36 |  $\pm$0.002 |  $\pm$0.023 |  $\pm$3.2 |  $\pm$1.8 | fixed | … | …
17157$-$0949 | Ba,Bb | 5.257 | 2011.08 | 0.46 | 0.033 | 52.3 | 69.0 | 138.6 |  $-$0.436 | …
|  |  $\pm$0.052 |  $\pm$0.59 |  $\pm$0.11 |  $\pm$0.008 |  $\pm$11.7 |  $\pm$22.5 |  $\pm$24.9 |  $\pm$0.040 | …
17157$-$0949 | A,B | 137.1 | 2016.09 | 0.357 | 0.344 | 26.5 | 280.5 | 141.4 | … | …
|  |  $\pm$1.8 |  $\pm$0.26 |  $\pm$0.007 |  $\pm$0.002 |  $\pm$1.2 |  $\pm$1.7 |  $\pm$1.1 | … | …
19453$-$6823 | Ba,Bb | 4.40 | 2017.22 | 0.850 | 0.0544 | 148.2 | 218.9 | 127.4 |  $-$0.344 | …
|  |  $\pm$0.09 |  $\pm$0.06 | fixed |  $\pm$0.0052 |  $\pm$6.4 |  $\pm$10.3 |  $\pm$7.6 |  $\pm$0.032 | …
19453$-$6823 | A,B | 84.4 | 2001.55 | 0.252 | 0.475 | 204.0 | 336.2 | 140.0 | … | …
|  |  $\pm$1.8 |  $\pm$0.86 |  $\pm$0.016 |  $\pm$0.027 |  $\pm$2.6 |  $\pm$1.9 | fixed | … | …
22300$+$0426 | Ba,Bb | 2.113 | 2016.60 | 0.193 | 0.0412 | 266.9 | 151.3 | 113.1 |  $-$0.456 | …
|  |  $\pm$0.017 |  $\pm$0.06 |  $\pm$0.027 |  $\pm$0.0012 |  $\pm$2.0 |  $\pm$9.0 |  $\pm$1.7 |  $\pm$0.021 |
22300$+$0426 | A,B | 123.53 | 2035.38 | 0.515 | 0.714 | 117.1 | 214.6 | 89.94 | … | …
|  |  $\pm$1.26 |  $\pm$0.92 |  $\pm$0.016 |  $\pm$0.009 |  $\pm$0.1 |  $\pm$1.5 |  $\pm$0.04 | … | …

Elements of visual orbits (including the wobble factor) and their errors are
listed in Table 2. The last column gives the RV amplitudes when the RVs were
used; the systemic velocities are given in the discussion of individual
systems. As only published RVs are used, they are not duplicated here. The
position measurements, their adopted errors, and residuals to the orbits are
assembled in Table 3, published in full electronically.

Table 3Position measurements and residuals (fragment)  WDS | Syst. | Date | $\theta$ | $\rho$ | $\sigma_{\rho}$ | (O$-$C)θ | (O$-$C)ρ | Ref.aa A: adaptive optics; C: CCD measurement; G: Gaia; H: Hipparcos; J: Finsen’s eyeiece micrometer; M: visual micrometer measurement; P: photographic measurement; S: speckle interferometry at SOAR; s: speckle interferometry at other telescopes.
---|---|---|---|---|---|---|---|---
|  | (JY) | (deg) | (″) | (″) | (deg) | (″) |
00568+6022 | Aa,Ab | 1994.7183 | 161.9 | 0.0302 | 0.0020 | 1.8 | 0.0009 | s
00568+6022 | Aa,Ab | 1998.7731 | 98.3 | 0.0200 | 0.0020 | 9.7 | 0.0032 | s
00568+6022 | Aa,Ab | 1998.7760 | 69.8 | 0.0160 | 0.0020 |  $-$19.2 |  $-$0.0009 | s
00568+6022 | Aa,Ab | 2000.8744 | 228.1 | 0.0250 | 0.0020 |  $-$6.2 |  $-$0.0013 | s
00568+6022 | Aa,Ab | 2000.8770 | 228.9 | 0.0250 | 0.0020 |  $-$5.5 |  $-$0.0013 | s
00568+6022 | Aa,B | 1994.7183 | 339.7 | 0.2720 | 0.0050 |  $-$0.2 | 0.0011 | s
00568+6022 | Aa,B | 1994.7183 | 339.7 | 0.2725 | 0.0050 |  $-$0.2 | 0.0016 | s

##  3\. Individual Systems

###  3.1. 00568$+$6022 (ADS 784, HR 266)

Figure 1.— Outer and intermediate orbits of ADS 784 (periods 84 yr and 5 yr).
In this and following figures, the wavy line marks the outer orbit. Measured
positions (asterisks and crosses) are connected to the expected positions by
short dotted lines. Scale in arcseconds, north up, east left, primary
component at coordinate origin. The inner orbit is plotted at the center on
the same scale (magenta ellipse and triangles). The blue graph shows the
hierarchical structure of the complete system, with approximate periods and
masses in italics.

HR 266 (ADS 784) is an early-type quadruple system. The outer pair is a 84 yr
visual binary BU 1099. Its orbit, fully covered and quite accurate, is only
slightly updated here. Cole et a. (1992) published a detailed study of this
system, including spectroscopic orbits of the inner 4.2 day double-lined
spectroscopic binary and its 5 yr orbit in an intermediate subsystem, which
was also discovered independently by the wavy motion of A,B. Therefore, this
is a 3+1 hierarchy where a close binary has a tertiary companion and this
triple is orbited by another, more distant visual companion (Figure 1).

Schoeller et al. (1998) reported direct resolution of the intermediate pair
Aa,Ab at 25 mas separation, near the diffraction limit of the 6 m telescope.
While Cole et al. attributed the spectroscopic triple to the secondary visual
component B, in fact the component A has been resolved. Balega et al. (1999)
revise the system model where the inner triple belongs to the component A and
show that it does not contradict the spectroscopy of Cole et al. However,
their model identifies the inner spectroscopic binary with Aa and predicts
zero wobble in the unresolved measurements of A,B because the mass and light
ratios of the 5 yr subsystem are nearly equal (about 0.5), canceling the
photo-center wobble. I repeated their analysis and reached the same conclusion
that contradicts the observed wavy motion of A,B. Docobo & Andrade (2006)
computed a crude visual orbit of the 5 yr pair (without using the RVs) and
also estimated masses from the absolute magnitudes. They found that most of
the mass is concentrated in the inner triple, but still call this component B.

Here I attempt to clarify the confusion regarding this system by attributing
the inner 4-day pair to the secondary component Ab and denoting it as Ab1,Ab2.
Speckle observations published since the last analysis by Balega et al. and
the RVs of the center-of-mass of Ab from Cole et al. are used to determine the
combined orbit of Aa,Ab jointly with the outer orbit of A,B.

The only resolved measurements of Aa,Ab were taken at the 6 m telescope on
three epochs, 1994.7, 1998.8, and 2000.9 (I swapped the published quadrant in
2000.9). Micrometric and speckle measurements of the outer pair A,B (without
resolving the subsystem) from 1889 to 2010 cover 131 yr or 1.56 outer periods.
The photo-center wobble is assumed to be the $f^{*}/f=0.32$ fraction of the
full wobble to fit the wave in the unresolved measurements. The first speckle-
astrometric orbit of the 5-yr pair computed by Cole et al. was based only on
the unresolved measurements made before 1990 and had a wobble amplitude of 4
mas.

Differential speckle photometry by Schoeller et al. (1998) in the $V$ band
indicates that the magnitudes of Aa and B are similar within 0.1 mag, while Ab
is 0.7 mag fainter. The inferred magnitude difference between A and B, 0.45
mag, agrees with its multiple measurements. So, Aa and B have similar
luminosities and masses and, together, dominate the combined spectrum. This
explains why Cole et al. could not detect the lines of the 5 yr secondary and
why they attributed the 4-day binary to B. Using the 50 Myr PARSEC isochrone
(Bressan et al., 2012), I found that the magnitude difference between Ab1 and
Ab2 of 1.4 mag matches the spectroscopic mass ratio $q_{\rm Ab1,Ab2}=0.67$.
With this assumption, the visual magnitudes of all components are defined
(Table 4). Furthermore, I adopt the dynamical parallax of 5.6 mas derived from
the A,B orbit and the expected mass sum of $\sim$11 ${\cal M}_{\odot}$ . The
corresponding spectral types, also listed in Table 4, agree with the spectral
classification by Cole et al., except the A7V component Ab2 which they
classified as A1V. The two most massive members of this system, B and Aa, have
spectral type B7V and are fast rotators.

Table 4Parameters of Components of ADS 784  Parameter | Ab1 | Ab2 | Aa | B
---|---|---|---|---
$V$ (mag) | 7.53 | 8.83 | 6.57 | 6.57
${\cal M}$ (${\cal M}_{\odot}$) | 2.44 | 1.64 | 3.39 | 3.39
Spectral type | B9V | A7V | B7V | B7V

The adopted parallax of 5.6 mas leads to the inner mass sum of 7.1 ${\cal
M}_{\odot}$ , in reasonable agreement with 7.5 ${\cal M}_{\odot}$ estimated
from the isochrone. The RV amplitude of Ab and its mass, 4.1 ${\cal
M}_{\odot}$ , correspond to the 3.1 ${\cal M}_{\odot}$ mass for Aa that
matches the isochrone mass approximately. Cole et al. noted the large mass of
the unseen secondary in the 5-yr subsystem and suggested that it could be a
pair of low-mass stars. This hypothesis is no longer viable because speckle
interferometry resolved the subsystem Aa,Ab.

The system model in Table 4 implies the mass ratio in the 5-yr subsystem
$q_{\rm Aa,Ab}=1.2$ which, together with the measured light ratio $r=0.5$,
results in the photo-center wobble factor $f^{*}=0.20$ and the wobble
amplitude of 6.2 mas, similar to the amplitude estimated by Cole et al.
However, the calculated wobble factor for the resolved Aa,B positions,
$f=0.55$, disagrees with the measured $f=0.35\pm 0.07$. I suspect that the
measurement of $f$ based on only three epochs is questionable.

The outer orbit and the estimated masses correspond to the RV amplitudes in
the outer orbit of $K_{3}=4.3$ and $K_{4}=9.5$ km s-1 . No RV trend associated
with the outer orbit is obvious in the RVs measured by Cole et al. Adopting
$K_{3}=4.3$ km s-1 , I obtain the systemic velocity of $-7.6$ km s-1 or $-9.5$
km s-1 , depending on the node of A,B. The choice of the outer node also
defines the mutual inclination. The model proposed here attributes the 4-day
subsystem to Ab, hence $\omega_{\rm Ab}=279\fdg 5$ refers to the secondary
component of the 5-yr subsystem. If $\omega_{B}=327\fdg 9$ in the outer orbit
also refers to the secondary, the mutual inclination between the orbits of A,B
and Aa,Ab is $\Phi=20\arcdeg\pm 2\arcdeg$, otherwise the orbits are nearly
perpendicular, $\Phi=90\arcdeg$. Continued RV monitoring will define the sign
of the long-term RV trend and hence the correct node. The inclination of the
4-day subsystem is $i_{\rm Ab1,Ab2}=63\arcdeg$.

Relation between ADS 784 and $\gamma$ Cas (WDS J00567+6043, HD 5394, HR 264),
the archetype of Be stars, was noted by Mamajek (2017). The Hipparcos parallax
of $\gamma$ Cas is 5.32$\pm$0.56 mas and its RV is $-7.4$ km s-1 ; the PMs of
these systems also match. Note that $\gamma$ Cas is a triple system consisting
of the 203 day spectroscopic pair, a 60 yr astrometric subsystem, and a faint
physical companion at 2$\farcs$05\. The projected distance between HR 266 and
HR 264 is about 1 pc, too large for a bound pair (its orbital period would be
$\sim$20 Myr). More likely, these young and massive hierarchical systems are
linked together by their common origin.

###  3.2. 01198$-$0031 (42 Cet, ADS 1081)

Figure 2.— Orbits of 42 Cet (ADS 1081), periods 1000 yr and 27 yr.

The bright star HR 385 (42 Cet, ADS 1081) consists of an evolved G8III primary
and the tight pair of nearly equal A7V stars B and C (FIN 337 BC). The inner
27 yr orbit is very well defined (Mason et al., 2010), unlike the long-period
outer orbit of STF 113 A,BC that lacks coverage despite measurements available
since 1829. Gaia gives rather accurate and consistent parallaxes of stars A
and BC, and their average, 7.76 mas, defines the distance to the system.

The differential photometry at SOAR gives $\Delta y_{\rm AB}=0.96\pm 0.15$
mag, hence the $V$ magnitudes of B and C are 7.50 and 8.46 mag. The orbit of
B,C gives a mass sum of 3.96 ${\cal M}_{\odot}$ . The wobble factor
$f=-0.466\pm 0.016$ implies $q_{\rm BC}=0.87$, hence the measured masses of B
and C are 2.11 and 1.84 ${\cal M}_{\odot}$ , somewhat larger than 1.81 and
1.46 ${\cal M}_{\odot}$ estimated from the absolute mangitudes.

The latest outer orbit with $P=650$ yr (Zirm, 2015) yields an unrealistically
large outer mass sum of 11.8 ${\cal M}_{\odot}$ vs. 6.5 ${\cal M}_{\odot}$
estimated from the absolute magnitudes. The new outer orbit in Figure 2 has a
period arbitrarily fixed at 1000 yr and the fixed eccentricity chosen to
obtain the target mass sum. Historic observations that deviated by more than
0$\farcs$2 from the orbit were removed from the final fit. A free fit has huge
errors, indicating that the data do not really constrain the outer orbit. The
tentative outer orbit serves mostly as a reference to measure the wobble
factor, and the formal errors of its non-fixed elements are only lower limits.
The two mutual inclinations are almost equal, about 40°.

###  3.3. 01350$-$2955 (GJ 60)

Figure 3.— Orbits of GJ 60 AB,C (BU 1000, 121 yr) and A,B (DAW 31, 4.56 yr).

The well-studied, solar neighborhood quadruple system GJ 60 consists of four
K-type dwarfs with comparable masses (Figure 3). The Gaia parallax of star C,
45.83$\pm$0.18 mas, defines the system’s distance (the parallax of AB, 38.4
mas, is strongly biased). Star B is an eclipsing binary (designated BB Scl)
with a period of 0.4765 day. This bright, nearby, and chromospherically active
star is mentioned in 149 references.

This is a rare case where both the outer 121 yr and the inner 4.56 yr visual
orbits are well constrained and of good quality. New, slightly updated, orbits
are obtained by the full unconstrained fit of 15 elements. Accurate SOAR data
from 2008–2020 define the wobble factor $f=0.640\pm 0.012$, which clearly
shows that B is more massive than A, $q_{\rm A,B}=1.78$. The inner mass sum is
2.33 ${\cal M}_{\odot}$ , hence the measured masses of A and B are 0.84 and
1.49 ${\cal M}_{\odot}$ , respectively. The chromospherically active eclipsing
subsystem Ba,Bb consists of almost equal mass stars with $q_{\rm Ba,Bb}=0.97$
(Watson et al., 2001). The estimated mass of C, 0.56 ${\cal M}_{\odot}$ ,
leads to the total system mass of 2.89 ${\cal M}_{\odot}$ , which agrees well
with 2.75 ${\cal M}_{\odot}$ deduced from the outer orbit and the parallax. If
the eclipsing nature of B were not known, the subsystem Ba,Bb could be
independently discovered from the excessive mass of B.

The period ratio of the visual pairs is 26.5$\pm$0.2. The crude RVs of
component A plotted in Figure 5 of Watson et al. (2001) indicate that the node
of the 4.6 yr orbit listed here corresponds to the component A. The true node
of the outer 121 yr orbit remains unknown, and the small estimated RV
amplitude of 1.3 km s-1 makes such prospect unlikely. This leaves two options
for the mutual inclination, either 28°$\pm$2° or 47°. The first choice appears
more likely, considering the moderate inner and outer eccentricities (0.31 and
0.23). This hierarchical system resembles other planetary type low-mass
hierarchies (Tokovinin, 2018b), but its inner eclipsing system is definitely
not coplanar with the low-inclination middle and outer orbits. Therefore, the
eclipsing subsystem BB Scl should be precessing and this might be detectable
by the variable eclipse depth. Analysis of the eclipse time variation can
bring additional information on the orbits, but it is outside the scope of
this study.

###  3.4. 02291$+$6724 ($\iota$ Cas, ADS 1860)

Figure 4.— Orbits of $\iota$ Cas (ADS 1860). A fragment of the outer (2400 yr)
orbit and the inner (49 yr) orbit.

$\iota$ Cas (HR 707, ADS 1860) is a well-known quintuple system (Figure 4)
with an A5Vp magnetic primary. Three stars A, B, and C were resolved by W.
Struve in 1828 (STF 262) at comparable separations and earlier by W. Hershel
(his inaccurate measurements are not used here). The 47 yr astrometric
subsystem Aa,Ab was detected from the wavy relative motion of A,B (Heintz,
1996) and directly resolved for the first time by speckle interferometry in
1982 as CHR 6. In 2002, star C was also resolved into a 0$\farcs$4 pair Ca,Cb
(CTU 2). The period of Ca,Cb estimated from its separation is $\sim$60 yr and
its orbit is not known yet.

Gaia measured concordant parallaxes of A, B, and C; the latter, most accurate
(22.22$\pm$0.08 mas), is adopted here. The orbit of A,B with $P=620$ yr
(Heintz, 1996), updated here to $P=2400$ yr, remains essentially
unconstrained. Drummond et al. (2003) reached the same conclusion and modeled
the observed segment of this orbit by a linear motion. They combined the
wobble with direct resolutions of Aa,Ab, computed its first orbit with $P=47$
yr, and estimated masses of Aa and Ab; their results are refined here (Figure
4).

The orbit of Aa,Ab is now very well constrained and, together with the Gaia
parallax, defines the mass sum of Aa,Ab, 2.91 ${\cal M}_{\odot}$ . The wobble
factor $f=0.322\pm 0.013$ implies the mass ratio of $q_{\rm Aa,Ab}=0.47$,
hence the masses of Aa and Ab are 1.98 and 0.93 ${\cal M}_{\odot}$ ,
respectively. The absolute magnitudes of main-sequence stars Aa and Ab (visual
magnitudes 4.65 and 8.63 mag) agree with these masses. The mass of B estimated
from its magnitude is 1.28 ${\cal M}_{\odot}$ , hence the outer mass sum
should be 4.2 ${\cal M}_{\odot}$ .

Comparable separations between A, B, and C raise concerns about dynamical
stability of this system and thus constrain somewhat the orbit of A,B. On the
one hand, its semimajor axis must not be too big compared to the 7″ projected
separation between A and C, on the other hand the periastron separation must
be at least three times larger than the semimajor axis of Aa,Ab, i.e.
$>$1$\farcs$3, which rules out large eccentricities. I fixed the eccentricity
of A,B at 0.4 and also fixed the period (2400 yr) and the semimajor axis
(6$\farcs$5) to values that yield the expected mass sum, 4.2 ${\cal
M}_{\odot}$ . Without these restrictions, the errors of $P,e,a$ from the free
fit are huge. The tentative orbit of A,B proposed by Heintz (1996) has a
smaller semimajor axis of 2$\farcs$9 and a shorter period and yields the mass
sum of 5.9 ${\cal M}_{\odot}$ .

The 6$\farcs$5 semimajor axis of A,B is in tension with the 7″ projected
separation between A and C. A circular orbit of AB,C with $P^{*}\sim 2$ kyr,
estimated from the projected separation, corresponds to the relative orbital
speed of $\mu^{*}=18$ mas yr-1 . I computed the observed speed of the relative
motion between AB (center of mass) and C using historic measurements of A,C,
subtracting the orbital motion of A, and accounting for the precession in
angle. The resulting relative motion speed is only 4.1 mas yr-1 . It is
directed at an angle of 31° to the vector AB,C (C approaches AB). The slow
relative motion is a sign that the actual separation between AB and C in space
is substantially larger than the projected separation. In such case, the large
semimajor axis of A,B is plausible. However, even if the ratio of the
semimajor axes of AB,C and A,B is compatible with the dynamical stability, it
is still moderate, suggesting dynamical evolution of this system.

Although the orbit of A,B is poorly constrained, the relative inclination
between A,B and Aa,Ab is almost insensitive to the choice of $P,e,a$ and has
values of either 46° or 108°. Kozai-Lidov cycles (Naoz, 2016) in the inner
triple are expected, and the large inner eccentricity of 0.64 may be a
manifestation of these cycles.

###  3.5. 04375$+$1509 (vB 102)

Figure 5.— Outer orbit of vB 102 (125 yr) with a 2-yr wobble. The inner pair
Aa,Ab remains unresolved.

This triple system is a G0V star in the Hyades cluster, often referred to as
vB 102. It is a single-lined spectroscopic binary with a period of 734.2 day
(Griffin, 2012) with a faint visual companion discovered in 1989 at
0$\farcs$25 separation and designated as CHR 153. Thirty years of
interferometric coverage resulted in the first, still tentative visual orbit
with $P=128$ yr (Tokvinin et al., 2019). The Gaia parallax of 20.61$\pm$0.46
mas is used here, although it is likely biased by the multiplicity; the PM
anomaly is large (Brandt, 2018).

The estimated inner semimajor axis is 37 mas, but the inner subsystem remains
unresolved owing to the large contrast; however, the wobble in the observed
motion of the outer pair is detectable. Two orbits fitted to the speckle
measurements and RVs (Figure 5) have a wobble amplitude of 8.5 mas, or
$f=0.22\pm 0.04$ and $q_{\rm Aa,Ab}=0.29$ (the inner semimajor axis is
computed from the third Kepler’s law). The mass of the G0V star Aa is
estimated at 1.13 ${\cal M}_{\odot}$ , hence the mass of Ab is about 0.32
${\cal M}_{\odot}$ . Most measurements come from SOAR. Griffin (2012)
determined the negative linear RV trend caused by the outer system and
suggested that $\omega_{\rm A,B}$ is about 270°, as is indeed the case. Note
that the RV coverage starts in 1959; it exceeds the speckle coverage and helps
to constrain the outer orbit. In the final fit I fixed the outer eccentricity
at 0.2 and obtained the expected outer mass sum of 2.16 ${\cal M}_{\odot}$ .
The outer RV amplitude is 2.23$\pm$0.23 km s-1 and the system velocity is
$41.57\pm 0.41$ km s-1 .

The RV amplitude in the inner orbit is small, 3.85 km s-1 , and corresponds to
the minimum mass of 0.17 ${\cal M}_{\odot}$ for Ab. Bender & Simon (2008)
apparently detected weak lines of Ab in the infrared spectra and deduced the
Ab mass of 0.17 ${\cal M}_{\odot}$ from the three RVs, implying a large
inclination of the inner orbit. However, their RVs deviate from the orbit more
than allowed by the claimed errors (rms 6.6 km s-1 ), and Griffin considered
them as spurious. The wobble amplitude measured here also disagrees with the
infrared RVs.

The most unusual result is the apparent counter-rotation in the inner and
outer orbits (Figure 5). The free fit gives the inner inclination of $i_{\rm
Aa,Ab}=153\arcdeg\pm 60\arcdeg$. I fixed the inclination to 144° for
consistency between wobble and RV amplitudes; both correspond to the Ab mass
of 0.33 ${\cal M}_{\odot}$ . Attempts to fit a co-rotating inner orbit fail.
Mutual inclination between the orbits is 72° with fixed inner inclination or
$82\arcdeg\pm 47\arcdeg$ with the free fit (both nodes are known).

Admittedly, the wobble signal is weak and the astrometric inner orbit is still
preliminary. Future Gaia data releases will define this orbit much better if
the triple nature of this system can be properly accounted for in the data
reduction of this mission.

###  3.6. 04397$+$0952 (GJ 173.1)

Figure 6.— Outer orbit of GJ 173.1 Aa,Ab ($P=285$ yr) with a 1.7 yr wobble.

This nearby (29 pc) quadruple system is composed of low-mass stars (Figure 6).
The primary (GJ 173.1, G 83-28) has a spectral type K0V. The 14th magnitude
star B (G 83-29) at 34″ has a common PM. The Gaia parallax of B,
34.04$\pm$0.12 mas, defines the distance to the system better than the biased
parallax of A, 36.80$\pm$0.39 mas. Another visual companion Ab, 3.4 mag
fainter and at 0$\farcs$75 separation, was detected by Hipparcos (HDS 601).
Most observations of this pair Aa,Ab come from the speckle program at SOAR.
Star Aa is also a single-lined spectroscopic binary Aa1,Aa2 with $P=610.43$
day (Halbwachs et al., 2018; Sperauskas et al., 2019). The estimated semimajor
axis of the innermost orbit, 46 mas, favors detection of the wobble signal.

The orbit of Aa,Ab, including wobble, was fitted to the eight available
position measurements of Aa,Ab and 40 published RVs of Aa1 covering the period
from 1986 to 2014 (Figure 6). The outer orbit is not fully constrained, and
its period is fixed to 285 yr to yield the expected outer mass sum of 1.4
${\cal M}_{\odot}$ ; the published preliminary orbit had $P=204$ yr
(Tokovinin, 2017a). Even with the fixed period, the errors of some elements
are large. A negative RV trend caused by the outer orbit is obvious
($K_{3}=1.9\pm 0.6$ km s-1 ), fixing its correct node. The system velocity is
$\gamma=-26.2\pm 1.0$ km s-1 .

The wobble factor $f=0.17\pm 0.03$ corresponds to $q_{\rm Aa1,Aa2}=0.20$, so
the estimated mass of Aa1, 0.75 ${\cal M}_{\odot}$ , leads to the mass of 0.15
${\cal M}_{\odot}$ for Aa2. The RV amplitude $K_{1}=4.47\pm 0.10$ km s-1
corresponds to the secondary mass of 0.17 ${\cal M}_{\odot}$ , hence the two
estimates match within errors. The mass of the pair Aa is 0.90 ${\cal
M}_{\odot}$ , and the outer RV amplitude implies the mass of 0.53 ${\cal
M}_{\odot}$ for Ab The absolute magnitude of Ab corresponds to a star of 0.48
${\cal M}_{\odot}$ , in agreement with its spectroscopically estimated mass.

This system resembles the previous one. The orbital nodes are known, and the
mutual inclination deduced from the wobble is large, 51°$\pm$10°. This result
is a consequence of the different node angles $\Omega$, while the inclinations
of the inner and outer orbits are similar (both are large). An attempt to
force approximate orbit coplanarity fails. The weighted rms residuals of
positional measurements are $\sim$1 mas if the wobble is fitted and 5 mas
without wobble.

###  3.7. 04400$+$5328 (ADS 3358)

Figure 7.— Outer (STF 566, 660 yr) and inner (BU 1295, 26 yr) orbits of ADS
3358.

The classical resolved visual triple system 2 Cam (HR 1466, ADS 3358, spectral
type A8V) has been the subject of several studies (e.g. Heintz, 1996). Here
both orbits are updated using recent speckle measurements (Figure 7). The
eccentric orbit of the inner 26 yr pair A,B (BU 1295) is now fully covered by
the speckle data and of excellent quality (the weighted rms residuals are 5
mas). Several measurements of the visual magnitude difference in this pair
average at $\Delta V_{\rm A,B}=1.29$ mag. I adopt the masses of 1.94 and 1.45
${\cal M}_{\odot}$ deduced from the absolute magnitudes of these stars and the
dynamical parallax of 13.0 mas. The Gaia parallax of 15.3$\pm$0.4 mas is
obviously inaccurate and biased (in 2015 the inner pair, unresolved by Gaia,
moved rapidly through the periastron at 0$\farcs$07 separation) and implies an
unrealistically small inner mass sum of 2.1 ${\cal M}_{\odot}$ .

The outer orbit of AB,C (STF 566) is not so well defined, despite its numerous
measurements available since 1828 that cover a 144° arc. I fixed the outer
period to 660 yr and fitted the remaining elements with wobble. The wobble
amplitude is very accurate ($f=0.446\pm 0.009$) and the measured mass ratio
$q_{\rm A,B}=0.805$ is close to the ratio of the adopted masses, $q_{\rm
A,B}=0.77$. The outer mass sum computed with the dynamical parallax is 4.8
${\cal M}_{\odot}$ , approximately matching the mass of C estimated from its
absolute magnitude.

Two possible values of mutual inclination deduced from the orbits are close to
each other: 57$\fdg$3$\pm$2$\fdg$2 or 61$\fdg$7\. Therefore, the orbits are
highly inclined. Note the large inner eccentricity of 0.85 suggestive of
Kozai-Lidov cycles in this system.

###  3.8. 15440$+$0231 (23 Ser, GJ 596.1)

Figure 8.— Orbits of GJ 596.1 (ADS 9763), periods 900 yr and 6.6 yr.

The bright star 23 Ser (HR 5853, GJ 596.1, ADS 9763), located within 15 pc
from the Sun, has been known to host a faint physical satellite B since 1910,
when this pair was first measured by R. Aitken (A 2230). The latest orbit of
A,B with $P=529$ yr and $a=5\farcs 04$ computed by Gatewood & Mason (2013) is
only preliminary, despite the observed arc of 85°. Other companions mentioned
in the WDS are optical. The bright and nearby G3V star A has been studied from
various angles, including non-detection of planets.

The M3V star B was resolved into a 0$\farcs$2 pair Ba,Bb by Rodriguez et al.
(2015) in 2009 and re-discovered at approximately the same position in 2015 by
Tokovinin & Horch (2016). They suggested that Ba,Bb has made one full
revolution between these observations and its period is about 6 yr. Three
additional observations made at SOAR (the last one in 2020.1) now fully
constrain this orbit with $P=6.6$ yr and $a=0\farcs 19$. The Gaia parallax of
star A, 67.71$\pm$0.07 mas, leads to the inner mass sum of 0.50 ${\cal
M}_{\odot}$ . The magnitude difference between Ba and Bb measured by Tokovinin
& Horch (2016), 0.3 mag, tells us that stars Ba and Bb are similar (a twin),
so their masses are 0.26 and 0.24 ${\cal M}_{\odot}$ . The photometry by
Rodriguez et al. (2015) defines the $K$ magnitude of Ba, 8.08 mag, which
corresponds to the mass of 0.27 ${\cal M}_{\odot}$ according to the standard
relations.

The estimated mass of A, 0.98 ${\cal M}_{\odot}$ , implies the outer mass sum
of 1.48 ${\cal M}_{\odot}$ . I fixed the outer elements $P=900$ yr and
$a=7\farcs 2$ to match this mass sum and fitted the remaining elements (Figure
8). The orbital motion changes the PM of A. According to Brandt (2018), the
Gaia PM anomaly of star A is $(+1.3,+3.8)$ mas yr-1 . The difference between
the orbital speed of B in 2015.5 and its mean speed between 1991.25 and 2015.5
is $(-3.1,-7.2)$ mas yr-1 , in qualitative agreement with the PM anomaly. The
RV amplitude in the outer orbit is $\sim$1 km s-1 , explaining the linear RV
trend of $-3.04$ m s-1 per year detected by Butler et al. (2017).

Only one resolved measurement of A,Ba is available, hence the wobble amplitude
remains unknown; I fixed it to $f=-0.5$. Opposite rotation directions in the
inner and outer pairs are evident. The relative inclinations are 75°$\pm$3° or
141°. This triple system is definitely not co-planar.

###  3.9. 16057$-$0617 (ADS 9918)

Figure 9.— Fragment of the the outer orbit (300 yr) of ADS 9918. The inner
period of Aa,Ab is 5 yr.

This is a classical visual triple system (ADS 9918, HR 5989) with two known
orbits and the fourth distant (260″) and bright component E (HIP 78822, HD
144309) with common PM and parallax. The Gaia parallax of E, 12.03$\pm$0.05
mas, defines the distance better than the less accurate and potentially biased
parallax of A, 12.2$\pm$0.3 mas.

The outer pair A,B (BU 948), discovered in 1879 at 1$\farcs$5 separation, is
slowly closing down; the observed arc is 75° (Figure 9). The inner subsystem
Aa,Ab discovered by W. Finsen in 1964 (FIN 384) has an orbital period of
exactly 5 yr. Components Aa and Ab are equal and their separation never
exceeds 70 mas. The orbit of Aa,Ab has a good coverage by speckle
interferometry. Measurements of Aa,Ab at close separations in 2008 at SOAR and
in 2017 at Gemini (Horch et al., 2019), at phases not covered before, further
constrain the orbit. The inner mass sum of 3.26 ${\cal M}_{\odot}$ matches the
absolute magnitude of its equal components of spectral type F2IV.

The outer orbit with $P=459$ yr and $e=0.65$ was computed by Zirm (2014). The
data do not constrain the period, so I fixed it to 300 yr and, furthermore,
fixed the outer inclination to 155° to obtain the expected outer mass sum, 4.3
${\cal M}_{\odot}$ (the free fit gives $i_{\rm AB}=158\arcdeg\pm 11\arcdeg$).
The wobble factor $f=0.49\pm 0.05$ confirms that the masses of Aa and Ab are
equal. However, these stars were sometimes swapped and some outer measurements
referred to Ab,B instead of Aa,B; these cases are rectified here. Moreover,
the measurement of Aa,B at Gemini in 2017.4 revealed inaccurate calibration of
the scale and was given a low weight, while the tight inner pair Aa,Ab was
measured well at 27.5 mas separation. The mutual inclination between the
orbits is 32° or 78°; the smaller inclination appears more likely.

###  3.10. 17066$+$0039 (ADS 10341)

Figure 10.— The outer orbit (900 yr) with wobble and the inner 6.3 yr orbit of
ADS 10341.

The outer pair of ADS 10341 (BU 823) is known since 1881. The latest visual
orbit of A,B with $P=532$ yr computed by Hartkopf & Mason (2000) is still
preliminary, although the observed arc slightly exceeds 180°(Figure 10). The
main G0V star A has an estimated mass of 0.98 ${\cal M}_{\odot}$ and its Gaia
parallax is 17.90$\pm$0.07 mas.

The secondary component B was resolved at SOAR in 2009 into a close pair Ba,Bb
(TOK 52). The orbit of Ba,Bb with $P=6.3$ yr is well constrained (Tokovinin,
2016). It is slightly updated here using the latest measurements (e.g. Horch
et al., 2019). The mass sum of Ba,Bb, 1.50 ${\cal M}_{\odot}$ , agrees with
masses estimated from the absolute magnitudes of Ba and Bb. The magnitude
difference between Ba and Bb is small, $\sim$0.1 mag, and the wobble factor
$f=-0.52\pm 0.03$ confirms the equality of masses. No wobble is therefore
expected in the positions of A,B, and only the resolved measurements of A,Ba
at SOAR show characteristic loops with a 6 yr period.

The period of A,B is between 350 yr and 1000 yr. The curvature of the observed
arc constrains the mass sum to values between 2 and 2.5 ${\cal M}_{\odot}$ .
The period in the free fit has a large error, hence it can be selected within
the error range. Orbits with shorter periods have smaller eccentricities and
inclinations and the mass sum around 2 ${\cal M}_{\odot}$ . I fix $P_{\rm
A,B}=900$ yr and $i_{\rm A,B}=40\arcdeg$ to obtain the mass sum of 2.5 ${\cal
M}_{\odot}$ , in agreement with the estimated mass of A.

The two possible values of mutual inclination, 60° or 46°, are similar. If a
shorter outer period is enforced, the mutual inclination is reduced to
$\sim$33°. The inner and outer orbits are thus aligned, but only approximately
so.

###  3.11. 17157$-$0949 (ADS 10423)

Figure 11.— Orbits of ADS 10423 (137 yr and 5.26 yr).  Figure 12.— Components
of ADS 10423 in the $V,V-K$ CMD (squares) and the 2 Gyr isochrone for solar
metallicity (Bressan et al., 2012). Numbers and crosses mark masses from 1.3
to 1.7 ${\cal M}_{\odot}$ on the isochrone.

The 7th mag star HD 156034 was resolved by R. Aitken in 1913 as a 0$\farcs$3
pair with components of similar brightness (ADS 10423, A 2592). Its orbit has
been calculated and re-calculated many times. Now most of the outer orbit with
$P=137$ yr is covered and its elements are quite accurate (Figure 11).

The inner subsystem Ba,Bb (TOK 53) was discovered at SOAR in 2009. Its
preliminary 5 yr orbit (Tokovinin et al., 2015b) is well defined now, after
two complete revolutions have been monitored at SOAR. The weighted rms
residuals of Ba,Bb positions are 1.5 mas. The latest measurement of Ba,Bb made
in 2020.2 at 27 mas separation (below the diffraction limit) has a large
residual in angle and is assigned a low weight. The inner semimajor axis is 33
mas and the Ba,Bb pair is not resolved at SOAR near periastron. The wobble
amplitude is 14.5 mas ($f=-0.44\pm 0.04$).

This triple system is lucky in having both orbits well constrained, but its
analysis is nevertheless tricky because the distance is not accurately
measured and the components are evolved. The original and revised Hipparcos
parallaxes are 5.7$\pm$1.3 mas and 4.9$\pm$0.9 mas, respectively. Gaia
normally does not measure parallaxes and PMs of close binaries (A,B was at
0$\farcs$18 in 2015.5), but it did so for ADS 10423; the Gaia parallax of
2.8$\pm$0.9 mas is obviously erroneous. All three quoted trigonometric
parallaxes correspond to unrealistically large masses. I adopt the dynamical
parallax of 7.9 mas based on the orbits and estimated masses.

The spectral type F5V quoted in Simbad disagrees strongly with the combined
color $V-K=2.40$ mag that corresponds to the spectral type K3, matching the
low effective temperature of 4933 K estimated by Gaia. This contradiction
suggests that star A is redder and more evolved compared to the less massive
stars Ba and Bb. The light of B dominates at short wavelengths and its
estimated spectral type is indeed F5V. I propose the system model where star A
has a mass of 1.705 ${\cal M}_{\odot}$ and stars Ba and Bb have masses of 1.38
and 1.30 ${\cal M}_{\odot}$ , respectively. The age of the system is about 2
Gyr.

Figure 12 compares absolute magnitudes and colors of the components with the 2
Gyr PARSEC isochrone for solar metallicity (Bressan et al., 2012). However,
the $V-K$ colors of stars are guessed rather than measured because no
differential photometry in the $K$ band is available. I use the magnitude
difference between Ba and Bb of 0.32 and 0.30 mag measured by Horch et al.
(2019) at wavelengths of 692 nm and 880 nm (roughly in the $R$ and $I$ bands)
and confirmed independently by the differential speckle photometry at SOAR,
e.g. $\Delta y_{\rm Ba,Bb}=0.33$ mag. According to Horch et al. (2019), the
magnitude difference between A and Ba is 1.30 and 1.66 mag in the $R$ and $I$
bands, and SOAR complements this with $\Delta y_{\rm A,Ba}=1.11$ mag.
Photometry confirms that Ba is bluer than A. The two-color photometry of A and
B provided by Tycho is discrepant and I ignore it, while the Hipparcos
measurement $\Delta Hp_{\rm A,B}=0.27$ mag appears reliable. The $V$
magnitudes of A, Ba, and Bb deduced from the combined and differential
photometry are 7.51, 8.61, and 8.94 mag, respectively. I assign the $K$
magnitudes of Ba and Bb to match their positions on the isochrone (and the
spectral type F5), and obtain the $K$ magnitude of A from the combined
photometry; the result is 4.71, 7.57, and 7.85 mag.

The proposed system model defines the masses of the components quoted above
and matches the differential photometry in the $V$ and $I$ bands. The outer
mass sum of 4.4 ${\cal M}_{\odot}$ deduced from the model and the outer orbit
correspond to the dynamical parallax of 7.9 mas. With this parallax, the less
accurate inner orbit gives the mass sum of 2.6 ${\cal M}_{\odot}$ for Ba,Bb,
in good agreement with the model. The model thus matches both the orbits and
the photometry. The inner mass ratio $q_{\rm Ba,Bb}=0.93$ corresponds to
$f=0.48$, which agrees, within errors, with the measured wobble factor
$f=0.44$. If additional multi-color photometry and high-resolution
spectroscopy are secured in the future and properly interpreted, this system
might offer a valuable test of the isochrones because Aa is just leaving the
main sequence.

Mutual inclination between the orbits is either 29°$\pm$14°or 77°; the first,
smallest value seems more likely. It is worth mentioning that the RV of this
star (apparently dominated by the late-type spectrum of A) has been monitored
by Tokovinin & Smekhov (2002) and found to be constant.

###  3.12. 19453$-$6823 (HIP 97196)

Figure 13.— Orbits of HIP 97196 (84 and 4.4 yr). The insert shows the speckle
ACF recorded in 2018.8.

An inconspicuous 10th magnitude K-type dwarf star HIP 97196 has been resolved
by Hipparcos into a 0$\farcs$3 pair (HDS 2806) with unequal components,
$\Delta Hp=2.63$ mag. The binary has been observed for the second time at SOAR
in 2014.3, and its secondary component was resolved into a 0$\farcs$05 pair
Ba,Bb (TOK 425) with $\Delta I=1.2$ mag. As expected, the inner pair is a fast
mover, and its first orbit with $P=4.1$ yr was published by Tokvinin et al.
(2019). The insert in Figure 13 shows a typical speckle auto-correlation
function (ACF) to illustrate the difficulty of measuring this tight and faint
triple system with substantial contrast. All observations at SOAR were made in
the $I$ filter, the last one in 2020.8.

About half of the outer orbit is now covered, and the first set of its
elements is computed here. However, the coverage is uneven, with a large gap
between 1991 and 2014. The orbit is not fully constrained: shorter periods
with smaller eccentricity also fit the data. I fixed the outer inclination to
140° to match the expected mass sum. The eccentric ($e_{\rm Ba,Bb}=0.85$)
inner orbit is also not very accurate because its part near the periastron is
below the resolution limit. The quadrants of Ba,Bb are defined, so the
alternative low-eccentricity orbit with double period is not allowed. The
closest resolved measurement of Ba,Bb at 36 mas separation was made in 2017.4
at Gemini (Horch et al., 2019), at 880 nm wavelength. Their simultaneous
measurement at 692 nm has discordant photometry and is given a low weight, so
this object is a difficult one even for an 8 m telescope.

The Gaia parallax of 21.02$\pm$0.39 mas is not very accurate and possibly
biased, although it agrees with the revised Hipparcos parallax of 21.3$\pm$1.9
mas (van Leeuwen, 2007). I adopt the Gaia parallax and estimate masses from
the standard relations and absolute $I$ magnitudes. The combined $I=8.83$ mag
is deduced by interpolation from other bands (no direct measurement of $I$ is
found in the literature), and I adopt $\Delta I_{\rm A,B}=2.0$ mag and $\Delta
I_{\rm Ba,Bb}=1.2$ mag. The estimated masses of A, Ba, and Bb are 0.78, 0.54,
and 0.40 ${\cal M}_{\odot}$ , respectively. Main-sequence stars with these
masses match the combined magnitudes in the $V$ and $K$ bands and the
magnitude differences, the orbits, and the Gaia parallax. The model implies an
inner mass ratio of $q_{\rm Ba,Bb}=0.74$ and $f=-0.43$, while the measured
wobble factor is $f=-0.34\pm 0.03$.

The outer orbit matches the PM anomaly of $(17.8,7.6)$ mas yr-1 in 1991.25
determined by Brandt (2018) because the corresponding orbital motion of B
relative to A, $(-31.3,-20.5)$ mas yr-1 , has approximately opposite direction
and is two times faster. The photo-center wobble factor $f^{*}$ in the outer
orbit in the $V$ band, computed using the adopted masses, is 0.47. The Gaia PM
anomaly, $(-1.2,-19.2)$ mas yr-1 , does not agree so well with the orbital
motion of AB, $(9.3,37.73)$ mas yr-1 , possibly because of the shorter time
span of Gaia data and the larger effect of the inner subsystem on the photo-
center motion in the $G$ band. Astrometric signal caused by both orbits should
be recovered correctly in the future Gaia data releases that will also yield
the unbiased parallax.

Both inner and outer subsystems of HIP 97196 rotate clockwise, but the
relative inclination is substantial, $\Phi=42\arcdeg\pm 7\arcdeg$ or
$\Phi=80\arcdeg$; the period ratio is about 20. The current linear apparent
configuration of these stars on the sky suggests coplanar and highly inclined
orbits, but this impression is misleading. It is likely that this triple
system goes through Kozai-Lidov cycles and currently is in the low-inclination
and high-$e$ phase of the cycle. Evidently, this hierarchical system differs
from the quasi-coplanar “dancing twins” with comparably small masses and short
periods (Tokovinin, 2018b) in several important respects: large mutual
inclination, large inner eccentricity, and unequal masses of the components.

The spatial velocity of this system $U,V,W=(45,4,-46)$ km s-1 strongly
suggests that it is old and possibly metal-poor. Its further detailed study
will be interesting for various reasons, e.g. to measure accurate masses and
luminosities and to probe the long-term dynamical evolution. Continued speckle
monitoring, relative photometry in the infra-red, and high-resolution
spectroscopy are needed.

###  3.13. 22300$+$0426 (ADS 15988)

Figure 14.— Visual orbit of the outer subsystem in ADS 15988 (STF 2192). Top:
orbit on the sky, bottom: separation vs. time.

The bright star 37 Peg (HR 8566, ADS 15988) was resolved as a 1″ visual pair
by W. Struve in 1825 and is known as STF 2192. This pair slowly closed down
during the 19th century, went through the periastron around 1911, opened up
again and started to close down in the 2nd half of the 20th century. From 1976
on, accurate speckle interferometry replaced visual micrometer measurements,
providing excellent coverage of the decreasing angular separation. This bright
pair was frequently observed by speckle interferometry, possibly as a
convenient calibrator. Its latest orbit by Söderhjelm (1999) has a grade 2,
i.e. is reliable and accurate by current standards. The inclination is 90°, so
the orbit is based only on the measured separations (Figure 14). The WDS
database contains 335 measurements of its relative position.

The parallax of 37 Peg was measured by Hipparcos at 18.93$\pm$1.23 mas,
revised to 19.1 mas by Söderhjelm (1999) and to 19.58$\pm$0.58 by van Leeuwen
(2007). The Gaia parallax of 18.84$\pm$0.39 (53 pc distance) is only slightly
more accurate, possibly because this star is bright and its astrometry is
disturbed by the orbital acceleration. Fortunately, motion in the outer orbit
between 1991 and 2015 was almost linear and the PM anomaly is small (Brandt,
2018). All parallax measurements are mutually consistent, within their errors.
The orbital elements and the masses estimated here favor a slightly smaller
dynamical parallax of 18.4 mas.

This $V=5.5$ mag star attracted attention of observers for various reasons. It
was monitored spectroscopically by Abt & Levy (1976) in a survey of bright
solar-type stars. They found a spectroscopic subsystem with a period of 372.4
day and a semi-amplitude of 8.2 km s-1 . However, this orbit, together with
other orbits from that paper, has not been confirmed by subsequent
observations. The star is no longer listed as a spectroscopic binary in the
current catalog. And yet it is triple!

ADS 15988 was observed by speckle interferometry at SOAR for the first time in
2008.7 when it still approached the conjunction. Shortly after the
conjunction, in 2015, the pair was re-visited at SOAR twice. On the second
visit in 2015.9, star B was clearly resolved into a close 46 mas subsystem
(Figure 15) with nearly equal components Ba and Bb (Tokovinin et al., 2016).
We can only guess why the subsystem Ba,Bb has not been noted much earlier in
numerous speckle observations with 4 m telescopes. Maybe the fact that it is
such a well-known binary prevented observers from noting that the secondary
component is itself a close pair.

The small separation of Ba,Bb implied a short orbital period. For this reason,
the object was frequently observed at SOAR. The subsystem played “hide and
seek”: it disappeared in 2016 but was resolved again in 2017–2020. In 2017,
the subsystem was also measured with the 8 m Gemini telescope (Horch et al.,
2019). Finally, the measurements accumulated to date allow calculation of the
orbit of Ba,Bb with a period of 2.1 yr (Figure 15). The orbit of Ba,Bb
successfully models the measurements (the weighted rms residuals are 0.9 mas)
and predicts small separations for the dates when the subsystem was not
resolved, e.g. in 2008.7. The observations cover two orbital periods of Ba,Bb.
The Gaia parallax of 18.84$\pm$0.39 mas corresponds to the inner mass sum of
2.34$\pm$0.25 ${\cal M}_{\odot}$ .

Figure 15.— Fragment of the outer orbit of ADS 15988 covered at SOAR. The wavy line is the position of Ba relative to A (the red circle at the coordinate origin) affected by the wobble, asterisks show the measured positions of Ba. Small crosses and the blue dash line show the unresolved measurements of B without wobble. The inner orbit is over-plotted on the same scale (the magenta ellipse and triangles).  Figure 16.— Location of the components of ADS 15988 (squares) on the color-magnitude diagram. The lines are PARSEC isochrones (Bressan et al., 2012) for solar metallicity. Small triangles mark the masses of 1.70, 1.17, and 1.10 ${\cal M}_{\odot}$ on the 1.5-Gyr isochrone.  Figure 17.—  Cross-correlation function (line) of the spectrum of ADS 15988 taken on 2020 December 3 showing its triple-lined nature. The dotted curves are fitted Gaussians and the crosses are their sum. The RVs of Ba, Bb, and A are $-17.56$, 17.81, and 10.50 km s-1 , respectively.  Table 5Parameters of Components of ADS 15988  Parameter | AB | A | Ba | Bb
---|---|---|---|---
$V$ (mag) | 5.51 | 5.75 | 7.84 | 8.22
$V-K$ (mag) | 0.93 | 0.81: | 1.26: | 1.36:
${\cal M}$ (${\cal M}_{\odot}$) | 3.97 | 1.70 | 1.17 | 1.10

The edge-on orbit of A,B (Figure 14) computed by Söderhjelm (1999) fits the
latest measurements rather well and, in principle, does not require revision.
However, the errors of orbital elements are not given in the above paper, and
the system of weights is not known. Here the orbit is re-fitted using all
available data with appropriate weights and rejecting the outliers. The orbit
of A,B serves as a reference for determination of the inner mass ratio from
the resolved measurements of the subsystem. Both orbits were fitted
simultaneously by orbit4.pro using only speckle measurements and fixing all
outer elements except $T_{0}$, $\Omega$, and $i$ to the values determined from
the historic data set. The result (Figure 15) yields the wobble factor
$f=0.46\pm 0.02$, hence $q_{\rm Ba,Bb}=0.83\pm 0.07$.

Table 5 gives the combined and individual magnitudes of ADS 15988 in the $V$
and $K$ bands based on the available data. The magnitude difference between A
and B has been measured by Davidson et al. (2009) ($\Delta V=1.50\pm 0.05$
mag, $\Delta R=1.35\pm 0.07$ mag), by Fabricius & Makarov (2000) ($\Delta
V=1.51$ mag, $\Delta B=1.78$ mag), and by Hipparcos ($\Delta Hp=1.539$ mag).
It is clear that B is slightly redder than A. The mean of six measurements of
the magnitude difference between Ba and Bb at SOAR in the $y$ band (close to
$V$) is 0.38 mag, with an rms scatter of 0.05 mag. The differential photometry
and the combined $V_{\rm AB}=5.51$ mag define the components’ individual
magnitudes. However, no differential photometry in the $K$ band is available,
and only the combined $V-K$ color is known. This leaves two unknowns for the
three $V-K$ colors of the components. The colors of Ba and Bb in Table 5 are
chosen to place them on the main sequence in the CMD (Figure 16), thus also
defining the color of A. Obviously, this is only a plausible guess, not a real
measurement.

The most massive star A in this system appears to be slightly evolved off the
main sequence; it is located on the 1.5 Gyr isochrone. The masses of the stars
derived from this isochrone and the absolute $V$ magnitudes are listed in the
last line of Table 5. Adopting these masses, the magnitudes in other
photometric bands can be deduced from the isochrone. These estimates lead to
the magnitude difference between A and B of 1.74 and 1.38 mag in the $B$ and
$R$ bands, respectively, in agreement with the observed differential colors.

The masses of Ba and Bb estimated from the isochrone correspond to the mass
sum of 2.27 ${\cal M}_{\odot}$ which agrees with the measured mass sum of
2.34$\pm$0.25, within the error. The isochrone masses imply the inner mass
ratio of 0.94, somewhat larger than 0.83$\pm$0.07 derived from the wobble
amplitude. The total mass sum deduced from the isochrone, 3.97 ${\cal
M}_{\odot}$ , is slightly larger than measured (3.56$\pm$0.28 ${\cal
M}_{\odot}$ ) and corresponds to the dynamical parallax of 18.4 mas.

A high-resolution spectrum taken on 2020 December 3 (Figure 17) shows lines of
all three components. The RVs match qualitatively their expected values and
allow correct identification of the orbital nodes. They are not used in the
orbital fit, pending additional observations. Future joint analysis of RVs and
resolved measurements will allow us to measure masses without relying on the
isochrone and parallax.

The mutual inclination is 37$\fdg$5$\pm$2$\fdg$0\. Inclination of the outer
orbit, 89$\fdg$9$\pm$0$\fdg$06, suggests possible eclipses during the
conjunctions. However, star B is a close pair, so only eclipses involving its
components can take place. The next conjunction will happen around 2042.

##  4\. Summary

In most hierarchical systems studied here the outer binaries are bright
classical pairs with ADS numbers (Aitken, 1932) observed for a century or
longer. Their inner subsystems are either discovered recently or also known
from the era of visual observations. The published orbital elements are
revised here using recent observations and appropriate weights and accounting
for the wobble. Some long-period orbits remain poorly constrained even when
the expected mass sum is used as additional input. Joint fitting of the inner
and outer orbits using all available information is the main result of this
paper. The orbits, together with the estimated masses, will enable detailed
study of the internal dynamics in these systems (e.g. Xia & Fu, 2015; Hamers,
2020). The ratios of inner and outer separations are moderate, so some
interaction between the orbits is expected.

Our small and random sample of hierarchical systems is not statistically
relevant, but it does illustrate the diversity of their architectures, ranging
from approximately aligned configurations with small eccentricities to highly
inclined or even counter-rotating systems where eccentric inner orbits likely
result from the Kozai-Lidov cycles. This study enlarges the small collection
of systems with resolved inner and outer orbits. Previous analysis of such
systems (Tokovinin, 2017b) indicated that a mixture of hierarchies aligned
within 70° and those with uncorrelated orbits, in a 8:2 proportion, matches
the observed distribution of mutual inclinations $\Phi$.

Some data used here were obtained at the Southern Astrophysical Research
(SOAR) telescope. This work used the SIMBAD service operated by Centre des
Données Stellaires (Strasbourg, France), bibliographic references from the
Astrophysics Data System maintained by SAO/NASA, and the Washington Double
Star Catalog maintained at USNO; I thank B. Mason for extracting historic
measurements from the WDS database. This work has made use of data from the
European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia),
processed by the Gaia Data Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. Work of the author is funded
by the NSF’s NOIRLab.

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Facility: SOAR