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Astrophys Space Sci (2016) 361:46
DOI 10.1007/s10509-015-2630-y
ORIGINAL ARTICLE
A3DmodelforαGem AB: orbits and dynamics
José A. Docobo1,2 ·Manuel Andrade1,3 ·Pedro P. Campo1·Josefina F. Ling1,2
Received: 28 July 2015 / Accepted: 13 December 2015
© Springer Science+Business Media Dordrecht 2015
Abstract The well-known multiple star system, Castor, and
particularly, the [(Aa, Ab), (Ba, Bb)] subsystem, was stud-
ied in detail. After a rigorous analysis of the quality con-
trols, a new solution for the visual orbit yielded new values
for the different physical and orbital parameters of the sys-
tem. In addition, a comprehensive investigation of the orbital
configuration of the quadruple system allowed us to provide
both accurate individual masses and orbital inclinations of
the spectroscopic subcomponents, as well as a new value of
its orbital parallax. Finally, by means of a numerical analysis
of the long-term dynamics, we obtained the most probable
values of the nodal angles of the two spectroscopic subsys-
tems for the first time.
Furthermore, the full characterization of this remarkable
system allowed us to discuss some interesting features of its
present configuration. In this way, we concluded that the ec-
centric (Aa, Ab) subsystem is yet undergoing Kozai-Lidov
cycles whereas, on the contrary, the circular (Ba, Bb) sub-
system has probably already reached a stable stage.
Keywords Astrometry ·Celestial mechanics ·Binaries:
spectroscopic ·Binaries: visual ·Stars: fundamental
parameters ·Stars: αGem AB (Castor)
BM. Andrade
manuel.andrade@usc.es
1R.M. Aller Astronomical Observatory, University of Santiago
de Compostela, Santiago de Compostela 15782, Galiza, Spain
2Departamento de Matemática Aplicada, Facultade
de Matemáticas, University of Santiago de Compostela, Santiago
de Compostela 15782, Galiza, Spain
3Departamento de Matemática Aplicada, Escola Politécnica
Superior, University of Santiago de Compostela, Lugo 27002,
Galiza, Spain
1 Introduction
When W.D. Heintz published his results regarding Castor
(αGeminorum) in 1988, he wrote that this system did not
really need to be introduced, and he was correct. Castor is
probably one of the most studied hierarchical multiple star
systems. Consequently, ample scientific literature exists that
concerns it.
As is well known, the visual system consists of three
stars: A, B, and C, in such a way that A and B consti-
tute a long-period binary with brilliant components, white
in colour. Nevertheless, the eclipsing binary, YY Gemino-
rum, is gravitationally associated with the system and is,
therefore, the component C. The three stars: A, B, and C are
single-lined binaries, for which the system is a sextuplet.
These six components: Aa, Ab, Ba, Bb, Ca, and Cb (see
Fig. 1) form a stable dynamic system that is defined by five
orbits. Three of them have a very short period that corre-
sponds with the spectroscopic pairs (Aa, Ab) (9.2128 days),
(Ba, Bb) (2.9283 days), and (Ca, Cb) (0.8143 days). These
orbital elements have been well determined for a long time
(Vinter Hansen 1940 for the systems A and B, and Sé-
gransan et al. 2000 for system C). They are included in the
SB9: The Ninth Catalogue of Spectroscopic Binary Orbits
(Pourbaix et al. 2004).
Lastly, the orbit of the AB pair (actually, (Aa, Ab), (Ba,
Bb)) has now moved 300 degrees since Bradley and Pound
discovered Castor to be a double star at the beginning of the
18th century having been permanently tracked by means of
different observation techniques.
The resolution of the A and B components in X-ray light
by the Chandra X-ray Observatory has informed us that
both are highly variable and are frequently subject to flar-
ing (Stelzer and Burwitz 2003).Duetothesimilarityof
these flares as compared with those in dMe flares stars, it
46 Page 2 of 12 J.A. Docobo et al.
Fig. 1 Mobile diagram of Castor
is commonly accepted that this X-ray emission is caused by
the late-type companion in each subsystem (Pallavicini et al.
1990).
Although the eclipsing YY Geminorum subsystem is of
special astrophysical interest and, as such, is being studied
by numerous scientists, its small mass and its significant dis-
tance (as compared with the rest of the system) hardly af-
fects the dynamics of the other components. Moreover, be-
cause the elements of the very long period orbit of the distant
companion, C, remain unknown, it is not practical to initiate
a global study of αGeminorum. For that reason, we have fo-
cused on the quadruple system, [(Aa, Ab), (Ba, Bb)], in this
report and conducted an exhaustive investigation regarding
the dynamics of the same.
Following this Introduction, we present a new orbit of
system AB in Sect. 2, having utilised all of the observational
material available: visual, photographic, CCD, and speckle
measurements. The proposed solution, that has better qual-
ity controls as compared to previously calculated orbits, will
be that which is used in the rest of this study. We call this or-
bit, Orbit 2. Orbit 1 will be that of the spectroscopic pair:
(Aa, Ab). Orbit 3 corresponds to the other spectroscopic
pair: (Ba, Bb). Alternatively, we also give the name “inner
pairs” to the orbits 1 and 3 and “outer pair” to orbit 2.
In Sect. 3, using an ad-hoc version of the methodol-
ogy designed by Docobo and Andrade (2006) for the study
of stellar systems with spectroscopic components, we con-
tribute the values of the semimajor axes and inclinations of
orbits 1 and 3, as well as the parallax of the system and,
consequently, of the masses and spectral types of the four
components. In addition, the dynamical analysis carried out
in Sect. 4will allow us to estimate the most probable val-
ues of the nodal angles of orbits 1 and 3 as well as indicate
some constraints with respect to the long-term evolution of
the quadruple system.
2 A new orbit for the AB system
The first orbit of the [(Aa, Ab), (Ba, Bb)] system (hence-
forth, AB) was calculated by John Herschel in 1832. Since
then numerous solutions have been contributed, the latest
being those of Muller (1956), Rabe (1958), Walbaum and
Duvent (1983), Docobo and Costa (1987), Heintz (1988),
De Rosa et al. (2012), Docobo et al. (2014), Matvienko et al.
(2015).
Table 1contains the elements of these orbits. It also in-
cludes the grade assigned to each one of them according to
the criteria utilized by USNO in The Sixth Catalog of Orbits
of Visual Binary Stars (Hartkopf et al. 2001)aswellasthe
values of the quality controls (in θand ρ) in terms of the
root mean square (RMS). In light of the results contained in
these tables, one may conclude that, except for the orbital
period (probably between 450 and 470 years), the rest of the
elements are now satisfactorily defined.
With the idea of obtaining an orbit that best fits the set of
available observations, we have proceeded to compile all of
Table 1 Orbital elements (with uncertainties, if available) for Castor AB, USNO grades (G), and orbital quality controls in RMS (the best numbers
are highlighted in bold)
Author(s) P[yr] Te a[]i[◦]Ω[◦]ω[◦]GRMS
θ[◦]ρ[]
[1]Muller (1956) 511.3 1950.65 0.360 7.369 112.94 41.65 239.81 3.9 1.55 0.156
[2]Rabe (1958) 420.07 1965.3 0.330 6.295 115.94 40.47 261.43 3.8 2.62 0.141
[3]Walbaum and Duvent
(1983)
393.46 1970.09 0.33 5.948 116.9 39.58 269.18 3.4 4.02 0.166
[4]Docobo and Costa
(1987)
444.95 1960.10 0.323 6.593 114.6 41.5 253.3 3.4 1.10 0.137
[5]Heintz (1988) 467.0 1958.0 0.343 6.805 114.5 41.3 249.5 3.4 0.99 0.138
[6]De Rosa et al. (2012) 466.8+6.3
−6.11957.3±0.30.333+0.007
−0.006 6.78 ±0.05 113.56 ±0.09 41.2±0.1 249.3+0.6
−0.53.2 1.32 0.143
[7]Docobo et al. (2014) 459.8 1958.7 0.336 6.755 114.6 41.2 250.5 3.1 0.99 0.138
[8]Matvienko et al. (2015) 453.6±0.8 1958.1±0.40.326 ±0.002 6.69 ±0.03 114.2±0.240.3±0.2 248.8±0.5 3.2 1.17 0.150
[9]This work 459.8±4.2 1959.1±0.90.337 ±0.004 6.732 ±0.043 114.7±0.341.3±0.2 251.1±1.43.10.98 0.138
A 3D model for αGem AB: orbits and dynamics Page 3 of 12 46
Table 2 The percentage of weight contributed by the different obser-
vations
Visual Photographic CCD Speckle
Weight [%] 65.4 7.4 11.2 16.0
Table 3 The last observations
used in the calculation of the
new orbit
tθ[◦]ρ[]
2014.3680 55.0 5.02
2014.3980 54.8 5.02
2015.1839 54.7 5.05
2015.1865 54.5 5.06
Observer: J.A. Docobo
Technique: visual (micrometer)
Telescope: 0.62 m (OARMA)
the existing observational material which has been provided
to us by the USNO (Hartkopf 2014). In total, we have made
use of 1416 observations of which 1120 are visual, 165 are
photographic, 55 are CCD images, and 76 are speckle inter-
ferometric measurements.
As a function of the number of nights that correspond
to each observation, the technique employed, the observer,
and the telescope used, each measurement was assigned a
weight according to the criteria established by USNO itself
(Hartkopf 2014). Table 2presents the percentage of weight
contributed by the measurements obtained with each tech-
nique. The last observations used were performed in 2014
and 2015 by one of the authors of this article (J.A.D.), using
a micrometer attached to the 0.62 m telescope of the Ramón
María Aller Astronomical Observatory (OARMA) (see Ta-
ble 3). All of the observations were corrected by precession
for the epoch, J2000.0.
The method used to obtain a new orbit was proposed
by Docobo (1985,2012) with which more than 300 or-
bits have been calculated in recent decades. As is well
known, this method consists of establishing an application:
V→(P,T,e,a,i,Ω,ω) from the interval (0, 2π)into
the set of elliptical Keplerian orbits whose corresponding
apparent orbits pass through three previously fixed points
(θi,ρ
i;ti)i=1,2,3. These points must obviously belong to ar-
eas with maximum of observation evidence in their favour.
From the set of generated orbits, the selection can be
made according to different criteria: a minimum RMS
(square root of the mean of the squares of the residuals)
and/or a minimum AM (arithmetic mean of the residuals) in
θand ρ, the computed parallax which most closely approx-
imates that measured by Hipparcos, deduced masses that
correspond to the spectral types, etc. Using this method, it is
not necessary to calculate the areal constant as occurs, e.g.,
in the Thiele-Innes-Van den Bos method.
After analysing the complete collection of the observa-
tions, we proceeded to choose several sets of three points
and applied the mentioned orbit calculation method. We
obtained different orbits with periods between 443 and
468 years that reasonably well fit the observations. A rig-
orous selection accomplished using the minimum RMS/AM
criterion led us to a concrete orbit. Finally, we minimized
their residuals using a gradient descent algorithm in order
to improve the solution. The orbital elements along with the
corresponding standard uncertainties are listed in Table 1.
The new orbit, that is quite similar to another that we
calculated last year, will be used in the following sections.
In accordance with the USNO criterion, the correspond-
ing grade of this orbit is 3.1 (Hartkopf 2015). In Fig. 2,
we present the apparent orbit of the new solution with the
visual, photographic, CCD, and speckle measurements, re-
spectively.
3 A quasi-3D model
As mentioned earlier, we are studying a quadruple stellar
system which comprises a visual binary wherein both com-
ponents are spectroscopic binaries. Although visual (outer),
as well as spectroscopic (inner) orbits are well known as we
have seen in Sects. 1and 2, many astrometric and astrophys-
ical parameters remain unknown.
Nevertheless, we can overcome this problem by applying
a procedure (Docobo and Andrade 2006) which allows us
to obtain many astrometric and astrophysical parameters in
order to provide a more detailed description of this system.
This approach takes advantage of many implicit relations
that arise in a three-body stellar system.
In this case, however, we must take into consideration
that Castor AB is a hierarchical quadruple stellar system,
therefore, we will use a modified version of the aforemen-
tioned procedure. The major change will consist of using the
Hipparcos parallax instead of an intermediate mean parallax
for each triple system throughout the procedure.
First, we take the orbital elements of both spectroscopic
orbits (PA,kA,1,eA;PB,kB,1,eB) from the SB9, the com-
bined apparent magnitudes of each subcomponent of the vi-
sual pair (mA≡mAaAb ,mB≡mBaBb) from the WDS, and
the trigonometric parallax from The Hipparcos Catalogue.
The former allows us to calculate the mass functions and
thus to estimate minimum masses and later spectral types of
the second components in both spectroscopic subsystems.
The next step is to estimate the most appropriate range
for the absolute magnitudes of the spectroscopic subcom-
ponents in each subsystem by using Jaschek’s criterion
(Jaschek and Jaschek 1987) according to which, in the case
of SB1, only stars that differ by less than about 1 magnitude
are observable as a composite spectrum. Then, mean values
of the absolute magnitudes for each component (MAa ,MAb,
MBa,MBb) can be estimated by applying the procedure to
46 Page 4 of 12 J.A. Docobo et al.
Fig. 2 Apparent orbit with visual (top left), photographic (top right), CCD (bottom left) and speckle (bottom right) measurements
each spectroscopic subsystem independently and consider-
ing only those values compatible with the Hipparcos paral-
lax by means of the expression
logπ=M1−m12 −5
5−1
2log1+10−0.4M ,(1)
where πis the parallax corresponding to each value of the
absolute magnitude of the first component in the spectro-
scopic binary, M1(=MAa or MBa), and to each value of
the difference between the magnitudes of both components,
M =M2−M1(=MAb −MAa or MBb −MBa). In ad-
dition, m12 is the combined visual apparent magnitude of
the spectroscopic binary (=mAor mB). Then, for the mean
values of M1and πHip, we calculate the corresponding val-
ues of M2(obtained from M) and M3(inferred from m3,
the visual apparent magnitude of the distant companion,
and πHip).
Moreover, individual masses and spectral types for each
component can be calculated using the suitable calibrations
(see Sect. 3.1).
At this point, well-determined visual orbital elements of
the AB system (namely, period and semimajor axis) are cru-
cial to calculate the orbital parallax as well as the inclina-
tions and the semimajor axes of both spectroscopic orbits
A 3D model for αGem AB: orbits and dynamics Page 5 of 12 46
Fig. 3 The flowchart of the
Docobo and Andrade (2006)
methodology as applied to this
particular case
(see Sect. 3.2). A flowchart that describes this methodology
is shown in Fig. 3. Furthermore, a detailed description with
examples can be seen in Docobo and Andrade (2006).
In the end, the only orbital elements that will remain un-
known will be the inner angles of the nodes, Ω1and Ω3,of
the spectroscopic subsystems. For that reason, we call this a
quasi-3D model. Further on in this paper, we will offer a few
clues regarding the estimation of plausible values for these
orbital elements.
3.1 Computation of the individual masses and spectral
types
This procedure also provides individual masses and spec-
tral types which are listed in Table 4. We conclude that each
binary subsystem comprises a massive component of spec-
tral type A along with a cool dwarf companion of spectral
type M, the latter accounting for a fifth of the total mass of
each subsystem.
Table 4 Spectral types and
derived individual masses Sp M[M]
Aa A2V 2.57 ±0.11
Ab M0V 0.53 ±0.09
Ba A5V 2.13 ±0.10
Bb M1V 0.49 ±0.09
Tot a l 5 .72 ±0.20
These individual masses are somewhat larger than those
obtained in previous studies (Heintz 1988; Torres and Ribas
2002) but this is easy to understand if we take into account
that they used larger values of the parallax. In reality, the
new total mass obtained in this work from the sum of indi-
vidual masses (5.72 ±0.20 M) matches well with the dy-
namical mass obtained from orbital elements together with
the Hipparcos parallax, that is, 5.47 ±0.97 M. Actually,
this match is even better if we take into consideration the
old Hipparcos reduction (5.70 ±0.36 M).
46 Page 6 of 12 J.A. Docobo et al.
Table 5 Semimajor axes,
inclinations, largest angular
separations, and differences of
magnitudes of the spectroscopic
subsystems
Subsystem a[au] a[mas] i[◦]ρmax [mas] m
(Aa, Ab) 0.127 ±0.007 8.17 ±0.68 25.6±4.311.1±1.07.6
(Ba, Bb) 0.0562 ±0.0033 3.61 ±0.30 53.9±13.03.61 ±0.30 7.5
Indeed, the fractional mass given by
f=MB
MA+MB
,(2)
where MAand MBare the masses of the A and B spec-
troscopic subsystems, respectively, turns out to be 0.458 ±
0.017. That is, it is only slightly larger than that given by
Heintz (1988), 0.436 ±0.003, obtained from relative posi-
tions of stars A and B with respect to star C (YY Gem).
3.2 Computation of the semimajor axes and
inclinations of the spectroscopic orbits
By using the new visual orbit (this work) in conjunction
with the two spectroscopic orbits Vinter Hansen (1940) and
the Hipparcos parallax, we have calculated the most proba-
ble values of both the separations between their components
as well as of the spectroscopic binary inclinations using the
above-mentioned Docobo and Andrade (2006) methodology
according to the following relationships:
a12 =aP12
P2M1+M2
M1/3
,(3)
a1sin i12 (in km)=13751K1P121−e2
121/2,(4)
where a12,i12 ,e12 , and P12 are the semimajor axis, the incli-
nation, the eccentricity and the period of each spectroscopic
subsystem, respectively. On the other hand, M,M1, and M2
are the total mass and the masses of each component, re-
spectively. In addition, a1=a12 M2
M1+M2.
Furthermore, regarding an eventual observation of the
spectroscopic subcomponents by means of optical tech-
niques, we have also estimated their largest angular sepa-
rations. Table 5summarizes these results along with the dif-
ferences of magnitudes between the components according
to the Straižys and Kuriliene (1981) calibration.
These semimajor axes well match those estimated by
M. Güdel, 0.121 and 0.052 au, respectively, in a private
communication quoted by Stelzer and Burwitz (2003). Like-
wise, the inclinations are in reasonable agreement with those
roughly estimated by Torres and Ribas (2002)forthe(Aa,
Ab) and (Ba, Bb) subsystems, (∼28◦) and (∼90◦), respec-
tively.
3.3 Computation of the parallax
Regarding the system parallax, we have computed the or-
bital parallax (derived from the new visual orbit, AB, along
with the new estimation of the total mass). This is listed in
Table 6 Comparison of parallaxes
Parallax [mas]
Trigonometric (I) (Heintz 1988)73
Trigonometric (II) (Heintz 1988) 65.5
Trigonometric (ESA 1997)63.27 ±1.23
Trigonometrica(Torres and Ribas 2002)66.90 ±0.63
Trigonometric (van Leeuwen 2007)64.12 ±3.75
Orbital (this work) 63.19 ±0.92
aThis was re-reduced from the Hipparcos transit data (ESA 1997)ac-
counting for the orbital motion of the visual binary
Table 6together with the trigonometric parallaxes measured
by the Hipparcos mission (old and new reductions) and with
those given by Heintz (1988) and Torres and Ribas (2002).
We note that the old Hipparcos trigonometric parallax is
slightly smaller than the new one but, in comparison, the
former almost exactly matches the accurate orbital parallax
calculated in this work. This type of discrepancy between
the parallaxes given by the old and the new reductions of the
Hipparcos data when the old value seems to be more accu-
rate, has occurred previously (Docobo and Andrade 2013).
4 Dynamical analysis and the 3D model
4.1 Aims and models
Considering that we have no initial knowledge of the angles
of the nodes of the spectroscopic subsystems in addition to
the fact that the ratios between the orbital period of the outer
orbit and each of the smallest orbital periods of the inner or-
bits are huge (roughly 6 ·104), the implementation of many
direct integrations of the motion equations of this four-body
problem is not practical. From another standpoint, such huge
ratios have led some authors (Anosova et al. 1989; Beust
2003) to conclude that the nearly circular binary (Ba, Bb) is
very stable. In fact, as suggested by Schmitt et al. (1994), it
would be in synchronous rotation.
Taking all of this into account, in order to accomplish an
analysis of the long-term dynamics of αGem AB, we will
consider that each binary subsystem operates as a one-body
perturber on the other binary subsystem. Thus, we will focus
our attention on the following systems:
1. [(Aa, Ab), B] and
2. [(Ba, Bb), A].
A 3D model for αGem AB: orbits and dynamics Page 7 of 12 46
Fig. 4 System of Jacobi coordinates for a three-body system (dis-
tances and sizes are not to scale)
During this investigation, we hope to eventually be able
to determine those regions of the orbital elements space
with the most probable values for the inner angles of the
nodes.
Assuming the above-mentioned facts, our model will be
that of a hierarchical three-body system characterized by an
Ab body in an elliptical inner orbit around the Aa body and
a third body, B, moving in an outer elliptical orbit around the
centre of mass of the subsystem (Aa, Ab) as shown in Fig. 4
(and similarly in case (ii)). As usual, we will work with the
system of Jacobi coordinates.
According to the Mardling and Aarseth (2001) stability
criterion valid for non-coplanar systems, one body may es-
cape from a hierarchical three-body system if it obeys:
rout
p
ain <2.8(1+qout)1+eout
(1−eout)1/22/51−0.3I[◦]
180 ,(5)
where eout and rout
pare the outer eccentricity and perias-
tron separation, qin =M2/M1and qout =M3/(M1+M2)
are the inner and outer mass ratios, respectively, ain is the
inner semimajor axis, and I[◦]is the mutual inclination in
degrees.
Regarding the stability of the αGem [(Aa, Ab), B] and
[(Ba, Bb), A] systems, this criterion is not satisfied by a
very wide margin independent of the value considered for
the inner angles of the nodes. Other modern criteria such
as that of Valtonen and Karttunen (2006) yield very similar
results.
A summary of the entire set of orbital data concerning
this stellar system used in the calculations is presented in
Table 7.
With the aim of distinguishing the main dynamical fea-
tures of these three-body systems and, afterwards, to deter-
mine the most probable values of the nodes, we will study
some aspects of their secular dynamics.
4.2 Constraints in the inner angles of the nodes
First, we will ascertain the set of allowed mutual inclina-
tions determined by the remaining orbital elements using the
well-known expression:
cosI=cos iin cosiout +sin iin sin iout cos(Ωout −Ωin), (6)
with Ibeing the mutual inclination between both the inner
and outer orbital planes.
Plots displaying the dependencies of the mutual inclina-
tion on the inner angle of the node for each subsystem are
showninFig.5. These plots allow us to know what combi-
nations of inclinations and angles of the nodes can give rise
to the well-known Kozai-Lidov cycles (hereafter, KL cycles)
(Kozai 1962;Lidov1962), a mechanism that transfers angu-
lar momentum between the inner and the outer orbits which
can happen only if the mutual inclination is large enough
(between 39.
◦2 and 140.
◦8).
By means of a numerical analysis of Eq. (6), we learn
that the [(Aa, Ab), B] system can undergo KL cycles for
whichever value of its angle of the node. In contrast to that,
the [(Ba, Bb), A] system may or may not undergo KL cycles
depending on its angle of the node. Results are shown in
Table 8.
When KL cycles are present, long-period oscillations of
the inner eccentricity and inclinations are observed. The
time scale of these KL cycles, tKL, can be estimated (see
Table 7 Summary of the
orbital parameters of αGem AB
aP,T,e,andω(Vinter Hansen
1940); a,i,Ω,M1,andM2
(this work)
Subsystem 1 (Aa, Ab)aSubsystem 2 (A, B) Subsystem 3 (Ba, Bb)a
P[yr] 2.52224 ·10−2459.8±4.28.017180 ·10−3
T1934.288668 ±0.000077 1959.1±0.9 1934.173037 ±0.000309
e0.499 ±0.010 0.337 ±0.004 0.002 ±0.004
a[au] 0.127 ±0.007 105.0±6.20.0562 ±0.0033
i[◦]25.6±4.3 114.7±0.353.9±13.0
Ω[◦](seeTable10)41.3±0.2(seeTable10)
ω[◦] 266.4±1.8 251.1±1.494.7±13.9
46 Page 8 of 12 J.A. Docobo et al.
Fig. 5 Plots of the mutual inclinations against the angles of the nodes. The regions where Kozai-Lidov cycles can eventually occur are those
shown in light red. In contrast, regions where Kozai-Lidov cycles cannot occur are shown in green
Table 8 Angles of the nodes, ΩKL, which lead to Kozai-Lidov cycles
along with their time scales, tKL
Subsystem ΩKL [◦]tKL [yr]
(Aa, Ab) (0.
◦0, 360.
◦0) 106
(Ba, Bb) (85.
◦2,177.
◦4)∪(265.
◦2,357.
◦4)107
Table 8)using(Lietal.2014):
tKL Min
Mout aout
ain 31−e2
out3/21−e2
in1/2Pin,(7)
with Min and Mout being the mass of the inner binary and
the mass of the outer perturber, respectively. The in and out
subindices indicate inner and outer orbits, respectively.
Therefore, almost circular although highly inclined orbits
can become very eccentric over time. In accordance with the
assumption that the initial inner eccentricity is small (nearly
zero), the quadrupole order expansion of the hierarchical
three-body Hamiltonian allows us to calculate the maximum
eccentricity, emax, that can be induced by the KL cycles de-
pending on the mutual inclination:
emax =1−5
3cos2I. (8)
On the other hand, since the mutual inclination can only
vary depending on the inner angle of the node, we can solve
Eq. (8) in order to know the maximum eccentricities allowed
for the angles of the nodes. These curves are shown in Fig. 6.
Additionally, we must take into account that such max-
imum eccentricities are limited due to the fact that the dis-
tance in the periastron should, in any case, be less than the
Roche limit. Otherwise, the motion could end in the merging
of the system. Thus, in practice, considering the small sep-
arations between companions in both subsystems of αGem
AB, the actual maximum eccentricities must be less than
those obtained from Eq. (8). In the computation of these up-
per limits, eupp
max, we have estimated the effective radii of the
Roche lobes, RL, using the well-known expression (Eggle-
ton 1983):
RL=0.49q2/3
0.6q2/3+ln(1+q1/3)a, (9)
where q=M1/M2. Therefore, we establish the merging
condition that the periastron distance must be larger than the
Roche limit and, subsequently, that
eupp
max <1−RL
a.(10)
In addition, we must also investigate if each system has
a lower boundary, elow
max, determining the existence of mini-
mums in the curves shown in Fig. 6. The results are listed in
Table 9.
Again, solving Eq. (8) in an implicit manner, and taking
into account the above-mentioned constraints, we obtain the
angles of the nodes allowed.
Regarding motion in the (Aa, Ab) subsystem, it under-
goes KL cycles for whichever value of its angle of the node.
A 3D model for αGem AB: orbits and dynamics Page 9 of 12 46
Fig. 6 Plots of the maximum eccentricities, due to the Kozai-Lidov
cycles, against the angles of the nodes. Regions where the maximum
eccentricity is less than eupp
max and, therefore, motion is allowed, are
shown in green. In contrast, those where motion is not allowed are
shown in light red
Table 9 Stellar radii (R), distances in the periastra (rp), Roche limits (RL), and boundaries for the maximum eccentricities due to the Kozai-Lidov
cycles (elow
max) and to the merging condition (eupp
max)
Subsystem Ra+Rb[R]arp[R]RL[R]elow
max eupp
max
(Aa, Ab) 2.0+0.60 13.8±0.814.2±0.4 0.116 0.482 ±0.015
(Ba, Bb) 1.7+0.54 12.1±0.76.1±0.2 0.000 0.491 ±0.016
aEstimated using the Straižys and Kuriliene (1981) calibration from masses
This should be roughly between 178.
◦9 and 263.
◦7 (direct
motion) or 358.
◦9 and 83.
◦7 (retrograde motion). In fact, any
other value would lead to extreme KL cycles that could
eventually result in the merging of both components.
In regard to the (Ba, Bb) subsystem, we must distinguish
two cases: if it is undergoing KL cycles, its angle of the node
would vary from 167.
◦0to177.
◦3, and from 265.
◦3 to 275.
◦6
(direct motion) or from 85.
◦3to95.
◦6, and from 347.
◦0to
357.
◦3 (retrograde motion). Otherwise, it could vary from
177.
◦3 to 265.
◦3 (direct motion), and from 357.
◦3to85.
◦3
(retrograde motion).
The windows of feasible values of the angles of the nodes
obtained from a dynamical analysis can be used to determine
the expected values for such orbital elements.
4.2.1 Expected angle of the node for (Aa, Ab)
Taking into account the aforementioned merging condition,
the allowed interval for the maximum eccentricity caused
by KL cycles is also constrained by the current value of the
eccentricity. In fact, the latter cannot be larger than its up-
per boundary for a given KL cycle. As a consequence, in
this case, the emax corresponding to the present configura-
tion must be confined to the interval (elow
max,eupp
max), that is,
(0.116, 0.482).
In this respect, the actual eccentricity closely matches
its upper boundary. This suggests that this binary would
be in the maximum eccentricity stage corresponding to the
present KL cycle. In this way, four possibilities around each
extreme value arise for the angle of the node: two in direct
motion and another two in retrograde motion (see Table 10).
4.2.2 Expected angle of the node for (Ba, Bb)
Under the same assumptions, emax must be confined to the
interval (0.000, 0.491). Taking into account that the current
eccentricity is practically zero as well as that this binary
is likely undergoing tidal locking (Schmitt et al. 1994), we
suggest that the KL mechanism is no longer driving this sys-
tem. Thus, we propose the means corresponding to each re-
46 Page 10 of 12 J.A. Docobo et al.
Table 10 The most probable angles of the nodes
Subsystem Direct motion Retrograde motion
(Aa, Ab) 180.
◦2±1.
◦90.
◦2±1.
◦9
262.
◦4±1.
◦982.
◦4±2.
◦1
(Ba, Bb) 221.
◦3±31.
◦041.
◦3±31.
◦0
Table 11 Combinations of allowed mutual inclinations
System Direct motion Retrograde motion
[(Aa, Ab), B] 132.
◦3±2.
◦647.
◦7±3.
◦5
[(Ba, Bb), A] 168.
◦6±13.
◦011.
◦4±13.
◦0
Fig. 7 Contour plot of the mutual inclinations depending on the angle
of the node and the inclination of each inner orbit. Positions related to
the (Aa, Ab) and (Ba, Bb) subsystems are coloured in black and white,
respectively. Bars show standard uncertainties. Regions that are free
of Kozai-Lidov cycles are those inside the green lines.Theupper one
corresponds to binaries with direct motion, whereas the lower one is
where the binaries exhibit retrograde motion
gion free of KL cycles as the most probable values for the
angle of the node. As a consequence, uncertainties are larger
than in the (Aa, Ab) case. Results are also listed in Table 10.
4.3 Long-term mutual inclinations
Taking into consideration the complete set of combinations
among these angles of the nodes as well as Eq. (6), we can
calculate what mutual inclinations are allowed for the entire
system. Those are summarized in Table 11 and their values
along with the corresponding angles of the nodes are shown
using a contour plot in Fig. 7.
In agreement with the analysis carried out earlier, we can
assert that the [(Aa, Ab), B] system would be undergoing KL
cycles whether the motion is direct or not. On the contrary,
the [(Ba, Bb), A] system would have passed this phase and,
in any case, its mutual inclination would be relatively low at
present.
Regarding more bizarre features such as an eventual flip
of the orbital plane of any of these systems, we have tested
the analytical criterion given by Li et al. (2014). According
to that, an eccentric inner orbit with an initial near-coplanar
configuration in a hierarchical three-body system with an ec-
centric outer perturber can flip its orientation by about 180◦
if the parameter, , measuring the relative significance of the
octupole to quadrupole term (Naoz et al. 2013),
=Ma−Mb
Ma+Mb
ain
aout
eout
1−e2
out
,(11)
where Maand Mbare the masses of the inner companions
and in and out subindices indicate inner and outer orbits,
respectively, is larger than c,
c=8
5
1−e2
in
7−ein(4+3e2
in)cos(ωin +Ωin).(12)
The /cratios of the both [(Aa, Ab), B] and [(Ba, Bb),
A] systems are on the order of 10−3and 10−4, respectively.
Thus, we should not expect such behaviour in the future evo-
lution of αGem AB.
4.4 Full orbits for the (Aa, Ab) and (Ba, Bb) subsystems
Due to the above-mentioned constraints concerning the mo-
tion of the (Aa, Ab) and (Ba, Bb) binaries, we can comple-
ment their sets of orbital elements (see Table 7) with reli-
able values of the angles of the nodes (see Table 10). Never-
theless, ambiguity concerning the relative direction of mo-
tion cannot be eliminated. The apparent orbits are shown in
Fig. 8. Note that both directions (direct and retrograde) are
possible.
5 Conclusions
1. After an investigation of the most recently calculated or-
bits for the system, Castor AB, we proceeded to make
use of the complete set of observations in order to deter-
mine a new solution that generally improves the quality
controls of previous orbits, keeping in mind the assigned
weights of each observation according to USNO criteria.
2. Using the application of a new ad hoc version of the
methodology developed by Docobo and Andrade (2006),
we present a quasi-three dimensional model of the four-
star system [(Aa, Ab), (Ba, Bb)] at the same time that
we offer the most probable values of the masses, spectral
types, and magnitudes for each component as well as the
parallax.
A 3D model for αGem AB: orbits and dynamics Page 11 of 12 46
Fig. 8 Apparent orbits for αGem AaAb (left)andαGem BaBb (right). The scale on both axes is in arc seconds and the dashed line is the line of
nodes. Retrograde motion in the αGem AaAb orbit is shown using a dotted line
3. An exhaustive analysis of the dynamics of the system
also permitted us to be able to evaluate the most prob-
able values of the angles of the node for the orbits of the
spectroscopic subcomponents (Aa, Ab and Ba, Bb) and,
at the same time, to estimate their maximum angular sep-
arations as concerns possible optical resolution.
4. Close binaries (with orbital periods of a few days) tend
to be circular because of tidal interactions that efficiently
circularise initially close eccentric binaries. Taking this
into account, we suggest that the (Ba, Bb) subsystem,
probably after a phase of high eccentricities caused by
KL cycles, has already reached this stage. Therefore, this
system would be in a stable circular orbit resistant to the
minor perturbations caused by the distant (Aa, Ab) sub-
system.
5. On the contrary, the (Aa, Ab) subsystem is still subject to
KL cycles that, until now, have avoided an effective cir-
cularisation of its orbit. Moreover, its current eccentricity
has led this binary to practically cross its Roche limit dur-
ing the periastron passages. Although we cannot discard
that tidal effects as well as general relativity precession
could finally suppress KL cycles in a shorter time scale,
the dynamical evolution of this binary could end in the
merging of both companions.
Acknowledgements The authors thank W. Hartkopf of USNO for
providing the data and other relevant information necessary for this
research.
This paper was supported by the Spanish Ministerio de Economía
y Competitividad under Project AYA 2011-26469 as well as by the
IEMath-Galicia Network (FEDER-Xunta of Galicia).
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