Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.sao.ru/hq/balega/PUBL_BAL/PUB_2003/aa3732.ps Äàòà èçìåíåíèÿ: Wed Feb 24 16:01:49 2010 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 09:51:51 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: blue stragglers

A&A 409, 245--250 (2003)
DOI: 10.1051/0004­6361:20031064
c
# ESO 2003
Astronomy
&
Astrophysics
Fundamental parameters and origin of the very
eccentric binary 41 Dra #
A. Tokovinin 1 , Y. Y. Balega 2 , E. A. Pluzhnik 2 , N. I. Shatsky 3 , N. A. Gorynya 4 , and G. Weigelt 5
1 Cerro­Tololo Inter­American Observatory, Casilla 603, La Serena, Chile
2 Special Astrophysical Observatory, Nizhnij Arkhyz, Zelenchuk region, Karachai­Cherkesia 369167, Russia
3 Sternberg Astronomical Institute, 13 Universitetskii pr., 119992 Moscow, Russia
4 Institute of Astronomy of Russian Acad. Sci., 48 Pyatnitskaya St., 109017 Moscow, Russia
5 Max­Planck­Institut f˜ur radioastronomie, Auf dem H˜ugel 69, 53121 Bonn, Germany
Received 17 March 2003 / Accepted 28 May 2003
Abstract. The evolutionary status and origin of the most eccentric known binary in a quadruple system, 41 Dra (e = 0.9754,
period 3.413 yr), are discussed. New observations include the much improved combined speckle­interferometric orbit, resolved
photometry of the components and their spectroscopic analysis. The age of the system is 2.5 ± 0.2 Gyr; all four components
are likely coeval. The high eccentricity of the orbit together with known age and masses provide a constraint on the tidal
circularization theory: it seems that the eccentric orbit survived because the convective zones of the F­type dwarfs were very
thin. Now as the components of 41 Dra are leaving the Main Sequence, their increased interaction at each periastron passage
may result in detectable changes in period and eccentricity.
Key words. stars: binaries: visual -- stars: binaries: spectroscopic -- stars: formation -- stars: individual: HD 166866, HD 166865
1. Introduction
Multiple stars remain an active research topic, despite having
been studied for more than 200 yrs. Nowadays the emphasis
has shifted from determination of orbits and masses (although
masses accurate to few percent are still needed to check stellar
models) to investigation of specific and rare objects, to statistics
of binary populations and to binary star formation. The multi­
ple system studied here fits these criteria: it is a unique object
with well­defined fundamental parameters that holds promise
for elucidating some aspects of multiple­star formation.
Two 6th magnitude stars, 40 and 41 Dra (respectively,
HR 6809 and 6810, HIP 88127 and 88136, HD 166865
and 166866), are 19 ## from each other on the sky and form
a physical common proper motion couple # 2308 discovered
by V. Struve in 1832 and also known as ADS 11061. Moreover,
each is itself a spectroscopic binary, making the whole sys­
tem quadruple. Hereafter we denote the visual components
as A = 41 Dra and B = 40 Dra, and the spectroscopic com­
ponents as Aa, Ab, Ba, and Bb.
The orbit of Aab published by Tokovinin (1995, here­
after T95) is very eccentric -- the most eccentric among all
known spectroscopic binaries. The Aab system was resolved by
Send o#print requests to: A. Tokovinin,
e­mail: atokovinin@ctio.noao.edu
# Tables 1, 2, and 3 are only available in electronic form at the CDS
via anonymous ftp to
cdsarc.u­strasbg.fr (130.79.128.5) or via
http://cdsweb.u­strasbg.fr/cgi­bin/qcat?J/A+A/409/245
speckle interferometry (designated as BAG 6 in the WDS cata­
logue) and its preliminary interferometric orbit was computed
by Balega et al. (1997). There are now enough speckle data
to derive a new good­quality combined orbit. Radial velocities
measured during the last passage through periastron in May--
June 2001 permit us to establish the orbital period very accu­
rately and even to address the issue of orbit evolution.
The summary of interferometric measurements of the mag­
nitude di#erence of Aab in di#erent pass­bands, from visible
to the infrared, is provided by Balega et al. (2001b) who show
that #m is practically constant over a wide range of wave­
lengths, with the average value of #m = 0.426 ± 0.028. It
points to the fact that Aa and Ab have very similar e#ective
temperatures and di#erent radii. Low­resolution spectroscopy
reported in that work was used to fit standard photospheric
models in an attempt to find the best­matching e#ective temper­
atures and radii of the components Aa and Ab. A similar analy­
sis of both 40 and 41 Dra was done by Al Wardat (2002). There
is some disagreement in the temperature estimates for 41 Dra
(6500 and 6100 K) between those two works, neither of which
quotes errors on these numbers. High­resolution spectroscopy
revealed that 41 Dra has a slight (+0.2 dex) over­abundance
of iron compared to the Sun (Balega et al. 2003). X­ray
flux measured by Pizzolato et al. (2000) seems to be normal
for F7V dwarfs.
The visual secondary component, 40 Dra, did not at­
tract much attention, being a typical ``garden variety'' double­
lined spectroscopic binary with 10.5 d period and moderately
eccentric (e = 0.38) orbit which was first computed by

246 A. Tokovinin et al.: Eccentric binary 41 Dra
Table 4. New combined orbit of 41 Dra.
Element Value Error
Period P, days 1246.680 0.004
Periastron epoch T , JD 2 449 571.037 0.008
Eccentricity e 0.9754 0.0001
Semi­major axis a, arcsec 0.0706 0.0014
Position angle of
node# a , deg 1.9 1.7
Argument of periastron # a , deg 127.31 0.13
Inclination i, deg 49.7 2.9
Primary amplitude K 1 , km s -1 44.62 0.10
Secondary amplitude K 2 , km s -1 48.06 0.11
System velocity V 0 , km s -1 5.76 0.05
Primary mass, M # 1.28 0.15
Secondary mass, M # 1.20 0.14
Orbital parallax, mas 23.0 2.2
Boothroyd (1922) and re­determined with greater accuracy
in T95. Here we come back to this orbit to check whether its
elements are stable over time.
2. New combined orbit of 41 Dra
The first double­lined spectroscopic orbit published in T95
was based on the periastron passage in August 1994 ob­
served by N.I.S and N.A.G. The following periastron around
January 1, 1998, was not covered because of cloudy sky.
However, N.I.S and N.A.G did obtain the data on the next oc­
casion, in May--June 2001. The observations were made with
the 70­cm telescope at Moscow University campus using the
correlation radial velocity meter (Tokovinin 1987). As in T95,
the velocities were derived from fitting Gaussian curves to the
correlation dips; partially blended dips were split by fixing the
width and contrast of the fitted Gaussians. The individual ve­
locities and their residuals to the new orbit are given in Table 1,
published electronically. Table 2 lists the observations of the
Bab binary obtained on this occasion.
All speckle interferometry of 41 Dra comes from the
Russian 6 m telescope. We add to the published data another
point taken in 2001.27 and provide in Table 3 all speckle mea­
surements and their residuals.
Merging new and old data in the combined speckle­
spectroscopic orbital solution yields the elements and their for­
mal errors derived by weighted least­squares fitting (Table 4).
Initially the weights were taken to be inversely proportional to
the squares of observational errors, assumed to be 2 mas for
speckle­interferometry. It turns out that speckle data are more
precise, while the errors of radial velocities are larger than their
formal estimates. The final solution was computed by equal­
izing the relative weights of interferometry and velocities, so
that # 2 /N ratio is close to 1 for all kinds of data. The result
did not change significantly compared to the initial weight­
ing. The weighted rms residuals to the orbit are 0.53 km s -1
and 0.59 km s -1 for Aa and Ab velocities, 0.9 # and 1 mas for
interferometry. One interferometric measurement (1994.71)
0.99 1 1.01 1.02 1.03
Phase
­60
­40
­20
0
20
40
60
80
V
R
,
km/s
Aa
Ab
Fig. 1. Radial velocities of 41 Dra during the periastron passage
in 2001. The primary component is denoted by filled squares and full
line, secondary component by empty squares and dashed line. The de­
scending portion of the Aa radial velocity curve lasts only 3 days.
with 25 mas separation -- at the limit of 6 m telescope reso­
lution -- was excluded, but the overall quality of interferom­
etry is excellent. This orbit represents the state­of­the­art in
interferometry: a better orbit could be obtained only with long­
baseline interferometers, but none of the current interferom­
eters yet reaches 6­th magnitude stars. Note that two perias­
tron observations define the 3.41 yr period with an error of
only 6 min., which might be the most accurate period among
resolved binaries. The timing of periastron passages is accurate
to #10 min. because the component acceleration in this eccen­
tric orbit is high (some 3 km s -1 per hour near periastron), de­
spite the long period. In Fig. 1 we plot the radial velocities for
the last periastron passage and in Fig. 2 we plot the interfero­
metric observations.
The masses and orbital parallax are computed directly from
the combined orbit. Estimates of their errors take into ac­
count correlations between the orbital elements. The mass ratio
q = M 2 /M 1 = 0.928 ± 0.003 is determined with a much better
accuracy than the masses themselves, hence any system model
must precisely fit the mass ratio. The Hipparcos parallax (ESA
1997) was measured independently for A and B (18.8 ± 1.8
and 19.6 ± 3.8 mas respectively). It is manifestly wrong, being
distorted by the orbital motion of Aab as noted by Shatskii &
Tokovinin (1998) for this and other similar systems.
The semi­major orbital axis is 3.1 AU The components
approach each other at periastron to a distance of 0.078 AU
or 16.7 R # .
3. Evolution of the orbital elements?
New spectroscopic data on the systems of Aab and Bab ob­
tained in 2001 can serve to study the possible evolution of those
orbits.
We determined the eccentricity and semi­amplitudes
of Aab from the 2001 velocities only, in the hope of detect­
ing any change of orbital elements that might have occurred

A. Tokovinin et al.: Eccentric binary 41 Dra 247
Fig. 2. Interferometric observations of 41 Dra (filled circles) are con­
nected to the corresponding points on the orbit ellipse. The position of
the primary is marked by the asterisk in the upper right corner.
Table 5. Orbital elements of 41 Dra in 1994 and 2001.
Element 1994 2001
Value Error Value Error
T , JD 49 571.047 0.009 52 064.3929 0.0025
e 0.9754 0.0001 0.9752 0.0001
#, deg 127.6 0.2 127.0 0.2
K 1 , km s -1 44.79 0.14 44.72 0.13
K 2 , km s -1 47.88 0.17 47.75 0.13
V 0 , km s -1 5.84 0.05 5.75 0.09
since the periastron passage in 1994. The result is presented in
Table 5. Compared to 1994, the changes are below the signif­
icance level. But they are, curiously enough, going in the ex­
pected direction: the eccentricity is diminishing, accompanied
by a decrease in both semi­amplitudes! Observations of further
periastron passages will show if such fast eccentricity variation
is indeed taking place.
A more sensitive test for orbit evolution is o#ered by pe­
riod changes. If circularization proceeds with approximately
constant periastron distance (Goldman & Mazeh 1994), the
quantity P(1 - e) 3/2 is constant, so any change of eccentric­
ity corresponds to the change of period, dP/P = 1.5de/(1 - e).
Hence, if e decreased by 0.0002, the period would be shortened
by 15 d. Although P is known to within 0.004 d, we have not
yet observed three periastron passages as needed to detect pe­
riod variation. At this level of precision, an eccentricity change
of 5 â 10 -8 is observable. Precise timing of future periastron
passages is the best way to detect ongoing orbit circularization.
It was conjectured in T95 that the larger semi­amplitudes
of Bab measured by Boothroyd in 1920 compared to the or­
bit in 1986--94 could result from the precession engendered by
Table 6. Orbital elements of 40 Dra in 1986--1994 and 2001.
Element 1986--1994 2001
Value Error Value Error
P, days 10.52785 0.0020 10.52785 *
T , JD 48 000.008 0.011 52 053.260 0.040
e 0.374 0.003 0.380 0.008
#, deg 246.2 0.4 244.35 1.33
K 1 , km s -1 39.14 0.14 39.01 0.39
K 2 , km s -1 42.91 0.17 42.86 0.43
V 0 , km s -1 5.76 0.07 5.31 0.19
Table 7. Photometry of 41 and 40 Dra.
Comp. Source V B - V U - B V - R
A Tycho 5.682 0.508 -- --
JM53 5.68 0.50 -0.01 --
WBVR 5.699 0.506 -- 0.432
B Tycho 6.022 0.513 -- --
JM53 6.04 0.51 -0.01 --
WBVR 6.066 0.507 -- 0.435
an invisible more distant companion, like in other such cases
detected by Mayor & Mazeh (1987). The 24 new observa­
tions of Bab obtained in 2001 were used to re­calculate the
orbit. We adjusted all elements except the period (Table 6).
The comparison with previous elements shows that there was
no significant change, although both semi­amplitudes did de­
crease very slightly. The change in the systemic velocity V 0
is only marginal. If there is indeed a fifth component orbiting
around Bab, its period should be longer than 15 yrs covered
by our observations. It cannot be longer than #100 yr, other­
wise this sub­system would be dynamically unstable against
perturbations from A. Existence of the fifth companion seems
unlikely.
4. Physical parameters of the components
We now know from the di#erential photometry of Aab that
the colors of Aa and Ab are very similar. The combined col­
ors of visual components Aab and Bab are also very close,
despite the magnitude di#erence of #m = 0.34. Photometry
from the Tycho space mission (ESA 1997), Johnson & Morgan
(1953, JM53) and Kornilov et al. (1991, WBVR) is summa­
rized in Table 7.
Processing of the high signal­to­noise correlation dips mea­
sured in 2001 enabled us to find the equivalent widths of Aa
and Ab dips as 1.20 ± 0.01 and 0.77 ± 0.01 km s -1 and the
projected rotation velocities V sin i of 5.7 ± 0.3 and 4.4 ±
0.5 km s -1 , confirming the T95 results. The estimates of mag­
nitude di#erences #m were made in T95 under the assumption
that all components are on the Main Sequence (MS). Now we
know that the e#ective temperatures of all components are sim­
ilar, which permits us to derive #m directly from the ratio of
dip equivalent widths: #m = 0.48 ± 0.02 for Aab (compare to

248 A. Tokovinin et al.: Eccentric binary 41 Dra
Table 8. Physical parameters of the components.
Comp. M/M # R/R # M V
Aa 1.39 1.93 2.99
Ab 1.30 1.63 3.42
Ba 1.32 1.57 3.28
Bb 1.20 1.32 3.85
0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.6
B­V
2.5
3
3.5
4
4.5
M V
Aa 1.39
Ba 1.32
Ab 1.30
Bb 1.20
2.51 Gyr
2.24 Gyr
2.82 Gyr
Fig. 3. Components of ADS 11061 on the isochrones of Girardi
et al. (2000). The absolute magnitudes M V are measured, whereas the
B - V colors are ``assigned'' to match the combined colors and to make
the components coeval. The numbers show component's masses.
#m = 0.43 from speckle­interferometry) and #m = 0.56 ±0.03
for Bab. This refers approximately to the photometric band V.
The orbital parallax corresponds to the distance modulus
m - M = 3.19 ± 0.25 (refined to m - M = 3.25 ± 0.05 be­
low). This distance and the photometry lead to the absolute
magnitudes M V in the last column of Table 8. Thus the observa­
tions constrain luminosities and colors of all 4 components to
a narrow region of the Hertzsprung­Russel (H--R) diagram ly­
ing above the MS. The components are slightly evolved F­type
stars; we show below that they can be coeval.
In Fig. 3 we place all 4 components on the evolutionary
tracks computed by Girardi et al. (2000). We use the tracks for
solar metallicity (cf. Balega et al. 2003) with over­shooting.
Theoretical tracks are conveniently related to the observed val­
ues, B - V and M V , removing any additional uncertainty of
bolometric corrections and e#ective temperature scale.
At the age of a few Gyr, as appropriate in this case, the
stars evolve with fairly constant absolute magnitude M V . This
greatly simplifies our task: independently of the exact age, the
mass of Aa component can be found directly from its M V
as 1.39 M # . The mass of Ab follows from the measured mass
ratio. These ``evolutionary'' masses of Aa and Ab are compati­
ble with the directly measured masses. The same procedure is
applied to Ba and Bb, leading to the mass estimates given in
Table 8. With the modeled mass sum of Aa+Ab M = 2.7 M #
we obtain parallax # dyn = aP -2/3 M -1/3 = 22.4 ± 0.5 mas (here
the semi­major axis a is in arcseconds and period P is in years).
Its error comes exclusively from the error of the orbital semi­
major axis of Aab; it is the best current estimate of the distance
to ADS 11061.
We do not know the individual B - V color indices of
the components with su#cient accuracy, but we can select an
isochrone that matches the known combined colors of A and B.
Such a fit is shown in Fig. 3. The B - V colors of components
were ``assigned'' to be on the isochrone. According to the model
(Table 8), the combined B - V colors of Aab and Bab are both
equal to 0.51, their U - B colors are -0.01. The modeled mag­
nitude di#erence of the Aab pair is 0.39 in the V band and 0.43
in the K band. Selecting adjacent isochrones would lead to dis­
crepant combined colors. So, the age of ADS 11061 can be
established as 2.5 ± 0.2 Gyr. The e#ective temperatures of all
components are similar, from the coolest 6280 K for Bb to the
hottest 6380 K for Ba. These estimates depend on the stellar
models used.
Is the axial rotation of components a#ected by their close
neighbors? The angular velocity of the Aa­Ab vector at peri­
astron is (1 - e 2 ) 1/2 (1 - e) -2 = 364 times faster than the av­
erage orbital angular velocity. If Aa were synchronized at pe­
riastron, it would have the equatorial velocity of 28.2 km s -1 ,
and V sin i = 21.6 km s -1 , assuming that rotational axes are
perpendicular to the orbital plane and hence i = 50 # . The mea­
sured V sin i (5.7 and 4.4 km s -1 ) show that both Aa and Ab ro­
tate sub­synchronously. The inclination of the Bab system es­
timated from its spectroscopic mass sum M sin 3 i = 0.273 M #
and the model mass sum 2.52 M # is 35 # . The synchronous ro­
tational velocity is then expected to be V sin i = 10.2 km s -1 ,
to be compared with 7.2 and 6.7 km s -1 reported for Ba and Bb
in T95. Hence, none of the four components is synchronized
with the orbits.
5. Orbit of the AB system
With the ``evolutionary'' mass sum 5.21 M # and the distance to
the system 44.6 pc, we compute the orbit of AB by the dynam­
ical method of Apparent Motion Parameters (AMP), as done
previously by Kiselev & Romanenko (1996) and T95. This
method gives a family of orbits which depend on one unknown
parameter, the distance between the components along the line
of sight r z . For elliptic orbits, the distance between the com­
ponents r obeys the inequality r < r max = 8# 2 M/V 2 , where
the distance is in AU, V is in AU/yr, M is in solar masses.
The relative velocity V of the components and their projected
distance r xy are known and we can write r z = z # r 2
max - r 2
xy ,
so that z = ±1 corresponds to r = r max (parabolic orbits). In
Table 9 we list some of those orbits. Apart from new mass
sum and distance, we used the same input data (relative po­
sition and proper motion) as in T95 and the velocity di#er­
ence V A - V B = 0.05 km s -1 as follows from the spectro­
scopic orbits of Aab and Bab. Because of the short time base of
the Hipparcos experiment, it measured the relative proper mo­
tion of AB with much less precision than ground­based photo­
graphic astrometry. The orbits in Table 9 sensitively depend on
input data, e.g. on parallax, hence they should be considered as
being only indicative of the possible orbital motion of AB.

A. Tokovinin et al.: Eccentric binary 41 Dra 249
Table 9. Orbits of the AB pair computed by the AMP method.
z P e
a# B # i #
yr AU # # # #
--0.9 494 060 0.88 10834 19 302 108 154
--0.7 77 030 0.71 3139 18 320 113 158
--0.3 15 171 0.75 1062 18 358 135 167
0.0 10 019 0.84 806 232 150 178 129
0.3 15 171 0.73 1062 200 174 133 85
0.7 77 030 0.69 3139 200 134 112 64
0.9 494 060 0.87 10834 199 117 108 60
In the last column of Table 9 we give the relative angle #
between the angular momenta of the outer system AB and the
inner system Aab computed as
cos # = cos i out cos i in + sin i out sin i in
cos(# out
-# in ). (1)
It is interesting that all AMP orbits correspond to large relative
inclinations between the orbits of AB and Aab.
6. Dynamics and origin of the multiple system
In this section we try to understand why the orbit of Aab is
so eccentric. First we evaluate the chances of orbit circulariza­
tion to see how such high eccentricity could have survived over
the life­time of this system. Then we investigate the dynamical
origin of high eccentricity.
The very high eccentricity of the Aab orbit and its known
age o#er an interesting possibility to check theories of dissi­
pative orbital circularization. Such a check was attempted for
Gliese 586A (Goldman & Mazeh 1994) with e = 0.9752 and
P = 890 d. Less strict constraints were obtained from an­
other eccentric binary, HD 2909, with e = 0.949, P = 2128 d
(Mazeh et al. 1995). In these works the semi­major axis evolu­
tion time T a is estimated as
T a # 5# -1 T 0 (P/P 0 ) 16/3 (1 - e) 15/2 , (2)
where the constants T 0 = 2 â 10 10 yr, P 0 = 20 d, and the vis­
cosity constant # = (# s /P 0 ) n depends on the prescription as
explained by Goldman & Mazeh (1994): n = 0 for unreduced
viscosity, n = 1 and n = 2 for di#erent theories. The convective
time # s # 0.27 d is estimated from the angular velocity at peri­
astron, so # = 0.014 n . Straightforward application to Aab gives
T a # 3.3 â 10 8 yr for n = 0 and T a # 2.4 â 10 9 yr for n = 1.
The theory of tidal circularization is not yet free of un­
certainties. The case of 41 and 40 Dra is particularly tricky
because their components, while on the MS, were close to
the border­line between convective and radiative stars. Even a
small di#erence in mass has a large influence on the thickness
of convective zones, especially for Aa and Ab. We speculate
that the orbit of Bab is more circular (though still not quite so,
e = 0.38) because Ba and Bb have lower masses and thicker
convective zones than Aa and Ab, whereas periastron distances
in both sub­systems are equal. In short, it seems plausible that
the highly­eccentric orbit of Aab escaped tidal circularization,
and the still eccentric orbit of Bab confirms this hypothesis.
Sub­synchronous rotation is an additional symptom of the in­
e#ciency of tidal friction in Aab and Bab.
If tidal orbit evolution preserves the periastron distance, the
final orbital periods of Aab and Bab after complete circulariza­
tion can be computed as P(1 - e) 3/2 . They turn out to be re­
markably similar, 4.8 and 5.1 days respectively. Such periods
are very frequent among spectroscopic sub­systems in late­type
multiple stars (Tokovinin & Smekhov 2002).
Multiple systems can be chaotic or hierarchical; for
ADS 11061 either may be true, with hierarchical configura­
tions being more probable (T95). If ADS 11061 were chaotic,
we must admit that it survived over #10 6 crossing times, a very
unlikely hypothesis. So we believe that this multiple system is
hierarchical and dynamically stable. The ratio of the periastron
distance of AB to the semi­major axis of Aab a out (1 - e out )/a in
that follows from Table 9 is over 40, confirming this statement.
The high eccentricity of an inner orbit in a hierarchical mul­
tiple system can be attained by the Kozai mechanism (Kozai
1962). If the inner orbit were initially almost perpendicular to
the orbital plane of the outer system, it will change its incli­
nation and eccentricity periodically while preserving the Kozai
invariant # = (1 - e 2 ) cos 2 #, where # is the angle between the
inner and outer orbital angular momentum vectors. The period
of the Kozai cycle is estimated as
T Kozai # P 2
out /P in (1 - e out ) 3/2 . (3)
Kozai cycles are perturbed by the apsidal rotation in the inner
system caused by a component's structure and relativistic pre­
cession. The period of relativistic precession is
T rel = 3.36 â 10 7 (1 - e 2
in )P a/M yr, (4)
where P is in years, a in AU, mass M in solar masses (Holman
et al. 1997). For the Aab system T rel = 6 Myr. If T rel < T Kozai ,
Kozai cycles can no longer occur. Given that T rel depends on e in
and T Kozai does not, the condition T rel # T Kozai will be reached
if e in grows su#ciently during the first Kozai cycle. When this
happens, the relativistic rotation of the line of apsides averages
out the Kozai e#ect, so e in will not evolve periodically and will
remain high. This consideration shows that very similar peri­
astron distances in Aab and Bab may be not a coincidence but
rather a consequence of the interplay between Newtonian and
relativistic dynamics in this multiple system.
Now we estimate T Kozai . The outer period P out is quite un­
certain, however. The values of z = 0, 0.3, 0.7 from Table 9
correspond to T Kozai of 1.9, 10, and 300 Myr, respectively. So,
the condition T Kozai # T rel can be satisfied for some plausible
elements of the AB orbit. It seems thus likely that the high ec­
centricity of Aab is indeed a result of the dynamical evolution
outlined above which occurred in the first 10 Myr after forma­
tion of the multiple system.
The most likely evolutionary scenario of ADS 11061 is as
follows (Fig. 4). This quadruple system was formed 2.5 Gyr
ago from a small gas cloud. Initially, the orbits of Aab and Bab
had long periods, were not very eccentric and almost perpen­
dicular to the orbit of the outer system AB. Within #10 Myr

250 A. Tokovinin et al.: Eccentric binary 41 Dra
Formation
Bab Bab
Aab
Aab
Bab
Aab A*
Bab
Eccentric
orbits orbits
Initial
state
Present
of Aab?
Merger
Fig. 4. Possible evolution of the ADS 11061 multiple system.
both inner orbits acquired high eccentricity, the latter be­
ing limited by the relativistic precession. During the follow­
ing 2.5 Gyr the Bab sub­system was partially circularized, de­
creasing its period to 10.5 d, while the circularization of Aab
did not occur because its components had thinner convec­
tive zones. If both sub­systems were less massive, they would
have been completely circularized now with orbital periods
around 5 d.
Presently the components of Aab increase their radii, evolv­
ing o# the MS. Their strong interaction at each periastron
leads to significant orbital circularization which may even be
detectable by a careful timing of future periastron passages.
Further expansion of Aa and Ab may result in a merger, leav­
ing an unusual triple with apparently non­coeval components
until Bab follows the same route and only a wide pair of blue
stragglers remains.
7. Conclusions
The scenario outlined above can be checked with modern tools
for joint modeling of nuclear and dynamical stellar evolution.
New constraints on the still obscure physics of tidal interac­
tion may be obtained from this system when it is compared to
models.
On the observational side, 40 and 41 Dra will be excellent
candidates for long­baseline interferometers when they are
able to observe 6 m stars. Improved accuracy of the Aab orbit
will lead to very precise mass and distance determination.
By resolving Bab, it will be possible to determine the incli­
nation of its orbit relative to AB and Aab, thus achieving a
complete dynamical description of this interesting quadruple
system.
Acknowledgements. We thank R. Zhuchkov for a careful reading of
the manuscript and for pointing out some errors. The comments of the
anonymous referee helped to improve the article.
References
Al Wardat, M. A. 2002, Bull. Spec. Astrophys. Obs., 53, 51
Balega, Y. Y., Leushin, V. V., & Weigelt, G. 2003, Bull. SAO, 55, in
press
Balega, I. I., Balega, Y. Y., Hofman, K.­H., et al. 2002, A&A, 385, 87
Balega, Yu. Yu., Leushin, V. V., & Pluzhnik, E. A. 2001b, Bull. Spec.
Astrophys. Obs., 51, 61
Balega, I. I., Balega, Y. Y., Hofman, K.­H., & Weigelt, G. P. 2001a,
Astron. Lett., 27, 95
Balega, I. I., Balega, Y. Y., Maksimov, A. F., et al. 1999, A&AS, 140,
287
Balega, I. I., Balega, Y. Y., Falcke, H., et al. 1997, Astron. Lett., 23,
172
Boothroyd, S. L. 1922, Publ. Dom. Astrophys. Obs., 1, 246
ESA 1997, The Hypparcos and Tycho Catalogues. ESA SP­1200
Girardi, L., Bressan, A., Bertelli, G., & Chiosi, C. 2000, A&AS, 141,
371
Goldman, I., & Mazeh, T. 1994, ApJ, 429, 362
Holman, M., Touma, J., & Tremaine, S. 1997, Nature, 386, 254
Johnson, H. L., & Morgan, W. W. 1953, ApJ, 117, 313 (JM53)
Kiselev, A. A., & Romanenko, L. G. 1996, Astron. Rep., 40, 795
Kornilov, V. G., Volkov, I. M., Zakharov, A. I., et al. 1991, Catalog of
WBVR magnitudes of Bright Northern­Sky Stars. Trudy Astron.
Inst. Sternberg (Moscow: Moscow State University), 63
Kozai, Y. 1962, AJ, 67, 591
Mayor, M., & Mazeh, T. 1987, A&A, 171, 157
Mazeh, T., Zucker, S., Goldberg, D., et al. 1995, ApJ, 449, 909
Pizzolato, N., Maggio, A., & Sciortino, S. 2000, A&A, 361, 614
Shatskii, N. I., & Tokovinin, A. A. 1998, Astron. Lett., 24, 673
Tokovinin, A. A. 1987, Sov. Astron., 31, 98
Tokovinin, A. A. 1995, Astron. Lett., 21, 250 (T95)
Tokovinin, A. A., & Smekhov, M. G. 2002, A&A, 382, 118