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IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
Dynamic evolution of resonance orbits of exoplanets
in system 47 Uma
B. R. Mushailov, A. A. Kaloshin
Sternberg Astronomical Institute, Moscow, Russia
The orbits of the major and minor planets of the Solar system indicate that
their dynamic evolution is essentially influenced by the effects of orbital reso­
nance interactions. Among dynamic systems the most amplitudes are caused by
the resonances of Lindblad -- orbital commensurabilities of the first order with
different multiples k:
jkn 1 \Gamma (k + 1)n 2 j џ n 2
p ЇO(1)
This inequality executes at initial moment of time. Here n 1
, n 2
, are average
motions (frequencies) of material points P 1 , P 2 with masses m 1 = Їff 1 , m 2 = Їff 2 ,
in units of mass of the `central body P 0
' (the star), Ї Ь 1, k 2 N , and ff 1
, ff 2
are
some real constants.
In the row of events (under libration type of motion) the resonance effects can
lead to stable orbital motions. One may expect that the orbital resonances can
exist not only in the Solar system, but also in other star systems. But presently,
there is no due attention to this question in celestial mechanics studies.
The improvement of observing facilities and methods of search permitted to
find significant number (ё 100) of exoplanets outside the Solar system. The rate
of discovery of exoplanets steadily increases. All exoplanets were discovered by
indirect methods based on analysis of the interference for stars with expected
presence of planetary systems (accuracy of the method based on the variations of
beam velocities approaches to 1 m/s in these measurements). For eight discovered
exoplanet systems there exist more than one planet and in all these cases there
are orbital resonances.
In preceding works [1­3] an analytical theory of the duel--frequency dynamic
systems has been elaborated allowing to interpret the evolution of orbital elements
of gravitational bodies for time lags of order 1=Ї. The use of rigorously motivated
asymptotic methods results in mathematically correct analytical solution in terms
of double periodic elliptic functions of Weierstrass. High sensitivity to initial
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conditions involves dynamical unstability and the possibility of arising of chaos
preventing to determine realistically--significant result for time lags t AE Ї \Gamma1 .
The developed theoretical technique was applied to predetermine the exo­
planet system 47 Ursa Majoris (resonance 2:1 ) in which the first component was
discovered eight years ago, but the second (external) one was discovered on Au­
gust 15, 2001. At present, among all known exoplanets this external component
possesses the largest semi--major axis a 2 = 3; 73 a.e., so the system 47 Uma looks
like our Solar system.
Within inaccuracy in values of initial orbital elements (resulted from obser­
vations) and values of masses of the gravitational bodies one may conclude that
in system 47 Ursa Majoris there exist three families of stationary solutions in the
configuration (phase) space. One of them is unstable in the sense of Lyapunov
(the type `turn gray'), but two other represent the `centers of stability'. The
trajectories of motion in the considered system can be located in vicinities of a
stable stationary point in accordance with the circulation nature of motion. The
evolution periods of changing of the semi--major axes of the orbits of components
P 1
and P 2
in system 47 Ursa Majoris have been evaluated permitting to perform
qualitative studies of the orbital features of these components.
Thus, the orbital resonances are widespread not only in the Solar system, but
also in exoplanet systems and represent an important stage in the orbital evolu­
tion. The developed analytical techniques for major and minor planets, comets,
objects of Kuiper belt, satellite systems of the bodies of the Solar system can be
successfully applied in the case of exoplanets as well.
This work is supported by grant RSSI 00­02­17744.
References
1. Mushailov B. R. Solution of analytical for predetermined resonance system
of the first order in the three--body problem. Solar System Research., 1995,
1--29, 47--57.
2. Gerasimov I. A., Mushailov B. R. Analytical solution of the three--body
problem in case of the commensurability of the first order. Solar System
Research., 1995, 1--29, 67--71.
3. Gerasimov I. A., Mushailov B. R. Orbital evolution of the system of Saturn's
satellites Enceladus and Dione. Solar System Research, 1996, 4--30, 355--367.
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