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Galushina T., Bykova L., Letner O., Baturin A. Tomsk State University Russia
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Investigation Dynamics Asteroid
of

of

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Solving problems
construction of numerical model of asteroid motion with




account chosen perturbed factors; asteroid orbit improvement according to positional observations; construction of initial probabilistic domain by linear and nonlinear methods; study of evolution of initial probabilistic domain; revealing of close encounters and possible collisions of investigated asteroid with planets, the Moon and Pluto; estimation of asteroid collision probability with planets and the Moon; revealing of commensurabilities of mean motions of asteroid and planets; estimation of predictability time of asteroid by average MEGNO parameter; demonstration of asteroid and planets motion in different coordinate frames on computer screen.
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2012 MF7
The collision probability of this object with the Earth

in June 2046 is about 10-4. D = 15 m

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The algorithms
Perturbed two body problem in the heliocentric coordinate



frame related to equator 2000.0 Perturbations from major planets, Pluto, the Moon, Ceres, Pallas, Vesta, the Earth and the Sun oblateness, the solar pressure and relativistic effects from the Sun. Fundamental ephemeris DE405, DE406 or DE408. Everhart method Least squares method, earth-based and space positional observations Linear or non-linear methods of construction of initial probabilistic domains
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Revelation of close encounters of clones with planets bases on







calculation of minimum of square distance function which approximated by Lagrange cubic expression relatively time. The impact probability is estimated as relation of collided clone number to all ones. The resonance argument = k11 - k22 - (k1 - k2)1 - (k1 - k2)1 The resonance band = k1n1 - k2n2 n1, n2 are mean motions, 1, 2 are mean longitudes of asteroid and planet correspondingly, 1 is argument of pericenter of asteroid, 1 is longitude of the ascending node of asteroid, k1, k2 are integers.

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MEGNO (Mean Exponential Growth of Nearby Orbit)

parameter represents the time-weighted integral form of the Lyapunov characteristic number (LCN)

2 Y t t

t ( s) 0 ( s )



sds

1 Y t t

t

0

Y s ds

is the tangent vector, which measures the evolution of the

initial infinitesimal deviation between the solution of motion equations and a very close orbit. For chaotic orbits with the exponential divergence of close trajectories increases linearly and reaches values > 2. For quasi-periodic (regular) orbits with linear divergence of close trajectories oscillates around 2, and for stable orbits harmonic oscillator type = 0.
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The structure of the application suite "IDA"
subsystem "Assol" allows to study orbital evolution for nominal





orbit and to demonstrate the asteroid and planets motion on a computer screen; subsystem "Observations" intends to asteroid orbit improvement according positional observations and construction of initial probabilistic domain by non-linear methods; subsystem "Distribution" developed for the visualization of distribution of observations along an asteroid orbit; subsystem "Clone ensemble" allows to construct an initial probability domain by the linear method; subsystem "Evolution" designed for the study of the orbital evolution of an ensemble of asteroid clones ; subsystem "Megno" intends to estimate of predictability time of asteroid motion by means of average MEGNO parameter.
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Subsystem "Assol"

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Subsystem "Assol"

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Subsystem "Assol"

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The orbit of 2012 MF7

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Subsystem "Distribution"
2012 MF7 N = 27 t = 29 days

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Subsystem "Observations"

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The results of orbit improvement of the asteroid 2012 MF7

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Subsystem "Clone ensemble"

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Subsystem "Megno"

The average MEGNO parameter evolution for asteroid 2012 MF7
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Subsystem "Evolution"

2012 MF7: close encounter with Venus (a), the Earth (b) and Mars (c),

evolution of semimajor axis (d), eccentricity (e) and inclination (f )

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The data about passing clones through gravitation sphere of the Earth

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Conclusion
Thus this report contains description of software "IDA"

which developed for study of asteroid dynamics. The application suite consists from several subsystems which allow to carry out the comprehensive study of asteroid motion. The opportunities of software have been demonstrated on example of asteroid 2012 MF7 motion investigation.

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Thank you for attention!

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The maximum distance of clones from nominal

particle for asteroid 2012 MF7
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2012 MF7: the projection of probability domain on

equator plane in 2037 (a), 2039 (b) and 2042 (c) years
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