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PROGRAM GALACTICA FOR THE SOLUTION OF
THE SPACE DYNAMICS TASKS :
Dynamics and evolution of Solar system bodies;
Passive motion of spacecrafts;
The evolution of the assembly of bodies, including galaxies.
J. J. Smulsky
Institute of Earth's Cryosphere SB RAS, Tyumen, E-mail: [email protected],
webpage: http://www.smul1.newmail.ru/.
Year of Russia in Spain
Exhibition "scientific, technical and innovative
achievements of Russia"
12 -15 May 2011
Madrid
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CONTENTS
1. Differential equations of the celestial bodies motion and the
method of their solution.
2. Illustration of the motion of bodies at the integration of
differential equations by Galactica.
3. The evolution of the orbits for 100 million years.
4. Precession of the orbits.
5. Compound model of the Earth’s rotation.
6. Compound model of the Sun’s rotation.
7. The motion of the asteroid Apophis.
8. Optimizing passive orbit using gravimanevr
9. Multilayer circular structures
10. References
11. Acknowledgments
12. Some information
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View of planshet
at the exhibition
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1. Differential equations of the celestial bodies motion and the
method of their solution
At the interaction of bodies under the Newton's law of gravitation their motion
are described by differential equations [1] :


n
d ri
mk rik
 G  3 ,
2
dt
k i rik
2
(1)

r
where i - the radius-vector of a body with mass mi relative to the center of mass of
bodies;
  
rik  ri  rk - the radius vector from the body with mass mk to the body with a mass mi;
G – the gravitational constant.
The equations for the Solar system
2. Illustration of the motion of bodies
at the integration of differential equations by Galactica
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3. The evolution of orbits for 100 million years
motionless barycentric
equatorial coordinate
system
ieq
The parameters of the planet orbit in a motionless barycentric equatorial coordinate
system xyz : A0A0’ – Earth's equatorial plane at the epoch of JDs; E0E0’ – Earth's orbital
plane (the ecliptic plane) at the epoch of JDs, for example, at the epoch of 2000.0;
MaMa’ – the planet's orbital plane (Mars) at the epoch of Т; B – position of the
perihelion of the orbit; 0 – vernal equinox at the epoch of JDs; angles of the orbital
plane  = 0D and its inclination ieq = ADA0’; perihelion angleр= DB.
Ascending node
Eccentricity
ieq
Inclination
Perihelion
ΔP
The calculations agree with observations.
The secular changes of the
Earth's orbital elements for
seven thousand years: e eccentricity; iea – angle of
inclination;  – the angle of
the ascending node of the
orbit; р0 – the angle of the
perihelion; Δa – deviation in
meters under the mean semimajor axis for 7 thousand
years; ΔP - deviations in years
under the mean orbital period
for 7 thousand years.
Angles are in radians, and the
time T - in a centuries.
1 – numerical integration of
the program Galactica; 2 –
secular variations of S.
Newcomb [6]; 3 – secular
variations of Simon J.L. et al
[7].
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Earth
Eccentricity
Ascending node
Inclination
Perihelion
The evolution of the Earth's orbit
for 3 million years: е eccentricity; φΩ - the angular
position of the ascending node; i
- angle of inclination of the
orbital plane; φр - the angular
position of the perihelion; Т –
time in millions of Julian years
from 1950.
Angles are in radians, and the
time - in millions of years.
Obtained
periods
and
amplitudes of all oscillations of
the Earth’s orbital parameters.
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Earth
50 Myr evolution of Earth’s orbital
elements: es is eccentricity sliding
average; φp is perihelion longitude; ys
and Ss are the sliding average precession
and nutation angles, respectively, of the
Earth’s orbital axis ( S). р and ys are in
radians and Ss is in degrees.
The evolution of the Earth's orbital
parameters for -100 Myr ≤ T ≥ -50 Myr.
Angles φр and yS are counted from the
epoch Т = -50 Myr.
Earth's orbit is stable and steady.
Mars
The evolution of the orbit of Mars for 100
million years. The orbit of Mars and other
planets are stable and steady.
The Solar system is stable and steady.
4. Precession of the orbits

Precession of the axis of the Earth’s orbit
 S for 50 million years around the
angular momentum M of Solar system.
a) 1 – celestial sphere; 2 - plane of the Earth's equator at 1950; 3 - Earth's orbital plane
in 1950; 4 - Earth's orbital plane in a different epoch, 5 - Earth's orbit in another epoch,
6 - Line of intersection of the orbital plane with the equatorial plane.
b, c) the precession of the axis of the Earth's orbit on the plane yMxM (solid line - for
400 thousand years) and in the plane zMxM (points - for 50 million years). The interval
between the points of the graphics - 10 thousand years. Large dots indicate the positions
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of the axis of the Earth's orbit in the corresponding time.
Threedimensional
precession
orbits axes for
3.76 Myr. The
vertical axis is
parallel shifted
relative
momentum
vector, and the
top axis offcenter
coordinates.
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1. Evolution of the orbits of the planets and the Moon is the result of four movements:
1) the precession of the orbit, 2) nutation oscillation of the orbit axis, 3) the oscillations
of eccentricity of the orbit, 4) rotation of the orbit in its plane (the rotation of the
perihelion).
2. The axis of the Moon's orbit precesses relative moving Earth's orbit axes, i.e. as the
axis of the Earth’s rotation precesses.
These results have allowed to investigate the evolution of Earth's axis by
different way, as well as create a compound model of the rotating Earth.
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5. Compound model of the Earth’s rotation
We consider the n-bodies symmetrically located in the equatorial plane.
Parameters of a compound model of a rotating Earth are determined by the following
conditions:
1. The total mass of the peripheral and central bodies equal to the mass of the Earth.
2. Peripheral bodies revolve on a circle with angular velocity of rotation of the Earth.
3. Moments of inertia of the Earth and the body system for axes x and y are equal.
The investigated
models
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Illustration of motion of model of the Earth’s rotation
and other bodies of Solar system
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Motion of the peripheral bodies of model relative to central body .
The duration of one revolution - 1 day, and issuance of the images on the screen - after
about 20 revolutions.
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Is the similarity of the outlines of the western frontier of the American continent with
the coastline of Africa and Europe due to the multiple convergences?
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Dynamics of
parameters of
peripheral body in
the first model for
0.5 years
Eccentricity
Ascending node
Inclination
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Dynamics of
parameters of the
first model for 50
years
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Dynamics of
parameters of the
first model for 1000
years
Period
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Nutation oscillations of the axis
of rotation of the third
compound model of the Earth
at five time intervals: graphics
4 shows the two time intervals:
T and T1; Δε – the deviation of
the nutation angle from the
sliding average.
The dynamics of the
compound model of the Earth’s
rotation in the direction of
precession and the precession
axis is agreed with
observations. The nutation
oscillation periods of 14 days,
0.5 years and 18.6 years are
also consistent with
observations. 24
6. Compound model of the Sun’s rotation
As a result of integrating the equations of motion of the planets, Moon and Sun we
received perihelion velocity of Mercury 529.9" per century relative to the motionless
space.
We have found that according to observations [11-12] the perihelion velocity relative
to the motionless space is equal to 582.3“ in a century. Therefore, the difference
between them is 52.4" per century, and not as it was accepted in the early 20-th century,
41“ per century.
If inside the orbit of Mercury is the planet a certain mass, it could make the rotation
of the perihelion of Mercury in 52.4“ per century and at the same time did not exert
significant influence on other planets. This planet is not. But the Sun rotates around its
axis, and the moving mass of its substances can affect the Mercury as well as the above
assumed planet.
For this purpose, a compound model of the Sun rotation is developed [11-12].
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Compound model of the Sun rotation
and its parameters for the mass of the
Sun MS = 1.98892 1030 kg and its
radius RS = 6.97113 108 m: 1 – a
compound model of the Sun; 2 – the
central body; 3 – peripheral body of
model No. 5; 4 – Mercury; 5 – Venus;
6 – Earth and Moon; 7 – Mars; other
planets are outside the figure. The
positions of the bodies are given at
12.30.1949, the lines of the bodies
represent the vectors their velocities.
No
Model
2
3
4
5
n
5
5
5
10
m1 10-23
kg
604.8
1.966
1.565
0.782
a 10-10
m
2.528358
2.528449
2.528449
2.528449
The secular changes of the orbits of Mercury and Venus under the influence of the
planets and the compound model of the Sun No. 4
The angle of Mercury's perihelion coincided with the observations and the orbit of
Venus and other planets has not changed.
Angles are in radians, and the time T in a century. 1 – results of the numerical solution
by the program Galactica; 2 – secular variations of S. Newcomb [6]; 3 –27secular
variations on Simon J.L. et al [7].
7. The motion of the asteroid Apophis
The authors of several papers, e.g. [13-14] and others have shown that the
asteroid Apophis at April 13, 2029 will be approach at a distance from the center of the
Earth in the range of 5.62 to 6.3 of its radius, and because of the chaotic change of orbit
the further prediction of motion becomes impossible. These authors believe that there is
some probability of collision with Earth in 2036.
We analyzed the problem and found that the uncertainty in the trajectory of
Apophis due to imperfections in the methods of its determination. We have integrated
the differential equations of motion of Apophis, the planets, Moon and Sun for 1000
years by Galactica [15] and studied its evolution of the orbit.
At April 13, 2029 Apophis will approach at a distance Rmin = 38 ÷ 39
thousand km from the center of the Earth and its closer passage will not be during 1000
years.
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Evolution of the distance R between Apophis and Earth for 100 years. Influence of
initial conditions (IC): 1 – IC of 30.0 November 2008; 2 – IC of 04.0 January 2010.
Calendar dates in approaching points: A – 13 April 2029; H – 13 April 2036 г.; F1 – 13
April 2067 г.; F2 – 14 April 2080 г.; T - time in Julian centuries from 30.11.08.
RA = 38907 km, RH = 50 Million km, RF = 622000 km .
Such encounters of Apophis with the Earth as April 13, 2029 will be no more.
We can use this chance and can transform Apophis in the satellite.
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The motion of the asteroid over 2 years, when it approaches to Earth 13.04.2029; the
view from the south pole.
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Possible use of Apophis-satellite
Many of the pioneers of cosmonautics had thought the exploration of space
using large manned space stations. However, to deliver from the Earth such a large
mass poses a serious technical and environmental problems. Therefore, thanks to a
happy occasion, we can create a habitable station, turning the asteroid Apophis in the
satellite.
There are other possible applications of such satellites. It can serve as the basis
for the Space lift. It can be used as a "shuttle" to deliver goods to the Moon. In this case,
the satellite must have an elongated orbit with a perigee radius close to the radius of the
geostationary orbit, and apogee radius, approaching to radius of lunar orbit. Then the
cargo from the geostationary orbit at perigee would be board on Apophis-satellite and
then these goods could be delivered at the apogee to the Moon. The last two
applications are possible if the satellite motion coincides with direction of the Earth
rotation and the direction of the Moon orbiting.
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The trajectory of Apophis
(1) in the geocentric
equatorial
coordinate
system yrxr: а – the normal
scale, б – a larger scale at
the time approach of
Apophis with the Earth
(2); 3 – position of
Apophis at the time of its
rapprochement with the
Earth after correction its
trajectory with the velocity
reduction ratio k = 0.9992
in p. Ap1; coordinates xr
and yr are given in AU
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The transformation of Apophis in the satellite with the necessary direction of
orbiting
For the transformation of Apophis in the satellite with the necessary direction of its
motion, it is necessary before 0.443 years to Apophis approach with the Earth to reduce
its velocity on 2.54 m/c.
Approach of Apophis
with the Earth after
correction its velocity (in
the coordinates relative to
the Earth).
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If the Apophis velocity will be reduced on 3.5 km/s, then it become a satellite with the
same direction of revolution around the Earth as the Moon. Our studies have given that
the satellite orbit is stable. Therefore, it can perform its task for a long time.
Apophis
satellite
orbiting around the
Earth in the same
direction
as
the
Moon.
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The transformation of an asteroid in the satellite is a very
difficult task. However, the solution to this problem greatly increases the
opportunity to prevent a serious asteroid hazard. Therefore, if the society
undertakes efforts for its actualization, it would indicate the transition
from purely theoretical research to practical work in protect the Earth
from asteroid hazard.
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8. Optimizing passive orbit using gravimanevr
The trajectories and
orbits of the spacecraft
during launch
20.01.2001, with
different initial
velocities v. Flight
passive. After the
approaches to Venus (at
the site of intersection
of its orbit) the
spacecraft goes to an
elliptical orbit. 1 – the
orbit of Venus.
Using the gravity of Venus allows in 1.7 times to get closer to the Sun at the same initial
velocity of the spacecraft, and at the same approach to reduce the initial velocity from –
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18.2 km / s to -15 km / s [16].
9. Multilayered circular structures
Evolution of multi-ring structures are important for understanding the
problems of existence and stability of rings of the planets, globular star clusters and
galaxies. Creating ring structures based on the following two conditions [17].
1. For body, located outside of the ring structure, the exerted force equals to the force,
which would create a body located in the center of the structure and has a mass equal to
the mass of the entire structure.
2. For body, located inside the ring structure, the total force exerted by all its bodies is
equal to zero.
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The ring structure
symmetrically
arranged peripheral
bodies Bk,ik relative to
the central body B0.
1 – elliptical orbit of
the first body on the
first ring.
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Table. 1. Parameters of the models of ring structures: mk – the masses of peripheral
body on the first ring; Rp,k – pericentre radii in astronomical units (AU); P – orbital
period in years. Dn – characterization of the dynamics for 30 years: Stable - there are no
visible changes in 30 years; Unstable – destroys after specified number of years.
Parame
ters
nk
mk
Rp,k
ek
Pk
Dn
The parameters for each ring for different models of the ring
structures
1
5
1
1
0
1.4
Mod. 1, m0 = MS
2
3
7
8
2
3
2
3
0
0
4
7.3
Stable
Mod. 10, m0 = 0.5MS
1
2
3
6
7
7
1
2
3
1
10
16
0
0
0
1.4
39.9
70.3
Unstable, 23
Two group of models, stable and unstable, were investigated. Table presents one
model for each group. Model 1 with small peripheral masses of bodies - is stable. In the
model 10 the changes begin with the interaction between the bodies of two outer rings,
which later lead to formation of a common ring and after 23 years of motion the
destruction of the inner ring is happened.
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Dynamics of 10-th model of the ring structure: m0 = 0.5mS.
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10. References
1. Smulsky, J.J. The Theory of Interaction. - Ekaterinburg, Russia: Publishing house
"Cultural Information Bank", 2004. - 304 p.
http://www.ikz.ru/~smulski/TVfulA5_2.pdf.
2. Duboshin, G.N. (Editor), 1976. Celestial Mechanics and Astrodynamics: A
Handbook [in Russian]. Nauka, Moscow.
3. Stendish E.M. JPL Planetary and Lunar Ephemerides, DE405/LE405.//Interoffice
memorandum: JPL IOM 312. F - 98-048. August 26. 1998.
(ftp://ssd.jpl.nasa.gov/pub/eph/export/DE405/).
4. Melnikov,V.P., Smulsky,J.J., Krotov O.I., Smulsky,L.J. Orbits of The Earth And The
Sun And Possible Their Influences On of The Earth Cryosphere (Statement of a
Problem And The First Results)// Cryosphere of the Earth. - 2000. - Vol. IV, No. 3, Pp.
3-13. (In Russian).
5. Laskar J. Marginal stability and chaos in the Solar System/ Ferraz Mello S. et al.
(eds.) Dynamics, ephemerides and astrometry of the Solar System. – IAU: Netherlands.
– 1996. Pp. 75 – 88.
6. Newcomb S. The elements of the four inner planets and the fundamental constants of
astronomy. Washington: Government printing office. 1895. –202 p.
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7. Simon J.L., Bretagnon P., Chapront J. et. al. Numerical Expression for Precession
Formulae and Mean Elements for the Moon and the Planets // Astron. and Astrophys. –
1994, vol. 282, p. 663-683.
8. Cohen C.J., Hubbard E. C.; Oesterwinter C. Planetary Elements for 10,000,000
Years//Celestial Mechanics. - 1973. - No. 3.-Pp. 438-448.
9. Brouwer D., Van Woerkom A. J. J. The secular variation of the orbital elements of the
principal planets// Astr. Pap. - 1950. – 13, 2.
10. Laskar J., Correia A. C. M., Gastineau M., Joutel F., Levrard B. and Robutel P.
Long term evolution and chaotic diffusion of the insolation quantities of Mars//Icarus.2004.-Vol. 170, Iss. 2.-P. 343-364.
11. Smulsky J.J. Compound model of rotation of the Sun and displacement of Mercury
perihelion / The Fundamental and Applied Problems of the Mechanics: Proceeding of
the VI All-Russian scientific Conference, devoted 130-th anniversary of Tomsk state
university and 40-th anniversary NII of Applied Mathematics and the Mechanics of
Tomsk State University. Tomsk, September 30 - October 2, 2008 - Tomsk: University
Publishing House. – 2008 - Pp. 433-434. (In Russian).
12. Smulsky J.J. Gravitation, Field and Rotation of Mercury Perihelion// Proceedings of
the Natural Philosophy Alliance. 15th Annual Conference 7-11 April 2008 at the
University of New Mexiko, Albuquuerque, USA. Vol. 5, No. 2. Published by Space
Time Analyses, Ltd. Arlington, MA, USA.- 2009. - Pp. 254-260.
http://www.ikz.ru/~smulski/Papers/08Smulsky2c.pdf.
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13. Giorgini J.D., Benner L.A.M., Ostro S.I., Nolan H.C., Busch M.W. Predicting the
Earth encounters of (99942) Apophis // Icarus. 2008 v.193, pp. 1-19.
14. Ryhlova L.V., Shustov B.M., Pol’ V.G., Suhanov K.G. Scientific Problems of the
Aasteroid Hazard / Near-Earth Astronomy 2007 / / Proceedings of International
Conference, September 3-7, 2007, Terskol. International Centre for Astronomical,
Medical and Environmental Research of the National Academy of Sciences of Ukraine
and the Institute of Astronomy. Nalchik, 2008, pp. 25-33.
15. Smulsky J.J., Smulsky Ya.J. Evolution of Movement of Asteroids Apophis and 1950
DA for 1000 Years and their Possible Use / Institute of the Earth Cryosphere SB RAS. Tyumen, 2011. - 36 p. - Fig.: 10. Refer.: 27. - Russian. - Dep. In VINITI 25.01.11. No.
21-V2011. (In Russian).
16. Smulsky J.J. Optimization of Passive Orbit with the Use of Gravity Maneuver //
Cosmic Research, 2008, Vol. 46, No. 5, pp. 456–464. Original Russian Text ©, 2008,
published in Kosmicheskie Issledovaniya, 2008, Vol. 46, No. 5, pp. 484–492.
17. Smulsky J.J. Designing of ring structures // The Fundamental and Applied Problems
of the Mechanics: Proceeding of the VI All-Russian scientific Conference, devoted 130th anniversary of Tomsk state university and 40-th anniversary NII of Applied
Mathematics and the Mechanics of Tomsk State University. Tomsk, September 30 October 2, 2008 - Tomsk: University Publishing House. – 2008 - Pp. 431-432. (In
Russian).
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11. Acknowledgments
This work was supported by grants from the Governor of the Tyumen region
2003 and 2004 and by integration program of the Presidium of RAS № 13 in 2008 –
2011.
Calculations were performed on the supercomputers of the Siberian
Supercomputing Centre SSCC SB RAS at the ICMMG SB RAS.
In work on different stages involved were: L.I. Smulsky, Ya.I. Smulsky, P.A.
Apasev, O.I. Krotov, M.E. Chikmareva, M.A. Ponomarev, I.V. Bineev, E.N.
Nevidimova, N.J. Apohin, V.S. Botvin, E.A. Kovrizhkina, A.A. Pavlova, I.A.
Shabolina, M.L. Panova and E.F. Safina.
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12. Some information
1. Results of equation (1) solutions and planets orbit evolution for 100 million
years are accessible on site: : http://www.ikz.ru/~smulski/Data/OrbtData/.
2. Now there are several our books:
2.1. Smulsky J.J. The Theory of
Interaction. - Ekaterinburg, Russia:
Publishing house "Cultural Information
Bank", 2004. - 304 p.
http://www.ikz.ru/~smulski/TVfulA5_2.
pdf.
2.2. Grebenikov E.A., Smulsky J.J.
Evolution of the Mars Orbit on Time
Span in Hundred Millions Years /
Reports on Applied Mathematics.
Russian Academy of Sciences: A.A.
Dorodnicyn Computing Center.
Moscow. - 2007. 63 p.(In Russian).
http://www.ikz.ru/~smulski/Papers/Ev
Ma100m4t2.pdf.
2.3. Melnikov V.P., Smulsky J.J. Astronomical theory of ice ages: New approximations.
Solutions and challenges. – Novosibirsk: Academic Publishing House “GEO”, 2009. – 182
p. http://www.ikz.ru/~smulski/Papers/AsThAnR.pdf.
4. Scientific sate: http://www.smul1.newmail.ru/
5. Popular sate: http://zhurnal.lib.ru/s/smulxskij_i_i/
Thank you for your attention!
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