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Russian Academy of Sciences
Central Astronomical Observatory at Pulkovo

SIXTH US/RUSSIAN
SPACE SURVEILLANCE WORKSHOP

August 22-26, 2005

Proceedings
Edited by P. Kenneth Seidelmann and Victor K. Abalakin

St. Petersburg, 2005


ON A NEW THEORY OF THE GEOSTATIONARY SATELLITE MOTION AND ITS APPLICATIONS

R. I. Kiladze

Abastumani Astrophysical Observatory, Georgia (FSU)

1. INTRODUCTION

Launching artificial Earth satellites into the geostationary orbits (GO) began about 40 years ago. At the present time the geostationary satellites (GS) moving in a sharp resonance with the Earth rotation have the most wide applications in resolving many practical problems, mainly those related to the TV-communications, intercontinental communication and military purposes.

After expiration of its active period a GS becomes a passive object and is transferred onto a higher or lower orbit in the neighbourhood of the GO. According to the statistics by January 2005 there were 346 GS working in the GO and their total number was 1124.

Besides of observable objects there are ten thousand small fragments, smaller than 1 m in size, which are small space debris (SD) moving in the vicinity of the GO. They are accompanying every launch of the GS and are created as a result of the GS explosions.

According to the recent investigations the number of such explosions is 12; 2 of them were the Russian GS (Ekran 2 and Ekran 4) and 10 were the U.S. Transtages. Maybe this number is underestimated; therefore, in addition to elaboration of reliable methods for the detection of explosions by direct observations, it is necessary to dispose of the program for the search of all lost objects, and especially, of potentially dangerous ones for working GS.

These fragments are endangering the existence of the GS because of the possibility of collisions. The number of small fragments is growing with time (their existence time-span in the region of the GO is unlimited) and the probability of their collision with the SD grows. Recent investigations of the evolution of 456 uncontrolled GS observed more than 1000 sharp changes in their drift, which may be explained in terms of collisions with small SD [1].

The main problem of the GS motion control is still related to the cataloguing of all observed small objects moving in the GO region.

For elaborating a model of choking up the GO region by the SD it is necessary to have a reliable theory of the GS motion by which the available observations and orbital elements might be represented on the long time-span. It is also necessary to compile the software for calculation of the orbital evolution for hundreds of fragments in a short time.
2 THE THEORY OF A GEOSTATIONARY OBJECT MOTION
2.1. Statement of the problem

Our task was to construct a simple method to calculate the long-period evolution of the real GS orbits. The elaborated method, based on observations made on the time-span of 1000 through 2000 days with the mean square error σ in the range of 0.025 through 0.045, allows to represent the orbital evolution on the time-span of 7000 through 10000 days with an error of 0.1 0.2.

The theory has been tested by means of representation of long series of Two-Line Orbital Elements (TLE) of the NASA Goddard Space Flight Center and of the osculating elements (OE) determined by us directly from astrometric observations [2].

The principal peculiarity of the motion of a GS is the exact 1:1 commensurability of its mean daily orbital motion n with the Earths rotation velocity S1. In the case of absence of perturbations due to the nonsphericity of the Earths shape and to the luni-solar attraction, the GS once launched at some geographical longitude would always rest there. However, under the influence of ellipticity of the Earths equator, it begins to move like a pendulum near two stable points of libration with longitudes 75 and 255. Because of the asymmetry of Earths equator the periods of the small oscillations around these points differ by 110 days, being equal to 840 and 950 days, respectively.

To simplify the representation of such a motion Gedeon [3] has introduced a special system of coordinates, called by him stroboscopic.

In this system instead of the continuous time the discrete time-time-moments are used as the argument of the equations of motion separated by daily intervals, and the phase plane of the stroboscopic longitude λ and drift dλ/dt is introduced in terms of orbital elements as follows:

          

where M is the mean anomaly, Ω is the longitude of the ascending node, ω is the argument of the perigee, S is the Greenwich sidereal time, Ω1 and ω1 are secular changes of the node and perigee longitudes.

In the stroboscopic system of coordinates, in the case of an exact commensurability of motions, the central body and a satellite seem to be motionless, and in the case of an inexact commensurability the satellite would seem to slowly move along a trajectory. Such a picture may be observed if the system, the Earth a satellite, is in the darkness, being periodically illuminated by short light-flashes.

The stroboscopic system has properties of the inertial and noninertial (rotating) coordinate systems. Introduction of this system essentially simplifies the equations describing the long-period motion of the GS, but results in a loss of the short-period oscillations.

To describe the GS motion A. Sochilina in 1985 has used the Laplace plane [4], being first applied by H. Struve [5] to the satellite motion. It passes through the vernal equinox and is inclined by some angle Λ (depending on the semi-major axis of an orbit) to the geoequator. For the GS the angle Λ is equal to 7.34188.

The physical meaning of the Laplace plane is the following. Due to the equatorial bulge of the Earth a couple of forces are created which are trying to make the plane of the GS orbit to coincide with that of the geoequator; as a result the plane of the orbit (in the case of absence of other forces) precesses, being constantly inclined to the geoequator. On the other hand the common attractions of the Moon and the Sun try to make this plane to coincide with the ecliptic.

The resultant of these two couples of forces compels the GS orbit plane to precess by the constant inclination to a plane lying between the geoequator and the ecliptic and called the Laplaces plane.

The choice of the Laplace plane gives some advantages over other planes of reference: relative to it the inclination of orbit i changes (because of inclination of lunar orbit to ecliptic) only in limits of 0.5 and the longitude of the node Ω and the perigee argument ω become practically linear functions of time. The resonance equation for the stroboscopic longitude λ also becomes simpler.

As a result the problem may be resolved by use of the first order theory with small corrections for the second order only.

Usually the precision of some osculating orbital elements is not satisfactory, especially for the calculation of satellites drift dλ/dt but the derivatives of λ with respect to time may be improved by use of the data of preliminary orbits combined with the long-period motion theory.

In the process of work with the Catalog of improved orbital elements [6] in 1996 it was pointed out that on the 1000-day span some GS show considerable systematic differences in the longitude (in some cases reaching several degrees) between observed O and calculated C values.

After estimation of the theory accuracy R. Kiladze [7] assumed that these discrepancies could result from accidental changes in dλ/dt which in their turn are caused by the collisions with small particles.
2.2. Perturbation forces

Perturbations of the GS orbital elements may be found from the Lagrange equations. The perturbing function due to the geopotential U, the luni-solar perturbations RL, RS and the radiation pressure R may be expressed in terms of orbital elements referred to the Laplace plane [8]:

          (1)

where GE is the gravitational constant, ae , the mean equatorial radius of the Earth, Glm = Clm -1 Slm, Clm, Slm, the geopotential parameters, Elmk(Λ), Dlkp(i), the inclination functions and Xl-2p+q-l-1,l-2p(e) , the Hansen coefficients calculated in [9], M, Ω, ω, e, i, a, the GS orbital elements.

The perturbing function due to the Moon is expressed as follows:

          (2)

where       ε is the inclination of the ecliptic to the geoequator, ΩL, ωL, eL, iL, aL, the orbital elements of the Moon.

One can obtain the perturbing function due to the Sun by changing the index L in (2) to the index S.

According to Keplers third law

          

The mean radiation pressure function is as follows:

          (3)

where λS is longitude of the Sun, the ratio of the surface area to the mass expressed in cm2/gr).

The equation for calculation of Λ may be obtained from the condition that the sum of terms of perturbing functions U, RL, RS, with argument Ω, be equal to zero:

          (4)

where are the mean motions of a satellite, the Moon and the Sun, respectively, expressed in radians per day:

          

At last, all resonance perturbations are accounted for by the differential equation for λ [2]:

          (5)

where Almkpq is the function of a, e, i and of the geopotential parameters, λlmkpq, the phase angle depending on λlm, and the geopotential parameters Clm = Jlmcoslm, Slm = Jlmsinlm, LSP meaning the luni-solar perturbations.

The solution of this equation is described in [10].
2.3. Integration of the equations of motion

For simplification sake let us write the equation (5) as follows:

          (6)

where

          (7)

Choosing the Laplace plane as the basic one for the coordinate system guarantees that the change of coefficients i, mi, φi and si in (6) with time is very slow (i.e. that they are almost constant).

If these coefficients and the longitude of node Ω are constants, in the case of absence of the luni-solar perturbations, the equation (6) allows for the first integral:

          (8)

where 1 is the integration constant, being analogous to the Jacobis constant in the restricted problem of three bodies and in such a case the problem can be solved in terms of the quadrature:

          (9)

          (10)

The behavior of the function Π(λ) is shown on Fig.1. It is like the asymmetric double sinusoid with maximums and minimums of different heights.

Accordingly, depending on the value of parameter 1, the GS can move in accordance with one of the three modes the simple libration around one from the two stable points 75E or 105W, the compound libration (around both points), or in the mode of the circulation (the drift) around the Earth.

Fig.1. The function Π(λ).

Because of the smallness of dΩ/dt, the solution of equation (6) may be expressed in the form (8), if one considers 1, as a slowly changing function of time.

          (11)

In such a case one can obtain the derivative of 1 with respect to time: introducing a new variable of integration instead of dt by means of (8) we get:

          (12)

The exact calculation of the right-hand side of the equation (12) is not feasible since the dependence of variables C1 and Ω on λ is unknown; but, with an accuracy acceptable for practical applications, they may be considered on short time intervals as constants equal to the average values of the quantities C1 and Ω, respectively.

The variability of C1 and inaccuracy of calculations by means of (10), connected with the variability of Ω with time, may be significantly diminished by introduction of the oscillating system of coordinates, attached to the Π(λ) function.

On Fig. 2 different types of motion in the phase plane are schematically shown. If the value of the drift is less then 0.3/day, then the GS move in the mode of a libration around one of the stable points with the amplitude depending on the initial longitude.

The GS having maximum amplitudes of libration may periodically change their motion modes because of the luni-solar perturbations.

Fig 2. Different types of the GS motion in the phase plane longitude λ vs. rate of drift dλ/dt.
2.4. Formation of the intermediate orbit, the Lagranges equations

The method for calculation of the longitude just described allows to calculate ephemerides for a majority of the GS for the time intervals of several years with quite satisfactory accuracy for observers.

Further improvement of the ephemerides accuracy is possible by improving the GS orbital elements. For all this it is important to have the intermediate orbit which is maximally close to the real one.

The solution of this problem is published in [11] and [12]. Keeping the main terms in the right-hand sides, the Lagrange equations for the 4 orbital elements e, i, ω and Ω, in terms of the coordinates used, may be reduced to the form:

          (13)

          (14)

          (15)

          (16)

where

          (17)

ω, i, Ω and e denote here the orbital elements (the argument of the latitude, the inclination, the longitude of the ascending node, and the eccentricity), ωL, ΩL, the argument of the latitude and the longitude of the node of the Moon on the ecliptic, respectively, λS, the longitude of the Sun. D stands here for the light pressure, k, A, m, x, ΩL and R are considered as constants.
2.5. Solution of the Lagranges equation system

The equation system (13-16) is actually subdivided into two systems, each containing two equations.

Multiplying equations (13) and (14) by expressions respectively, and summing them up, one can obtain the equation in complete derivatives with respect to time, allowing for the first integral:

          (18)

Using the integral (18) the solution of system (13) and (14) may be received in the form of the elliptic integral:

          (19)

In order to resolve the system (15)-(16) let us introduce new coordinates p and q by:

          (20)

and also

          (21)

where

          (22)

Then the system (15)-(16) becomes:

          (23)

The solution of the linear system (23) isnt difficult, because the free terms in the right- hand sides of these equations are the variables found from the equation of system (18)-(19).

At last, by integrating the expression (21) the time can be calculated as the function of variable τ:

          (24)

The computer software compiled on the basis of the described solution of the problem copes successfully with the task of the orbit construction and of following its evolution on the time-span of tens of years for every known GS.
2.6. A new motion integral the Kren integral

It is necessary to mark the scientific meaning of the first integral (18) obtained by integration of the Lagrange equations. After the epoch of resolving the two-body problem by Newton, in the scientific armory of celestial mechanics there appeared only a single first integral, i.e. that of Jacobi found in XIX century.

By means of the integral (18) the inclination of the GS orbit to the Laplace plane is given as the function of its longitude of the node and of the orientation parameters of the lunar orbit.

Because the oscillations of the inclination of the GS orbit are described by (18), we named this integral by the Russian word Kren (the tilt, the banking angle) and the constant C2 by the Kren constant, respectively.

It is shown in [11] and [12] that in the general case the change of the value of C2 is not greater than C2×10-8, i.e. C2 is practically constant. Consequently, (18) is always highly accurate.

Today the constants C1 and C2 are used for the identification of unknown GS discovered in the process of the sky surveillance with objects lost in the past. But with the science progress each of the known first integrals has found a new, sometimes unexpected application. We should expect that the Kren integral will not be an exception of this rule.
3. THE MODEL OF CHOKING THE NEIGHBORHOOD OF GEOSTATIONARY ORBIT BY FRAGMENTS OF EXPLODED SATELLITE
3.1. Explosions on geostationary orbits

As it was mentioned above the cosmic space developing is accompanied by the choking of it with different fragments following the satellite launches. This phenomenon has caused anxiety since a long time, not only because of damage to working satellites by collisions, but also due to the potential start of a cascade process of their destruction.

An especially tense situation is created on the GO zone, where the quantity of objects inaccessible for observation because of their smallness is estimated as hundreds of thousands.

One of the sources of the GO choking are explosions of geostationary objects. From time to time this information appears in the scientific literature [13, 14]. The explosion of only one satellite may generate several thousands of small fragments thrown out from the satellite with great or small velocities and remaining in the vicinity of the GO for a practically unlimited time-span. The velocity of the GS changes during this time-span by several m/s. The drift of GS changes, therefore, more than by 0.05/day.

Because of the smallness of these fragments they can be observed but with great technical difficulties, and the ascertainment of the explosion of each single GS is, as a rule, possible only by indirect methods, a long time after the event [15, 16].

The method of determination of the time-time-moments and of dynamic characteristics of the explosion of an GS by using its orbital element changes is elaborated by the authors [17, 18].

Today there are revealed 12 exploded GS by means of this method. Their real quantity can be greater.

For these objects their designation numbers, the time-time-moments of the explosions T0, the orbital elements e, i, Ω, ω, λ and the drifts dλ/dt before and after the events are given in Table 1. The sharp change of the drift Δ(dλ/dt) given in the last column of Table 1 is considered as the principal indication of an explosion: if its value exceeds 0.1/day, the event is considered to be an explosion.

The data about first 10 exploded objects are given in [18].

Table 1. The orbital elements of 12 exploded GS

NOS. T0 (MJD) e i Ω ω λ dλ/dt (/day) Δ(dλ/dt) (/day)
66053J 47071.688587 0.010312 11.5253 9.4976 281.7312 288.3453 22.53757 0.67682
03/10/1987 .016240 11.5321 9.5757 281.2806 288.9544 23.21439
67066G 49397.408163 .005317 11.6745 25.3957 25.8977 6.2241 32.02443 -0.93796
14/02/1994 .008096 11.6578 25.4061 5.6721 6.6536 31.08647
68081E 48673.397616 .008545 11.9100 21.7275 76.5843 196.7101 4.27995 0.20669
21/02/1992 .008862 11.9100 21.7541 71.3055 196.8043 4.48664
73040B 44671.200700 .004358 5.8669 62.8461 19.1543 145.2013 -2.32077 -0.21648
08/03/1981 .002713 5.8728 62.8123 328.2317 144.8817 -2.53725
73100D 48718.887352 .027538 13.3263 45.5479 165.4079 215.9878 -18.79458 -0.19387
06/04/1992 .026787 13.3121 45.4283 163.3701 216.0936 -18.98845
75118C 46867.643023 .002005 8.5729 57.4781 25.9386 301.1890 0.80057 0.17468
13/03/1987 .001049 8.5710 57.0155 154.5391 300.6946 0.97525
76023F 43060.207900 .013845 25.3482 10.9980 215.4257 226.6138 -7.22838 *
76023J 09/10/1976 .014202 25.2918 10.6278 215.9062 226.5970 -7.25248
77092A 43680.632778 .003366 0.1407 77.3145 256.1496 98.8366 0.04767 -0.13279
21/06/1978 .000195 0.1356 74.7306 -50.7829 98.5127 -0.08512
78113D 50744.547145 .028236 14.1715 38.2444 177.1164 163.1494 -22.90318 -0.55491
23/10/1997 .027325 14.1604 38.1593 166.2476 163.8886 -23.45809
79087A 45121.755000 .000987 1.6575 90.9652 196.5378 52.5730 0.07580 -0.08803
01/06/1982 .000451 1.6578 92.3283 83.5444 52.5333 -0.01223
82019B 45960.349103 .000518 0.3705 143.1092 301.5961 201.9020 3.06907 0.47435
17/09/1984 .001376 0.3440 138.3305 35.1682 201.7803 3.54342
84129B 50074.650093 .000938 5.9226 57.9216 294.3247 359.9551 -4.06615 -0.80244
87095A 23/12/1995 .005996 5.8441 55.5646 203.3469 359.6639 -4.86859 **

(*) Orbital elements of GS 76023F and of its fragment GS 76023J.
(**)In different sources this GS is designated in a various way.

For GS 76023F we hadnt the orbital elements before the explosion, therefore the explosion time-time-moment is determined by establishing the identity of the object itself and its fragment GS 76023J.

The check for correctness of the time-time-moment T0 of an explosion consists in identifying the GS positions in the three-dimensional space as calculated by two orbital element systems (before and after the explosion). The mean discrepancy depends on the accuracy of the orbital elements used; for the TLE data it is about 1-2 km.

To get a clear view of the field of the space debris spreading after an explosion it is useful to construct the manifold of orbits for the fragments in the framework of the two-body problem for the spherically symmetrical explosion. The study of the orbital evolution allows to determine the size and the shape of the region of motion of the fragments.

For determination of the shape of the space occupied by the fragments it is important to know the ratio Earths attraction/ the luni-solar attraction and the radiation pressure. The orbital evolution of the fragments with a cross-section up to 40 m2/kg has been studied by means of the numerical integration on the time-span of 100 yrs [19].
3.2. Determination of orbits of fragments generated after an explosion

We use the Cartesian system of coordinates with equator of date as a fundamental plane; X-axis is directed to the point of equinox of date [20]; the position of the GS at the time-time- moment of an explosion t0 is defined by its radius-vector r0 and the vector of its velocity V0. Suppose that at this time-time-moment all fragments are ejected with the velocity ΔV.

Fig. 3. Determination of the intersection point of orbits

On Fig. 3 the arc LL' represents the celestial equator. The arcs Ω0S and ΩS are projections of orbits of the GS and of its fragment on the celestial sphere; S is their intersection point. By Ω0, i0, u0, Ω, i and u are denoted the longitudes of nodes, the inclinations of orbits and the arguments of latitude of GS and its fragment, respectively; α and δ are the right ascension and declination of the point S.

The equations relating arcs of great circles (Fig. 3) to the orbital elements of the GS and its fragment are given in [17].

For a symmetrical explosion the absolute value of ΔV is in limits of 1 250 m/sec depending on the fragment masses. The angle l is measured from the direction of the transversal velocity of the GS from 0 to 360, the angle φ is measured from its orbital plane in limits of 90.

Small values of eccentricities are natural for the GS, and the ratio ΔV/Vτ is less than 1/12. Furthermore, one can use simple formulae [17] to estimate the change of the orbital parameter Δp and that of Δi and ΔΩ as well:

          (25)

          (26)

where ν0 is the true anomaly, ω0, the perigee argument, e0, the eccentricity, a0, the semi-major axis, p0, the orbit parameter, Ω0, the longitude of the ascending node, i0, the inclination of the GS orbit at the time-time-moment of an explosion.

In the process of investigation of the orbital evolution of the GS fragments it is necessary for each of them to use its own Laplace plane, turning the coordinate system around the X-axis by the angle Λ-Λ0.

The data for the explosions of 9 objects and for their 5 fragments are given in Table 2 containing the value of ΔV, the longitude l relative to the GS transversal velocity and the latitude with respect to the orbital plane of the motion in the orbital system of coordinates, the argument of latitude u0, the geocentric distance r0 of the GS and the discrepancy Δr in distances expressed in km.

Table 2. Changes in velocities and directions of motion for exploded objects and differences in distances Δr, calculated from data of Table 1

NOS. T(MJD) ΔV, m/sec l φ u0 r0, (in the RE units ) Δr, km
66053J 47071.689 18.27 96.56 -1.16 177.43 6.36488 2.0
67066G 49397.408 10.56 76. 49 -5.50 271.40 6.26042 -2.5
68081E 48673.398 2.70 257.35 4.92 109.21 6.51187 -6.8
73040B 44671.201 10.28 273.54 1.98 319.84 6.62452 -1.5
73100D 48718.887 3.77 80. 27 7.74 326.37 7.02865 0.2
77092A 43680.633 9.93 272.27 -0.29 158.40 6.61330 -1.3
781 13D 50744.547 16.48 84.01 1.62 354.05 7.10098 -1.0
79087A 45121.755 4.06 273.81 21.97 123.20 6.60815 3.9
82019B 45960.349 4.79 248.83 39.31 180.88 6.57548 2.7
Fragments
68081G 48673.398 6.60 314.07 -21.52 109.21 6.51187 7.9
68081H 48673.398 21.93 96.58 -6.93 109.21 6.51187 -15.4*
76023J 43060.208 3.89 354.55 89.01 309.98 6.70681 0.6
77092H 43680.633 11.38 275.84 -14.12 158.40 6.61330 -5.2
79087C 45121.755 10.47 101.35 2.04 123.20 6.60815 0.3

The detailed investigation of the GS 68081E explosion [13] shows that for the fragments of 18 20 magnitude the average velocity change ΔV is 70km/sec. Furthermore, ΔV is about 250m/sec. for smaller particles which are invisible for optical instruments.

The example of calculation of the initial orbital elements of the fragments of the GS 76023F referred to their own Laplace planes for the ejection velocities of 250 km/sec, is illustrated by the Table 3.

The Table 3 clearly shows that the inclination of the Laplace plane to the equator Λ exceeds 12 for the fragments having the great negative drift. The total inclinations of the orbits of small fragments to the equator, calculated as a sum of the orbit inclinations to the Laplace plane and Λ, may reach 27 in the evolution process. For the GS moving in the equatorial plane the relative velocities of such fragments may reach 1.5 km/sec.

It should be mentioned that at the time-time-moment of an explosion the longitude of the ascending node of the GS 76023F orbit was 25. In the case of the explosion at the longitude of 180 the inclinations of the fragments orbits to the equator would reach 40.

Table 3. Initial orbital elements of fragments generated by the explosion of the GS 76023F for ΔV = 250 km/sec

76023F Transtage          (MJD)= 43060.2079
N L φ ω Ω i e ν Λ dλ/dt /d r(RE)
0 210.64 25.46 17.84 .014 94.56 7.70 -7.23 6.699
1 0 0 248.34 4.55 20.75 .015 67.15 7.89 -10.90 6.746
2 0 27 300.65 14.12 14.09 .155 5.48 12.78 -86.54 7.745
3 72 27 0.52 11.08 17.66 .075 308.74 9.19 -33.91 7.021
4 144 27 118.15 8.70 21.82 .120 193.59 5.09 55.31 5.925
5 216 27 156.64 8.71 21.77 .128 155.08 5.13 53.96 5.925
6 288 27 247.59 11.15 17.54 .098 61.60 9.31 -35.93 7.021
7 36 -27 307.69 27.35 13.10 .128 345.66 11.71 -71.97 7.541
8 108 -27 69.56 20.47 17.65 .072 230.96 6.65 14.75 6.404
9 180 -27 127.47 18.75 19.52 .144 174.89 4.61 70.64 5.747
10 252 -27 181.76 20.57 17.57 .094 118.65 6.75 12.63 6.404
11 324 -27 267.02 27.52 13.03 .138 26.15 11.79 -73.17 7.541
12 0 -90 223.66 30.45 15.41 .015 67.15 7.89 -10.90 6.746
3.3. Model calculations of the orbital evolution of the fragments after an explosion

The problem of the study of the fragment dynamics is interesting from the point of view of choosing the program of search for table (i.e. of 18 20 mag.) objects.

For this purpose let us investigate the evolution of an explosion point position in time and space. Obviously, at the time-time-moment of the explosion the orbits of all fragments are intersecting in this point.

For representation of the dynamics of fragments generated by the GS explosion by use of the method described in [17] there were investigated the behavior of 32 GS fragments ejected with the same velocities in different directions directed to the vertices and along the sides of a rectilinear icosahedron (i.e. the polyhedron with 20 sides). Moreover, also the model consisting of 300 fragments with the same ejection velocities was investigated.

For the time-time-moments of each GS explosion the initial orbits of their fragments referred to the corresponding Laplace planes were constructed. It is postulated that at this time-time-moment the modules of the vectors of fragment velocity changes are the same, not exceeding 250 km/sec.

After this time-time-moment each fragment begins to move along its orbit intersecting the orbital planes of other bodies, including the orbital plane of the paternal body because it may be considered as the greatest fragment, in the point of the explosion.

The inclination of the fragment orbit to the initial one is

          (27)

where Δi is expressed in radians, Vp is the polar component of the total velocity of the fragment and Vτ is the tangential velocity of the GS.

Accordingly, the initial poles of all fragment orbits are displaced along the great circle of the celestial sphere 90 from the point of explosion (Fig. 4).

On the other hand, the semi-major axes of the fragment orbits are differing from the semi-major axis of the initial orbit by:

          (28)

After an explosion the motion of each fragment is referred to its own Laplace plane. Their poles are disposed along the great circle of the celestial sphere passing through the celestial pole and initial pole of the relevant Laplace plane (Fig.4).

Fig. 4. The location of the poles of fragment orbits and of those of their Laplace planes just after the explosion.

After the explosion the orbits of all fragments begin to precess around the poles of their Laplace planes with different speeds. As a result, after some time-interval these poles will be located along the curve slightly differing from a great circle. Consequently, the corresponding orbital planes intersect each other in almost the same line. We have called this phenomenon the regularization of fragment orbits.

The problem concerned with calculation of the regularization time-time-moment (the most favorable epoch for observations of the fragments) and its numerical description by computation of spherical coordinates A and D of the intersection point of trajectories of the fragments and their dispersion σ is studied in [18].

The evolution of the ensemble of 32 fragments ejected from the GS with velocity of 75 m/sec is studied on the basis the theory of motion [21 27] for all 12 events, described in Table 1, over the time-span of 30,000 days (about 80 yrs).

The results of these calculations are given in Table 4 containing designations of the exploded GS, the data for the most compact location of the intersection points of orbits of the fragments, the geocentric coordinates A, D of these points and the values of the average deviations σ of the fragment trajectories from the intersection points.

Table 4. The time-time-moments, equatorial coordinates of points of orbits compact intersection and dispersions for 12 fragments of the exploded GS

NOS MJD Year A D σ
66053J 56821 2014 9h.41 0.9 0.57
67066G 51322 1999 17.11 12.2 0.02
68081E 52748 2003 3.49 1.3 0.07
73040B 53496 2005 2.04 0.4 0.21
73100D 61693 2027 21.50 3.2 0.31
75118C 52361 2002 3.88 12.8 0.09
76023F 52835 2003 21.37 1.0 0.34
77092A 54355 2007 14.13 -9.5 0.19
78113D 64119 2034 21.22 2.2 0.52
79087A 54446 2008 14.59 -9.9 0.16
82019B 48400 1991 6.65 3.4 0.03
84129C 52539 2002 6.63 8.3 0.03

For an illustration, the behavior of the GS 68081E fragments on the time- spans of 2000 days is shown in Fig. 5.

Fig. 5. The evolution of the poles of the GS 68081E fragments on the intervals of 2000 days.

It is seen from Fig.5 that the poles of the fragment orbits 2000 days after the explosion occupy a significant area on the sky, but after 3899 days they are aligned along the great circle. Subsequently they occupy again more and more area on the sky.

A majority of the GS investigated by us show the similar process of the orbit regularization with other characteristic times.

In several cases (GS 66053J, 78113D), however, the process of the orbit regularization is faintly presented.

Fig. 6. The evolution of the poles of the GS 66053J fragments on the time- intervals of 3000 days.

As an example the orbital evolution of the GS 66053J fragments on the time-intervals of 3000 days is represented on Fig 6. The minimum of the dispersion is reached 9630 days later after the explosion time-time-moment but it is so faintly presented that Fig.6 is hardly differing from the picture of the stochastic distribution of the points on the plane.

The cases for other ejection velocities (250, 20 and 10 m/sec) of fragments are also investigated but they dont influence significantly the behavior of the GS fragment orbits: the configurations on Figs. 5 and 6 are invariant with respect to the change of the fragment ejection conditions, and only the scale of each Figure changes proportionally to ΔV.
3.4. Evolution of fragments orbits after explosion

It is clear from above that the regularization process of the GS fragment orbits is mainly depending on the angles φ1 and φ2 represented on Fig. 4, i.e. the angles between the great circle passing through the poles of the initial orbit and the Laplace plane and the great circles on which the poles of the orbits of fragments and their Laplace planes are located.

To clarify this dependence a numerical experiment has been made: it has been calculated for each exploded GS what the orbital evolution would be if the explosion occurs with some delay with respect to its real time-time-moment (i.e. if it would happen at another point of the GS orbit).

Some results of this experiment are represented in Table 5. In the first column of Table 5 the differences of mean anomalies ΔM with the step of 30 (which corresponds to the explosion delay of 2h) are given, in other columns the time-intervals between the time-time- moments of explosions and those of regularizations and corresponding values of dispersions σ for the intersection points of trajectories are given.

Table 5 shows that for each GS there are two values of ΔM (separated by approximately 180); for them the values of σ reach minimum. These values are written in the bold type.

If the time-time-moments of the real explosions of the GS and the values of ΔM are known, it is easy to find that the GS whose right ascensions are about 9 or 21 hours at the time-time-moment of the explosion are characterized by the minimal values of σ.

Table 5. The regularization models of the orbital planes for 12 exploded GS. The configurations with minimal values of σ are written in the bold type.

\NOS.
\ΔM
66053J 67066G 68081E 73040B 73100D 75118C
ΔT σ ΔT σ ΔT σ ΔT σ ΔT σ ΔT σ
0 9630 0.573 1295 0.023 3899 0.070 9472 0.211 11041 0.307 5494 0.091
30 9372 .816 5152 .154 7810 .225 12662 .409 12833 .573 9129 .250
60 17177 .695 8307 .391 10991 .449 14631 .657 12752 .820 11986 .483
90 16559 .532 9445 .668 11755 .720 13895 .854 20439 1.015 13080 .743
120 4477 .110 8022 .863 19671 .838 1726 .023 1826 .022 ~ ~
150 7944 .321 17013 .871 19451 .637 5937 .090 6211 .104 1256 .018
180 9572 .595 1443 .026 3814 .066 9565 .213 10466 .318 5547 .092
210 9141 .827 5285 .162 7603 .213 12758 .408 12805 .601 9190 .252
240 17125 .675 8405 .397 10902 .431 14686 .651 12156 .851 12030 .484
270 16478 .513 9462 .671 11665 .708 14100 .849 21052 1.017 13136 .742
300 4443 .103 8166 .861 19728 .852 1650 .022 2577 .030 ~ ~
330 7889 .306 16990 .867 19482 .648 5819 .087 6848 .123 1191 .045
\NOS.
\ΔM
76023F 77092A 78113D 79087A 82019B 84129C
ΔT σ ΔT σ ΔT σ ΔT σ ΔT σ ΔT σ
0 6908 0.340 11012 0.192 12784 0.520 9295 0.161 2393 0.033 2465 0.031
30 8079 .712 14388 .348 13384 .782 12667 .310 4925 .070 6254 .095
60 7954 .948 16614 .564 ~ ~ 15454 .517 7783 .115 9996 .212
90 ~ ~ 2299 .030 21814 .863 15963 .752 11087 .208 13380 .390
120 15375 .876 4699 .065 5363 .077 2671 .037 14460 .360 15421 .627
150 2098 .029 7883 .112 9708 .265 6008 .084 16483 .578 15106 .838
180 6744 .300 11101 .195 12707 .536 9318 .161 2413 .034 2493 .031
210 7930 .682 14450 .350 12877 .814 12696 .311 4937 .070 6248 .094
240 7789 .944 772 .011 21318 1.025 15436 .516 7789 .115 9996 .210
270 ~ ~ 2350 .030 1219 .014 15930 .751 11091 .207 13361 .389
300 15703 .889 4643 .064 6206 .097 2645 .037 14453 .360 15400 .625
330 2617 .045 7812 .111 10560 .267 5981 .084 16450 .577 15119 .837

The dynamics of this phenomenon are as follows.

At the time-moment of explosion of the GS all fragment orbits intersect each other in two opposite points of the celestial sphere.

Because of perturbations their orbits change and, according to the initial conditions, the intersection points of the visual fragment trajectories create two streams the centers of which are lying apart by 180 in the longitude. After a sufficient time the fragments are intersecting with the basic orbit practically at every point. If the ejection velocity of the fragments is 75 m/sec, the inclinations of their orbits dont exceed 5.

After several (about ten) years of the evolution the orbit planes intersect each other nearly in one line, i. e. they have the common node.

Such ordered configuration exists during 2 4 years after which the orbital planes take different orientations anew.

The values of σ as a function of time passed after the explosion and its delay for the GS 66053J and 73040B are shown on Figs. 7 and 8.

Fig.7. Dependence of σ on the time passed after the explosion and its delay for the GS 66053J

Fig. 8. Dependence of σ on the time passed after the explosion and its delay for the GS 73040B

Figure 7 shows that σ may reach minimum not only once, but every successive minimum always is less deep than previous one. This happens because the line of the poles of the fragment orbits placed eventually becomes more and more different from the great circle (Fig. 9).

Fig. 9. Displacement of the poles of the GS 68081E fragment orbits at the second minimum of σ, 19890 days after the explosion

The regularization phenomenon of the GS fragment orbits creates the most favorable situation for observations of fragments near their common node which all of them pass once a day.

From this affinity of the GS fragment orbits is founded the barrier method of search for the fragments: in the years favorable for their detection (when σ has a minimum, according to Table 4) the permanent observations in the vicinity of both nodes should be made as long as possible. By monitoring the environments of these points (of the radius σ) during a day it is possible to observe all fragments generated by an explosion of the same GS.

By the way, the small particles ejected by the explosion with great velocities spread themselves along the GS orbit faster than those having the smaller relative velocities, the evolution of which runs slower.

In Table 4 the geocentric values of the data (in particular those of σ) are given. When observing from the Earths surface the deviation of individual fragments from the common center may grow because of the differential parallax. This effect of the vertical stretching depends linearly on the ejection velocity by the explosion. For velocity of 75 m/sec it is equal to 0.83 sin z and 0.89 sin z (where z is the zenith distance) for the upper and lower directions, respectively.

At last, the studies of the evolution of fragment orbits show that in their intersection points the distances between the orbits periodically become zero, which makes the favourable conditions for collisions. As it is shown in [15, 16] many uncontrolled GS had repeated collisions, the exploded objects being frequently met among them.
4. CONCLUSIONS

For elaboration of efficient precaution measures for safety of the controlled satellites in the GO zone it is necessary to clear up primarily the real situation about the littering and to process all orbital data in order to know the real quantity of explosions.

The new long-period theory of the GS motion allows to analyze the observation data of the uncontrolled objects for long time-spans and to discover the accidental changes of orbits depending on explosions or collisions with the space debris.

The model of creation of fragments for 12 exploded GS is constructed and their orbits evolution on the time-span of 80 yrs is investigated, the time-moments and dynamic parameters of explosions being determined.

The maximal change of the inclination of fragment orbits (when the ejection velocity is 75 m/sec) is about 5 depending on the argument of latitude and the location of the explosion point on the GS orbit. In the process of the evolution the inclination of small particle orbits to the geoequator may reach 40. Such fragments are the most dangerous for the Earth satellites, because they can provoke the cascade process of continuous fragmentation [28, 29].

The search for the fragments becomes easier at time periods when the planes of the fragment orbits intersect themselves in the same line. During this period all fragments once a day pass two isolated points (the antipodes) on the sky.

Our results may be used for the organization of observations of the spreading area of the small-size debris using the ground-based instruments and for identification of the litter sources as well.

The process of filling-up the cosmic space with the explosion-generated fragments depends on relative velocities ΔV: the velocities ΔV < 75 m/sec are typical for greater fragments which intersect the initial orbit of the GS at two parts about 10 long, lying apart by 180. When the distances between the orbits in the intersection points become zero, the favorable conditions for collisions occur.

The small-sized debris with great velocities (about 250 m/sec) fills up the neighborhood of the exploded GS and soon begin to intersect all orbits. Moreover, there are created objects having masses of about 10 g and relative speeds of 1-1.5 km/sec. They can not only collide with the GS but also can provoke the cascade process of fragmentation.

From this point of view also the process of permanent burial of the GS near the GO (staying 200 300 km afar) which ceased to work, because the exploded GS can create fragments with great orbital eccentricities and inclinations of order of 30.

In order to have the permanent and reliable information about the choking of the GO area it is necessary to put into operation more powerful means of ground-based technics for observations, and also to launch a satellite, with a telescope aboard, in the environments of the GO with the means for the permanent communication with the Earth.
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