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ON ORBITAL EVOLUTION OF EXPLOSION FRAGMENTS


A. Sochilina1, R. Kiladze2, K. Grigoriev1, I. Molotov1, A. Vershkov1

1 Central Astronomical Observatory, Pulkovo road 65/1, 196140 St.-Petersburg, Russia
2 Abastumani Astrophysical Observatory, 383762,Observatory, Abastumani, Georgia

Abstract

The dynamic explosion model and its use for the investigation of geostationary object breakups are considered. It is assumed that the fragment velocity directions have a spherically symmetrical distribution. The initial fragment orbits are determined under the condition that their distances from the Earth's center are equal and the increments of their velocities differ because of their directions. At the moment of breakups the fragment orbits and the orbit of the exploded object intersect in one point of space. The study of the long-term evolution of the intersection points allowed to discover the compact distributions of these points along the orbit of the exploded object. Observations of the vicinity of such point of accumulations, which fragments have to intersect, help the search for the fragments. The method has been tested for orbital analysis of observed explosions and applied to objects, which had a sharp change in their drift rate. The results of these investigations will be presented.

INTRODUCTION

The number of geostationary objects observed by optical means is more than 900. A large number of unobserved small objects is moving in the geostationary orbit (GEO) and its vicinity eternally. An important source of space debris in GEO are explosions of geostationary objects (GS) (Jehn, 1997).

In the geostationary ring several objects have been discovered, which sharply changed their rates of drifts in the limits (0.15-1.0) /day. Four objects have observable fragments, which permit with confidence to calculate the moments of their possible explosions. The subject of this paper is the long-term orbital evolution of fragments and their search. The time of breakup of all suspected objects have also been calculated. Today ten such objects are known. Among them there are 2 Ekrans and 8 Transtages. It is possible that the real number of explosions in GEO is more than 10.
DETERMINATION OF FRAGMENT ORBITS

Based on the formulae of the paper (Sochilina et al., 2000), new expressions have been derived, which permit to determine after the explosion of GS the direction of its fragment motion if their orbital elements are known.

It has to be noted that GS after explosion can be considered as a fragment, too. The parameters characterizing the magnitude of ΔV and the direction of V are calculated by the following formulae:
                 (1)

where
                 (2)

where u = v + ω, v is the true anomaly of fragment, ω - argument of perigee, e - eccentricity, a - semi-major axis, p - orbital parameter, Vτ - tangential velocity, κ - Gauss constant. The orbital elements of GS are denoted by the index "0".

Let's take a system of coordinates, in which the x axis is pointing to the vernal equinox point of date and the plane of the equator is taken as the basic plane. The position and motion of GS at the epoch T0 is determined by the position vector r0 and velocity V0. We assume, that at the moment of explosion, all fragments receive the increase of velocity ΔV, which is in the limits of 1 - 250 m/s, depending on the fragment mass; the longitude l is an angle, counted from the direction of the tangential velocity Vτ from 0 to 360, the angle φ is counted from the plane of satellite's orbit in the limits of 90.

Fig. 1. The definition of the intersection place of orbits.

In Figure 1 the arc LL' corresponds to the celestial equator. The arcs Ω0S and ΩS are the projections of the orbits of GS and a fragment on the celestial sphere, S is the point of their intersection. Ω0, i0, u0 Ω, i, u denote the longitudes of nodes, inclinations of orbits and arguments of latitude of GS and fragment, accordingly. The equatorial coordinates α and δ of the intersection point S are determined from the rectangular spherical triangles ΩS and Ω0S (Figure 1) from the following equations:
                 (3)

The time of the explosion is computed from the coincidence of the satellite coordinates determined with both orbital systems. The mean error of discrepancy depends on the accuracy of the used elements. For Two-Line-Elements (TLE) the error may be 5 - 10 km. In case of absence of the orbital elements during a long time (several years) the error may be larger.

In Table 1 the orbital elements of objects which may have exploded are given for the epoch of each event. The main characteristic of an explosion is a sharp change of the rate of drift, the value of which is given in the last column. These changes are in the limits of 0.1 - 1.

For GS 76023F we did not have the orbital elements before the possible explosion, therefore the moment of explosion was calculated from the position coincidence of the object and its fragment - GS 76023J.

Table 1. Orbital elements of 10 satellites before and after possible explosion

NN T0 (MJD) e i Ω ω λ dλ/dt (/day) Δdλ/dt (/day)
66053J 47071.688587 .010312040 11.5253 9.4976 281.7312 288.3453 22.53757 0.67682
  03/10/1987 .016240500 11.5321 9.5757 281.2806 288.9544 23.21439  
67066G 49397.408163 .005317158 11.6745 25.3957 25.8977 6.2241 32.02443 -0.93796
  14/02/1994 .008095644 11.6578 25.4061 5.6721 6.6536 31.08647  
68081E 48673.397616 .008544623 11.9100 21.7275 76.5843 196.7101 4.27995 0.20669
  21/02/1992 .008861994 11.9100 21.7541 71.3055 196.8043 4.48664  
73040B 44671.200700 .004358301 5.8669 62.8461 19.1543 145.2013 -2.32077 -0.21648
  08/03/1981 .002713236 5.8728 62.8123 328.2317 144.8817 -2.53725  
73100D 48718.887352 .027537580 13.3263 45.5479 165.4079 215.9878 -18.79458 -0.19387
  06/04/1992 .026786840 13.3121 45.4283 163.3701 216.0936 -18.98845  
76023F 43060.207900 .013845130 25.3482 10.9980 215.4257 226.6138 -7.22838 *
76023J 09/10/1976 .014202330 25.2918 10.6278 215.9062 226.5970 -7.25248  
77092A 43680.632778 .003366272 0.1407 77.3145 256.1496 98.8366 0.04767 -0.13279
  21/06/1978 .000195225 0.1356 74.7306 -50.7829 98.5127 -0.08512  
78113D 50744.547145 .028235910 14.1715 38.2444 177.1164 163.1494 -22.90318 -0.55491
  23/10/1997 .027324530 14.1604 38.1593 166.2476 163.8886 -23.45809  
79087A 45121.755000 .000987337 1.6575 90.9652 196.5378 52.5730 0.07580 -0.08803
  01/06/1982 .000451044 1.6578 92.3283 83.5444 52.5333 -0.01223  
82019B 45960.349103 .000517985 0.3705 143.1092 301.5961 201.9020 3.06907 0.47435
  17/09/1984 .001376090 0.3440 138.3305 35.1682 201.7803 3.54342  

* Orbital elements of GS 76023F and its fragment 76023J.

Table 2 gives for 9 GS and 5 fragments the magnitude and direction of vector ΔV due to the explosions in orbital system of coordinates longitude l, latitude φ , arguments u0 , distances r0 from the Earth' center and differences r - r0 in km.

Table 2. The change of velocities and directions of motion of objects as a result of explosion and the difference of distances Δr at this moment, calculated from the data of Table 1

NN T (MJD) ΔV (m/s) l φ u0 r0 (RE) 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
78113D 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 fragment 68081H has maximum of discrepancy in Δr. The variation of the rate of drift is in the limits from -0.0015 to -0.0024/day and the lack of orbital data during 1800 days after explosion is the main reason for these residuals.

MODEL CALCULATION OF FRAGMENT ORBIT EVOLUTION AFTER EXPLOSION

For the investigation of fragment's dynamics after the explosion it is necessary to construct the manifold of fragment orbits assuming a spherically symmetric distribution of their velocity increase ΔV.

The realization of such condition can be done in case of fragment ejection with the equal initial rates into 32 different directions situated on apexes and centers of sides of icosahedron. Beyond the model of 300 fragments was used, with random distribution of directions.

For the moment of each explosion the initial fragment orbits are constructed with the conditions that r=r0 and V=V0+ΔV. These orbits are referred to the Laplace planes with inclinations Λ to equator and their evolutions are calculated. For GS 76023F initial orbits are calculated with ΔV = 250 m/s and in Table 3 the initial data of twelve fragments are given only.

Table 3. The initial orbits of fragments created by explosion GS 76023F for ΔV equal to 250 m/s 76023F Transtage = 43060.207900 MJD

NN l φ ω Ω i e ν Λ dλ/dt
(/day)
r
(RE)
0 0 0 210.637 25.460 17.842 .0138 94.557 7.704 -7.228 6.6994
1 0 90 248.338 4.553 20.751 .0150 67.147 7.895 -10.896 6.7460
2 0 27 300.652 14.116 14.095 .1554 5.477 12.777 -86.537 7.7446
3 72 27 0.520 11.076 17.657 .0748 308.745 9.191 -33.912 7.0208
4 144 27 118.151 8.696 21.817 .1199 193.591 5.090 55.313 5.9251
5 216 27 156.638 8.713 21.773 .1285 155.085 5.135 53.960 5.9251
6 288 27 247.587 11.150 17.537 .0984 61.600 9.313 -35.931 7.0208
7 36 -27 307.686 27.349 13.103 .1283 345.664 11.709 -71.970 7.5410
8 108 -27 69.562 20.469 17.655 .0717 230.960 6.654 14.749 6.4036
9 180 -27 127.473 18.752 19.519 .1436 174.894 4.609 70.645 5.7472
10 252 -27 181.761 20.573 17.567 .0942 118.651 6.748 12.634 6.4036
11 324 -27 267.025 27.520 13.026 .1385 26.150 11.795 -73.167 7.5410
12 0 -90 223.657 30.446 15.410 .0150 67.147 7.894 -10.896 6.7459

The orbital evolution of fragment 2 (Table 3) has shown that the maximum inclination of its orbit with respect to the equator in the process of its evolution can be up to 27. If at the moment of explosion the node of GS 76023F were 180 instead of 25, the inclination of the orbit would reach 40. In case of collision with GS the relative velocity would be about 2 km/s.

During the orbital evolution, the inclination of fragments with respect to the equator can reach 30-40. Such fragments are especially dangerous for satellites because the collision with them can cause further fragmentation (Kessler, 1993), (Jehn and Flury, 1996). Detailed investigation of the GS 68081E explosion (Pensa et al., 1996) shows that for a fragment of 18 - 20 magnitudes the mean value of ΔV is about 70 m/s. Therefore, it is interesting to study the dynamics of fragments for their search as objects of 18 - 20 magnitude.

In Figures 2 - 5 the trajectories of intersecting points of GS orbits (with respect to latitude arguments of primary body) by their fragment orbits during about 30 years after the explosion are given (the black lines).

The evolution of the opposite points of intersection (gray lines) is done in a similar way. After several years these points became equivalent because in both points the mutual distances of orbits were reaching many thousands of kilometers.

It has to be noted that most objects during some period have points of intersection consisting of two compact groups at opposite sides of the celestial sphere.

  

Fig. 2.
The evolution of the intersection points of the orbital plane of GS 68081E with the orbital planes
of its 32 fragments and ΔV = 75 m/s.
Fig. 3.
The evolution of the intersection points of the orbital plane of GS 68081E with the orbital planes
of its 300 fragments and ΔV = 75 m/s.

  

Fig. 4.
The evolution of the intersection points of the orbital plane of GS 79087A with the orbital planes
of its 32 fragments and ΔV = 75 m/s.
Fig. 5.
The evolution of the intersection points of the orbital plane of GS 79087A with the orbital planes
of its 300 fragments and ΔV = 75 m/s.

In Table 4 the date and equatorial coordinates of such groups are given for several GS. As a rule the length of arc of orbits intersected by fragment during these periods is 20 - 30 . The cases of very long arcs are denoted by asterisks (*).

Table 4. Date and equatorial coordinates of the events of compact intersection GS orbits by fragment orbits.

Object MJD Date α1 δ1 α2 δ2
66053J* 51500 1999 0h.7 +8 12h.7 -8
67066G 51000 1998 5.6 12 17.6 -12
68081E 52800 2003 3.5 11 15.5 -11
73040B 52600 2002 2.5 11 14.5 -11
73100D 57400 2016 1.0 12 13.0 -12
77092A 53500 2005 3.1 11 15.1 -11
78113D* 61000 2025 3.9 9 15.9 -9
79087A 54500 2008 2.6 10 14.6 -10
82019B* 49000 1993 5.9 4 19.7 -4

From Figures 2 - 5 it is clear that for GS 79087A intersection points will exist during MJD 54300-54700, and for GS 68081E - during MJD 52600-53000 (about 26 and 11 years after explosions, respectively).

The search of GS fragments is recommended at the intersection of projection of its orbit on the celestial sphere (Figures 2 - 5). The investigation of such orbits shows that in the intersecting points the mutual distances between orbits periodically became zero, which is the condition for collision. As it is shown in Kiladze et al. (1997) and Sochilina et al. (1999), many uncontrolled satellites collided repeatedly and among them there are also the exploded objects.
CONCLUSIONS

For elaborating effective methods for the safety of controlled satellites in GEO the first step is to study the real situation in this region of a number of exploded objects. It has been shown, that fragments can be found 11-26 years after satellite explosion.
ACKNOWLEDGMENTS

This work was partly supported by INTAS-01-0669.
REFERENCES

E-mail address of A. Sochilina

Manuscript received 11 November 2002; revised 12 March 2003, accepted 30 September 2003

6 2006.

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