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

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.

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.

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, V_τ - 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 V_τ from 0° to 360°, the angle φ is counted from the plane of satellite's orbit in the limits of 90°.

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

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.

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 Т = 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.

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.

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.

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

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Размещен 6 ноября 2006.