Catalogue GOCKU-2000 of positions and orbital elements of the geosynchronous space objects observed in 2000. Observations of uncontrolled space objects for the purpose of possible collisions detection.

¹- Space Research Laboratory of the Uzhgorod National University.

²- Main Astronomical Observatory of the National Academy of Science of Ukraine.

We describe the Catalogue GOCKU-2000 (Geosynchronous Objects Catalogue: Kyiv-Uzhgorod-2000) containing the topocentric equatorial coordinates and orbital elements of geosynchronous satellites obtained by the photographic method at the Main Astronomical Observatory of the National Academy of Science of Ukraine (MAO NASU) and at the Space Research Laboratory of the Uzhgorod National University (SRL UNU) in 2000:

Using the Electronic catalogue we determined the satellite ephemerides, the equatorial coordinates, time angle and the longitude of subsatellite point as a function of MJD for uncontrolled satellites.The corresponding calculations are carried out for the drifting uncontrolled satellite 95045D.

The regular space objects observations in 2000 at MAO NASU and SRL UNU have been performed by the methods described in [1], [2]. The survay carried out in SRL UNU covered the longitudes from 32°W to 26° E on February 29, May 2-3, September 25, 2000, and in MAO NANU - from 19° E to 81° E on February 1,4, May 1,2, August 21,23, Oktober 1,2. The observations are performed without ephemerides. The observational instruments are placed in equator, where as usual the active gestationary satellites are founded. The total number of observational objects is shown in Table 1. 121 space objects were observed in the longitude intervale Dl=113°. 10 space objects from them (Dl=7°) are observed in two stations.

Tables 2, Tables 3 presents the satellite positions reduced using the PPM Star Catalogue in J2000.0 reference frame. Time instants are given in the UTC (SU) scale.

The object name, its Cospar designation, the object motion type and its longitude l(°), longitude drift l^. (°/day) and orbital inclination i(°) towards the Earths' equator are provided for the identified objects. For the unidentified objects the word 'xxxxx' is used instead of the object name and for some of these objects (if the number of observations was more two), l^., i, W - longitude of the ascending node (°), u -argument of perigee (°) are given. The satellites orbital elements are calculated using the method [3].

All observed objects are identified according to international catalogues of geosynchronous objects [4]. Not all objects are available in these catalogues. So, for 2000 the international catalogues contain about 750 geosynchronous satellites. These objects are more than 1m in size and may be observed by photographic method.

In [5] we are discussed the question of the uncontrolled geosynchronous objects observations. A necessity of the investigation of uncontrolled objects evolution is evoked by the problem of risk to collision of the controlled space objects with other objects. It is possible to define the probability of collisions in geostationary orbits using the method applied to low orbits [6].

The essence of this method is to calculate the motion equation of all objects and to detect the objects' approachment in every moment of time. As a result of this way the more convenient time of uncontrolled satellites observations is determined. The ephemerides for these observations, the time angle and declination are calculated too.

The Electronic version of the geosynchronous objects catalogue for 1996 may be used for this work [7]. There are orbital elements e, i, W, u, W_lapl , u_lapl , l_lapl , referred to equatorial and Laplace planes in this catalogue for uncontrolled satellite during the time in MJD 49000-51000 from 20 to 30 days interval. All these orbital elements and the longitude may be represented by the sum of linear and periodic terms [8], the values of which changed in a moment while i crossed zero-point. The functions representation of the orbital elements allowed to extrapolate their values in two- three years ahead.

Using the satellite orbital elements from the Electronic catalogue we obtained its rectangular coordinates and then the equatorial coordinates and time angle as function of MJD. Fig.1 and Fig.2 show the satellite 95045D declination and longitude as a functions of MJD. Shown in Fig.2 in dotted line are the satellite time angles for Uzhgorod station.

Fig.1. Declination values versus time for uncontrolled satellite 95045D.	Fig.2. Longitude values (curve) and time angle values (points) in Uzhgorod versus time for uncontrolled satellite 95045D.

The analysis of these dependences and corresponding tables showed that the satellite 95045D may be observed in Uzhgorod and Kyiv stations during about 3 monthes approximately every 2.2 years. The more favourable conditions for observations in Uzhgorod station will be during MJD 52640-52744 in the declination region from -6.3° to -3.3° and in Kyiv station during MJD 52534-52652 in the declination region from -9.1° to -6.0°.

Shown in Fig.3 is a drift l^. as a function of time expressed in MJD.

Fig.3. Drift values versus time for uncontrolled satellite 95045D.

The values obtained from Electronic catalogue [7] are shown as points in Fig.3. Using these points the dotted periodic curve is obtained by the least square method. This curve is a sum of four garmonics. Four points, marked in Fig.3, are obtained using the observations [2], [5].

The method above-mentioned allows to increase the number of uncontrolled satellites observations, to correct the prognosis of satellites visibility, to control their motion, to appreciate the risk of collisions in geostationary orbits.

Because of asymmetry of the gravitational field due to the tesseral and sectorial harmonics of the third degree the uncontrolled geostationary satellites can move in the following way depending on the initial conditions [9]:

1. Simple libration around the nearest stable position (the satellites of the type l₁, l₂).
2. Complex libration around both stable positions (l₃).
3. Circulation with different drift rates (d₁, d₂, d₃).

There are stable positions of libration satellites about 75° and 255° in longitude. Due to the luni-solar perturbations the total energy of geostationary satellites is changing and some satellites change their regimes of motion, passing from the complex libration to a simple one or to circulation. The opposite changes also take place.

According to Electronic catalogue [7], the number of the satellites moving in 1, 2, 3 regimes are accordingly 80, 5, 279.

In case of the simple libration the change of the subsatellite longitude l is near to sinusoidal law

(1)

with a period of T, changed within the interval of 700 - 1900 days. The libration is take place around the position l₀.

In Fig.4 the Electronic catalogue [7] the values of the l₁ type of 95063D satellite longitude are given and the curve (1) was obtained by the least squares method, where A=66.85° , T=1166 days, j=183.8°, l₀=68.4°. The horizontal solid lines in Fig.4 are separated the time intervals of the possible objects observations in Kyiv, the horizontal dotted line - the time intervals of the observations in Uzhgorod. The separate point is the result of the observation in Uzhgorod. The real curves of the l changes for all satellites, which move in the simple libration regime are changed from the function (1) more smoothed peaks.

Fig.4. Longitude change of subsatellite point of the satellite 95063D during MJD 48645-50973 and prognosis up to MJD 53000.

The satellite drift may be determined by differentiate of the function (1):

The satellite motion in 2 regime, the complex libration, is more complicated. In Fig.5 the longitudes of the satellite 91064A (l₃) are shown by points according to Electronic catalogue [7].

Fig.5. Libration for l₃ type satellite.

Approximately to the moment of MJD 50200 the longitude is changed by sinusoidal law, the libration about l₀ point:

(2).

Then, approximately to MJD 51400 the satellite motion is the libration about L₀ point:

(3).

Approximately in the point of MJD 51400 the return take place to the change of the longitude by equation (2), but another phase. The functions (2) and (3) are given in Fig.5 as a dotted lines. The function parameters for satellite 91064A obtained by the least squares method are next:

The change of the satellite longitude l may be shown as a function:

(4)

The value d=43.8 days is a time, which is necessary for satellite to turn from the libration relatively one stable position to the libration relatively another stable position. The distance in longitude l, which the satellite must be passed for this time (the distance between sinusoides) is 7.2° .

As a initial moment may be taken MJD₀ approximately equal MJD 48600. For accurate definition of this time we determined n (n=31.95) from the equation

The function (4) is given in Fig.5 by solid line and is recurring function with a period of T₁+T₂+2d. The resonance period of the l₃ type satellite is approximately 2 times more than the period of the libration satellites another types. The satellite drift may be obtained from (4) :

The longitude as a function of time of the drift satellites contains the linear and periodic components :

(5)

This dependence for drift satellite 95045D (d₃) is given by graphically in Fig.2. All coeffitients in (5) are determined by least squares method. The angle coefficient (a) must be such value, that the axis relatively which the periodic component is constructed, was horizontal. This one may be obtained using once more the least squares method for the linear component to the data which the main garmonic was excluded from. The second garmonic changes only the amplitude of oscillations, l₀₂ for it approximately equal 1; A2<A1; T2 approximately equal 2T1. The main garmonic has parameter l₀₁ approximatly equal zero. We are given the values of the parameters of the distribution (5) for the satellite 95045D obtained by least squares method: a=-0.4057; b=19995.59;

The d type satellites drift ( Fig.3) is a periodic curve with an axis l^. approximately equal a. The periodic components may be neglected for the satellites with a large drift (the type d1, d2); approximately l^. equal a.

The prognosis of l and l^. according to determined functions allowed to determine the time intervals of the satellite visibility for given observational station,to identify the satellites with a large drift. It is possible to define more precise the obtained functions using the result of the observations.

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