We describe the GOCK-2002 Catalogue (Geosynchronous Objects Catalogue: Kyiv 2002) containing the topocentric equatorial coordinates and orbital elements of geosynchronous satellites obtained by photographic method at the Main Astronomical Observatory of the National Academy of Science of Ukraine in 2002. (http://www.mao.kiev.ua).
The results of the identification of 30 objects from 193 observations among the total 259 observations of 52 objects are presented. The method of uncontrolled geostationary objects identification on the base of analytic theory of their motion is discussed. General character of the declinations and time angles changes are presented for uncontrolled and controlled satellites observed at Kyiv station.
The regular space objects photographic observations in 2002 at MAO NASU have been performed by the method described in [3,8]. The observations covered the longitudes from 28° E to 68.6 °E and from 70° E to 82° E.
From the total number of 52 observed objects only 30 controlled objects were identified according to international catalogues of geosynchronous objects [6,7]. The 22 observed objects are not presented in these catalogues.
GOCK-2002 Catalogue presents the satellite positions reduced using the PPM Star Catalogue in J2000.0 reference frame. Time instants are given in the UTC scale. The object name, its COSPAR designation, the object motion type and its longitude λ , longitude drift λׂ , orbital inclination i towards the Earth`s equator are provided for the identified objects. For the unidentified objects the word XXXXX is used instead of the object name.
For some of these objects (if the number of observations was more two), λ , i , the longutude of the ascending node W and argument of perigee u are given. The satellites orbital elements are calculated using the method [4].
The total number of objects in the geostationary region (status of December 2002) is 934, out of wich 323 are controlled and 611 are uncontrolled [7].
The uncontrolled geostationary satellites are difficulty identified because these satellites with a large longitude drift are observed frequently only during one night. Using these observations the orbital elements and longitude drift λׂ have been calculated with a large error. But from the observations we obtained the accurate satellite right ascention and declination.That's why these coordinates should be use for the uncontrolled satellites identification. The main criterion may be: the observed satellite coordinates a0 , δ0 for the time t are small different from the catalogue coordinates ac , δc of the satellite nearest to the satellite with the a0 , δ0 coordinates, e.g. |Δ α | < 1º, |Δδ | < 0.5º. Δ α may be rather large only in special case, e.g. for the unstable types of satellites or under satellite collision with a space debris. The additional conditions may be : |Δλׂ |< 1º and |Δi |< 1º.
First of all it is need to determine the uncontrolled objects from all observations. For this it should be use the condition |λׂ|>0.01º/day for uncontrolled satellites. The geostationary satellite distance is large as compared with an Earth radius. That's why it may be use the topocentric satellite drift, the error should be not more than 15%:
where α1, α2 - the satellite right ascention obtained from observations for the time moments t 1 , t 2; the whole numbers m and n takes into account the cycles (revolution) of α and t change; ω 0 - the velocity of the Earth rotation into star coordinate system (ω 0 =1.0027379 rev/day). During one night observations the longitude drift is determined with an accuracy of 0.2-0.3º/day.
Fig.1. Orbit inclination of
the uncontrolled geosynchronous satellites
observed in Kyiv during 1994-2000 and in Uzhgorod during 1989-2000
Using the catalogue data [5], the right ascention αc, declination δc, the longitude of the subsatellite point and longitude drift, the orbital inclination ic are calculated on the base of analytical motion theory of the geosynchronous satellites [1]. The Moon and Sun influence and the ununiform of the gravitation Earth field are taken into account. Then the observed coordinates are compared with calculated ones and the programm determined the satellites satisfied the above mentioned criterium and additional conditions.
Fig.1 showed the orbital inclination i versus longitude for the identified satellites, observed in Kyiv and Uzhgorod. The differences (o-c)α= -0.07º±0.57º ; (o-c)δ= 0.03º ±0.33º. This method, obtained for uncontrolled satellites identification, may be used for calculation of the declinations and time angles needed for observations.
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Fig.2. Prognosis of the satellite 93062 (l1)
declination change |
Fig.3. Prognosis of the satellite 93062 (l1)
time angle change |
Fig.2,3 presents the prognosis of the δ and t changes for the 93062D (l1) satellite during 1-5 January 2004, observed in Kyiv on May 2, 1995. Solid lines shows the δ and t changes during the time interval 0h±6h. The change of the inclination has a short periodical component, sinusoid, with one day period. This component is due to the orbit inclination. The amplitude of this garmonic is approximately equal the satellite orbit inclination.
The 20 satellites were used for determination of the declination change. The result was: the period of the declination change is T δ=0.997±0.001 of the solar day, the error due to present the δ change by sinusoid garmonic is σδ=0.05º±0.02º. The amplitude of garmonic is correlates with an orbital inclination, the coefficient of regression is:
Aδº=1.025 iº - 0.178º. (1)
The coefficients of this equation confirmed the idea that the amplitude of the curve and the orbital inclination are closed. But, it is exist the large mean square deviation, σ =±0.80º, for points from the curve. Fig 4. shows the orbital inclination i change during two years for 830065A satellite. The curve is almost stright line, its ordinate is to the right. Two curves marked in δ are the maximum (upper part) and minimum (low part) declinations on the sinusoids. The distance between the points of two δ curves in one vertical line is double amplitude of the day change of the orbital inclination. The curve O in Fig.4 is the totality of the mean δ values.
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Fig.4. Change of declination for maximum |
Fig.5. Mean values δ0 versus t0 for controlled satellites in Kyiv (K) and Uzhgorod (U) latitudes |
The change of time angle versus time is the curve wich garmonic has the line and periodic components. The line component for 93062D satellite is: t = -0.083º MJD+4426.0º, the mean square deviation of the points from the line component is due to periodic component is σ =±0.17º. The coefficient of MJD with an accuracy of 0.01º is coincided with a mean longitude drift, that is with a coefficient of the line component of the longitude λ change and has the opposite sign.
The periodic component of the time angle change with an accuracy of 0.0004º is a sum of two garmonics with a period of P1=0.997 and P 2=P1 / 2. Thus, during the small time interval, up to one month, the time angle change for uncontrolled satellites may be shown with a hight accuracy (mean square deviation ±0.01º) as :
where A1 ,φ1 -the amplitude and phase of the first half-day garmonic; A2 ,φ2 - these for second garmonic, P- the period of the Earth day rotation in the star coordinate system. The prognosis for the satellite 93062D on January 2004 is : A1 = 0.192º, A2 = 0.132º.
The amplitude of the first garmonic of t change depends on the orbital inclination i and has a 1/2 day period. The line of regression obtained using the 20 satellites
has correlation coefficient ρ=0.986 and the mean square deviation σ=±0.036º. The main garmonic may be absent if the orbital inclination is small.The amplitude of the second garmonic with one day period depends on the eccentricity of the geostationary satellite orbit. This one is not more than maximum amplitude of the first garmonic but sometimes is generally absent.
The main peculiarity of the
controlled satellites is a small orbital inclination and a small drift. It
should be considered to unchange the orbital elements for these satellites and the drift is zero. This model for controlled satellites is a small changed, during the small time interval, of more real controlled geostationary satellites. Using the formula (1) we obtained, that the declination may changed only by sinusoid with a small amplitude and the period about one day. As in
(2) λׂ=0, the line component of the time angle change is absent.
It may be seen from the equation (3) that the amplitude of the first garmonic of the
curve of t change, with a period of 0.5 day, is near to zero, that is this
garmonic practically absent.We observed it for the controlled satellites with a
small orbital inclination to the equator. There is only second garmonic with a
one day period in the equation (2), but the ordinate t0 for
it isn't zero. This consideration is confirmed by the calculation for 91003A
controlled geostationary satellite with orbital elements : e=0.0003,
i=0.047º, λׂ=0.017º/day during 5 days on September 2003. The
declination change for this time may be shown as sinusoid with such
parameters: P-the period, A-the amplitude, δ0 - the axis of
garmonic:
P=0.997 day, A=0.051º , δ0 = -7.145º,
and the change of the time angle - by sinusoid with parameters:
P=0.997 day, A=0.040º, t0 =10.34º.
The mean square deviation of the calculated δ and t from the corresponding
curves which are shown their change versus time is the same and equal ±0.0003º.
The mean period of the declination change obtained on using the 20 satellites,
with an accuracy of 0.000005 day, is coincided with an Earth rotation period
in the star coordinate system and is 0.99727 day. The period of
the time angle change is the same, but the error of it determination is more
large. There is appeared a garmonic with a period of 0.5 day for the orbital
inclination i >1.0º. The error resulted from the δ and t
changes as sinusoid shows a tendency to increase when the orbital inclination
increases, that's why
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for i: |
0º - 0.4º |
1º |
3º |
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the errors of Δδ: |
0.0003º |
0.0007º |
0.0035º |
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the errors of Δt: |
0.0003º-0.0005º |
0.0030º |
0.0308º |
The increase of Δ t is more quickly and caused by the appearence some garmonic with a period of 0.5 day.
The orbital controlled satellite inclination is influenced on the amplitude of the curve characterised of the δ change. The correlation dependence is : Aδ =1.0703360·i +0.0018444 and the correlation coefficient ρ=0.99994. The mean square deviation σ = ±0.0096º. Aδ amplitude is always more than the orbital inclination i.
The amplitude of a curve characterised the change of the time angle is depended on the eccentricity of the satellite orbit and is caused by it. The regression line At =127.68385º ּe-0.00059º, has a large correlation coefficient ρ=0.99995. The mean square deviation is ±0.00076º.
The middle meanings of the free term δ0 of garmonic (the ordinate of garmonics axes) are depended on the analogous term t0 of the curve of time angle change. To evaluate this dependence, the 20 controlled satellites were used and the curves of the change δ and t and so δ0 and t0 were obtained. Using these data the correlation dependence δ0 versus t0 was determined (see Fig.5):
δ0= a + bּ t 0 + cּ t02.
The coefficients calculated for Kyiv and Uzhgorod are :
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b |
c |
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Kyiv: |
-7.30380 |
-3.237 10-5 |
8.978 10-5 |
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Uzhgorod: |
-7.14928 |
-1.786 10-5 |
9.444 10-5 |
The mean square deviation of the points from the curves are Δδ0=± 0.008º. As see from Fig.5 the declination of the geostationary orbit is δ≈-7.3º for the time angle t=0º and δ ≈-6.8º for t=55º in Kyiv geographic latitude. The geometric conclusions confirm it.
REFERENCES
