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The Orbital Evolution of Two Sounding Satellites and Analysis of the Accuracy of Orbit Determination
CHINESE ASTRONOMY AND ASTROPHYSICS
ELSEVIER
Chinese Astronomy and Astrophysics 31 (2007) 420–429
The Orbital Evolution of Two Sounding Satellites and Analysis of the Accuracy of Orbit Determination WU Gong-you1,2∗ ZHAO Chang-yin1 ZHANG Rong-zhi2 WANG Jia-song2 WANG Hong-bo1 1
Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008 2 Xi’an Satellite Observation and Control Center, Xi’an 710043
Abstract The satellites TC-1 and TC-2 are the two Chinese satellites with great elliptical orbits which are still in orbit around the earth at present. Since the launch the orbits of the two satellites have continuously evolved, which has a certain effect on the orbit determination and prediction precision. The regularities of the orbital evolution of the two sounding satellites are qualitatively and quantitatively analyzed. Under the current tracking mode the corresponding prediction precision of orbit determination is analyzed based on the different stages of the orbital evolution, thereby providing the basis for the adjustment of planning mode by the satellite application departments and the guarantee of normal satellite payload. Finally, the orbital lifetimes of the two satellites are predicted through the trend of the orbital evolution. Key words: satellite—TC-1 and TC-2—celestial mechanics
1. INTRODUCTION The two sounding satellites are the only two with great elliptical orbits in the satellites of China which are in orbit around the earth, consisting of Satellite TC-1 and TC-2. TC-1, moving around the equator with the eccentricity of 0.85 and the apogee distance of more than 60000 km, is the furthest one from the earth at present in the satellites launched by
∗
Received 2006-04-03; revised version 2006-06-22 A translation of Acta Astron. Sin. Vol. 48, No. 3, pp. 355-363, 2007 miky [email protected]
0275-1062/07/$-see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.chinastron.2007.10.006
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China, while TC-2 with the eccentricity of 0.74 and the furthermost distance from the earth of more than 40000 km, is running around the South Pole and North Pole of the earth. The main purpose of the two sounding satellites is to understand the change in the earth magnetosphere and the factors which have effect on the occurrence of great magnetic storms of the earth, and so forth. Through the close coordination between them and the 4 satellites of the “Cluster-II Spacecraft” of the European Space Agency the omni-bearing sounding system for the terrestrial space, composed of the satellites distributed in depth and in scope, has been formed, which is the first 6-point solid sounding of the terrestrial space in the history of mankind. Owing to the influence of perturbations there exist the long-term and long-periodic variations in the orbits of the two satellites. It is seen from the current situation that the orbit of TC-2 has much more pronounced change and after the launch for a year its height of perigee descended to 240 km from 700 km at the initial stage. With the variation in the satellite orbit, the orbit determination and prediction precision of the satellite exhibit changes and have a certain effect on the satellite-borne payloads. It is needed to make an analysis of the reason why the satellite orbit evolves, to predict the future variation trend and to ascertain the prediction precision of orbit determination, thereby being advantageous to the adjustment of the planning mode of the satellite-borne payloads by the application departments and to the guarantee of the normal running of the payloads. 2. ANALYSIS OF EVOLUTIONARY REGULARITIES OF ORBITS OF TC-1 AND TC-2 TC-1 and TC-2 are the two satellites with great elliptical orbits and from the angle of both the payload application and the orbit determination, the height of perigee is one of very important factors. As the observation and control unit of TC-1 and TC-2, the Xi’an Satellite Observation and Control Center (XSCC) makes arrangements for the tracking of the two satellites for one or two circles every day by means of the USB equipment, makes the orbit determination by utilizing the data segmental arcs of 3 days and analyzes the precision of orbit determination and the regularities of changes. After the launch of TC-2 in August of 2004, the height of perigee of its orbit descended all the way, reached the lowest point 230 km in September, 2005, and then began to ascend gradually (see Fig.1). After the launch of TC-1 in December, 2003, the period of change in the orbital perigee was a year, with the lowest height being about 350 km and the highest height being about 700 km, and the height of perigee began to show the ascending trend in October of 2005 (see Fig.2). Among the 6 orbital elements, only the semi-major axis and eccentricity have correlation to the height of perigee and the changing trend of the height of perigee can be determined through the analysis of the law of variation in the semi-major axis and eccentricity. Figs.3 and 4 demonstrate the curves of change in eccentricities of TC-2 and TC-1, respectively, and Figs.5 and 6 display the curves of change in semi-major axes of TC-2 and TC-1, respectively. It can be seen that the variation in eccentricity corresponds to that in perigee. Among the comparatively remarkable disturbing forces exerting on TC-1 and TC-2 there are the terrestrial non-spherical perturbation, solar gravitational perturbation, lunar gravitational perturbation, solar light pressure perturbation, atmospheric damping perturbation and relativistic effect perturbation[1] . The maximum values of the perturbation accelerations in a day at the height of perigee about 600 km and 250 km, respectively, are
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Fig. 1 Curve of the perigee evolution of TC-2
Fig. 2
Curve of the perigee evolution of TC-1
listed in Table 1. When the height of perigee of a satellite is about 600 km, the atmospheric damping perturbation, light pressure perturbation and relativistic effect perturbation are all relatively small, being in the order of 10−8 , and have a limited effect on the orbit. The non-spherical perturbation, solar perturbation and lunar perturbation have comparatively great effects on the satellite orbit. Based on the orbital perturbation theory, the terrestrial non-spherical perturbation has no long-term and long-periodic effects on the semi-major axis and eccentricity of the satellite. Therefore, when the problem of the evolution of the height of perigee is discussed, the direct effect of the terrestrial non-spherical perturbation can be excluded. When the satellite height of perigee is about 250 km, the atmospheric damping perturbation becomes great and the maximum perturbation acceleration surpasses the luni-solar gravitational perturbation, to which close attention needs be paid when the orbit determination and prediction are made.
Fig. 3 Curve of the eccentricity evolution of TC-2
Table 1
Fig. 4
Curve of the eccentricity evolution of TC-1
Perturbation accelerations of TC-1 and TC-2 for different heights of perigee
Satellite TC − 1 TC − 2 TC − 1 TC − 2
Perigee Nonspherical Solar Luna Drag Sunlight Relativistic 600 km 9.9911e-3 4.3758e-6 9.6659e-6 3.5026e-8 8.4383e-8 1.3282e-8 600 km 1.8540e-2 2.7117e-6 3.7097e-6 5.7013e-8 8.5528e-8 1.3150e-8 250 km 1.4043e-2 4.8657e-6 1.8921e-5 5.0226e-5 8.2739e-8 1.6252e-8 250 km 1.4048e-2 3.4591e-6 1.1440E-5 8.0075E-5 8.4642E-8 1.5448E-8
When TC-1 and TC-2 came into their orbits, the heights of perigee were relatively higher, about 700 km, and the atmospheric damping perturbation was comparatively small.
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Fig. 5 Curves of change in eccentricity of TC-2 under different three-body perturbations (L1: without the luni-solar perturbation, L2: with solar perturbation, L3: with lunar perturbation and L4: with luni-solar perturbation)
Table 2 exhibits the attenuation ratios of the semi-major axis of TC-2 at different orbital heights obtained under different simulation conditions of space environment. The conclusion can be drawn that when the height exceeds 600 km, the attenuation of the orbital semi-major axis owing to the effect of atmosphere is below one order of meter. But, the effects of the solar and lunar gravitational perturbations on the eccentricity are comparatively great and cause the orbital height of perigee to have radical changes. According to Ref.[2], the long-term perturbation of the semi-major axis and eccentricity of the satellite can not be produced by the luni-solar perturbation, but the long-periodic perturbation can be produced. Then one may have al (t) = 0 el (t) = −
H1 =
3 3 βα n 2
5 e 2
(1) 1 − e 2 H1 ,
1 1 1 [ (1 − cos i)2 K1 + (1 + cos i)2 K2 + sin2 iK3 + 16 2 2 2 sin i(1 − cos i)K4 + 2 sin i(1 + cos i)K5 ,
(2)
(3)
where K1 , K2 , K3 , K4 and K5 stand for the slowly varying coefficients related to the sun and the moon. As for the concrete calculations, please see Ref. [2]. The curves of variation in the eccentricity of TC-2 obtained from the simulation calculation of the long-periodic perturbation are shown in Fig.5. It can be seen that in the case without the luni-solar perturbation the eccentricity remains to be stable, while the effect of the solar and lunar perturbations, besides the long-periodic term for about 3 years and also the long-periodic term for about a year, causes the eccentricity of TC-2 to fluctuate between 0.725 and 0.745.
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Table 2
Attenuation rate (in units of m/day) of the semi-major axis of TC-2 for different atmospheric simulation conditions of space environment Simulant space environment 700 km 500 km 300 km F 10.7 = 150, KP = 2.4 1.45 m/day 8.6 m/day 582 m/day F 10.7 = 90, KP = 2.0 0.46 m/day 1.9 m/day 274 m/day F 10.7 = 70, KP = 1.0 0.33 m/day 0.7 m/day 180 m/day
F 10.7 is the 10.7 cm radiant flux of the sun, in units of 10−22 w/m2 , and KP is the geomagnetic index. As for TC-1, before 2007, the height of perigee was always above 400 km, therefore the effect of the atmospheric damping perturbation on the attenuation of the semi-major axis of the satellite was not great. After June of 2007, since the orbital height of perigee will descend below 200 km, the atmospheric damping acts remarkably and makes the semi-major axis attenuate intensely (see Fig.6), while as for TC-2, its orbital height of perigee is below 400 km at half time in every period. Therefore the atmospheric damping perturbation is relatively greater and has comparatively remarkable effect on the semi-major axis, with the attenuation of the semi-major axis in every period being about 130 km (see Fig.7)
Fig. 6 TC-1
Curve of variation in the semi-major axis of
Fig. 7 Curve of variation in the semi-major axis of TC-2
3. ANALYSIS OF THE PREDICTION PRECISION OF ORBIT DETERMINATION AT DIFFERENT HEIGHTS OF PERIGEE For TC-1 and TC-2, since different heights of perigee result in different amounts of effect of perturbation factors and the tactics for the orbit determination prediction also different, the prediction precision of orbit determination will also change. According to the characteristics of the satellite orbit at the initial stage of launch and the requirements of the payload application, the requirements for the orbital precision are that the positional error of perigee does not surpass 4 km and the positional error of apogee is 20 km. It is corresponding to this precision that the satellites are tracked for a circle per day and for more than one hour per circle, and the national USB mode of observation and control network is utilized (for the distribution of the observation and control network and the precision, please refer to Table 3). Next we shall determine the prediction precision of orbit determination at different stages
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based on the tracking data under this mode. Since there are no observed data with relatively higher precision, the orbital determination precision is determined by taking advantage of the method of cross segmental arc[3]. The so-called cross segmental arc method lies in that the continuous data are independently divided into two segments in chronological sequence. The data of the first segment are used to make the orbit determination and predict the position of the satellite outside the time interval of the segment data, and then the data of the second segment are utilized to make the orbital determination and give the satellite position within the segmenta1 arc of orbit determination. Because there is the cross segmental arc between the prediction segmental arc of the first segment and the orbit determination segmental arc of the second segment, the differences of the two orbital positions are compared within the cross segmental arc, by which the prediction precision is judged. The orbit determination data are regarded as the primary standard when the prediction precision is determined. Table 3
Distribution of the observation and control network of TC-1 and TC-2 and the precision of orbit determination station longitude latitude altitude precision Weinan 109◦ 34◦ 600 m Qindao 120◦ 36◦ 0m Range: 10 m Xiamen 112◦ 25◦ 20 m Range rate: 5 cm/s 29◦ 1000 m Azimuth: 50 Kashi 76◦ Nanning 108◦ 23◦ 100 m Elevation: 50
3.1 Tactics for orbit determination at different heights of perigee For different heights of perigee because of different amounts of various perturbation factors we determine relevant tactics for orbit determination in order to be able to increase the precision of orbit determination to the maximum limit (see Table 4). Since only a station tracks the satellites every day, the selected segmental arc should be long enough, 4 or 5 days on the whole and in the light of the atmospheric factors at different heights of perigee it needs to be determined whether the atmospheric damping coefficient is found. Here, the atmospheric model adopts MSIS90 [4,5] and the solar radiant flux adopts the predicted value in the international internet. Table 4
Tactics catalog of orbit determination at different heights of perigee Perigee Height Drag coefficient Earth Gravitation arc under 500 km fixed 50×50 4 − 5 days 300 − 500 km solved 70×70 4 days below 300 km Solved every 2 days 70×70 4 days
3.2 Prediction precision of orbit determination at the height of perigee of about 700 km At the initial stage of the launch of the satellite TC-2, the orbital height of perigee was about 700 km and we employed the data of two time intervals from Aug.22 to Aug.25 and from Aug.24 to Aug.27 in the observed data of tracking from Aug.22 to Aug.27 in 2004 (for the observed data, please refer to Table 5) to determine the orbit determination precision of the cross segmental arc from Aug.24 to Aug.25 (see Fig.8). It can be seen that when the orbital height of perigee is about 700 km, the perigee’s positional precision of orbit determination is in the order of 100 meters and the apogee’s positional precision
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is in the order of a kilometer and they completely meet the requirements for the perigee positional error of 4 km. The trajectory prediction of 7 days (or 216 hours, where the 48 hours are the segmental arc of orbit determination, the same below) determined from the orbit determination carried out by taking advantage of the data obtained from Aug.22 to Aug.25 is compared with the trajectory of orbit determination in the time interval and for the positional error, please refer to Fig.9. It can be seen that the apogee error of the satellite prediction is in the order of one kilometer and the perigee error is within 1 km. For the observed data of different positions of tracking orbits, there is a small difference between the determined orbit’s determination precision and prediction precision, but the errors are roughly equal to each other. Table 5
Observed data obtained from Aug.22 to Aug.27 in 2004
arc 1 2 3 4 5 6
date 8/22/2004 8/23/2004 8/24/2004 8/25/2004 8/26/2004 8/27/2004
Start time 7:40:23.200 6:50:31.200 5:50:51.200 6:45:39.200 6:26:10.200 1:26:13.200
Fig. 8 Orbit determination precision at the height of perigee of about 700 km
arc length Location in arc 6670.676s near apogee 5706.800s near apogee 11748.716s near apogee 8769.800s midway 7374.800s midway 4335.800s near apogee
Fig. 9 Prediction precision at the height of perigee of about 700 km
3.3 Prediction precision of orbit determination at the height of perigee of about 500 km In January of 2005 or so, the orbital height of TC-2 descended to about 500 km. We respectively utilized the data of the two time intervals from Jan.6 to Jan.9 and from Jan.8 to Jan.11 in the observed data of tracking obtained from Jan.6 to Jan.11 in 2005 to make the orbit determination and found the orbit determination precision of the cross segmental arc from Jan.24 to Jan.25 (see Fig.10). The result obtained from the comparison of the trajectory prediction of 7 days determined from Jan.6 to Jan.9 with the orbit determination trajectory shown in Fig.11. It can be known from the above two figures and much more calculated results that when the height of perigee is about 50 km, the positional precision of perigee is 2 or 3 km and the positional precision of apogee is within 10 km. In comparison with the positional
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precision when the perigee is about 700 km, the positional precision of perigee obviously becomes poor and the positional precision of apogee is equivalent to that when the height of perigee is about 50 km. If the outward prediction of 7 days is made, the perigee positional precision is also within 5 km and the apogee positional precision is 20 km.
Fig. 10
Orbit determination precision when the
height of perigee is about 500 km
Fig. 12
Orbit determination precision when the
height of perigee is about 300 km
Fig. 11 Orbit prediction precision when the height of perigee is about 500 km
Fig. 13 Orbit prediction precision when the height of perigee is about 300 km
3.4 Orbit determination prediction precision at the height of perigee of about 300 km In June of 2005 or so, the orbital height of TC-2 descended to about 300 km. We respectively employed the data of the two time intervals from Jun.24 to Jun.27 and from Jun.26 to Jun.29 in the observed data of tracking obtained from Jun.24 to Jun.29 in 2005 to carry out the orbit determination and determined the orbit determination precision of the cross segmental arc from Jun.26 to Jun.27 (see Fig.12). The result obtained from the comparison between the trajectory prediction of 7 days and the orbit determination trajectory is shown in Fig.13. From the above two figures it can be seen that when the height of perigee is about
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300 km and if the number of the tracking circles made for one circle per day is used to conduct the orbit determination, the orbital positional deterministic precision of perigee is about 10 km, which can not meet the needs for the precision. As for the prediction of 7 days, the perigee is about 10 km and the apogee positional precision possibly reaches 80 km.
Fig. 14 TC-1
Prediction curve of the height of perigee for
Fig. 15 Prediction curve of the height of perigee for TC-2
4. LIFETIME ESTIMATE OF ORBITS OF TC-1 AND TC-2 As for the two sounding satellites TC-1 and TC-2, the main factor affecting the orbital lifetime is the height of perigee. Owing to the solar and lunar gravitational perturbations there exists the long-periodic variation in the orbital eccentricity, which causes the increase or decrease in the orbital height of perigee. If the height of perigee descends below 100 km, the effect of the atmospheric damping and the attenuation of the semi-major axis become intense. So, we respectively used the orbits of TC-1 and TC-2 determined on 2005 November 2 to carry out the extrapolation to post-2009 and took into account the dynamic models, such as the atmospheric damping perturbation, light pressure perturbation, solar, lunar and planetary perturbations and terrestrial non-spherical perturbation, etc.[7] . For the space environment parameter we adopt the predicted value. The results are shown in Figs.14 and 15. It can be seen that on 2007 October 2 or so, TC-1 will enter the atmosphere when its height of perigee comes down to 100 km and will drop onto the ground on October 28 or so when its height of perigee is zero. In the process of prediction, if the restriction factor that the near-earth distance is greater than the radius of the earth is not considered, the height of perigee will descend to -200 km and then ascend again. But after 2009 the height of perigee of TC-2 will be always above 200 km and will not drop. It is known that the long-periodic effect is caused mainly by the lunar and solar gravitational perturbations, and though the variation in solar activities and the different atmospheric models have effect on the satellite height of perigee, they do not change the whole trend of its variation[7].
5. CONLUSIONS
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From the orbit determination prediction made by means of the theoretical analysis and observed data it can be known that the effect of the solar and lunar perturbations result in the long-periodic variation in the eccentricity of the two sounding satellites TC-1 and TC-2, thereby causing the long-periodic variation in the orbital height of perigee, lowering in one time interval and rising in another time interval. When the height of perigee descends to a certain height, the effect of the atmospheric damping perturbation leads to the attenuation of the semi-major axis of the satellite, thereby decreasing the prediction precision of the orbit determination. When the height of perigee is 700 km, the positional error of perigee can be in the order of a hundred meters and when the height of perigee is about 500 km, the perigee positional error is in the order of one kilometer which can meet the needs for the precision. However, when the height of perigee of the satellite is about 300 km, the perigee positional error reaches 10 km, therefore the number of circles of tracking need to be increased so that the orbit determination precision may be guaranteed. In addition, under the condition of no the orbital adjustment, the lifetime of TC-2 can be prolonged after 2009 and it is estimated that TC-1 will drop onto the ground in October of 2007. References 1
Li J.S., Precision Orbit Determination for Artificial satellites, Army Press, 1995
2
Liu L., Orbit Theory of Spacecraft, Beijing: National Defense Industry Press, 2000
3
Hu X.G., Huang C., Huang Y., AcAsn, 2005, 46(2), 186
4
Huang Y., Hu X.G., Huang C., ChA&A, 30(3), 318
5
Huang Y., Hu X.G., Huang C., AcAsn, 2006, 47(1), 82
6
Chen J.R., Wang J.S., Journal of Spacecraft TT&C Technology, 2005, 98(6), 34
7
Wang J.S., Chen J.R., Ma P.B., Journal of Spacecraft TT&C Technology, 2006, 99(1), 31
ELSEVIER
Chinese Astronomy and Astrophysics 31 (2007) 420–429
The Orbital Evolution of Two Sounding Satellites and Analysis of the Accuracy of Orbit Determination WU Gong-you1,2∗ ZHAO Chang-yin1 ZHANG Rong-zhi2 WANG Jia-song2 WANG Hong-bo1 1
Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008 2 Xi’an Satellite Observation and Control Center, Xi’an 710043
Abstract The satellites TC-1 and TC-2 are the two Chinese satellites with great elliptical orbits which are still in orbit around the earth at present. Since the launch the orbits of the two satellites have continuously evolved, which has a certain effect on the orbit determination and prediction precision. The regularities of the orbital evolution of the two sounding satellites are qualitatively and quantitatively analyzed. Under the current tracking mode the corresponding prediction precision of orbit determination is analyzed based on the different stages of the orbital evolution, thereby providing the basis for the adjustment of planning mode by the satellite application departments and the guarantee of normal satellite payload. Finally, the orbital lifetimes of the two satellites are predicted through the trend of the orbital evolution. Key words: satellite—TC-1 and TC-2—celestial mechanics
1. INTRODUCTION The two sounding satellites are the only two with great elliptical orbits in the satellites of China which are in orbit around the earth, consisting of Satellite TC-1 and TC-2. TC-1, moving around the equator with the eccentricity of 0.85 and the apogee distance of more than 60000 km, is the furthest one from the earth at present in the satellites launched by
∗
Received 2006-04-03; revised version 2006-06-22 A translation of Acta Astron. Sin. Vol. 48, No. 3, pp. 355-363, 2007 miky [email protected]
0275-1062/07/$-see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.chinastron.2007.10.006
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China, while TC-2 with the eccentricity of 0.74 and the furthermost distance from the earth of more than 40000 km, is running around the South Pole and North Pole of the earth. The main purpose of the two sounding satellites is to understand the change in the earth magnetosphere and the factors which have effect on the occurrence of great magnetic storms of the earth, and so forth. Through the close coordination between them and the 4 satellites of the “Cluster-II Spacecraft” of the European Space Agency the omni-bearing sounding system for the terrestrial space, composed of the satellites distributed in depth and in scope, has been formed, which is the first 6-point solid sounding of the terrestrial space in the history of mankind. Owing to the influence of perturbations there exist the long-term and long-periodic variations in the orbits of the two satellites. It is seen from the current situation that the orbit of TC-2 has much more pronounced change and after the launch for a year its height of perigee descended to 240 km from 700 km at the initial stage. With the variation in the satellite orbit, the orbit determination and prediction precision of the satellite exhibit changes and have a certain effect on the satellite-borne payloads. It is needed to make an analysis of the reason why the satellite orbit evolves, to predict the future variation trend and to ascertain the prediction precision of orbit determination, thereby being advantageous to the adjustment of the planning mode of the satellite-borne payloads by the application departments and to the guarantee of the normal running of the payloads. 2. ANALYSIS OF EVOLUTIONARY REGULARITIES OF ORBITS OF TC-1 AND TC-2 TC-1 and TC-2 are the two satellites with great elliptical orbits and from the angle of both the payload application and the orbit determination, the height of perigee is one of very important factors. As the observation and control unit of TC-1 and TC-2, the Xi’an Satellite Observation and Control Center (XSCC) makes arrangements for the tracking of the two satellites for one or two circles every day by means of the USB equipment, makes the orbit determination by utilizing the data segmental arcs of 3 days and analyzes the precision of orbit determination and the regularities of changes. After the launch of TC-2 in August of 2004, the height of perigee of its orbit descended all the way, reached the lowest point 230 km in September, 2005, and then began to ascend gradually (see Fig.1). After the launch of TC-1 in December, 2003, the period of change in the orbital perigee was a year, with the lowest height being about 350 km and the highest height being about 700 km, and the height of perigee began to show the ascending trend in October of 2005 (see Fig.2). Among the 6 orbital elements, only the semi-major axis and eccentricity have correlation to the height of perigee and the changing trend of the height of perigee can be determined through the analysis of the law of variation in the semi-major axis and eccentricity. Figs.3 and 4 demonstrate the curves of change in eccentricities of TC-2 and TC-1, respectively, and Figs.5 and 6 display the curves of change in semi-major axes of TC-2 and TC-1, respectively. It can be seen that the variation in eccentricity corresponds to that in perigee. Among the comparatively remarkable disturbing forces exerting on TC-1 and TC-2 there are the terrestrial non-spherical perturbation, solar gravitational perturbation, lunar gravitational perturbation, solar light pressure perturbation, atmospheric damping perturbation and relativistic effect perturbation[1] . The maximum values of the perturbation accelerations in a day at the height of perigee about 600 km and 250 km, respectively, are
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Fig. 1 Curve of the perigee evolution of TC-2
Fig. 2
Curve of the perigee evolution of TC-1
listed in Table 1. When the height of perigee of a satellite is about 600 km, the atmospheric damping perturbation, light pressure perturbation and relativistic effect perturbation are all relatively small, being in the order of 10−8 , and have a limited effect on the orbit. The non-spherical perturbation, solar perturbation and lunar perturbation have comparatively great effects on the satellite orbit. Based on the orbital perturbation theory, the terrestrial non-spherical perturbation has no long-term and long-periodic effects on the semi-major axis and eccentricity of the satellite. Therefore, when the problem of the evolution of the height of perigee is discussed, the direct effect of the terrestrial non-spherical perturbation can be excluded. When the satellite height of perigee is about 250 km, the atmospheric damping perturbation becomes great and the maximum perturbation acceleration surpasses the luni-solar gravitational perturbation, to which close attention needs be paid when the orbit determination and prediction are made.
Fig. 3 Curve of the eccentricity evolution of TC-2
Table 1
Fig. 4
Curve of the eccentricity evolution of TC-1
Perturbation accelerations of TC-1 and TC-2 for different heights of perigee
Satellite TC − 1 TC − 2 TC − 1 TC − 2
Perigee Nonspherical Solar Luna Drag Sunlight Relativistic 600 km 9.9911e-3 4.3758e-6 9.6659e-6 3.5026e-8 8.4383e-8 1.3282e-8 600 km 1.8540e-2 2.7117e-6 3.7097e-6 5.7013e-8 8.5528e-8 1.3150e-8 250 km 1.4043e-2 4.8657e-6 1.8921e-5 5.0226e-5 8.2739e-8 1.6252e-8 250 km 1.4048e-2 3.4591e-6 1.1440E-5 8.0075E-5 8.4642E-8 1.5448E-8
When TC-1 and TC-2 came into their orbits, the heights of perigee were relatively higher, about 700 km, and the atmospheric damping perturbation was comparatively small.
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Fig. 5 Curves of change in eccentricity of TC-2 under different three-body perturbations (L1: without the luni-solar perturbation, L2: with solar perturbation, L3: with lunar perturbation and L4: with luni-solar perturbation)
Table 2 exhibits the attenuation ratios of the semi-major axis of TC-2 at different orbital heights obtained under different simulation conditions of space environment. The conclusion can be drawn that when the height exceeds 600 km, the attenuation of the orbital semi-major axis owing to the effect of atmosphere is below one order of meter. But, the effects of the solar and lunar gravitational perturbations on the eccentricity are comparatively great and cause the orbital height of perigee to have radical changes. According to Ref.[2], the long-term perturbation of the semi-major axis and eccentricity of the satellite can not be produced by the luni-solar perturbation, but the long-periodic perturbation can be produced. Then one may have al (t) = 0 el (t) = −
H1 =
3 3 βα n 2
5 e 2
(1) 1 − e 2 H1 ,
1 1 1 [ (1 − cos i)2 K1 + (1 + cos i)2 K2 + sin2 iK3 + 16 2 2 2 sin i(1 − cos i)K4 + 2 sin i(1 + cos i)K5 ,
(2)
(3)
where K1 , K2 , K3 , K4 and K5 stand for the slowly varying coefficients related to the sun and the moon. As for the concrete calculations, please see Ref. [2]. The curves of variation in the eccentricity of TC-2 obtained from the simulation calculation of the long-periodic perturbation are shown in Fig.5. It can be seen that in the case without the luni-solar perturbation the eccentricity remains to be stable, while the effect of the solar and lunar perturbations, besides the long-periodic term for about 3 years and also the long-periodic term for about a year, causes the eccentricity of TC-2 to fluctuate between 0.725 and 0.745.
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Table 2
Attenuation rate (in units of m/day) of the semi-major axis of TC-2 for different atmospheric simulation conditions of space environment Simulant space environment 700 km 500 km 300 km F 10.7 = 150, KP = 2.4 1.45 m/day 8.6 m/day 582 m/day F 10.7 = 90, KP = 2.0 0.46 m/day 1.9 m/day 274 m/day F 10.7 = 70, KP = 1.0 0.33 m/day 0.7 m/day 180 m/day
F 10.7 is the 10.7 cm radiant flux of the sun, in units of 10−22 w/m2 , and KP is the geomagnetic index. As for TC-1, before 2007, the height of perigee was always above 400 km, therefore the effect of the atmospheric damping perturbation on the attenuation of the semi-major axis of the satellite was not great. After June of 2007, since the orbital height of perigee will descend below 200 km, the atmospheric damping acts remarkably and makes the semi-major axis attenuate intensely (see Fig.6), while as for TC-2, its orbital height of perigee is below 400 km at half time in every period. Therefore the atmospheric damping perturbation is relatively greater and has comparatively remarkable effect on the semi-major axis, with the attenuation of the semi-major axis in every period being about 130 km (see Fig.7)
Fig. 6 TC-1
Curve of variation in the semi-major axis of
Fig. 7 Curve of variation in the semi-major axis of TC-2
3. ANALYSIS OF THE PREDICTION PRECISION OF ORBIT DETERMINATION AT DIFFERENT HEIGHTS OF PERIGEE For TC-1 and TC-2, since different heights of perigee result in different amounts of effect of perturbation factors and the tactics for the orbit determination prediction also different, the prediction precision of orbit determination will also change. According to the characteristics of the satellite orbit at the initial stage of launch and the requirements of the payload application, the requirements for the orbital precision are that the positional error of perigee does not surpass 4 km and the positional error of apogee is 20 km. It is corresponding to this precision that the satellites are tracked for a circle per day and for more than one hour per circle, and the national USB mode of observation and control network is utilized (for the distribution of the observation and control network and the precision, please refer to Table 3). Next we shall determine the prediction precision of orbit determination at different stages
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based on the tracking data under this mode. Since there are no observed data with relatively higher precision, the orbital determination precision is determined by taking advantage of the method of cross segmental arc[3]. The so-called cross segmental arc method lies in that the continuous data are independently divided into two segments in chronological sequence. The data of the first segment are used to make the orbit determination and predict the position of the satellite outside the time interval of the segment data, and then the data of the second segment are utilized to make the orbital determination and give the satellite position within the segmenta1 arc of orbit determination. Because there is the cross segmental arc between the prediction segmental arc of the first segment and the orbit determination segmental arc of the second segment, the differences of the two orbital positions are compared within the cross segmental arc, by which the prediction precision is judged. The orbit determination data are regarded as the primary standard when the prediction precision is determined. Table 3
Distribution of the observation and control network of TC-1 and TC-2 and the precision of orbit determination station longitude latitude altitude precision Weinan 109◦ 34◦ 600 m Qindao 120◦ 36◦ 0m Range: 10 m Xiamen 112◦ 25◦ 20 m Range rate: 5 cm/s 29◦ 1000 m Azimuth: 50 Kashi 76◦ Nanning 108◦ 23◦ 100 m Elevation: 50
3.1 Tactics for orbit determination at different heights of perigee For different heights of perigee because of different amounts of various perturbation factors we determine relevant tactics for orbit determination in order to be able to increase the precision of orbit determination to the maximum limit (see Table 4). Since only a station tracks the satellites every day, the selected segmental arc should be long enough, 4 or 5 days on the whole and in the light of the atmospheric factors at different heights of perigee it needs to be determined whether the atmospheric damping coefficient is found. Here, the atmospheric model adopts MSIS90 [4,5] and the solar radiant flux adopts the predicted value in the international internet. Table 4
Tactics catalog of orbit determination at different heights of perigee Perigee Height Drag coefficient Earth Gravitation arc under 500 km fixed 50×50 4 − 5 days 300 − 500 km solved 70×70 4 days below 300 km Solved every 2 days 70×70 4 days
3.2 Prediction precision of orbit determination at the height of perigee of about 700 km At the initial stage of the launch of the satellite TC-2, the orbital height of perigee was about 700 km and we employed the data of two time intervals from Aug.22 to Aug.25 and from Aug.24 to Aug.27 in the observed data of tracking from Aug.22 to Aug.27 in 2004 (for the observed data, please refer to Table 5) to determine the orbit determination precision of the cross segmental arc from Aug.24 to Aug.25 (see Fig.8). It can be seen that when the orbital height of perigee is about 700 km, the perigee’s positional precision of orbit determination is in the order of 100 meters and the apogee’s positional precision
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is in the order of a kilometer and they completely meet the requirements for the perigee positional error of 4 km. The trajectory prediction of 7 days (or 216 hours, where the 48 hours are the segmental arc of orbit determination, the same below) determined from the orbit determination carried out by taking advantage of the data obtained from Aug.22 to Aug.25 is compared with the trajectory of orbit determination in the time interval and for the positional error, please refer to Fig.9. It can be seen that the apogee error of the satellite prediction is in the order of one kilometer and the perigee error is within 1 km. For the observed data of different positions of tracking orbits, there is a small difference between the determined orbit’s determination precision and prediction precision, but the errors are roughly equal to each other. Table 5
Observed data obtained from Aug.22 to Aug.27 in 2004
arc 1 2 3 4 5 6
date 8/22/2004 8/23/2004 8/24/2004 8/25/2004 8/26/2004 8/27/2004
Start time 7:40:23.200 6:50:31.200 5:50:51.200 6:45:39.200 6:26:10.200 1:26:13.200
Fig. 8 Orbit determination precision at the height of perigee of about 700 km
arc length Location in arc 6670.676s near apogee 5706.800s near apogee 11748.716s near apogee 8769.800s midway 7374.800s midway 4335.800s near apogee
Fig. 9 Prediction precision at the height of perigee of about 700 km
3.3 Prediction precision of orbit determination at the height of perigee of about 500 km In January of 2005 or so, the orbital height of TC-2 descended to about 500 km. We respectively utilized the data of the two time intervals from Jan.6 to Jan.9 and from Jan.8 to Jan.11 in the observed data of tracking obtained from Jan.6 to Jan.11 in 2005 to make the orbit determination and found the orbit determination precision of the cross segmental arc from Jan.24 to Jan.25 (see Fig.10). The result obtained from the comparison of the trajectory prediction of 7 days determined from Jan.6 to Jan.9 with the orbit determination trajectory shown in Fig.11. It can be known from the above two figures and much more calculated results that when the height of perigee is about 50 km, the positional precision of perigee is 2 or 3 km and the positional precision of apogee is within 10 km. In comparison with the positional
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precision when the perigee is about 700 km, the positional precision of perigee obviously becomes poor and the positional precision of apogee is equivalent to that when the height of perigee is about 50 km. If the outward prediction of 7 days is made, the perigee positional precision is also within 5 km and the apogee positional precision is 20 km.
Fig. 10
Orbit determination precision when the
height of perigee is about 500 km
Fig. 12
Orbit determination precision when the
height of perigee is about 300 km
Fig. 11 Orbit prediction precision when the height of perigee is about 500 km
Fig. 13 Orbit prediction precision when the height of perigee is about 300 km
3.4 Orbit determination prediction precision at the height of perigee of about 300 km In June of 2005 or so, the orbital height of TC-2 descended to about 300 km. We respectively employed the data of the two time intervals from Jun.24 to Jun.27 and from Jun.26 to Jun.29 in the observed data of tracking obtained from Jun.24 to Jun.29 in 2005 to carry out the orbit determination and determined the orbit determination precision of the cross segmental arc from Jun.26 to Jun.27 (see Fig.12). The result obtained from the comparison between the trajectory prediction of 7 days and the orbit determination trajectory is shown in Fig.13. From the above two figures it can be seen that when the height of perigee is about
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300 km and if the number of the tracking circles made for one circle per day is used to conduct the orbit determination, the orbital positional deterministic precision of perigee is about 10 km, which can not meet the needs for the precision. As for the prediction of 7 days, the perigee is about 10 km and the apogee positional precision possibly reaches 80 km.
Fig. 14 TC-1
Prediction curve of the height of perigee for
Fig. 15 Prediction curve of the height of perigee for TC-2
4. LIFETIME ESTIMATE OF ORBITS OF TC-1 AND TC-2 As for the two sounding satellites TC-1 and TC-2, the main factor affecting the orbital lifetime is the height of perigee. Owing to the solar and lunar gravitational perturbations there exists the long-periodic variation in the orbital eccentricity, which causes the increase or decrease in the orbital height of perigee. If the height of perigee descends below 100 km, the effect of the atmospheric damping and the attenuation of the semi-major axis become intense. So, we respectively used the orbits of TC-1 and TC-2 determined on 2005 November 2 to carry out the extrapolation to post-2009 and took into account the dynamic models, such as the atmospheric damping perturbation, light pressure perturbation, solar, lunar and planetary perturbations and terrestrial non-spherical perturbation, etc.[7] . For the space environment parameter we adopt the predicted value. The results are shown in Figs.14 and 15. It can be seen that on 2007 October 2 or so, TC-1 will enter the atmosphere when its height of perigee comes down to 100 km and will drop onto the ground on October 28 or so when its height of perigee is zero. In the process of prediction, if the restriction factor that the near-earth distance is greater than the radius of the earth is not considered, the height of perigee will descend to -200 km and then ascend again. But after 2009 the height of perigee of TC-2 will be always above 200 km and will not drop. It is known that the long-periodic effect is caused mainly by the lunar and solar gravitational perturbations, and though the variation in solar activities and the different atmospheric models have effect on the satellite height of perigee, they do not change the whole trend of its variation[7].
5. CONLUSIONS
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From the orbit determination prediction made by means of the theoretical analysis and observed data it can be known that the effect of the solar and lunar perturbations result in the long-periodic variation in the eccentricity of the two sounding satellites TC-1 and TC-2, thereby causing the long-periodic variation in the orbital height of perigee, lowering in one time interval and rising in another time interval. When the height of perigee descends to a certain height, the effect of the atmospheric damping perturbation leads to the attenuation of the semi-major axis of the satellite, thereby decreasing the prediction precision of the orbit determination. When the height of perigee is 700 km, the positional error of perigee can be in the order of a hundred meters and when the height of perigee is about 500 km, the perigee positional error is in the order of one kilometer which can meet the needs for the precision. However, when the height of perigee of the satellite is about 300 km, the perigee positional error reaches 10 km, therefore the number of circles of tracking need to be increased so that the orbit determination precision may be guaranteed. In addition, under the condition of no the orbital adjustment, the lifetime of TC-2 can be prolonged after 2009 and it is estimated that TC-1 will drop onto the ground in October of 2007. References 1
Li J.S., Precision Orbit Determination for Artificial satellites, Army Press, 1995
2
Liu L., Orbit Theory of Spacecraft, Beijing: National Defense Industry Press, 2000
3
Hu X.G., Huang C., Huang Y., AcAsn, 2005, 46(2), 186
4
Huang Y., Hu X.G., Huang C., ChA&A, 30(3), 318
5
Huang Y., Hu X.G., Huang C., AcAsn, 2006, 47(1), 82
6
Chen J.R., Wang J.S., Journal of Spacecraft TT&C Technology, 2005, 98(6), 34
7
Wang J.S., Chen J.R., Ma P.B., Journal of Spacecraft TT&C Technology, 2006, 99(1), 31