A.I. Nazarenko 1

Transcription

1 Model Study of the Possibilities of Space Debris Cataloging A.I. Nazarenko 1 anazarenko32@mail.ru The objective of this paper is the estimation of the characteristics of the measuring means and the software, providing the possibility of cataloging of smaller objects in the LEO region, in comparison with the existing catalog of space objects. Cataloguing conditions It is known that the cataloguing possibility depends not only on the quantity and accuracy of measurements, but also on the quantity of objects in space, as well as on the accuracy of the determination and prediction of orbital parameters. Estimation of orbit determination and prediction accuracy plays an important part in estimating the cataloguing possibilities, and the materials of known publications do not contain an accuracy estimation that is necessary for cataloguing the objects of various sizes. The figure below schematically shows the conditions under which the object designated by a red color can be catalogued. Figure 1. Cataloguing conditions The light red color flooded the region of the possible positions of a considered object during the prediction of its state vector. The necessary condition consists in the fact that the next new measurement of this space object (SO) would be first in time among the measured objects falling 1 Scientific Technological Center KOSMONIT, Roscosmos

2 in the mentioned region of possible SO positions. Otherwise, the measurement will be attributed to this SO erroneously, and the results of the orbital parameters determination based on measurements will appear unreal. Observations of space objects (SOs) To estimate the possibilities of various SOs observations by a radar network it is sufficient to specify the number of radars, the coordinates of points of their disposition in latitude (φ), and the relative position in longitude (L), as well as the aperture angle of a conical field of view (θ). Consider the possibility of observing various SOs in the altitude range up to 2000 km by a network of ground radar stations, in the interests of maximum cataloguing space debris of various sizes. We consider the hypothetical radars, which differ in places of location and size of the field of view, which is accepted to be conical. Such an approach is suitable, because the ultimate goal of our analysis is to determine the conditions, when the objects smaller than cm in size can be catalogued. Here, the data on radar performances are variable parameters. From the viewpoint of maximum cataloguing SOs, it is reasonable to direct the axes of the radar s field of view vertically upwards, by analogy with performing measurements of small-size space debris with using the Haystack locators. This allows one to minimize the distance to the SO and, accordingly, to increase the level of a reflected signal. In estimating the possibilities of SOs observation by some network of radars, we are interested in the mean time interval between the measurements of various SOs and in the mean time of the objects staying in the field of view. It is just these characteristics, which determine the possibility of the initial determination of the orbits and their subsequent updating, based on new measurements. Therefore, the mentioned characteristics form a basis for further analysis of the cataloguing possibilities, taking into account the estimates of the orbit accuracy in the catalog. The statistical model is developed for calculating the mean time between the measurements of the various SOs by a radar network. To obtain the stable estimates of the observation characteristics, the simulation modeling of the motion of each satellite is fulfilled over a rather long interval. Of course, in this case one takes into account the time variation of the sidereal time (s) of a radar position point. Thus, for each of the considered SOs it is sufficient to specify its altitude and inclination. The conditions of the SO entering into the considered radars field of view are verified in the course of the problem solution on each revolution. If entering occurs, the corresponding time is calculated. Thus, the obtained results allow one to calculate the time intervals between two successive measurements and the corresponding averaged estimates. The experiments have 2

3 shown that for obtaining stable average estimates, the number of random selections of the s Ω difference for each of SOs should be large enough. The examples of evaluation of the conditions of the various SOs observation by a radar network consisting of 4 stations, located at a latitude of 30º with a 30º shifting in longitude, are presented below. For 2 versions of the width of the field of view, θ =10º and 40º, the obtained estimates of the mean interval between measurements are presented in tables 1 and 2, respectively. Table 1. Intervals between measurements for θ =10º, days iº Altitude h, km For the conditions considered here and for the types of orbits, which differ in altitude and inclination, the values of the mean time interval between measurements lie in the range from 0.45 to 6.0 days. The maximum estimate differs from the minimum one as much as 13 times. As the SO inclination changes from 35º to 85º, the estimates of the mean time of the SO staying in the radar s field of view increase about 3 times, and as the SO altitude changes from 250 km to 1650 km, they decrease by a factor of 4.3. Table 2. Intervals between measurements for θ =40º, days iº Altitude h, km It is seen that, as the field of view increased 4 times, the estimates of the mean time between measurements decreased 4 times, respectively. In this case the ratio between maximum and 3

4 minimum estimates remained the same as that corresponding to the data of table 1 (i. e. it differed 13 times). The results of the calculation of the mean time of the SO staying in the radar s field of view have shown that, for the given values of the angle θ and the SO altitude, they do not depend on the satellite inclination. The results, obtained on a statistical model, are presented in table 3. Table 3. Estimates of the time of SO staying in radar s field of view, min θº Altitude h, km Estimation of orbit determination and prediction accuracy The problem of increasing the accuracy of determination and prediction of the SO orbits was considered in a series of publications issued by the author of the report. The basic results of his studies, fulfilled for the past 40 years, are as follows: 1. The technique, algorithm and software for a priori estimation of satellite motion prediction errors with accounting for the statistical characteristics of gravitational and atmospheric disturbances are developed. 2. The application of the maximum likelihood method (MLM) without parameterization (OFM) is shown to ensure minimizing the orbit determination and prediction errors, as well as estimating the errors with accounting for statistical characteristics of disturbances. 3. The technique, algorithm, and software for optimum filtering of measurements are developed. The application of MLM without parameterization was called the optimum filtering of measurements (OFM). For this method the traditional formula, applied for determining the state vector at the arbitrary time instant (t j ), is as follows: where X j T 1 T ( X j Pj X j ) X j Pj Z k xˆ =, (1) j = Zk x j is the matrix of partial derivatives, j P is the weighting matrix, which is calculated with regard to noise and measurement errors. A typical feature of estimate of Eq. 1 is the fact, that it is optimum for any time instant (both for the updating instant, and at prediction). The correlation matrix of errors of estimate (1) is calculated by formula 4

5 T ( ) ( ) 1 K t = X P X. (2) x j j j j The model for estimation the prediction accuracy was developed. In this model the state vector includes only the elements that characterize the motion in the near-circular orbit plane. Following this approach, we take the angular motion of a satellite in revolutions N as a parameter, and as the state vector components we use the following three elements: the time instant t i corresponding to the trajectory point with the given latitude argument (u=0), the period of one revolution T i, and the period variation during one revolution ΔT i. The following equations are valid for these variables: ti+ 1 = ti + Ti + 1, Ti + 1 = Ti + Ti,. (3) Ti = Tm + qi. Quantity is associated with the effect of atmospheric disturbances only. In this case, with Ti constant spacecraft s ballistic factor, the value of atmosphere. Quantity Tm is the mean value of parameter Ti process with the known correlation function K q ( t,τ ) 0. The mean value of the drag parameter Tm Ti is proportional to the current density of the, and q i is the Gaussian random, accepted for calculations, corresponds to the real conditions of the determination and prediction of orbits of catalogued objects with perigee altitudes from 300 to 700 m. This is seen from the data of figure 2, which presents the estimates of the drag characteristic October, T of various SOs calculated according to the TLE catalog data for 0-1 TLE Catalogue, October log(-del T, min) Perigee altitude, km Figure 2. Estimates of parameter T for SOs with various perigee altitudes 5

6 The particular modeling results testify to the necessity of applying the technique of optimum filtering of measurements (OFM), if we want to have realistic estimates of the motion prediction errors. Technique for the estimation of the possibilities of small objects cataloging This section outlines the technique of modeling using the technique of optimum filtering of the measurements OFM that was considered above. Figure 3 presents the flowchart of the model. Unit 1. The initial data are as follows: 1. For SOs with various altitudes (h) and inclinations (i) we have used time intervals del t (h, i, θ) between measurements with various values of the field of view (θ). Examples of these data are given in tables 1 and 2. Besides, we used the RMS of the radial ( σ ), along-track ( σ τ ) and cross-track ( σ ) errors of the measurements, which can change. Hereafter, we accepted σ r = σ τ = σ w. w 2. SDPA model [1]. The full range of object dimensions was divided into 8 groups (jd=1., 8). We have used the distribution of density of the objects of various sizes in altitude and latitude. Table 4 presents the example of these data for SOs of various sizes and various altitudes for the case of 30º latitude in r Table 4. Estimates of density for SOs of size cm and larger than 20 cm at latitude of 30 SO size Spatial density, km -3 for various altitudes, km d = cm 0.28E E E E E E E E-7 d > 20 cm 0.20E E E E E E E E-8 6

7 Figure 3. Flowchart of operations fulfilled on the model The SDPA model also contains statistical distributions of the values of ballistic parameters p(d,kb). These data underlay the calculation of the period change, under atmospheric drag effect, per revolution (ΔT) which is used in the model for estimating the prediction accuracy. Figure 4 presents the corresponding results of the calculation. These results are seen to agree rather well with the data of figure 2. Their use allows taking into account the diversity of the drag conditions for objects in the NES. 7

8 Figure 4. Calculated drag characteristics of catalogued SOs (of size > 20 cm) at various altitudes 3. Variable initial data include: =40º. - Field of view angle θ. The calculations below were carried out for values θ =10º and θ - RMS of along-track ( σ τ ) measurement errors. Below are considered two versions, corresponding to various values of the latitude crossing time measurement accuracy σ = V (Here V is the SO velocity). Calculations were carried out for the values z στ σ = min = 0.006sec and σ = 0.001min = 0.06sec. These data correspond to the z values σ τ 470 m and 47 m. z - Size of studied objects. Two versions of size were considered: cm and more than 20 cm. - Minimum and maximum number of measurements over a fit span (5 and 15), which is calculated on the basis of time intervals between measurements del t (h, i, θ), and should not be less than 2 days. Unit 2. The cycle is organized over inclination values of 35º, 45º, 55º, 65º, 75º and 85º. Unit 3. The cycle is organized over altitude values of 250 km, 450 km, 650 km, 850 km and 1050 km. Unit 4. The cycle is organized over 6 possible values of the ballistic parameter kb. Unit 5. Here the basic calculations are carried out: - Calculation of the interval (in revolutions) between the measurements dnz= round[del t(h,i,θ)/period]. - Calculation of a number of measurements nz on a fit span ((nz-1) dnz). 8

9 - Calculation of the drag characteristic ΔT, taking into account the orbital data, ballistic parameter values, and atmospheric density model (with mean values of solar and geomagnetic activity indices). - Calculation of RMS of atmospheric disturbancesσ = T. - Application of the OFM technique for calculating a priori RMS ( ( dnz) q σ of errors of prediction for dnz revolutions and corresponding RMS of the along-track error ( ( dnz) = σ t ( dnz) V σ τ ). - Calculation of the volume of the 3-dimensional ellipsoid, in which the considered SO will be located with the probability of at prediction for dnz revolutions. 2 Here the probability ( χ < 16.3) = U (4) 3 ( dnz) =.3 π στ ( dnz) σ r σ w P of the 2 χ distribution with 3 degrees of freedom is applied. In formula (4) the assumption is used, that the RMS of radial ( σ ) and crosstrack ( σ ) errors of measurements only differ slightly from the corresponding RMS of w prediction. This assumption results in some increase of the ellipsoid volume and goes into «reserve». - Calculation of a mean number of hits of objects of considered size into the ellipsoid of volume U ( dnz) ( U ) U ( dnz) ρ( h n =,. (5) ρ are density estimates according to the data of table 3. For small values the Here ( h, quantity (5) has a sense of probability P( h i, kb, = n( U ). of hitting other objects of considered size in the ellipsoid of volume U ( dnz). If this probability is low (for example, < ), then the considered object, with altitude h, inclination i and ballistic parameter k b, can be catalogued with a rather high reliability. Units 6, 7 и 8. Checkout of the termination of the corresponding cycles over the inclinations, altitudes, and ballistic parameters of the objects of a considered size. Unit 9. Formation and derivation of a set of calculation results. In particular, the mean value of the probability P ( h,i, of hitting the background SOs with various ballistic parameters into the corresponding ellipsoid of volume U ( dnz) is calculated as j j ( h,i, = P( h,i,k, p( h,k, P. (6) b b j t r 9

10 Here j p( h,k, b parameters. Basic results is the statistical density of objects distribution over possible values of ballistic Particular calculations of the probability P ( h i,. of hitting background SOs into the region of the possible positions of a considered SO were carried out in 8 versions, whose characteristics are given in table 5. Table 5. Characteristics of modeling versions Version θº d, cm > > > > σ, min/rev z The first 4 versions relate to the case of small field of view of a measuring instrument, and the other 4 versions to an essentially larger value of the field of view. For each of these cases we considered 4 similar versions, which differ in the object size and in the measurement accuracy. Calculation results for versions 2 and 4. It relates to the small field of view, to objects of size cm and to two versions of measurement accuracy. Results of calculation of probabilities P ( h. i,,%, are presented in tables 6 and 7. Remember that P ( h i,. is the probability of hitting of an unknown SO in the 3-dimensional region of space, where the given catalogued object can be situated. For P ( h i, object is fulfilled with the probability not less than Table 6. Version 2. Estimates of P ( h i,. <0.01% the condition of cataloguing the given.,%, (θ=10º, d=1 2.5 cm, σ z = min) h, P h. i, d,%,for SOs with various altitudes (h) and inclinations (i) km i=35º i=45º i=55º i=65º i=75º i=85º Estimates of ( ) For the 250 km altitude the estimates exceed 0.01% in the majority of cases. They are colored with a red tint. At altitudes of 450 km and higher the probability of the background SOs entering into the region of possible satellite positions does not exceed %. This implies that at these 10

11 altitudes the reliability of cataloguing SOs of size cm is acceptable. For SOs at the 250 km altitude the corresponding estimates do not exceed the value of 0.02 %; that is, the reliability of cataloguing SOs of size cm is insufficiently high. The values of a mean time interval between measurements lie in the range from 0.45 to 6.0 days. For the mean interval between measurements of 2 days, the number of measurements of SOs of a considered size equals per 1 day. About measurements per day fall on each of 4 measurement means. This implies that, if the mean duration of a SO staying in the measuring instrument s field of view equals 0.2 min (see Table 3), 7 objects, on the average, will be simultaneously located in this field of view. Table 7. Version 4. Estimates of P ( h i, Estimates of P ( h i,.,%, (θ =10º, d= cm, σ z =0.001 min) h,.,%, for SOs with various altitudes (h) and inclinations (i) km i=35º i=45º i=55º i=65º i=75º i=85º As compared to the corresponding results of calculations for version 2, these estimates are greater by 2 orders of magnitude. At the 250 km altitude the probability of background SOs entering into the region of the possible satellite positions reaches 2 %. At altitudes of 450 and 650 km it exceeds 0.01 % in the majority of cases. This implies that for the given version of the initial data the reliability of cataloguing SOs of size from 1.0 cm to 2 cm is insufficient. The inclination dependence of the P ( h i,. probability is explained by the variation of time intervals between the measurements, which is clearly seen from the data of Table 1. This results in changing the fit span duration and the forecasting interval before the next measurement. The altitude dependence of the P ( h i,. probability is explained by the changing of the value of atmospheric disturbances. The fit span values, which were applied in modeling for versions 2 and 4, are presented in table 8. Table 8. Versions 2 and 4. Fit span values, rev Fit span values for SOs with various altitudes and inclinations h, km i=35º i=45º i=55º i=65º i=75º i=85º

12 Here the red color indicates the same versions of calculations, as in table 7. In analyzing the model values of a fit span, it is useful to compare them with the corresponding data which are applied in practice. We shall make use of recommendations presented in the monograph by D. Vallado [2]. These estimates, which depend on the value of the SO drag in the atmosphere, are presented in table 9. The period change per 1 day, which is designated as P, is used as the drag characteristic here. Table 9. Recommended fit span values Drag characteristic P, min/day <-10-2 <-10-3 < > ΔT,min/rev < < < > Fit span days revolutions For the calculation version 2 and 4 the fit span duration values are presented in figure 5. The region of recommended fit span values is marked by a light-red color. It is seen from these data that for SOs with altitudes of 250 and 450 km the fit span values, applied in the model, are much greater than the recommended ones. In many cases they are greater for SOs with 650 km altitude as well. This implies that, for the 10º field of view and using the existing traditional techniques of updating orbital parameters based on measurements, the objects of size cm with altitudes lower than 700 km cannot be reliably catalogued under the considered conditions. 12

13 Figure 5. Fit span values in calculations by versions 2 and 4 (at θ =10º) Calculation results for versions 6 and 8. It relates to the large field of view, to the objects of size cm, and to the two versions of measurement accuracy. The results of the calculation of the probabilities P ( h i,., %, are presented in tables 10 and 11. As compared to the corresponding results of the calculations for version 2, these estimates decreased up to times. For SOs with altitudes higher than 250 km, the probability of background SOs entering into the region of possible satellite positions is small it does not exceed the value of %. For SO at the 250 km altitude the estimates of this probability do not exceed the value of %. Thus, under the considered conditions the reliability of cataloguing SOs of size from 1.0 to 2.5 cm is rather high. According to the data of table 6, the 4- fold increase of the field of view results in the 4-fold decrease of the mean time between measurements. For the mean interval between measurements of 0.5 days, the number of measurements of the SOs of the considered size equals /0.5= per 1 day. About measurements per day fall in each of 4 measurement means. This implies that, if the mean duration of the SO staying in the measuring instrument s field of view equals 0.8 min (see table 4), about 110 objects of the considered size, on the average, will be simultaneously located in this field of view. Table 10. Version 6. Estimates of P ( h i, Estimates of P ( h i, h, km.,%, (θ =40º, d = cm, σ z = min).,%, for SOs with various altitudes (h) and inclinations (i) i=35º i=45º i=55º i=65º i=75º i=85º

14 Table 11. Version 8. Estimates of P ( h i, Estimates of P ( h i, h, km.,%, (θ =40º, d = cm, σ z = min).,%, for SOs with various altitudes (h) and inclinations (i) i=35º i=45º i=55º i=65º i=75º i=85º As compared to the corresponding results of calculations for version 6, the maximum estimates increased 100 times. Nevertheless, the probability of the background SOs entering into the region of the possible satellite positions with 450 km altitude and higher does not exceed the value of 0.01 %. For SOs with the perigee altitude of 250 km, the probability estimates reach the value of 0.22 %, i.e. they are rather high. Thus, under the considered conditions the reliability of cataloguing SOs of the size from 1.0 to 2.5 cm is not high enough. This especially concerns SOs at low altitudes (250 km). The estimates of a number of measurements carried out by each of the stations are the same as for version 6. The fit span values, which were applied in modeling for versions 6 and 8, are presented in table 12. These estimates are 4 times lower as compared to the corresponding data for radars with a small field of view (table 8). Table 12. Versions 2 and 4. Fit span values, rev h, km Fit span values for SOs with various altitudes and inclinations i=35º i=45º i=55º i=65º i=75º i=85º

15 Calculation results for version 1 and 3. It relates to the small field of view, catalogued SOs, and the two versions of the measurement accuracy. The results of the calculation of the probabilities P ( h i,., %, are presented in table 13 and 14. Table 13. Version 1. Estimates of P ( h i, h, km Estimates of P ( h i,,,%, (θ=10º, d>20 cm, σ z = min).,%, for SOs with various altitudes (h) and inclinations (i) i=35º i=45º i=55º i=65º i=75º i=85º In all cases the probability of the background SOs entering into the region of the possible positions of satellites is small it does not exceed the value of %. This implies that under the considered conditions the reliability of cataloguing SOs of size larger than 20 cm is rather high. The general regularity consists in the fact that the estimates of the P ( h i,. probability increase with the growing inclinations and decrease with the growing altitude of objects. This is a natural consequence of the dependence of an interval between the measurement on satellite s inclination and altitude (see table 1). Under the considered conditions the values of the mean time interval between the measurements lie in the range from 0.45 to 6.0 days. For the mean interval between measurements of 2 days, the number of measurements of the catalogued SOs equals 13000/2=6500 per 1 day. About 1625 measurements per day fall on each of the 4 measurement means. Thus, under the conditions of the application of the measurement means with the field of view of θ = 10º, the time intervals between the measurements of particular SOs are rather large. This fact results in the necessity of assigning the fit span corresponding to an accepted minimum number of measurements (nz=5). Figure 6 and table 8 presents the fit span values, which were applied in the model in the calculations by versions 2, 4, 1 and 3. 15

16 Figure 6. Fit span values in calculations by version 1 (for θ =10º) The region of the recommended fit span values is marked in figure 6 by a light-red color. It is seen from these data that for SOs with altitudes of 250 and 450 km, the fit span values applied in the model are much greater than the recommended ones. This implies that, for the 10º field of view and using the existing traditional techniques of updating the orbital parameters based on the measurements, the objects of a size larger than 20 cm and with altitudes lower than 500 km cannot be catalogued. Table 14. Version 3. Estimates of P ( h i, h, km Estimates of ( ),,%, (θ=10º, d>20 cm, σ z =0.001 min) P h. i, d,%, for SOs with various altitudes (h) and inclinations (i) i=35º i=45º i=55º i=65º i=75º i=85º As compared to the corresponding results of the calculations for version 1, these estimates are greater by 2 orders of magnitude. At altitudes of 250 and 450 km the probability of background SOs entering into the region of the possible satellite positions exceeds % in the majority of the cases, and at the 250 km altitude it exceed 0.01 %. This implies that at these altitudes the reliability of cataloguing SOs of a size larger than 20 cm is insufficiently high. 16

17 Calculation results for versions 5 and 7. These versions relate to the large size of the field of view, to the catalogued SOs, and to two version of measurement accuracy. The results of the calculation of the probabilities P ( h i, h, km Table 15. Version 5. Estimates of P ( h i, Estimates of P ( h i,.,% for version 5, are presented in table 15..,%, (θ =40º, d> 20 cm, = min).,%, for SOs with various altitudes (h) and inclinations (i) i=35º i=45º i=55º i=65º i=75º i=85º As compared to the corresponding results of the calculations for version 1, these estimates are lower by 1 order of magnitude. In all cases the probability of the background SOs entering into the region of the possible satellite positions is small it does not exceed the value of %. This implies that under the considered conditions the reliability of cataloguing SOs of size larger than 20 cm is high. For the mean interval between the measurements of 0.5 days, the number of measurements of the cataloged SOs equals 13000/0.5=26000 per 1 day. About 6500 measurements per day fall on each of the 4 measurement means. This implies that, if the mean duration of the SO staying in the measuring instrument s field of view equals 0.8 min (see table 3), 3 4 objects, on the average, will be simultaneously located in this field of view. Figure 7. Fit span values of the calculations by version 5 (for θ =40º) 17

18 It is seen from the fit span values given in figure 7 that, as compared to the data of figure 6 (for version 1), these values decreased about 4 times. The region of the recommended values is marked by a light-red color. It is seen from these data that, for SOs with altitudes of 250 km, the fit span values applied in the model can be greater than the recommended ones. At other altitudes they either correspond, or are lower, than recommended values. This implies that, for the 40º field of view and using existing traditional techniques of updating the orbital parameters based on the measurements, the objects of size larger than 20 cm and with altitudes higher than 250 km can be catalogued with rather high reliability. Table 16. Version 7. Estimates of P ( h i, Estimates of P ( h i, h, km.,%, (θ =40º, d>20 cm, σ z =0.001 min).,%, for SOs with various altitudes (h) and inclinations (i) i=35º i=45º i=55º i=65º i=75º i=85º As compared to the corresponding results of the calculations for version 5, the maximum estimates increased by 100 times. Nevertheless, the probability of the background SOs entering into the region of the possible satellite positions is rather small: for SOs with 250 km altitude it does not exceed the value of %, and for satellites with 450 km altitude and higher, it does not exceed the value of %. This implies that under considered conditions the reliability of cataloguing SOs of size larger than 20 cm is rather high. The estimates of the number of measurements carried out by each of the stations are the same as for version 5. Taking into account that the considered conditions do not differ too highly from the conditions of the operation of Space Surveillance Systems (SSSs), this result seems to be rather important, since it indicates an acceptable correctness of the applied modeling technique. Conclusions. 1. The considered system of measurement means ensures acceptable conditions for cataloguing the objects of size from 1.0 to 2.5 cm at RMS errors of measurements of 50 m. For a smaller field of view of stations (θ=10º) the possibilities of SOs cataloguing at the 250 km altitude are essentially worse than for a larger field of view (θ =40º), but they are still 18

19 admissible. For SOs with altitudes of 450 km and higher the cataloguing reliability is high enough. In the former case (θ =10º) 7 objects are located simultaneously in the field of view of the stations, and in the second the case 100 objects are. 2. The estimation of the possibilities of cataloguing SOs of size from 1.0 to 2.5 cm, outlined in item 1, was obtained under the condition of applying the technique of optimum filtering of measurements (OFM), which is characterized by accounting for the random disturbances at "weighting" the measurements on a fit span and, therefore, it allows one to process the measurements over rather great time intervals (up to one month). The application of the traditional measurement processing techniques does not provide, under considered conditions, reliable cataloguing of the objects of a given size. 3. Using the OFM technique and for all versions of the initial data, the considered system of measurement means ensures the possibility of cataloguing SOs of size larger than 20 cm with a sufficiently high reliability. However, only for the field of view θ =40º and with RMS errors of measurements of 50 m, these objects can be catalogued with a sufficiently high reliability using the existing traditional techniques of updating the orbital parameters based on the measurements. Taking into account that the considered conditions do not differ too highly from the working conditions of the operational SSSs, this testifies to an acceptable correctness of the applied modeling technique. Directions of further investigations 1. Taking into account the last data about space debris environment [19]. Figure 8. Characteristics of space debris flux according to the data of various sources and the SDPA estimates accounting for mutual collisions 19

20 2. Estimate the probability of entering background SOs approximately 5-10 cm in size into the corresponding ellipsoid. 3. Calculate the corresponding probability for the use of the Least Square Technique. Acknowledgement The authors would like to acknowledge Professor Kyle T. Alfriend for support of this work. References 1. A.I. Nazarenko. Determination of satellite position for the given ephemerides time. Nablyudeniya Isk. Sputn. Astronomical Council of the USSR Acad. Sci., Moscow, No 49, Nazarenko, A.I. & Markova, L.G. (1973). Techniques of estimation and prediction of satellite orbits in the presence of errors in the mathematical description of motion. In: Applied problems of space ballistics. Moscow, Nauka. (There is English text: NASA Technical Report Server, Document ID= ). 3. A.I. Nazarenko, B.S. Skrebushevsky. Evolution and stability of satellite systems. Mashinostroenie, Moscow, Nazarenko, A.I. (1991). Apriori and A posteriori Orbit Prediction Errors Evaluations of Low Artificial Earth Satellite. Kosmicheskie Issledovaniya, Vol. 29, No. 4, Nazarenko, A.I. & Cherniavskiy, G.M. (1996). Evaluation of the Accuracy of Prediction Satellite Motion in the Atmosphere, 2nd U.S.-Russian Space Surveillance Workshop, Poznan, Poland, pp Nazarenko, A.I. (1998). Determination and Prediction of Orbits with Due Account of Disturbances as a "Color" Noise. AAS/AIAA Space Flight Mechanics Meeting. Monterey, CA. AAS D.A. Vallado. Fundamentals of Astrodynamics and Applications. Published jointly by Microcosm Press and Kluwer Academic Publishers, Nazarenko A.I., Yurasov V. S., Alfriend K. T., Cefola P.J. (2007). Optimal Measurement Filtering and Motion Prediction Taking Into Account the Atmospheric Perturbations, AAS/AIAA Conference, Mackinac, Paper AAS Nazarenko, A.I. (2007). Accuracy of Determination and Prediction Orbits in LEO. Estimate Errors Depending on Accuracy and Amount of Measurements, Seventh US/Russian Space Surveillance Workshop, Monterey, CA. 20

21 10. E.G. Stansbery & N.L. Johnson. Space Surveillance Network for a More Complete Catalog. 7th US/Russian Space Surveillance Workshop Paul W. Schumacher, Jr. US Naval Space Surveillance Upgrade Program th US/Russian Space Surveillance Workshop Felix R. Hoots et al. Cataloging with an upgraded space surveillance fence. 7th US/Russian Space Surveillance Workshop Heiner Klinkrad et al. Investigations of the Feasibility of a European Space Surveillance System. 7th US/Russian Space Surveillance Workshop Nazarenko, A.I. (2008). Accuracy of orbit determination and prediction for SOs in LEO. Dependence of estimate errors from accuracy and number of measurements. 26-th IADC, Moscow. 15. Holger Krag et al. The European Space Surveillance System Required Performance and Design Concepts. 8th US/Russian Space Surveillance Workshop A.I. Nazarenko. Results of updating the parameters of the space debris model in 2007 and in Fifth European Conference on Space Debris, Darmstadt, Germany, Nazarenko, A.I. (2010). Errors of prediction of satellite motion in the Earth gravitational field. Moscow, Space Research Institute of RAS. 18. A. Nazarenko. Space debris status for 200 years ahead and The Kessler effect. 29-th Meeting IADC, April A.I. Nazarenko. Estimation of the contribution of the effect of collisions of objects larger than 1 cm in size. 30 th IADC Meeting,

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