- Email: [email protected]
Estimating the number of debris in the geostationary ring
Acta Astronautica 59 (2006) 84 – 90 www.elsevier.com/locate/actaastro
Estimating the number of debris in the geostationary ring夡 Rüdiger Jehna , Shahram Ariafarb , Thomas Schildknechtc , Reto Muscic , Michael Oswaldd a European Space Operations Centre, Mission Analysis Office, Robert-Bosch-Street 5, D-64293, Darmstadt, Germany b Corona Space Surveillance Centre, Marselis gate 24, N-0551 Oslo, Norway c Astronomical Institute of the University of Bern, Sidlerstrasse 5, CH-3012, Bern, Switzerland d Institute of Aerospace Systems (ILR), Technische Universität Braunschweig, Hermann-Blenk-Street 23, D-38108, Braunschweig, Germany
Abstract Two thousand seven hundred and ninety uncorrelated targets brighter than magnitude 18.5 were detected by the European Space Agency (ESA) 1-m space debris telescope at Tenerife during more than 1000 observation hours between February 2001 and December 2004. The number of detections can be approximated by a Gaussian distribution. Probabilities to detect individual objects during an observation campaign are determined by propagating 10 debris clouds of the ESA MASTER model and two other fictitious background populations. Based on these probabilities, a 95% confidence interval for the total number of unknown debris in the geostationary ring is derived. It is estimated that there are between 450 and 540 uncatalogued objects brighter than visual magnitude 18.5. © 2006 Elsevier Ltd. All rights reserved.
1. Introduction Ground-based radars like Haystack and TIRA have observed the space debris environment in low-Earth orbit (LEO) for many years and our knowledge about the altitude and inclination distribution of centimetre-sized objects is well advanced. The space debris flux derived from MASTER [1] or EVOLVE [2] populations is pretty reliable in LEO today. The situation in the geostationary ring is different: Only objects larger than about 1 m are contained in the DISCOS database [3] and the models 夡 Prepared for the 56th International Astronautical Congress, Fukuoka, Japan, 2005. E-mail addresses: [email protected] (R. Jehn), [email protected] (S. Ariafar), [email protected] (T. Schildknecht), [email protected] (R. Musci), [email protected] (M. Oswald).
0094-5765/$ - see front matter © 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2006.02.053
for smaller objects are associated with very high uncertainties because little observational data is available. In this paper recent data from the ESA space debris telescope is used to characterize the space debris environment near the geostationary ring. 2. ESA space debris telescope observations The European Space Agency (ESA) is using its 1-m Zeiss telescope in Tenerife since 1999 to detect and track objects which are in the vicinity of the geostationary ring. Objects as small as about 15 cm are observed. In total more than 2200 h of observational data are collected up to now and a homogeneous subset of 1026 h between February 2001 and December 2004 were analysed in this paper to estimate the total number of objects. During these 1026 observation hours 2790 uncorrelated targets (UCTs) brighter than 18.5 mag were
R. Jehn et al. / Acta Astronautica 59 (2006) 84 – 90
85
Fig. 1. Right ascension and declination of a sample of uncorrelated targets brighter than 18.5 mag detected with the ESA space debris telescope between 2001 and 2004.
Fig. 2. Distribution of uncorrelated targets detected by the ESA space debris telescope between 2001 and 2004.
detected. Fig. 1 shows the right ascension and declination of those UCTs. Note, fainter objects and objects which could be correlated with objects contained in the DISCOS catalogue are not plotted. Although the telescope can detect objects as faint as magnitude 20 and occasionally even fainter, we focus in this paper on objects brighter than 18.5 mag because there is a high
probability (about 95 %) that no object of this brightness will be missed. If we were to include fainter objects we would need to estimate the percentage of missed targets as a function of the visual magnitude. The data is grouped in 38 campaigns (usually observation periods of 10–12 days always around New Moon). For instance in one observation during the
86
R. Jehn et al. / Acta Astronautica 59 (2006) 84 – 90
campaign in February 2001 the telescope was pointing 8.3 h to a right ascension of 9:30 h and a declination of 1.2◦ . 15 UCTs brighter than 18.5 mag were detected. Part of the campaigns used for this paper are listed in [4]. For Fig. 2 the right ascension of ascending node (RAAN) and inclination of the 2790 objects were calculated based on observations of a single night and assuming circular orbits. The observations do not directly reflect the debris population but are biased due to the selective “filter” of the observation scenario. Nevertheless the pronounced clustering along the well known RAAN-inclination line is also a feature for uncorrelated objects. The explanation is that fragmentations probably have taken place in common orbits along that line and the fragments just evolve over time staying close to the catalogued population. There are also objects that are known to be in GEO but that are not included in the TLE catalogue. In December 2004, there were 88 uncontrolled objects with a catalogue number but without TLEs (old payloads, rocket bodies and debris). Furthermore there were 65 uncatalogued objects (mainly mission related objects like telescope covers and three apogee kick motors). There were 32 objects with TLEs older than six months and there were 42 controlled but uncatalogued satellites [5]. All in all there were 227 objects that would show up as UCT but which were not created by explosions. However, there are regions, especially with RAAN between 120◦ and 240◦ , where UCTs are detected but where hardly any explosion fragment is expected to be. Most likely these observations can be attributed to objects in a geostationary transfer orbit or to objects with high area-to-mass ratios (a debris source described by Schildknecht et al. [6]). These two classes of objects were not filtered out in our data sets and slightly contaminate the results (although they are not too numerous).
populations which consist of several sub-populations (debris clouds) which have quite distinct distributions and probabilities to be “caught”. To describe the mathematical approach a population of fishes consisting of two species is used as an example.1 There are N1 pikes and N2 carps in a pond. A fisherman catches u1 fishes on the first day. Let pp1 be the probability to catch one specific pike in a given time period and pc1 be the probability to catch one specific carp, then the number of fishes caught during this time period is roughly Gaussian distributed (if we have a sufficiently large population). Let U1 be the random variable describing the number of caught fishes. U1 has the expectation value
3. Estimating the size of a population
If this result is not satisfactory more samples have to be taken. The fisherman goes to a better place! Let pp2 be the probability to catch one specific pike during the next fishing campaign and pc2 the probability to catch one specific carp. Then U = U1 + U2 describes the total number of caught fishes during the two campaigns. Again U is (approximately) Gaussian distributed with
Since the objects are not uniformly distributed in right ascension and declination and since most of the time the telescope was pointing at high-density areas a careful weighting of the statistical data is required in order to draw conclusions concerning the total population. The size of an unknown population can be estimated by drawing samples of the population and using the probability of an individual object to be “caught” or detected by a telescope. This approach was taken to derive confidence intervals for the number of centimeter-sized objects in LEO based on radar observations [7]. This idea is expanded here for the case of
(U1 ) = N1 pp1 + N2 pc1
(1)
and the standard deviation (U1 ) = N1 pp1 (1 − pp1 ) + N2 pc1 (1 − pc1 ).
(2)
With a given probability we can say that our sample u1 will fall within an interval: 1 (u1 − )/ 2 .
(3)
For instance we can be 95% sure that u1 will be inside the interval defined by 1 = −1.96 and 2 = 1.96 (95% corresponds to about 2-sigma confidence interval). Inserting Eqs. (1) and (2) in Eq. (3) gives constraints on the number of fishes in the pond: 1 f (N1 , N2 ) 2 . These inequality constraints are entered into a numerical algorithm that searches for the maximum and minimum of N1 + N2 . For instance, if pp1 = 0.2, pc1 = 0.4 and u1 = 20 then we can say with 95 % confidence that 36 (N1 + N2 ) 86.
(U ) = N1 (pp1 + pp2 ) + N2 (pc1 + pc2 )
1 This approach was not found in standard statistics textbooks (although it probably has appeared in some university publications) and therefore it is explained here in detail.
R. Jehn et al. / Acta Astronautica 59 (2006) 84 – 90
87
Fig. 3. Distribution of uncorrelated targets detected by the ESA space debris telescope between 2001 and 2004 and fictitious debris populations propagated to February 2001.
and 2 (U ) = N1 ppi (1 − ppi ) + N2 pci (1 − pci ). i=1
Again with a given probability we can say that our sample u = u1 + u2 will fall within an interval: 1 (u − )/ 2 . For pp2 = 0.3, pc2 = 0.45 and u2 = 25 we can derive with 95 % confidence that 42 (N1 + N2 )79. By taking a second sample, the width of the confidence interval has decreased, i.e. the uncertainty was reduced. Now the same approach will be taken to derive an estimate of the number of objects in GEO. Instead of two samples we will have 205 samples (i.e. the number of UCTs of the 205 sub-campaigns) and instead of two populations we will define 12 populations. 4. Fictitious debris populations in GEO For the upgrade of the MASTER model the Tenerife data is used to define the populations in the 10–100 cm
size range. A reasonable fit was achieved by simulating the explosion of 10 objects between 1978 and 1998 in near GEO orbits with inclinations between 0◦ and 12◦ . Three thousand six hundred and ninety five fragments larger than 10 cm were simulated [1]. Fig. 3 shows the inclination-right ascension distribution of these fragments after propagation to February 2001.2 It can be seen that the fragments cover most of the detected UCTs which are also plotted in Fig. 3. However, it must be noted that the observations and simulated objects should not be compared in such a simplified way. First of all the observations are taken over a period of 4 years whereas the simulated objects were propagated to the beginning of the observation period. For instance cloud 1 of the MASTER model would look significantly different a few years later. And furthermore, there are strong selection biases in the observational data due to the preferred observation scenarios as mentioned above. Nevertheless the 10 clouds from the MASTER population are very useful to reflect the orbital distribution
2 The debris population from MASTER used for this paper is about 25% smaller than the final MASTER 2005 population, which was not available at the time when this analysis was made. However, this has no consequences on the final results, because only the orbital distributions are required to calculate the detection probabilities. The number of objects is estimated with the method described in the previous section.
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R. Jehn et al. / Acta Astronautica 59 (2006) 84 – 90
of most of the observed UCTs. To account for the observations which are not following the “main-stream” RAAN-inclination pattern, two additional populations are defined. Population 1 has right ascensions randomly distributed between 50 and 260◦ and inclinations between 0 and 15◦ . Population 2 has right ascensions randomly distributed between 0 and 360◦ and inclinations between 0 and 35◦ . The mean motion is randomly distributed between 0.9 and 1.1 rev/day (i.e. they have a maximum deviation of 2800 km from GEO). The eccentricities of both populations were spread between 0 and 10−4 . The area-to-mass ratios were set to 0.05 m2 /kg for all objects.
object of cloud i during a campaign of t days: p=
ni t ∗ FOV, Ni
(4)
FOV is the diameter of the field-of-view of the telescope in degrees (0.7◦ for the ESA SDT). In Eq. (4), a linear relation between the detection probability and the FOV is assumed. This is justified for objects with low inclination because mainly the North–South extension of the telescope aperture is determining the detection rates. Having calculated the probability for each bin, for each cloud, and for each observation epoch, we can now apply the algorithm for the determination of the confidence interval for the total number of unknown objects as described above.
5. Detection probabilities 6. Results For each population the probability to detect an individual object during an observation campaign needs to be calculated. A 1◦ × 1◦ grid in right ascension and declination is defined and for each bin ni , the number of objects of cloud i passing through that bin is determined by propagating each object along one orbit. Dividing ni by the total number Ni of objects gives the probability to detect an individual object of this cloud if you were to observe that bin during 24 h. Fig. 4 shows these probabilities for cloud 6 where 1036 objects larger than 1 cm were spread out in right ascension and declination according to their orbital distribution. 1 cm was chosen as threshold to have a statistically significant number assuming that their distribution represents the distribution of the observable objects. It is now straightforward to calculate the probability p to observe one specific
Fig. 4. Percentage of objects of cloud 6 passing through a 1◦ × 1◦ observation bin in February 2001 (ni /Ni of Eq. (4)).
Using the 2790 UCTs detected during 1026 hours between February 2001 and December 2004 we derive with 95% confidence that the number of “uncataloged objects brighter than 18.5 mag” is between 450 and 540. Fig. 5 shows how the confidence interval is shrinking as more and more data are considered. 18.5 mag corresponds to a size of 29 cm if optimum observation conditions and an albedo of 0.08 is assumed. The MASTER fragment population consists of 860 objects larger than 29 cm. This difference can be explained by the fact that only a part of these objects are actually detected by the ESA space debris telescope when they pass through the FOV because the signal-to-noise ratio very often is below the detection threshold. A more direct comparison with the MASTER model is difficult because: (a) the observational data is “contaminated” by objects in GTO; and (b) since the objects change their brightness, there is no one-to-one relation between magnitude and size of the objects. The obtained interval gives more a formal uncertainty rather than a lower and upper bound for the number of objects of a physical population. For a limiting magnitude of 16 the number of debris in GEO is estimated between 182 and 258 (see Fig. 6). This is higher than the 140 objects estimated by Matney et al. [8]. Possible explanations may be an increase in the actual number of debris (the measurements with the NASA CDT were made from 1998 until the end of 2000) or the sensitivity roll-off effect that may be noticeable already at magnitude 16 (the limiting sensitivity of the NASA CDT being 17). The MASTER model contains 173 objects larger than 80 cm and 269 objects larger than 60 cm. These sizes correspond to about a magnitude of 16. Again our results agree reasonably well with the MASTER model.
R. Jehn et al. / Acta Astronautica 59 (2006) 84 – 90
89
Fig. 5. Ninety five percent confidence interval for the number of objects brighter than magnitude 18.5. The sizes on the right-hand side give the corresponding object diameter of the MASTER population (e.g. there are 500 objects larger than 40 cm).
Fig. 6. Ninety five percent confidence interval for the number of objects brighter than magnitude 16.
7. Conclusions The approach presented above provides results which are in a good agreement with the debris population models of MASTER. But rather than giving only a single value for the number of objects above a given size or brightness, this approach allows to calculate confidence intervals.
After more than 2200 observation hours out of which 1026 h were used for this analysis, it is possible to formally estimate the number of unknown objects in GEO brighter than magnitude 18.5 with an uncertainty below 10%. However, since objects change their brightness depending on attitude, phase angle and albedo, the results cannot directly be translated into a population corresponding to a given object diameter.
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R. Jehn et al. / Acta Astronautica 59 (2006) 84 – 90
The ultimate goal to get a better knowledge of the GEO debris environment is the establishment and maintenance of a GEO catalogue where objects are permanently tracked. ESA is currently studying the feasibility of such an endeavor. References [1] M. Oswald, S. Stabroth, C. Wiedemann, P. Vörsmann, T. Schildknecht, H. Klinkrad, Validation of the ESA MASTER2005 orbital debris model, AMOS Technical Conference, Wailea, Hawaii, 5–9 September 2005. [2] P.H. Krisko, EVOLVE historical and projected orbital debris test environments, Advance Space Research 34 (5) (2004) 975–980. [3] C. Hernández, F. Pina, N. Sánchez, H. Sdunnus, H. Klinkrad, The DISCOS database and web interface, in: Proceedings of the Third European Conference on Space Debris, ESA SP-473, 2001, pp. 803–807.
[4] T. Schildknecht, R. Musci, M. Ploner, U. Hugentobler, M. Serra Ricart, J. De Léon Cruz, L. Dominiguez Palmero, Geostationary orbit objects survey, Final report, ESA/ESOC Contract 11914/96/D/IM, September 2004. [5] I. Serraller, R. Jehn, Classification of Geosynchronous Objects, Issue 7, ESOC, Darmstadt, Germany, 2005. [6] T. Schildknecht, R. Musci, W. Flury, J. Kuusela, J. de Leon, L. de Fatima Dominguez Palmero, Optical observations of space debris in high-altitude orbits, in: Proceedings of the Fourth European Conference on Space Debris, ESA SP-587, 2005, pp. 113–118. [7] R. Jehn, Comparison of the 1999 beam-park experiment results with space debris models, Advance Space Research 28 (9) (2001) 1367–1375. [8] M.J. Matney, E. Stansbery, J. Africano, K. Jarvis, K. Jorgensen, T. Thumm, Extracting GEO orbit populations from optical surveys, Advance Space Research 34 (5) (2004) 1160–1165.
Estimating the number of debris in the geostationary ring夡 Rüdiger Jehna , Shahram Ariafarb , Thomas Schildknechtc , Reto Muscic , Michael Oswaldd a European Space Operations Centre, Mission Analysis Office, Robert-Bosch-Street 5, D-64293, Darmstadt, Germany b Corona Space Surveillance Centre, Marselis gate 24, N-0551 Oslo, Norway c Astronomical Institute of the University of Bern, Sidlerstrasse 5, CH-3012, Bern, Switzerland d Institute of Aerospace Systems (ILR), Technische Universität Braunschweig, Hermann-Blenk-Street 23, D-38108, Braunschweig, Germany
Abstract Two thousand seven hundred and ninety uncorrelated targets brighter than magnitude 18.5 were detected by the European Space Agency (ESA) 1-m space debris telescope at Tenerife during more than 1000 observation hours between February 2001 and December 2004. The number of detections can be approximated by a Gaussian distribution. Probabilities to detect individual objects during an observation campaign are determined by propagating 10 debris clouds of the ESA MASTER model and two other fictitious background populations. Based on these probabilities, a 95% confidence interval for the total number of unknown debris in the geostationary ring is derived. It is estimated that there are between 450 and 540 uncatalogued objects brighter than visual magnitude 18.5. © 2006 Elsevier Ltd. All rights reserved.
1. Introduction Ground-based radars like Haystack and TIRA have observed the space debris environment in low-Earth orbit (LEO) for many years and our knowledge about the altitude and inclination distribution of centimetre-sized objects is well advanced. The space debris flux derived from MASTER [1] or EVOLVE [2] populations is pretty reliable in LEO today. The situation in the geostationary ring is different: Only objects larger than about 1 m are contained in the DISCOS database [3] and the models 夡 Prepared for the 56th International Astronautical Congress, Fukuoka, Japan, 2005. E-mail addresses: [email protected] (R. Jehn), [email protected] (S. Ariafar), [email protected] (T. Schildknecht), [email protected] (R. Musci), [email protected] (M. Oswald).
0094-5765/$ - see front matter © 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2006.02.053
for smaller objects are associated with very high uncertainties because little observational data is available. In this paper recent data from the ESA space debris telescope is used to characterize the space debris environment near the geostationary ring. 2. ESA space debris telescope observations The European Space Agency (ESA) is using its 1-m Zeiss telescope in Tenerife since 1999 to detect and track objects which are in the vicinity of the geostationary ring. Objects as small as about 15 cm are observed. In total more than 2200 h of observational data are collected up to now and a homogeneous subset of 1026 h between February 2001 and December 2004 were analysed in this paper to estimate the total number of objects. During these 1026 observation hours 2790 uncorrelated targets (UCTs) brighter than 18.5 mag were
R. Jehn et al. / Acta Astronautica 59 (2006) 84 – 90
85
Fig. 1. Right ascension and declination of a sample of uncorrelated targets brighter than 18.5 mag detected with the ESA space debris telescope between 2001 and 2004.
Fig. 2. Distribution of uncorrelated targets detected by the ESA space debris telescope between 2001 and 2004.
detected. Fig. 1 shows the right ascension and declination of those UCTs. Note, fainter objects and objects which could be correlated with objects contained in the DISCOS catalogue are not plotted. Although the telescope can detect objects as faint as magnitude 20 and occasionally even fainter, we focus in this paper on objects brighter than 18.5 mag because there is a high
probability (about 95 %) that no object of this brightness will be missed. If we were to include fainter objects we would need to estimate the percentage of missed targets as a function of the visual magnitude. The data is grouped in 38 campaigns (usually observation periods of 10–12 days always around New Moon). For instance in one observation during the
86
R. Jehn et al. / Acta Astronautica 59 (2006) 84 – 90
campaign in February 2001 the telescope was pointing 8.3 h to a right ascension of 9:30 h and a declination of 1.2◦ . 15 UCTs brighter than 18.5 mag were detected. Part of the campaigns used for this paper are listed in [4]. For Fig. 2 the right ascension of ascending node (RAAN) and inclination of the 2790 objects were calculated based on observations of a single night and assuming circular orbits. The observations do not directly reflect the debris population but are biased due to the selective “filter” of the observation scenario. Nevertheless the pronounced clustering along the well known RAAN-inclination line is also a feature for uncorrelated objects. The explanation is that fragmentations probably have taken place in common orbits along that line and the fragments just evolve over time staying close to the catalogued population. There are also objects that are known to be in GEO but that are not included in the TLE catalogue. In December 2004, there were 88 uncontrolled objects with a catalogue number but without TLEs (old payloads, rocket bodies and debris). Furthermore there were 65 uncatalogued objects (mainly mission related objects like telescope covers and three apogee kick motors). There were 32 objects with TLEs older than six months and there were 42 controlled but uncatalogued satellites [5]. All in all there were 227 objects that would show up as UCT but which were not created by explosions. However, there are regions, especially with RAAN between 120◦ and 240◦ , where UCTs are detected but where hardly any explosion fragment is expected to be. Most likely these observations can be attributed to objects in a geostationary transfer orbit or to objects with high area-to-mass ratios (a debris source described by Schildknecht et al. [6]). These two classes of objects were not filtered out in our data sets and slightly contaminate the results (although they are not too numerous).
populations which consist of several sub-populations (debris clouds) which have quite distinct distributions and probabilities to be “caught”. To describe the mathematical approach a population of fishes consisting of two species is used as an example.1 There are N1 pikes and N2 carps in a pond. A fisherman catches u1 fishes on the first day. Let pp1 be the probability to catch one specific pike in a given time period and pc1 be the probability to catch one specific carp, then the number of fishes caught during this time period is roughly Gaussian distributed (if we have a sufficiently large population). Let U1 be the random variable describing the number of caught fishes. U1 has the expectation value
3. Estimating the size of a population
If this result is not satisfactory more samples have to be taken. The fisherman goes to a better place! Let pp2 be the probability to catch one specific pike during the next fishing campaign and pc2 the probability to catch one specific carp. Then U = U1 + U2 describes the total number of caught fishes during the two campaigns. Again U is (approximately) Gaussian distributed with
Since the objects are not uniformly distributed in right ascension and declination and since most of the time the telescope was pointing at high-density areas a careful weighting of the statistical data is required in order to draw conclusions concerning the total population. The size of an unknown population can be estimated by drawing samples of the population and using the probability of an individual object to be “caught” or detected by a telescope. This approach was taken to derive confidence intervals for the number of centimeter-sized objects in LEO based on radar observations [7]. This idea is expanded here for the case of
(U1 ) = N1 pp1 + N2 pc1
(1)
and the standard deviation (U1 ) = N1 pp1 (1 − pp1 ) + N2 pc1 (1 − pc1 ).
(2)
With a given probability we can say that our sample u1 will fall within an interval: 1 (u1 − )/ 2 .
(3)
For instance we can be 95% sure that u1 will be inside the interval defined by 1 = −1.96 and 2 = 1.96 (95% corresponds to about 2-sigma confidence interval). Inserting Eqs. (1) and (2) in Eq. (3) gives constraints on the number of fishes in the pond: 1 f (N1 , N2 ) 2 . These inequality constraints are entered into a numerical algorithm that searches for the maximum and minimum of N1 + N2 . For instance, if pp1 = 0.2, pc1 = 0.4 and u1 = 20 then we can say with 95 % confidence that 36 (N1 + N2 ) 86.
(U ) = N1 (pp1 + pp2 ) + N2 (pc1 + pc2 )
1 This approach was not found in standard statistics textbooks (although it probably has appeared in some university publications) and therefore it is explained here in detail.
R. Jehn et al. / Acta Astronautica 59 (2006) 84 – 90
87
Fig. 3. Distribution of uncorrelated targets detected by the ESA space debris telescope between 2001 and 2004 and fictitious debris populations propagated to February 2001.
and 2 (U ) = N1 ppi (1 − ppi ) + N2 pci (1 − pci ). i=1
Again with a given probability we can say that our sample u = u1 + u2 will fall within an interval: 1 (u − )/ 2 . For pp2 = 0.3, pc2 = 0.45 and u2 = 25 we can derive with 95 % confidence that 42 (N1 + N2 )79. By taking a second sample, the width of the confidence interval has decreased, i.e. the uncertainty was reduced. Now the same approach will be taken to derive an estimate of the number of objects in GEO. Instead of two samples we will have 205 samples (i.e. the number of UCTs of the 205 sub-campaigns) and instead of two populations we will define 12 populations. 4. Fictitious debris populations in GEO For the upgrade of the MASTER model the Tenerife data is used to define the populations in the 10–100 cm
size range. A reasonable fit was achieved by simulating the explosion of 10 objects between 1978 and 1998 in near GEO orbits with inclinations between 0◦ and 12◦ . Three thousand six hundred and ninety five fragments larger than 10 cm were simulated [1]. Fig. 3 shows the inclination-right ascension distribution of these fragments after propagation to February 2001.2 It can be seen that the fragments cover most of the detected UCTs which are also plotted in Fig. 3. However, it must be noted that the observations and simulated objects should not be compared in such a simplified way. First of all the observations are taken over a period of 4 years whereas the simulated objects were propagated to the beginning of the observation period. For instance cloud 1 of the MASTER model would look significantly different a few years later. And furthermore, there are strong selection biases in the observational data due to the preferred observation scenarios as mentioned above. Nevertheless the 10 clouds from the MASTER population are very useful to reflect the orbital distribution
2 The debris population from MASTER used for this paper is about 25% smaller than the final MASTER 2005 population, which was not available at the time when this analysis was made. However, this has no consequences on the final results, because only the orbital distributions are required to calculate the detection probabilities. The number of objects is estimated with the method described in the previous section.
88
R. Jehn et al. / Acta Astronautica 59 (2006) 84 – 90
of most of the observed UCTs. To account for the observations which are not following the “main-stream” RAAN-inclination pattern, two additional populations are defined. Population 1 has right ascensions randomly distributed between 50 and 260◦ and inclinations between 0 and 15◦ . Population 2 has right ascensions randomly distributed between 0 and 360◦ and inclinations between 0 and 35◦ . The mean motion is randomly distributed between 0.9 and 1.1 rev/day (i.e. they have a maximum deviation of 2800 km from GEO). The eccentricities of both populations were spread between 0 and 10−4 . The area-to-mass ratios were set to 0.05 m2 /kg for all objects.
object of cloud i during a campaign of t days: p=
ni t ∗ FOV, Ni
(4)
FOV is the diameter of the field-of-view of the telescope in degrees (0.7◦ for the ESA SDT). In Eq. (4), a linear relation between the detection probability and the FOV is assumed. This is justified for objects with low inclination because mainly the North–South extension of the telescope aperture is determining the detection rates. Having calculated the probability for each bin, for each cloud, and for each observation epoch, we can now apply the algorithm for the determination of the confidence interval for the total number of unknown objects as described above.
5. Detection probabilities 6. Results For each population the probability to detect an individual object during an observation campaign needs to be calculated. A 1◦ × 1◦ grid in right ascension and declination is defined and for each bin ni , the number of objects of cloud i passing through that bin is determined by propagating each object along one orbit. Dividing ni by the total number Ni of objects gives the probability to detect an individual object of this cloud if you were to observe that bin during 24 h. Fig. 4 shows these probabilities for cloud 6 where 1036 objects larger than 1 cm were spread out in right ascension and declination according to their orbital distribution. 1 cm was chosen as threshold to have a statistically significant number assuming that their distribution represents the distribution of the observable objects. It is now straightforward to calculate the probability p to observe one specific
Fig. 4. Percentage of objects of cloud 6 passing through a 1◦ × 1◦ observation bin in February 2001 (ni /Ni of Eq. (4)).
Using the 2790 UCTs detected during 1026 hours between February 2001 and December 2004 we derive with 95% confidence that the number of “uncataloged objects brighter than 18.5 mag” is between 450 and 540. Fig. 5 shows how the confidence interval is shrinking as more and more data are considered. 18.5 mag corresponds to a size of 29 cm if optimum observation conditions and an albedo of 0.08 is assumed. The MASTER fragment population consists of 860 objects larger than 29 cm. This difference can be explained by the fact that only a part of these objects are actually detected by the ESA space debris telescope when they pass through the FOV because the signal-to-noise ratio very often is below the detection threshold. A more direct comparison with the MASTER model is difficult because: (a) the observational data is “contaminated” by objects in GTO; and (b) since the objects change their brightness, there is no one-to-one relation between magnitude and size of the objects. The obtained interval gives more a formal uncertainty rather than a lower and upper bound for the number of objects of a physical population. For a limiting magnitude of 16 the number of debris in GEO is estimated between 182 and 258 (see Fig. 6). This is higher than the 140 objects estimated by Matney et al. [8]. Possible explanations may be an increase in the actual number of debris (the measurements with the NASA CDT were made from 1998 until the end of 2000) or the sensitivity roll-off effect that may be noticeable already at magnitude 16 (the limiting sensitivity of the NASA CDT being 17). The MASTER model contains 173 objects larger than 80 cm and 269 objects larger than 60 cm. These sizes correspond to about a magnitude of 16. Again our results agree reasonably well with the MASTER model.
R. Jehn et al. / Acta Astronautica 59 (2006) 84 – 90
89
Fig. 5. Ninety five percent confidence interval for the number of objects brighter than magnitude 18.5. The sizes on the right-hand side give the corresponding object diameter of the MASTER population (e.g. there are 500 objects larger than 40 cm).
Fig. 6. Ninety five percent confidence interval for the number of objects brighter than magnitude 16.
7. Conclusions The approach presented above provides results which are in a good agreement with the debris population models of MASTER. But rather than giving only a single value for the number of objects above a given size or brightness, this approach allows to calculate confidence intervals.
After more than 2200 observation hours out of which 1026 h were used for this analysis, it is possible to formally estimate the number of unknown objects in GEO brighter than magnitude 18.5 with an uncertainty below 10%. However, since objects change their brightness depending on attitude, phase angle and albedo, the results cannot directly be translated into a population corresponding to a given object diameter.
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The ultimate goal to get a better knowledge of the GEO debris environment is the establishment and maintenance of a GEO catalogue where objects are permanently tracked. ESA is currently studying the feasibility of such an endeavor. References [1] M. Oswald, S. Stabroth, C. Wiedemann, P. Vörsmann, T. Schildknecht, H. Klinkrad, Validation of the ESA MASTER2005 orbital debris model, AMOS Technical Conference, Wailea, Hawaii, 5–9 September 2005. [2] P.H. Krisko, EVOLVE historical and projected orbital debris test environments, Advance Space Research 34 (5) (2004) 975–980. [3] C. Hernández, F. Pina, N. Sánchez, H. Sdunnus, H. Klinkrad, The DISCOS database and web interface, in: Proceedings of the Third European Conference on Space Debris, ESA SP-473, 2001, pp. 803–807.
[4] T. Schildknecht, R. Musci, M. Ploner, U. Hugentobler, M. Serra Ricart, J. De Léon Cruz, L. Dominiguez Palmero, Geostationary orbit objects survey, Final report, ESA/ESOC Contract 11914/96/D/IM, September 2004. [5] I. Serraller, R. Jehn, Classification of Geosynchronous Objects, Issue 7, ESOC, Darmstadt, Germany, 2005. [6] T. Schildknecht, R. Musci, W. Flury, J. Kuusela, J. de Leon, L. de Fatima Dominguez Palmero, Optical observations of space debris in high-altitude orbits, in: Proceedings of the Fourth European Conference on Space Debris, ESA SP-587, 2005, pp. 113–118. [7] R. Jehn, Comparison of the 1999 beam-park experiment results with space debris models, Advance Space Research 28 (9) (2001) 1367–1375. [8] M.J. Matney, E. Stansbery, J. Africano, K. Jarvis, K. Jorgensen, T. Thumm, Extracting GEO orbit populations from optical surveys, Advance Space Research 34 (5) (2004) 1160–1165.