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Contribution of secondary ejecta to the debris population
Advances in Space Research 34 (2004) 944–950 www.elsevier.com/locate/asr
Contribution of secondary ejecta to the debris population J.-C. Mandeville *, M. Bariteau ONERA/DESP, 2 avenue E. Berlin, 31400 Toulouse, France Received 30 November 2002; received in revised form 20 March 2003; accepted 26 November 2003
Abstract When a micro-debris or a micrometeoroid impacts a spacecraft surface, secondary particles, called ejecta, are produced. These ejecta can contribute to a modification of the debris environment: either locally by the occurrence of secondary impacts on the components of complex and large space structures, or at great distances by the formation of a population of small orbital debris. This paper describes the ejecta production mechanism, and shows their orbital evolution. Then, the distribution of ejecta in low earth orbits is given. Some results are presented describing the number of ejecta as a function of size and altitude. 2004 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Space debris; Ejecta production
1. Introduction Most current orbital debris models, with the exception of MASTER99 (Sdunnus et al., 2001) do not take into account secondary fragments produced upon impact of high velocity particles on spacecraft. The purpose of this paper is to present the mechanism of production and some results concerning the distribution of the ejecta in orbit. Main sources of micro-debris identified until now are: • small fragments of spacecraft break-up (with a size from 200 lm to 1 cm), • Na/K droplets (leaks of Russian satellites Rorsat, 200 lm to 3–4 cm), • paint fragments and degradation products of spacecraft (from 5 to 200 lm), • propulsion residues (Al2 O3 particles, size <20 lm and from 5 mm to 3 cm), • secondary particles called ejecta (from 1 lm to 5 mm). These last particles are produced when a micro-debris or a micrometeoroid impacts a spacecraft surface. The
* Corresponding author. Tel.: +33-5-6225-2741; fax: +33-5-62252569. E-mail address: [email protected] (J.-C. Mandeville).
material ejected during the process become then an integral part of the space environment.
2. Surface degradation All materials exposed to the space environment can be impacted either by meteoroids or space debris. When an hypervelocity particle strikes the surface of a spacecraft, secondary particles called ejecta, are usually produced. The ejected mass is significantly larger than the mass of the projectile. It is about 100 times higher on a brittle target than on a ductile target because there is no plastic deformation and the fracture energy is lower. Three ejection processes can be identified: • Jetting: small and fast liquid particles ejected at grazing angles, representing less than 1% of the total ejected mass. • Cone of ejecta: small and fast particles ejected at constant elevation angle and with a revolution symmetry for a normal impact. The elevation angle is about 60 for a normal impact and is scattered on a few degrees. • Spall fragments: large fragments ejected at low velocity in a direction parallel to the target surface. The mass ejected through this process increases with projectile size and can reach 90% for mm-sized impacts on solar cells. For a ductile target, there are no spall fragments.
0273-1177/$30 2004 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2003.11.009
J.-C. Mandeville, M. Bariteau / Advances in Space Research 34 (2004) 944–950
Fig. 1 shows the three mechanisms involved during an hypervelocity impact on a brittle target. The materials that produce the greatest amount of ejecta upon an hypervelocity impact, are brittle surfaces and especially multilayered targets. The brittle materials currently in orbit are mainly solar arrays and painted surfaces (thermal control coating of satellites, antistatic paint used on rocket bodies). The ejecta model previously developed at ONERA/DESP (Rival and Mandeville, 1999) has been improved and extended to the case of painted surfaces (Bariteau et al., 2001). The total ejected mass (in SI units) has been given by (Gault, 1973): rffiffiffiffiffi qp 1:133 2 Me ¼ K 7:41 106 E ðcos hi Þ ; ð1Þ qt i Ei is impact kinetic energy, qp is projectile density, qt is target density, hi is impact incidence angle (from normal direction), hi < 60, and K depends on the target material and on the incident particle size. K is 1 for a projectile with a diameter larger than 10 lm impacting a brittle target. The size of cone fragments ejected from brittle surfaces varies approximately between 0.1 lm and the projectile size. The smallest particles are ejected with a velocity close to the impact velocity, and the largest particles are ejected with a velocity between 10 and 100 m/s. Spall fragments which are about 20 for each impact, are also ejected with a velocity between 10 and 100 m/s. For an impact on a painted surface, the model assumes that the incident kinetic energy is split in two parts (see Fig. 2). One produces the crater into the paint layer, and the other part creates the crater in the substrate. The paint fragments are quite large and ejected with a low velocity (Bariteau et al., 2001).
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Fig. 2. Impact on a solar cell and on a painted surface.
search of current printed sources (Jane’s, 1996, 1998; Kramer, 1996). The area of solar arrays is computed taking into account both sides of the panels (see Fig. 3). The entire surface of rocket bodies and 10% of the area of the satellites are assumed to be painted. The solar arrays surfaces in orbit are estimated at about 43,000 m2 and the painted surfaces at about 63,000 m2 . About 95% of these painted surfaces are located on rocket bodies. In LEO, there are about 14,000 m2 of solar arrays and 30,500 m2 of painted surfaces (see Figs. 4 and 5). About 65% of these surfaces have their inclination between 73 and 91.
4. Production and orbital evolution
By an extensive use of the database DISCOS (Klinkrad, 2000), satellites and rocket bodies currently in orbit have been identified, as well as their orbital parameters, their shape and size, whenever information was available. Determination of solar array and painted surfaces area has been additionally performed with a
In order to compute the contribution of ejecta as a space debris source, the method outlined in Fig. 6 is used. For every type of orbit, the surface area of solar arrays and painted surfaces are evaluated, and the primary flux received by these surfaces is computed. For low earth orbits, 19 altitude bands and nine inclination bands are considered. All these orbits are assumed as circular. For high earth orbits, 14 typical orbits are chosen: • Geostationary orbit (a ¼ 42,200 km, e ¼ 0:001, i ¼ 7), • Molniya orbit (a ¼ 25,550 km, e ¼ 0:7, i ¼ 65, x ¼ 270), • GPS orbit (a ¼ 26,560 km, e ¼ 0:005, i ¼ 55), • Glonass orbit (a ¼ 25,500 km, e ¼ 0:001, i ¼ 65),
Fig. 1. Ejecta production mechanism on a brittle target.
Fig. 3. Solar arrays and painted surfaces distribution.
3. Exposed surfaces evaluation
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J.-C. Mandeville, M. Bariteau / Advances in Space Research 34 (2004) 944–950
Fig. 4. Paint surfaces distribution in LEO.
Fig. 5. Solar arrays distribution in LEO.
• two GTO orbits (a ¼ 19,300 km, e ¼ 0:65, i ¼ 15 and a ¼ 24,200 km, e ¼ 0:72, i ¼ 15), and eight other types of orbits where are located a larger number of satellites (Bariteau, 2001). It is assumed that the meteoroids impact these surfaces with a velocity of 15 km/s and that the debris (that are taken into account only for low earth orbits) hit the targets with a velocity of 10 km/s. The models used for flux computation are the Gr€ un meteoroid model (Gr€ un et al., 1985) and the ORDEM 96 debris model (Kessler et al., 1997). As it is considered that the direction of impact and the orientation of the primary surface is
Fig. 6. Computation of the contribution of ejecta to the space debris environment.
random, it is supposed that the particles are ejected with a random angle. The argument of perigee (except for the orbit of the Molniya satellites), the right ascension of ascending node and the mean anomaly are assumed to be random. For each orbit, 400 representative particles are considered. Then, for each representative particle, its orbital evolution is computed. It is assumed that the number of the primary surfaces increases at a rate of 2% a year. The model used for the production of ejecta is similar to the one described by Rival and Mandeville (1999) and outlined earlier. The cone particles are ejected with a high velocity, so their initial orbit is very different from the one of the parent object and some particles (about 23%) are ejected on a re-entering or a hyperbolic trajectory. On the other hand, small particles (and paint flakes) are ejected with low velocity and their initial orbit is close to the parent body. The perturbing forces acting on a particle in orbit, in addition to the central gravitation, are: moon gravity, solar gravity, atmospheric drag, solar radiation pressure, electromagnetic forces, the force exerted by the non-spherical terms in the Earth’s gravitational field. In LEO, the dominant perturbing forces on a micro-particle are atmospheric drag, solar radiation pressure and the force exerted by the flattening of the Earth. In HEO, the main perturbing forces are the same except the atmospheric drag (Hamilton, 1993). By the use of a purposely developed semi-analytic model of orbital evolution (Hamilton, 1993; Borderies, 1980 and Anselmo et al., 1995), the lifetime of the micro-particles has been computed. For each perturbing force, the variation of orbital parameters is computed and averaged over one orbit. Computing time is thus dramatically reduced without a loss of accuracy (Bariteau, 2001). The Fig. 7 represents the lifetime calculated for a particle initially in circular orbit with a low inclination. Debris with a size smaller than 10 lm re-enter into the atmosphere quickly (in less than a year) whatever the initial orbit because of the solar radiation pressure.
Fig. 7. Lifetime of debris in orbit (in days) as a function of their size and their initial altitude.
J.-C. Mandeville, M. Bariteau / Advances in Space Research 34 (2004) 944–950
Indeed, solar radiation pressure, which is proportional to the ratio surface area to mass, induces an augmentation of the eccentricity, and the perigee decreases until the atmospheric re-entry. On the other hand, particles larger than 100 lm can stay in orbit for a long time. Therefore, there is an accumulation of large particles relative to small particles, on their orbit of ejection.
5. Spatial density Ejecta with a size smaller than 10 lm stay in orbit during a short time, but as they are produced in a large number and quasi continuously, they are the most numerous. The spatial density of ejecta is maximal between 800 and 1400 km of altitude. Below these altitudes, the spatial density decreases because of the increase of the atmospheric drag. Above, the spatial density diminishes, because primary surfaces are less numerous. The decreasing of spatial density at high altitudes of small particles is less important, because the ejection velocity disperses the initial orbits. The number of ejecta, in the millimetre size range, reaches 5% of the total debris density at 800 km of altitude. The spatial density of ejecta is maximal in LEO, at the altitude of navigation satellites (Glonass and GPS) and at the geostationary altitude. The number of ejecta, in the millimetre size range, represents about 1% of the total number of debris in GEO. The results shown in Fig. 8 are obtained by computing the production and orbital evolution of all secondary particles produced upon impact of meteoroids and orbital debris on all surfaces exposed to the particle environment. The models used for computation are the Gr€ un meteoroid model (Gr€ un et al., 1985) and the ORDEM96 debris model (Kessler et al., 1997). Spatial density given in the figure is representative for the year 2000.
functional forms. As in the ORDEM 96 model (Kessler et al., 1997), this model approximates the ejecta environment by a limited number of representative orbits: six inclination bands and two eccentricity families. It is assumed that the ejecta produced from LEO space vehicles have a circular orbit, and that the ejecta produced from HEO spacecraft, that have their perigee at low altitude, have an highly elliptical orbit with their apogee fixed at 20,000 km altitude. For each inclination band, the number N , of secondary particles in orbit with a diameter > d, within a 1 km band at the altitude h, is given by the following equations: Circular orbits N ðh; dÞ ¼
A/1 /2 /3 B/4 /5 þ ; /1 /2 þ /1 /3 þ /2 /3 /4 þ /5 d da
A ¼ Ka aa d da
ba ;
ð3Þ
d da
þ
ab þbb d db
B ¼ Kb ab d db
bb ;
ð4Þ
d db
þ
/1 ¼ 10ðhh1 Þ=a1 ;
ð5Þ
/2 ¼ 10ðhh2 Þ=a2 ;
ð6Þ
/3 ¼ 10ðhh2 Þ=a3 ;
ð7Þ
/4 ¼ 10ðhh3 Þ=a4 ;
ð8Þ
/5 ¼ 10ðhh3 Þ=a5 ;
ð9Þ
ai ¼ ai;1
The number of ejecta as a function of size, altitude and inclination is computed and approximated by
ð2Þ
aa þba
1 6. Ejecta functional forms
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1þ
d ai;3 d ai;3
2 2 þ ai;2 :
ð10Þ
Elliptical orbits N ðh; dÞ ¼
A/1 /2 /3 ; /1 /2 þ B/1 /3 þ /2 /3
ð11Þ
aa þba d da
A ¼ Ka aa d da
þ
ba ;
ð12Þ
d da
ab þbb d db
B ¼ Kb ab d db
Fig. 8. Spatial density of ejecta in LEO.
þ
bb ;
ð13Þ
d db
/1 ¼ 10ðhh1 Þ=a1 ;
ð14Þ
/2 ¼ 10ðhh1 Þ=a2 ;
ð15Þ
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J.-C. Mandeville, M. Bariteau / Advances in Space Research 34 (2004) 944–950
Fig. 9. Number of ejecta in LEO with a size larger than 1 lm as a function of altitude and inclination.
/3 ¼ 10ðhh1 Þ=a3 ; 1 ai ¼ ai;1 1þ
d ai;3 d ai;3
ð16Þ
2 2 þ ai;2 :
ð17Þ
The distribution of particles is shown in Figs. 9 and 10. The number of ejecta in circular orbits is maximum at 65, 82 and 98 inclination bands (almost 80% of the ejecta in circular orbits have their inclination included between 73 and 91). Concerning the ejecta in elliptical orbits, they are mainly located at 7, 28 and 65 inclination bands. The functional forms used to represent the number of ejecta, as a function of size and altitude, result from fitting the results of computation. The altitude distributions of ejecta vary with the particle size and the two variables cannot be separated. Therefore, the functional forms used in this model are more complicated than those used in the ORDEM 96 model. For circular orbit,
Fig. 10. Number of ejecta in LEO with a size larger than 1 mm as a function of altitude and inclination.
functional forms use 26 parameters for each inclination band. For elliptical orbits, there are 18 parameters. The value of coefficients used for computation are given in Tables 1 and 2. Using the collision probability equations given by Kessler et al. (1997), the flux of ejecta on a given orbit can be computed. As examples, the flux computed on ISS, Spot and Globalstar orbits are presented and compared with the total debris flux given by the model ORDEM 96. On ISS orbit (altitude: 360 km, inclination: 51.6), as seen in Fig. 11, the ratio of the ejecta flux by the total debris flux varies from 105 (for lm-sized particles) to 103 (for mm-sized debris). On Spot orbit (altitude: 825 km, inclination: 98.7), as seen in Fig. 12, the flux of ejecta, in the millimetre size range, reaches 2% of the total debris flux given by the model ORDEM 96. On Globalstar orbit (altitude: 1414 km, inclination: 52), as seen in Fig. 13, the flux of ejecta represents about 1.5% of the total debris flux for the debris with a size larger than 0.5 mm.
Table 1 Values of parameters used into analytic functions (circular orbits) 65
82
98
da db
1.02 103 1.78 105
1.03 103 5.65 104
5.30 104 4.18 104
Ka Kb
5.28 102 2.06 105
4.77 103 9.01 103
2.39 103 8.51 102
aa ab
)6.45 )1.11
)5.73 )1.05
)1.60 )0.98
ba bb
3.61 )2.15
)0.74 )4.11
)0.72 )3.61
h1 h2 h3
391 874 1376
326 842 1508
301 788 1381
a1;1 a1;2 a1;3
18.18 28.18 1.69 104
18.23 28.23 3.60 104
8.71 18.71 3.09 104
a2;1 a2;2 a2;3
85.72 280.12 6.65 106
130.50 311.49 6.30 106
102.88 270.72 7.76 106
a3;1 a3;2 a3;3
769.47 920.48 9.12 107
1112.49 1311.14 1.28 106
590.74 774.65 2.49 106
a4;1 a4;2 a4;3
)122.95 132.95 4.05 106
141.36 353.90 1.94 104
211.05 221.05 7.60 104
a5;1 a5;2 a5;3
444.60 603.57 1.38 106
301.79 408.77 2.57 106
409.40 571.66 209 106
J.-C. Mandeville, M. Bariteau / Advances in Space Research 34 (2004) 944–950
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Table 2 Values of parameters used into analytic functions (elliptic orbits) 7
28
65
da db
6.42 105 2.80 106
4.36 104 3.19 105
1.50 104 1.58 106
Ka Kb
3.74 104 8.29 100
2.69 104 2.81 102
3.57 105 2.06 101
aa ab
)1.16 )2.21
)4.42 )0.79
)0.43 )3.15
ba bb
3.65 0.15
1.95 )0.87
0.23 4.28
h1
280
482
493
a1;1 a1;2 a1;3
81.36 134.47 5.29 106
316.09 372.12 2.19 105
163.43 210.95 3.71 105
a2;1 a2;2 a2;3
816.09 826.09 4.37 104
)2759.15 4514.71 2.59 104
636.51 824.11 5.06 104
a3;1 a3;2 a3;3
1965.10 2513.44 4.95 105
2161.60 2351.28 3.01 104
)1027.10 3757.05 6.68 104
Fig. 12. Comparison of the ejecta flux and the total debris flux (ORDEM 96) on Spot orbit on a non-oriented surface.
Fig. 13. Comparison of the ejecta flux and the total debris flux (ORDEM 96) on Globalstar orbit on a non-oriented surface.
easily implemented into the current orbital debris models.
Fig. 11. Comparison of the ejecta flux and the total debris flux (ORDEM 96) on ISS orbit on a non-oriented surface.
7. Conclusions This study shows that the secondary particles ejected upon hypervelocity impact of meteoroids and debris contribute significantly to the overall population of small orbital debris. In 2000, the ejecta flux reaches about 3%, in the millimetre size range, of the total debris flux, at 800 km altitude, and 1% at 1500 km altitude. For GEO, the spatial density of ejecta represents about 1% of the spatial density of debris given by MASTER 99, in the millimetre size range. Since the lifetime of a millimetre size object is longer than 100 years for altitudes higher than 1500 km, the larger ejecta will continue to accumulate at high altitudes. This population is likely to increase in the long term. The analytic model can be
References Anselmo, L., Farinella, P., Pardini, C., Cordelli, A., Rossi, A. Study on long term evolution of earth orbiting debris. ESOC Contract 10034/92/D/IM, 1995. Bariteau, M. Proliferation des debris orbitaux: production et evolution des particules secondaires, in: These de doctorat, ENSAE, Toulouse, 2001. Bariteau, M., Mandeville, J.C., Sch€afer, F. Ejecta production mechanisms on painted surfaces, in: Proceedings of the 3rd European Conference on Space Debris, ESA SP-473, pp. 249–251, 2001. Borderies, N. Les perturbations d’orbite – Les effets, in: Le mouvement du vehicule spatial en orbite, Cours de technologie spatiale du CNES, pp. 131–158, 1980. Gault, D.E. Displaced mass, depth, diameter, and effects of oblique trajectories for impact craters formed in dense crystalline rocks. The Moon 6, 32–44, 1973. Gr€ un, E., Zook, H.A., Fechtig, H., Giese, R.H. Collisional balance of the meteoritic complex. Icarus 62, 244–272, 1985. Hamilton, D.P. Motion of dust in a planetary magnetosphere: orbitaveraged equations for oblateness, electromagnetic, and radiation forces with application to saturn’s E ring. Icarus 101, 244–264, 1993.
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Kessler, D.J., Zhang, J., Matney, M.J., et al. A computer-based orbital debris environment model for spacecraft design and observations in low-earth orbit, NASA TM-104825, 1997. Klinkrad, H. ESA DISCOS Database: User’s Manual. ESA, 2000. Kramer, H.J. Observation of the earth and its environment, Survey of Missions and Sensors, third ed. Springer, Berlin, 1996. Philip Clark (Ed.) Jane’s Space Directory, eleventh ed. (1995–1996), 1996.
Philip Clark (Ed.) Jane’s Space Directory, thirteenth ed. (1997–1998), 1998. Rival, M., Mandeville, J.C. Modeling of ejecta produced upon hypervelocity impacts. Space Debris 1, 45–57, 1999. Sdunnus, H., Bendisch, J., Klinkrad, H. The ESA MASTER’99 space debris and meteoroid reference model, in: Proceedings of the 3rd European Conference on Space Debris, ESA SP-473, pp. 299–307, 2001.
Contribution of secondary ejecta to the debris population J.-C. Mandeville *, M. Bariteau ONERA/DESP, 2 avenue E. Berlin, 31400 Toulouse, France Received 30 November 2002; received in revised form 20 March 2003; accepted 26 November 2003
Abstract When a micro-debris or a micrometeoroid impacts a spacecraft surface, secondary particles, called ejecta, are produced. These ejecta can contribute to a modification of the debris environment: either locally by the occurrence of secondary impacts on the components of complex and large space structures, or at great distances by the formation of a population of small orbital debris. This paper describes the ejecta production mechanism, and shows their orbital evolution. Then, the distribution of ejecta in low earth orbits is given. Some results are presented describing the number of ejecta as a function of size and altitude. 2004 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Space debris; Ejecta production
1. Introduction Most current orbital debris models, with the exception of MASTER99 (Sdunnus et al., 2001) do not take into account secondary fragments produced upon impact of high velocity particles on spacecraft. The purpose of this paper is to present the mechanism of production and some results concerning the distribution of the ejecta in orbit. Main sources of micro-debris identified until now are: • small fragments of spacecraft break-up (with a size from 200 lm to 1 cm), • Na/K droplets (leaks of Russian satellites Rorsat, 200 lm to 3–4 cm), • paint fragments and degradation products of spacecraft (from 5 to 200 lm), • propulsion residues (Al2 O3 particles, size <20 lm and from 5 mm to 3 cm), • secondary particles called ejecta (from 1 lm to 5 mm). These last particles are produced when a micro-debris or a micrometeoroid impacts a spacecraft surface. The
* Corresponding author. Tel.: +33-5-6225-2741; fax: +33-5-62252569. E-mail address: [email protected] (J.-C. Mandeville).
material ejected during the process become then an integral part of the space environment.
2. Surface degradation All materials exposed to the space environment can be impacted either by meteoroids or space debris. When an hypervelocity particle strikes the surface of a spacecraft, secondary particles called ejecta, are usually produced. The ejected mass is significantly larger than the mass of the projectile. It is about 100 times higher on a brittle target than on a ductile target because there is no plastic deformation and the fracture energy is lower. Three ejection processes can be identified: • Jetting: small and fast liquid particles ejected at grazing angles, representing less than 1% of the total ejected mass. • Cone of ejecta: small and fast particles ejected at constant elevation angle and with a revolution symmetry for a normal impact. The elevation angle is about 60 for a normal impact and is scattered on a few degrees. • Spall fragments: large fragments ejected at low velocity in a direction parallel to the target surface. The mass ejected through this process increases with projectile size and can reach 90% for mm-sized impacts on solar cells. For a ductile target, there are no spall fragments.
0273-1177/$30 2004 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2003.11.009
J.-C. Mandeville, M. Bariteau / Advances in Space Research 34 (2004) 944–950
Fig. 1 shows the three mechanisms involved during an hypervelocity impact on a brittle target. The materials that produce the greatest amount of ejecta upon an hypervelocity impact, are brittle surfaces and especially multilayered targets. The brittle materials currently in orbit are mainly solar arrays and painted surfaces (thermal control coating of satellites, antistatic paint used on rocket bodies). The ejecta model previously developed at ONERA/DESP (Rival and Mandeville, 1999) has been improved and extended to the case of painted surfaces (Bariteau et al., 2001). The total ejected mass (in SI units) has been given by (Gault, 1973): rffiffiffiffiffi qp 1:133 2 Me ¼ K 7:41 106 E ðcos hi Þ ; ð1Þ qt i Ei is impact kinetic energy, qp is projectile density, qt is target density, hi is impact incidence angle (from normal direction), hi < 60, and K depends on the target material and on the incident particle size. K is 1 for a projectile with a diameter larger than 10 lm impacting a brittle target. The size of cone fragments ejected from brittle surfaces varies approximately between 0.1 lm and the projectile size. The smallest particles are ejected with a velocity close to the impact velocity, and the largest particles are ejected with a velocity between 10 and 100 m/s. Spall fragments which are about 20 for each impact, are also ejected with a velocity between 10 and 100 m/s. For an impact on a painted surface, the model assumes that the incident kinetic energy is split in two parts (see Fig. 2). One produces the crater into the paint layer, and the other part creates the crater in the substrate. The paint fragments are quite large and ejected with a low velocity (Bariteau et al., 2001).
945
Fig. 2. Impact on a solar cell and on a painted surface.
search of current printed sources (Jane’s, 1996, 1998; Kramer, 1996). The area of solar arrays is computed taking into account both sides of the panels (see Fig. 3). The entire surface of rocket bodies and 10% of the area of the satellites are assumed to be painted. The solar arrays surfaces in orbit are estimated at about 43,000 m2 and the painted surfaces at about 63,000 m2 . About 95% of these painted surfaces are located on rocket bodies. In LEO, there are about 14,000 m2 of solar arrays and 30,500 m2 of painted surfaces (see Figs. 4 and 5). About 65% of these surfaces have their inclination between 73 and 91.
4. Production and orbital evolution
By an extensive use of the database DISCOS (Klinkrad, 2000), satellites and rocket bodies currently in orbit have been identified, as well as their orbital parameters, their shape and size, whenever information was available. Determination of solar array and painted surfaces area has been additionally performed with a
In order to compute the contribution of ejecta as a space debris source, the method outlined in Fig. 6 is used. For every type of orbit, the surface area of solar arrays and painted surfaces are evaluated, and the primary flux received by these surfaces is computed. For low earth orbits, 19 altitude bands and nine inclination bands are considered. All these orbits are assumed as circular. For high earth orbits, 14 typical orbits are chosen: • Geostationary orbit (a ¼ 42,200 km, e ¼ 0:001, i ¼ 7), • Molniya orbit (a ¼ 25,550 km, e ¼ 0:7, i ¼ 65, x ¼ 270), • GPS orbit (a ¼ 26,560 km, e ¼ 0:005, i ¼ 55), • Glonass orbit (a ¼ 25,500 km, e ¼ 0:001, i ¼ 65),
Fig. 1. Ejecta production mechanism on a brittle target.
Fig. 3. Solar arrays and painted surfaces distribution.
3. Exposed surfaces evaluation
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J.-C. Mandeville, M. Bariteau / Advances in Space Research 34 (2004) 944–950
Fig. 4. Paint surfaces distribution in LEO.
Fig. 5. Solar arrays distribution in LEO.
• two GTO orbits (a ¼ 19,300 km, e ¼ 0:65, i ¼ 15 and a ¼ 24,200 km, e ¼ 0:72, i ¼ 15), and eight other types of orbits where are located a larger number of satellites (Bariteau, 2001). It is assumed that the meteoroids impact these surfaces with a velocity of 15 km/s and that the debris (that are taken into account only for low earth orbits) hit the targets with a velocity of 10 km/s. The models used for flux computation are the Gr€ un meteoroid model (Gr€ un et al., 1985) and the ORDEM 96 debris model (Kessler et al., 1997). As it is considered that the direction of impact and the orientation of the primary surface is
Fig. 6. Computation of the contribution of ejecta to the space debris environment.
random, it is supposed that the particles are ejected with a random angle. The argument of perigee (except for the orbit of the Molniya satellites), the right ascension of ascending node and the mean anomaly are assumed to be random. For each orbit, 400 representative particles are considered. Then, for each representative particle, its orbital evolution is computed. It is assumed that the number of the primary surfaces increases at a rate of 2% a year. The model used for the production of ejecta is similar to the one described by Rival and Mandeville (1999) and outlined earlier. The cone particles are ejected with a high velocity, so their initial orbit is very different from the one of the parent object and some particles (about 23%) are ejected on a re-entering or a hyperbolic trajectory. On the other hand, small particles (and paint flakes) are ejected with low velocity and their initial orbit is close to the parent body. The perturbing forces acting on a particle in orbit, in addition to the central gravitation, are: moon gravity, solar gravity, atmospheric drag, solar radiation pressure, electromagnetic forces, the force exerted by the non-spherical terms in the Earth’s gravitational field. In LEO, the dominant perturbing forces on a micro-particle are atmospheric drag, solar radiation pressure and the force exerted by the flattening of the Earth. In HEO, the main perturbing forces are the same except the atmospheric drag (Hamilton, 1993). By the use of a purposely developed semi-analytic model of orbital evolution (Hamilton, 1993; Borderies, 1980 and Anselmo et al., 1995), the lifetime of the micro-particles has been computed. For each perturbing force, the variation of orbital parameters is computed and averaged over one orbit. Computing time is thus dramatically reduced without a loss of accuracy (Bariteau, 2001). The Fig. 7 represents the lifetime calculated for a particle initially in circular orbit with a low inclination. Debris with a size smaller than 10 lm re-enter into the atmosphere quickly (in less than a year) whatever the initial orbit because of the solar radiation pressure.
Fig. 7. Lifetime of debris in orbit (in days) as a function of their size and their initial altitude.
J.-C. Mandeville, M. Bariteau / Advances in Space Research 34 (2004) 944–950
Indeed, solar radiation pressure, which is proportional to the ratio surface area to mass, induces an augmentation of the eccentricity, and the perigee decreases until the atmospheric re-entry. On the other hand, particles larger than 100 lm can stay in orbit for a long time. Therefore, there is an accumulation of large particles relative to small particles, on their orbit of ejection.
5. Spatial density Ejecta with a size smaller than 10 lm stay in orbit during a short time, but as they are produced in a large number and quasi continuously, they are the most numerous. The spatial density of ejecta is maximal between 800 and 1400 km of altitude. Below these altitudes, the spatial density decreases because of the increase of the atmospheric drag. Above, the spatial density diminishes, because primary surfaces are less numerous. The decreasing of spatial density at high altitudes of small particles is less important, because the ejection velocity disperses the initial orbits. The number of ejecta, in the millimetre size range, reaches 5% of the total debris density at 800 km of altitude. The spatial density of ejecta is maximal in LEO, at the altitude of navigation satellites (Glonass and GPS) and at the geostationary altitude. The number of ejecta, in the millimetre size range, represents about 1% of the total number of debris in GEO. The results shown in Fig. 8 are obtained by computing the production and orbital evolution of all secondary particles produced upon impact of meteoroids and orbital debris on all surfaces exposed to the particle environment. The models used for computation are the Gr€ un meteoroid model (Gr€ un et al., 1985) and the ORDEM96 debris model (Kessler et al., 1997). Spatial density given in the figure is representative for the year 2000.
functional forms. As in the ORDEM 96 model (Kessler et al., 1997), this model approximates the ejecta environment by a limited number of representative orbits: six inclination bands and two eccentricity families. It is assumed that the ejecta produced from LEO space vehicles have a circular orbit, and that the ejecta produced from HEO spacecraft, that have their perigee at low altitude, have an highly elliptical orbit with their apogee fixed at 20,000 km altitude. For each inclination band, the number N , of secondary particles in orbit with a diameter > d, within a 1 km band at the altitude h, is given by the following equations: Circular orbits N ðh; dÞ ¼
A/1 /2 /3 B/4 /5 þ ; /1 /2 þ /1 /3 þ /2 /3 /4 þ /5 d da
A ¼ Ka aa d da
ba ;
ð3Þ
d da
þ
ab þbb d db
B ¼ Kb ab d db
bb ;
ð4Þ
d db
þ
/1 ¼ 10ðhh1 Þ=a1 ;
ð5Þ
/2 ¼ 10ðhh2 Þ=a2 ;
ð6Þ
/3 ¼ 10ðhh2 Þ=a3 ;
ð7Þ
/4 ¼ 10ðhh3 Þ=a4 ;
ð8Þ
/5 ¼ 10ðhh3 Þ=a5 ;
ð9Þ
ai ¼ ai;1
The number of ejecta as a function of size, altitude and inclination is computed and approximated by
ð2Þ
aa þba
1 6. Ejecta functional forms
947
1þ
d ai;3 d ai;3
2 2 þ ai;2 :
ð10Þ
Elliptical orbits N ðh; dÞ ¼
A/1 /2 /3 ; /1 /2 þ B/1 /3 þ /2 /3
ð11Þ
aa þba d da
A ¼ Ka aa d da
þ
ba ;
ð12Þ
d da
ab þbb d db
B ¼ Kb ab d db
Fig. 8. Spatial density of ejecta in LEO.
þ
bb ;
ð13Þ
d db
/1 ¼ 10ðhh1 Þ=a1 ;
ð14Þ
/2 ¼ 10ðhh1 Þ=a2 ;
ð15Þ
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Fig. 9. Number of ejecta in LEO with a size larger than 1 lm as a function of altitude and inclination.
/3 ¼ 10ðhh1 Þ=a3 ; 1 ai ¼ ai;1 1þ
d ai;3 d ai;3
ð16Þ
2 2 þ ai;2 :
ð17Þ
The distribution of particles is shown in Figs. 9 and 10. The number of ejecta in circular orbits is maximum at 65, 82 and 98 inclination bands (almost 80% of the ejecta in circular orbits have their inclination included between 73 and 91). Concerning the ejecta in elliptical orbits, they are mainly located at 7, 28 and 65 inclination bands. The functional forms used to represent the number of ejecta, as a function of size and altitude, result from fitting the results of computation. The altitude distributions of ejecta vary with the particle size and the two variables cannot be separated. Therefore, the functional forms used in this model are more complicated than those used in the ORDEM 96 model. For circular orbit,
Fig. 10. Number of ejecta in LEO with a size larger than 1 mm as a function of altitude and inclination.
functional forms use 26 parameters for each inclination band. For elliptical orbits, there are 18 parameters. The value of coefficients used for computation are given in Tables 1 and 2. Using the collision probability equations given by Kessler et al. (1997), the flux of ejecta on a given orbit can be computed. As examples, the flux computed on ISS, Spot and Globalstar orbits are presented and compared with the total debris flux given by the model ORDEM 96. On ISS orbit (altitude: 360 km, inclination: 51.6), as seen in Fig. 11, the ratio of the ejecta flux by the total debris flux varies from 105 (for lm-sized particles) to 103 (for mm-sized debris). On Spot orbit (altitude: 825 km, inclination: 98.7), as seen in Fig. 12, the flux of ejecta, in the millimetre size range, reaches 2% of the total debris flux given by the model ORDEM 96. On Globalstar orbit (altitude: 1414 km, inclination: 52), as seen in Fig. 13, the flux of ejecta represents about 1.5% of the total debris flux for the debris with a size larger than 0.5 mm.
Table 1 Values of parameters used into analytic functions (circular orbits) 65
82
98
da db
1.02 103 1.78 105
1.03 103 5.65 104
5.30 104 4.18 104
Ka Kb
5.28 102 2.06 105
4.77 103 9.01 103
2.39 103 8.51 102
aa ab
)6.45 )1.11
)5.73 )1.05
)1.60 )0.98
ba bb
3.61 )2.15
)0.74 )4.11
)0.72 )3.61
h1 h2 h3
391 874 1376
326 842 1508
301 788 1381
a1;1 a1;2 a1;3
18.18 28.18 1.69 104
18.23 28.23 3.60 104
8.71 18.71 3.09 104
a2;1 a2;2 a2;3
85.72 280.12 6.65 106
130.50 311.49 6.30 106
102.88 270.72 7.76 106
a3;1 a3;2 a3;3
769.47 920.48 9.12 107
1112.49 1311.14 1.28 106
590.74 774.65 2.49 106
a4;1 a4;2 a4;3
)122.95 132.95 4.05 106
141.36 353.90 1.94 104
211.05 221.05 7.60 104
a5;1 a5;2 a5;3
444.60 603.57 1.38 106
301.79 408.77 2.57 106
409.40 571.66 209 106
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Table 2 Values of parameters used into analytic functions (elliptic orbits) 7
28
65
da db
6.42 105 2.80 106
4.36 104 3.19 105
1.50 104 1.58 106
Ka Kb
3.74 104 8.29 100
2.69 104 2.81 102
3.57 105 2.06 101
aa ab
)1.16 )2.21
)4.42 )0.79
)0.43 )3.15
ba bb
3.65 0.15
1.95 )0.87
0.23 4.28
h1
280
482
493
a1;1 a1;2 a1;3
81.36 134.47 5.29 106
316.09 372.12 2.19 105
163.43 210.95 3.71 105
a2;1 a2;2 a2;3
816.09 826.09 4.37 104
)2759.15 4514.71 2.59 104
636.51 824.11 5.06 104
a3;1 a3;2 a3;3
1965.10 2513.44 4.95 105
2161.60 2351.28 3.01 104
)1027.10 3757.05 6.68 104
Fig. 12. Comparison of the ejecta flux and the total debris flux (ORDEM 96) on Spot orbit on a non-oriented surface.
Fig. 13. Comparison of the ejecta flux and the total debris flux (ORDEM 96) on Globalstar orbit on a non-oriented surface.
easily implemented into the current orbital debris models.
Fig. 11. Comparison of the ejecta flux and the total debris flux (ORDEM 96) on ISS orbit on a non-oriented surface.
7. Conclusions This study shows that the secondary particles ejected upon hypervelocity impact of meteoroids and debris contribute significantly to the overall population of small orbital debris. In 2000, the ejecta flux reaches about 3%, in the millimetre size range, of the total debris flux, at 800 km altitude, and 1% at 1500 km altitude. For GEO, the spatial density of ejecta represents about 1% of the spatial density of debris given by MASTER 99, in the millimetre size range. Since the lifetime of a millimetre size object is longer than 100 years for altitudes higher than 1500 km, the larger ejecta will continue to accumulate at high altitudes. This population is likely to increase in the long term. The analytic model can be
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