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Collision risk for high inclination satellite constellations
Planetary and Space Science 48 (2000) 319±330
Collision risk for high inclination satellite constellations A. Rossi a, b,*, G.B. Valsecchi c, P. Farinella d a
CNUCE-CNR, Area della Ricerca di Pisa, Via Al®eri 1, Loc. San Cataldo, 56010 Ghezzano (Pisa), Italy b Universite de Paris VII Ð Observatoire de Meudon, Paris, France c IAS-CNR, 00133, Roma, Italy d Dipartimento di Astronomia, UniversitaÁ di Trieste, 34131, Trieste, Italy Received 12 May 1999; received in revised form 17 December 1999; accepted 27 December 1999
Abstract We assess the collision hazard for a constellation of telecommunication satellites such as IRIDIUM, arising from the possible chance impact break-up of one of the satellites. The resulting swarm of fragments will orbit the Earth at about the same altitude as the surviving satellites, but will gradually spread due to orbital perturbations, so as to make possible impacts with satellites staying on orbital planes dierent from that of the parent satellite. We ®nd that at intermediate fragment masses of the order of 1 kg, sucient to trigger subsequent catastrophic impacts, the self-generated collision hazard for the constellation satellites exceeds the background level due to the overall debris population for several years. This is true, in particular, when dierential precession of the orbits leads the fragments to encounter satellites revolving around the Earth in the opposite sense, resulting both in higher impact speeds and in enhanced collision probabilities. We estimate that there is about a 10% chance that a ®rstgeneration fragment will trigger subsequent disruptive collisions in the constellation within a decade. 7 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction The increasing population of orbiting debris in Low Earth Orbit (LEO, up to 2000 km of altitude) is the cause of growing concern for the space agencies. Besides some 104 objects larger than 10±20 cm (mostly tracked by US Space Command sensors), the circumterrestrial space is populated by about 105 objects of diameter >1 cm and 40 million bodies exceeding 1 mm, with a total mass of some 3000 tons and a total cross-sectional area of about 40,000 m2 (Kessler et al., 1996). These bodies pose a signi®cant impact hazard to operational satellites and space stations: at the average collision velocity of 10 km/s, even centimetric projectiles are very dangerous, and their current average ¯ux per projectile particle (i.e., the expected impact rate per projectile and per unit target cross-section) of * Corresponding author. Tel.: +39-050-3156-2953; fax: +39-0503138-091. E-mail address: [email protected] (A. Rossi). 0032-0633/00/$20.00 7 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 2 - 0 6 3 3 ( 0 0 ) 0 0 0 0 5 - 2
about 3 10ÿ10 mÿ2 yrÿ1 (Rossi and Farinella, 1992) means that disruptive impacts have a signi®cant chance of occurring, especially on large structures remaining in orbit for comparatively long times. Over the long run, collisions will cause a kind of chain reaction, polluting in an irreversible fashion portions of near-Earth space (Kessler and Cour-Palais, 1978; Rossi et al., 1994). Most in danger from this point of view are the altitude regions between 800 and 1000 km and between 1400 and 1500 km, where there is already a high density of orbiting bodies and atmospheric drag is not eective in removing small fragments. A speci®c aspect of the space debris problem is related to satellite constellations, consisting of tens to hundreds of satellites orbiting at about the same altitude. Over the next decade, several such constellations will be inserted in LEO, mainly for global telecommunication purposes (van der Ha, 1998). One of them, the IRIDIUM system, is already operational. It consists of 66 LEO satellites (plus six spares) orbiting in six dierent orbital planes at an altitude of about
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780 km and with an inclination of 868.4 to the Earth's equator. The wet mass of each satellite is 667 kg. Although some preventive measures have been taken by the constellation owner in order to minimize the collision hazard (e.g., the satellites are deorbited at the end of their lifetime and station Ð keeping manoeuvres are routinely performed to control the satellite positions in their orbital slots), the possibility of disruptive collisions cannot be ruled out, especially because the selected altitude corresponds to a peak of the expected debris ¯ux (Su, 1997). As we will show, the peculiar architecture of such a multi-plane constellation will make a break-up event particularly dangerous, due to the spreading of the resulting fragment swarm, on a timescale intermediate between those analyzed in previous studies, i.e. several hours (Walker et al., 1998) and several decades (Rossi et al., 1998a). The reason is the following. Assume that a constellation satellite undergoes an explosion, possibly due to
an energetic collision by a piece of debris. The fragments are not very dangerous for the other satellites in the same orbital plane, due to the low relative speeds. However, the ejection velocities of the fragments send them into dierent orbits, where dierential perturbations (in particular, the precession of the orbital planes due to the Earth's oblateness and, for the lower-orbiting and smaller fragments, air drag) tend to spread the initial concentration in orbital element space. Since the fragments with longer survival times versus drag are those at higher altitudes, they have slower nodal precession rates than the constellation satellites, and, therefore, after some time, are ``reached'' by them. In most cases, the relative velocities will again be small, but there are some exceptions. In each constellation based on a set of equally spaced, near-polar orbital planes, there are necessarily two neighboring planes that contain satellites moving in opposite directions (plane 1 and 6 in Fig. 1); in the
Fig. 1. The orientation of the six orbital planes of an IRIDIUM-like satellite constellation, as seen from the celestial North pole. The sense of revolution of the satellites in the dierent planes is indicated by the arrows. Note the ``counter-rotating'' planes Nos 1 and 6.
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IRIDIUM case, they will be only 308 from each other when crossing the equator either northward or southward (see Fig. 1). If a satellite orbiting, for example, in the preceding one of these two planes is broken up, its fragments will be reached by the satellites in the following plane, and then they may cause very energetic head-on collisions, with higher than average collision speeds and impact probabilities. In this paper, we have simulated this scenario in order to assess the corresponding risk; the method used to do this is described in the next section. 2. Methodology We have simulated a disrupting collision involving an IRIDIUM satellite. We have assumed that a 650 kg satellite is impacted by a 1 kg projectile traveling at a relative speed of 10 km/s. The mass distribution of the fragments is a power law with an energy-dependent exponent (Rossi et al., 1998b). The average velocity increment DV of the fragments as a function of size is given by the relationship proposed by Su (1990), with the adoption of the ``intermediate'' model adopted by Reynolds (1990); from the average values DV the actual velocity increments DV have been obtained by generating a random number from a triangular distribution (Rossi et al., 1998b). Since the available data seem to con®rm the hypothesis that in many cases the break-up events are approximately isotropic (Wiesel, 1978), we have derived the orbital elements of the fragments generated by the explosion by adding vectorially to the velocity of the exploding object, a vector of magnitude DV with a random direction. A fundamental parameter determining the outcome of such a collision is the threshold speci®c energy for catastrophic break-up, that is the ratio between the projectile kinetic energy and the target mass at which extensive fragmentation of the target replaces localized crater-like damage. This parameter is relatively well known from laboratory experiments for targets made of rocky material, which have been used to simulate the behavior of meteoroids and asteroids, and for which values of the order of 103 J/kg have been measured (Fujiwara et al., 1989). On the other hand, some uncertainty remains for spacecraft-like structures, due to the few experimental data publicly available. It appears plausible that the threshold is about one order-of-magnitude larger for spacecraft than for rocky bodies, due to the dierent material and structure: in hollow structures, the impact-generated shock waves are transmitted less eciently than in nearlyhomogeneous rocks, and, therefore, the formation of large-scale fractures is more dicult. Actually, a few experiments discussed in the literature (McKnight et al., 1992) indicate that for satellite mock-ups the
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threshold speci®c energy for catastrophic break-up is 4.5 104 J/kg, and this is the value we have adopted here. With this choice, the ¯ux of potentially shattering fragments at 800 km height is about 10ÿ5 mÿ2 yrÿ1, and with a constellation cross-section of 1103 m2 there is an 110% hazard per decade of a catastrophic impact. Without our choices of projectile and target masses and impact velocity, a complete fragmentation of the target is predicted by the collision outcome model. A swarm of about 106 fragments heavier than 1 mg and 8000 heavier than 1 g is created as a consequence of the impact. The orbits of all the fragments and of the constellation satellites are then propagated over a time span of 12 years, taking into account all the main perturbing eects; for the very numerous small fragments sampling factors are adopted: 1000 for masses m such that 1 mg R m < 10 mg, 100 for 10 mg R m < 0.1 g, 10 for 0.1 g R m < 1 g and 5 for 1 g R m < 100 g. When impacts between fragments and satellites become possible, the corresponding projectile ¯ux is calculated as a function of time by applying OÈpik's (1976) expression for the intrinsic collision probability p per unit of time for a pair of orbiting bodies: P
2p2 a1:5
U , j Ux j sin I
1
where a is the semimajor axis of the projectile, I is the relative inclination between the orbits of the projectile and the target, and U and Ux are, respectively, the projectile velocity relative to the target, in units of the target's geocentric velocity, and its component along the radial direction. In terms of the orbital elements of the two bodies, these parameters are given by v s u u 2 aT a 1 ÿ e t ÿ2 2 cos I, U 3ÿ a aT s aT a 1 ÿ e2 ÿ , Ux 2 2 ÿ a aT
3
where aT is the semimajor axis of the target (assumed to have a circular orbit), and e, is the eccentricity of the projectile; that in the above expressions we pnote have put GM 1 (Valsecchi et al., 1999), with M the Earth's mass and G the gravitational constant. Taking into account all the fragments produced by the simulated collision event, the total projectile ¯ux as a function of altitude and mass is computed by adding up the contributions of all the relevant fragments. Eventually, this ¯ux of the simulated collision fragments is compared, for any given range of projectile masses (or energies), to the reference value corresponding to the background ¯ux resulting from the entire
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debris population currently present in space, as estimated from the NASA Ordem-96 model (Kessler et al., 1996). The dierence between these two ¯uxes will give a clear estimate of the extra collision risk faced by the constellation satellites due to the consequences of the simulated collision. 3. Results As pointed out in the previous section, our simulated collision produces about 106 particles, that we propagate for 12 years in order to obtain the ¯uxes shown in Figs. 4±8. But before discussing the results related to the impact hazard, let us analyze the dynamical properties of four ``typical'' fragments of dierent sizes, extracted from the above mentioned set of 106, over a time span of several years, with respect to the orbit of a potential target satellite. Table 1 shows the characteristics of these four particles. As shown by Figs. 2 and 3, the dierential precession of orbital planes is normally faster for smaller particles, whose ejection speeds are higher, yielding larger deviations from the orbit of their parent bodies. As a consequence of this precession, the expected impact velocity U and impact energy E can vary by large amounts, and so does the collision probability p (while Ux, which depends on the eccentricity, does not change much). The only dierence between Figs. 2 and 3 is in the choice of the reference orbit: in the former case it is an orbit 308 apart from those of the fragments, but revolving around the Earth in the same direction, whereas in the latter case the orbit of the target has a 308 shift, but is ``counter-rotating''. This implies that the collisions are almost head-on, and, therefore, both p and U (as well as E ), can reach higher values. In both Figs. 2 and 3 we see that the largest particle, 148 kg in mass (large full dots), is ejected with a small velocity change, and, therefore, remains basically on the same orbit as the parent object. The bottom panel of both ®gures shows that its Q/a0 ratio stays very close to 1. Its inclination with respect to the target's orbit increases very slowly due to dierential precession, up to about 108 over the 12-year time span. Table 1 Mass and initial post-break-up orbital elements of the four sample particles whose behavior is analyzed in detail in Figs. 2 and 3 Mass [kg]
a [km]
e
i [8]
O [8]
o [8]
M [8]
148.679 0.011552 0.001031 0.000105
7154.4 7206.2 7120.3 7581.5
0.0013 0.0223 0.0284 0.0723
86.39 87.01 84.94 88.86
147.31 146.23 149.83 143.04
179.37 47.76 18.55 161.64
300.8 69.89 98.06 323.81
Accordingly, U also increases by a small amount. The radial component Ux again stays almost constant and close to zero, due to the very small changes in a and e. As a consequence of the small changes in the orbital parameters, the intrinsic collision probability for this particle remains almost constant over the entire time span; due to the dierent impact velocities in the corotating and counter-rotating cases, the p value of Fig. 2 is lower than that in Fig. 3, and the same is true for the impact energy E. The smaller particles show a more complex behavior. For all of them there are periodic cycles in the plotted quantities. For example, examining the smallest fragment (0.105 g in mass, small open dots), we see that it is ejected to an orbit with an apogee height about 1.13 times the original one; this accounts for a signi®cantly dierent nodal precession, and in turn this is re¯ected in the fast change of the relative inclination, which completes a full cycle, from 0 to 1808, in about three years. This change in the orbital geometry gives rise to cycles of the relative velocity U, with the same periodicity and phase of I. Since Ux is almost constant (but now clearly dierent from zero, due to the fragment's semimajor axis and eccentricity), Eq. (1) implies that the changes in the intrinsic collision probability are driven by the cycles of U and I, and, therefore, p reaches an absolute maximum when I = 1808 and a small relative maximum when I = 08. Finally, the cycles of E clearly re¯ect the behavior of U, which is minimum for co-rotating and maximum for counterrotating particles. In Figs. 4 and 5 we have made the same choice for the reference orbits of the target and have plotted, as a function of time, the total ¯ux of fragments of impact energy, from 103 < 2E < 104 J (bottom panel) to 2E > 109 J (top panel). In the ``co-rotating'' case (Fig. 4), the fragment ¯ux grows during the ®rst several years, as dierential precession spreads the initial ring of fragments surrounding the parent's orbit, and then stays almost constant for a fairly long time (as drag is ineective in deorbiting fragments at this altitude). The impact ¯ux level reaches, or is comparable to, the background level (indicated in Figs. 4±8 by the horizontal dotted line Ð in the lower panels, where this line is not present, the background ¯ux is out of scale on the higher side) only in the middle energy range (2E between 106 and 108 J), corresponding to collisions which approach or even overcome the catastrophic fragmentation threshold. Even worse is the situation for the ``counter-rotating'' target orbit. Here, the ¯ux of small fragments initially decreases, but only over one year or so. For more energetic particles, the decrease is even slower, and the ¯ux remains, for several years, above the background level. For the largest bodies, the spikes in the ¯ux versus time curves are due to individual fragments whose orbits become tem-
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porarily coupled with the target one (i.e., potentially intersecting it) owing to changes in the orbital eccentricity; in such a case, the two orbits are nearly coplanar and tangent, and, according to Eq. (1), p becomes anomalously high. In discussing the results shown in Figs. 2±5, we have analyzed only two particular cases among all the possible parent±target geometries. The dynamics of the interaction between the collision debris cloud and the surviving constellation satellites shows many dierent,
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intermediate behaviors, ranging from the extreme case of the counter-rotating planes to the less hazardous case of two neighboring co-rotating planes. In Figs. 6± 8 we present a global view of the cloud±constellation interaction, using the same kind of plots as in Figs. 4 and 5. In Fig. 6 we show the eects produced by a breakup event occurring in plane 1 (see Fig. 1) on the satellites orbiting in all the constellation planes, including plane 1 itself. In every panel each one of the six curves
Fig. 2. Time evolution over about 12 years of the orbital properties of four particles produced by the simulated impact break-up of an IRIDIUM satellite; the six panels show, from bottom to top, the evolutions of the aphelion to initial semimajor axis ratio Q/a0, the angle I of the orbital plane to that of a selected target satellite, the relative velocity U and its radial component Ux, the logarithm of the intrinsic collision probability p and the impact kinetic energy E. The masses of the fragments are 148 kg (large full dots), 11.5 g (large open dots), 1.03 g (small full dots) and 0.105 g (small open dots). The target satellites orbit on planes shifted by 308 with respect to the orbit of the parent body of the fragments, in the same direction.
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shows the ¯ux with respect to one of the planes; in particular, looking at the bottom panel, the bottom curve (the one having the initial peak most on the right, at about 500 days) refers to the ¯ux on the same plane of the disrupted satellite (plane 1). Then, as the peaks shift to the left, the curves refer to the other planes, from 2±6. From the inspection of all the six curves in each panel we realize that, as we pointed out earlier, the neighboring counter-rotating plane, No. 6, faces the highest risk from the fragments coming from plane No. 1. On the other hand, due to dierential precession, later on the swarm of debris also reaches the other planes, but with ¯uxes reduced by the dis-
persion of the cloud, especially for the smaller particles. It is worth noting that, in the third panel from the top (corresponding to the range of energy between 107 and 108 J, i.e. of the order of fragmentation threshold for our target satellites), the ¯ux of particles stays above the background for several years for three planes (Nos 4±6). Moreover, spikes lasting for a time of about six months are apparent, implying that the entire constellation is anyway aected by the debris cloud, even eight years after the initial break-up event. In Fig. 7, the eects of a fragmentation occurring in plane no. 3 (see Fig. 1) are shown. Again, the six curves, with initial peaks increasingly shifted
Fig. 3. The same as in Fig. 2, but with the target satellites orbiting in the opposite direction with respect to the orbit of the parent body of the fragments.
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toward the origin of times, refer to dierent planes, from no. 3 (the plane of the initial event), to no. 4. Now the mutual inclinations are smaller, so that there are few high-energy particles. This shows that
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a fragmentation occurring in planes nos. 3 and 4 is the least prone to damage the whole constellation. Nonetheless, in the third and fourth panel from the top, planes 4 and 5 experience fragment ¯uxes
Fig. 4. The ¯ux of fragments vs time produced by the simulated impact break-up of an IRIDIUM satellite. The seven panels show, from bottom to top, the ¯uxes at increasing impact energies E: 103 < 2E < 104 J; 104 < 2E < 105 J; 105 < 2E < 106 J; 106 < 2E < 107 J; 107 < 2E < 108 J; 108 < 2E < 109 J; 2E > 109 J: As in Fig. 2, the target satellite is located in a neighboring orbital plane, revolving in the same sense as the parent body of the fragments. The horizontal dotted lines represent the background ¯uxes in the same energy ranges, computed from the overall space debris population according to Kessler et al. (1996) (in the lower panels, these ¯uxes are out of scale on the higher side).
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higher than the backgrond, from about 1.5 years up to about 8.5 years after the initial event. Finally, in Fig. 8 we show the results of our simulation for another extreme situation, similar to that of Fig. 6. Now the plots refer to a fragmentation occurring in plane No. 6. Again we have counterand co-rotating planes close to the one where the break-up is assumed to occur, and this re¯ects in
relatively high ¯uxes in the third and fourth panels from the top. It is worth stressing that, in the energy range between about 107 and 108 J, corresponding to potentially disruptive projectiles of mass of the order of 1 kg, the collision probability stays at approximately twice the background level for several years, at least in the most dangerous counter-rotating case
Fig. 5. The same as in Fig. 4, but with the target satellite located in a plane revolving in the opposite sense with respect to the parent body.
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(Fig. 5). We recall that, at this altitude, the background impact hazard is already one of the highest in circum-terrestrial space (Pardini et al., 1998; Rossi and Farinella, 1992), such that the situation is already critical in the sense that collisions produce new debris faster than air drag can remove it (Kessler, 1991; Rossi et al., 1998b). Our results imply that, after the initial
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break-up, there will be a probability of the order of 10% that a second one will follow within ®ve years, and eventually this may trigger a chain-reaction eect with a characteristic time scale of about one century, much less than the current estimates with the general debris population (some 300±500 yr) (Cordelli et al., 1998).
Fig. 6. Flux of fragments vs time, on all the constellation planes, produced by a simulated impact break-up on plane No. 1 (see Fig. 1). Each curve represents the ¯ux aecting one of the six constellation planes. The seven panels are the same as in Fig. 4.
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4. Conclusions Our results show that for IRIDIUM-like, high-inclination satellite constellations, the self-generated collision hazard is far from negligible, and lasts at levels higher than, or comparable to, the background debris risk for a time span of several years. During this long ``transient'' interval, the orbital planes of the fragments
are not yet randomized, so there may be shorter periods of enhanced risk (corresponding to lower I angles and higher values of U and p for numerous ``streams'' of fragments) or even impact hazard spikes due to individual massive fragments which stay temporarily in intersecting, nearly coplanar and tangent orbits with respect to a potential target satellite. This long-lasting period of increased impact hazard
Fig. 7. The same as Fig. 6, but for a break-up event occurring in plane No. 3.
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is apparent, in particular, at energy levels comparable to the threshold for catastrophic break-up (we have veri®ed that this result is independent of the particular mass/velocity distribution models for fragments that we have adopted Ð very similar results are derived from the alternative NASA/JSC EVOLVE break-up model of Reynolds (1991)). We have found that while these conclusions hold for any relative geometry
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between the fragment swarm and the potential target, the hazard is particularly acute in the case of the ``counter-rotating'' geometry, due to the higher impact velocity and collision probability in this case. While satellite constellations represent just a speci®c example of the problems which are going to be faced by LEO satellites in the next decades due to the debris population, their peculiar dynamical architecture makes
Fig. 8. The same as Fig. 6, but for a break-up event occurring in plane No. 6.
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them particularly vulnerable to the consequences of an initial chance break-up event.
Acknowledgements A.R. contributed to this paper in the framework of the Cooperation Agreement (1997±2001) between the CNUCE Institute of the National Research Council (CNR) and the Italian Space Agency (ASI).
References Cordelli, A., Farinella, P., Rossi, A., 1998. The in¯uence of the fragmentation threshold on the long term evolution of the orbital debris environment. Planet. Space Sci. 46, 691±699. Fujiwara, A., Cerroni, P., Davis, D.R., Ryan, E., Di Martino, M., Holsapple, K., Housen, K. 1989. Experiments and scaling laws on catastrophic collisions. In: Binzel, R.P., Gehrels, T., Matthews, M.S. (Eds.), Asteroids II. University of Arizona Press, Tucson, pp. 240±265. Kessler, D.J., 1991. Collisional cascading: limits of population growth in Low Earth Orbit. Adv. Space Res. 11 (12), 63±66. Kessler, D.J., Cour-Palais, B.G., 1978. Collision frequency of arti®cial satellites: the creation of a debris belt. J. Geophys. Res. 83, 2637±2646. Kessler, D.J. et al., 1996. A computer-based orbital debris environment model for spacecraft design and observation in Low Earth Orbit. NASA Technical Memorandum 104825. McKnight, D.S., Maher, R.L., Nagl, L., 1992. Fragmentation Algorithms for Satellite Targets (FAST) empirical breakup model. Version 2. Kaman Sciences Corporation. OÈpik, E.J., 1976. Interplanetary Encounters. Elsevier, New York.
Pardini, C., Anselmo, L., Rossi, A., Cordelli, A., Farinella, P., 1998. The 1997.0 CNUCE orbital debris reference model. Space¯ight Mechanics, Adv. in the Astronautical Sciences 99, 1041±1058. Reynolds, R.C., 1990. Review of current activities to model and measure the orbital debris environment in Low-Earth-Orbit. Adv. Space Res. 10, 359±372. Reynolds, R.C., 1991. Documentation of program evolve. System Planning Corporation, OD91-002-U-CSP. Rossi, A., Farinella, P., 1992. Collision rates and impact velocities for bodies in low Earth orbit. ESA Journal 16, 339±348. Rossi, A., Cordelli, A., Farinella, P., Anselmo, L., 1994. Collisional evolution of the Earth's orbital debris cloud. J. Geophys. Res. 99 (11), 23,195±23,210. Rossi, A., Pardini, C., Anselmo, L., Cordelli, A., Farinella, P. 1998a. Interaction of the satellites constellations with the Low Earth Orbit debris environment. In: van der Ha, J.C. (Ed.), Mission Design and Implementation of Satellite Constellations. Kluwer, Dordrecht, pp. 327±335. Rossi, A., Anselmo, L., Cordelli, A., Farinella, P., Pardini, C., 1998b. Modelling the evolution of the space debris population. Planet. Space Sci. 46, 1583±1596. Su, S.-Y., 1990. The velocity distribution of the collisional fragments and its eect on the future space debris environment. Adv. Space Res. 10, 389±392. Su, S.-Y., 1997. Impact of satellite constellation arrangement on the future orbital debris environment. Adv. Space Res. 19 (2), 351± 354. Valsecchi, G.B., Rossi, A., Farinella, P., 1999. Visualizing impact probabilities of space debris, submitted. van der Ha, J.C. (Ed.), 1998. Mission Design and Implementation of Satellite Constellations. Kluwer, Dordrecht. Walker, R., Crowther, R., Wilkinson, J., Stokes, P.H., Swinerd, G.G. 1998. Orbital debris collision risk to satellite constellations. In: van der Ha, J.C. (Ed.), Mission Design and Implementation of Satellite Constellations. Kluwer, Dordrecht. Wiesel, W., 1978. Fragmentation of asteroids and arti®cial satellites in orbit. Icarus 34, 99±116.
Collision risk for high inclination satellite constellations A. Rossi a, b,*, G.B. Valsecchi c, P. Farinella d a
CNUCE-CNR, Area della Ricerca di Pisa, Via Al®eri 1, Loc. San Cataldo, 56010 Ghezzano (Pisa), Italy b Universite de Paris VII Ð Observatoire de Meudon, Paris, France c IAS-CNR, 00133, Roma, Italy d Dipartimento di Astronomia, UniversitaÁ di Trieste, 34131, Trieste, Italy Received 12 May 1999; received in revised form 17 December 1999; accepted 27 December 1999
Abstract We assess the collision hazard for a constellation of telecommunication satellites such as IRIDIUM, arising from the possible chance impact break-up of one of the satellites. The resulting swarm of fragments will orbit the Earth at about the same altitude as the surviving satellites, but will gradually spread due to orbital perturbations, so as to make possible impacts with satellites staying on orbital planes dierent from that of the parent satellite. We ®nd that at intermediate fragment masses of the order of 1 kg, sucient to trigger subsequent catastrophic impacts, the self-generated collision hazard for the constellation satellites exceeds the background level due to the overall debris population for several years. This is true, in particular, when dierential precession of the orbits leads the fragments to encounter satellites revolving around the Earth in the opposite sense, resulting both in higher impact speeds and in enhanced collision probabilities. We estimate that there is about a 10% chance that a ®rstgeneration fragment will trigger subsequent disruptive collisions in the constellation within a decade. 7 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction The increasing population of orbiting debris in Low Earth Orbit (LEO, up to 2000 km of altitude) is the cause of growing concern for the space agencies. Besides some 104 objects larger than 10±20 cm (mostly tracked by US Space Command sensors), the circumterrestrial space is populated by about 105 objects of diameter >1 cm and 40 million bodies exceeding 1 mm, with a total mass of some 3000 tons and a total cross-sectional area of about 40,000 m2 (Kessler et al., 1996). These bodies pose a signi®cant impact hazard to operational satellites and space stations: at the average collision velocity of 10 km/s, even centimetric projectiles are very dangerous, and their current average ¯ux per projectile particle (i.e., the expected impact rate per projectile and per unit target cross-section) of * Corresponding author. Tel.: +39-050-3156-2953; fax: +39-0503138-091. E-mail address: [email protected] (A. Rossi). 0032-0633/00/$20.00 7 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 2 - 0 6 3 3 ( 0 0 ) 0 0 0 0 5 - 2
about 3 10ÿ10 mÿ2 yrÿ1 (Rossi and Farinella, 1992) means that disruptive impacts have a signi®cant chance of occurring, especially on large structures remaining in orbit for comparatively long times. Over the long run, collisions will cause a kind of chain reaction, polluting in an irreversible fashion portions of near-Earth space (Kessler and Cour-Palais, 1978; Rossi et al., 1994). Most in danger from this point of view are the altitude regions between 800 and 1000 km and between 1400 and 1500 km, where there is already a high density of orbiting bodies and atmospheric drag is not eective in removing small fragments. A speci®c aspect of the space debris problem is related to satellite constellations, consisting of tens to hundreds of satellites orbiting at about the same altitude. Over the next decade, several such constellations will be inserted in LEO, mainly for global telecommunication purposes (van der Ha, 1998). One of them, the IRIDIUM system, is already operational. It consists of 66 LEO satellites (plus six spares) orbiting in six dierent orbital planes at an altitude of about
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780 km and with an inclination of 868.4 to the Earth's equator. The wet mass of each satellite is 667 kg. Although some preventive measures have been taken by the constellation owner in order to minimize the collision hazard (e.g., the satellites are deorbited at the end of their lifetime and station Ð keeping manoeuvres are routinely performed to control the satellite positions in their orbital slots), the possibility of disruptive collisions cannot be ruled out, especially because the selected altitude corresponds to a peak of the expected debris ¯ux (Su, 1997). As we will show, the peculiar architecture of such a multi-plane constellation will make a break-up event particularly dangerous, due to the spreading of the resulting fragment swarm, on a timescale intermediate between those analyzed in previous studies, i.e. several hours (Walker et al., 1998) and several decades (Rossi et al., 1998a). The reason is the following. Assume that a constellation satellite undergoes an explosion, possibly due to
an energetic collision by a piece of debris. The fragments are not very dangerous for the other satellites in the same orbital plane, due to the low relative speeds. However, the ejection velocities of the fragments send them into dierent orbits, where dierential perturbations (in particular, the precession of the orbital planes due to the Earth's oblateness and, for the lower-orbiting and smaller fragments, air drag) tend to spread the initial concentration in orbital element space. Since the fragments with longer survival times versus drag are those at higher altitudes, they have slower nodal precession rates than the constellation satellites, and, therefore, after some time, are ``reached'' by them. In most cases, the relative velocities will again be small, but there are some exceptions. In each constellation based on a set of equally spaced, near-polar orbital planes, there are necessarily two neighboring planes that contain satellites moving in opposite directions (plane 1 and 6 in Fig. 1); in the
Fig. 1. The orientation of the six orbital planes of an IRIDIUM-like satellite constellation, as seen from the celestial North pole. The sense of revolution of the satellites in the dierent planes is indicated by the arrows. Note the ``counter-rotating'' planes Nos 1 and 6.
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IRIDIUM case, they will be only 308 from each other when crossing the equator either northward or southward (see Fig. 1). If a satellite orbiting, for example, in the preceding one of these two planes is broken up, its fragments will be reached by the satellites in the following plane, and then they may cause very energetic head-on collisions, with higher than average collision speeds and impact probabilities. In this paper, we have simulated this scenario in order to assess the corresponding risk; the method used to do this is described in the next section. 2. Methodology We have simulated a disrupting collision involving an IRIDIUM satellite. We have assumed that a 650 kg satellite is impacted by a 1 kg projectile traveling at a relative speed of 10 km/s. The mass distribution of the fragments is a power law with an energy-dependent exponent (Rossi et al., 1998b). The average velocity increment DV of the fragments as a function of size is given by the relationship proposed by Su (1990), with the adoption of the ``intermediate'' model adopted by Reynolds (1990); from the average values DV the actual velocity increments DV have been obtained by generating a random number from a triangular distribution (Rossi et al., 1998b). Since the available data seem to con®rm the hypothesis that in many cases the break-up events are approximately isotropic (Wiesel, 1978), we have derived the orbital elements of the fragments generated by the explosion by adding vectorially to the velocity of the exploding object, a vector of magnitude DV with a random direction. A fundamental parameter determining the outcome of such a collision is the threshold speci®c energy for catastrophic break-up, that is the ratio between the projectile kinetic energy and the target mass at which extensive fragmentation of the target replaces localized crater-like damage. This parameter is relatively well known from laboratory experiments for targets made of rocky material, which have been used to simulate the behavior of meteoroids and asteroids, and for which values of the order of 103 J/kg have been measured (Fujiwara et al., 1989). On the other hand, some uncertainty remains for spacecraft-like structures, due to the few experimental data publicly available. It appears plausible that the threshold is about one order-of-magnitude larger for spacecraft than for rocky bodies, due to the dierent material and structure: in hollow structures, the impact-generated shock waves are transmitted less eciently than in nearlyhomogeneous rocks, and, therefore, the formation of large-scale fractures is more dicult. Actually, a few experiments discussed in the literature (McKnight et al., 1992) indicate that for satellite mock-ups the
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threshold speci®c energy for catastrophic break-up is 4.5 104 J/kg, and this is the value we have adopted here. With this choice, the ¯ux of potentially shattering fragments at 800 km height is about 10ÿ5 mÿ2 yrÿ1, and with a constellation cross-section of 1103 m2 there is an 110% hazard per decade of a catastrophic impact. Without our choices of projectile and target masses and impact velocity, a complete fragmentation of the target is predicted by the collision outcome model. A swarm of about 106 fragments heavier than 1 mg and 8000 heavier than 1 g is created as a consequence of the impact. The orbits of all the fragments and of the constellation satellites are then propagated over a time span of 12 years, taking into account all the main perturbing eects; for the very numerous small fragments sampling factors are adopted: 1000 for masses m such that 1 mg R m < 10 mg, 100 for 10 mg R m < 0.1 g, 10 for 0.1 g R m < 1 g and 5 for 1 g R m < 100 g. When impacts between fragments and satellites become possible, the corresponding projectile ¯ux is calculated as a function of time by applying OÈpik's (1976) expression for the intrinsic collision probability p per unit of time for a pair of orbiting bodies: P
2p2 a1:5
U , j Ux j sin I
1
where a is the semimajor axis of the projectile, I is the relative inclination between the orbits of the projectile and the target, and U and Ux are, respectively, the projectile velocity relative to the target, in units of the target's geocentric velocity, and its component along the radial direction. In terms of the orbital elements of the two bodies, these parameters are given by v s u u 2 aT a 1 ÿ e t ÿ2 2 cos I, U 3ÿ a aT s aT a 1 ÿ e2 ÿ , Ux 2 2 ÿ a aT
3
where aT is the semimajor axis of the target (assumed to have a circular orbit), and e, is the eccentricity of the projectile; that in the above expressions we pnote have put GM 1 (Valsecchi et al., 1999), with M the Earth's mass and G the gravitational constant. Taking into account all the fragments produced by the simulated collision event, the total projectile ¯ux as a function of altitude and mass is computed by adding up the contributions of all the relevant fragments. Eventually, this ¯ux of the simulated collision fragments is compared, for any given range of projectile masses (or energies), to the reference value corresponding to the background ¯ux resulting from the entire
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debris population currently present in space, as estimated from the NASA Ordem-96 model (Kessler et al., 1996). The dierence between these two ¯uxes will give a clear estimate of the extra collision risk faced by the constellation satellites due to the consequences of the simulated collision. 3. Results As pointed out in the previous section, our simulated collision produces about 106 particles, that we propagate for 12 years in order to obtain the ¯uxes shown in Figs. 4±8. But before discussing the results related to the impact hazard, let us analyze the dynamical properties of four ``typical'' fragments of dierent sizes, extracted from the above mentioned set of 106, over a time span of several years, with respect to the orbit of a potential target satellite. Table 1 shows the characteristics of these four particles. As shown by Figs. 2 and 3, the dierential precession of orbital planes is normally faster for smaller particles, whose ejection speeds are higher, yielding larger deviations from the orbit of their parent bodies. As a consequence of this precession, the expected impact velocity U and impact energy E can vary by large amounts, and so does the collision probability p (while Ux, which depends on the eccentricity, does not change much). The only dierence between Figs. 2 and 3 is in the choice of the reference orbit: in the former case it is an orbit 308 apart from those of the fragments, but revolving around the Earth in the same direction, whereas in the latter case the orbit of the target has a 308 shift, but is ``counter-rotating''. This implies that the collisions are almost head-on, and, therefore, both p and U (as well as E ), can reach higher values. In both Figs. 2 and 3 we see that the largest particle, 148 kg in mass (large full dots), is ejected with a small velocity change, and, therefore, remains basically on the same orbit as the parent object. The bottom panel of both ®gures shows that its Q/a0 ratio stays very close to 1. Its inclination with respect to the target's orbit increases very slowly due to dierential precession, up to about 108 over the 12-year time span. Table 1 Mass and initial post-break-up orbital elements of the four sample particles whose behavior is analyzed in detail in Figs. 2 and 3 Mass [kg]
a [km]
e
i [8]
O [8]
o [8]
M [8]
148.679 0.011552 0.001031 0.000105
7154.4 7206.2 7120.3 7581.5
0.0013 0.0223 0.0284 0.0723
86.39 87.01 84.94 88.86
147.31 146.23 149.83 143.04
179.37 47.76 18.55 161.64
300.8 69.89 98.06 323.81
Accordingly, U also increases by a small amount. The radial component Ux again stays almost constant and close to zero, due to the very small changes in a and e. As a consequence of the small changes in the orbital parameters, the intrinsic collision probability for this particle remains almost constant over the entire time span; due to the dierent impact velocities in the corotating and counter-rotating cases, the p value of Fig. 2 is lower than that in Fig. 3, and the same is true for the impact energy E. The smaller particles show a more complex behavior. For all of them there are periodic cycles in the plotted quantities. For example, examining the smallest fragment (0.105 g in mass, small open dots), we see that it is ejected to an orbit with an apogee height about 1.13 times the original one; this accounts for a signi®cantly dierent nodal precession, and in turn this is re¯ected in the fast change of the relative inclination, which completes a full cycle, from 0 to 1808, in about three years. This change in the orbital geometry gives rise to cycles of the relative velocity U, with the same periodicity and phase of I. Since Ux is almost constant (but now clearly dierent from zero, due to the fragment's semimajor axis and eccentricity), Eq. (1) implies that the changes in the intrinsic collision probability are driven by the cycles of U and I, and, therefore, p reaches an absolute maximum when I = 1808 and a small relative maximum when I = 08. Finally, the cycles of E clearly re¯ect the behavior of U, which is minimum for co-rotating and maximum for counterrotating particles. In Figs. 4 and 5 we have made the same choice for the reference orbits of the target and have plotted, as a function of time, the total ¯ux of fragments of impact energy, from 103 < 2E < 104 J (bottom panel) to 2E > 109 J (top panel). In the ``co-rotating'' case (Fig. 4), the fragment ¯ux grows during the ®rst several years, as dierential precession spreads the initial ring of fragments surrounding the parent's orbit, and then stays almost constant for a fairly long time (as drag is ineective in deorbiting fragments at this altitude). The impact ¯ux level reaches, or is comparable to, the background level (indicated in Figs. 4±8 by the horizontal dotted line Ð in the lower panels, where this line is not present, the background ¯ux is out of scale on the higher side) only in the middle energy range (2E between 106 and 108 J), corresponding to collisions which approach or even overcome the catastrophic fragmentation threshold. Even worse is the situation for the ``counter-rotating'' target orbit. Here, the ¯ux of small fragments initially decreases, but only over one year or so. For more energetic particles, the decrease is even slower, and the ¯ux remains, for several years, above the background level. For the largest bodies, the spikes in the ¯ux versus time curves are due to individual fragments whose orbits become tem-
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porarily coupled with the target one (i.e., potentially intersecting it) owing to changes in the orbital eccentricity; in such a case, the two orbits are nearly coplanar and tangent, and, according to Eq. (1), p becomes anomalously high. In discussing the results shown in Figs. 2±5, we have analyzed only two particular cases among all the possible parent±target geometries. The dynamics of the interaction between the collision debris cloud and the surviving constellation satellites shows many dierent,
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intermediate behaviors, ranging from the extreme case of the counter-rotating planes to the less hazardous case of two neighboring co-rotating planes. In Figs. 6± 8 we present a global view of the cloud±constellation interaction, using the same kind of plots as in Figs. 4 and 5. In Fig. 6 we show the eects produced by a breakup event occurring in plane 1 (see Fig. 1) on the satellites orbiting in all the constellation planes, including plane 1 itself. In every panel each one of the six curves
Fig. 2. Time evolution over about 12 years of the orbital properties of four particles produced by the simulated impact break-up of an IRIDIUM satellite; the six panels show, from bottom to top, the evolutions of the aphelion to initial semimajor axis ratio Q/a0, the angle I of the orbital plane to that of a selected target satellite, the relative velocity U and its radial component Ux, the logarithm of the intrinsic collision probability p and the impact kinetic energy E. The masses of the fragments are 148 kg (large full dots), 11.5 g (large open dots), 1.03 g (small full dots) and 0.105 g (small open dots). The target satellites orbit on planes shifted by 308 with respect to the orbit of the parent body of the fragments, in the same direction.
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shows the ¯ux with respect to one of the planes; in particular, looking at the bottom panel, the bottom curve (the one having the initial peak most on the right, at about 500 days) refers to the ¯ux on the same plane of the disrupted satellite (plane 1). Then, as the peaks shift to the left, the curves refer to the other planes, from 2±6. From the inspection of all the six curves in each panel we realize that, as we pointed out earlier, the neighboring counter-rotating plane, No. 6, faces the highest risk from the fragments coming from plane No. 1. On the other hand, due to dierential precession, later on the swarm of debris also reaches the other planes, but with ¯uxes reduced by the dis-
persion of the cloud, especially for the smaller particles. It is worth noting that, in the third panel from the top (corresponding to the range of energy between 107 and 108 J, i.e. of the order of fragmentation threshold for our target satellites), the ¯ux of particles stays above the background for several years for three planes (Nos 4±6). Moreover, spikes lasting for a time of about six months are apparent, implying that the entire constellation is anyway aected by the debris cloud, even eight years after the initial break-up event. In Fig. 7, the eects of a fragmentation occurring in plane no. 3 (see Fig. 1) are shown. Again, the six curves, with initial peaks increasingly shifted
Fig. 3. The same as in Fig. 2, but with the target satellites orbiting in the opposite direction with respect to the orbit of the parent body of the fragments.
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toward the origin of times, refer to dierent planes, from no. 3 (the plane of the initial event), to no. 4. Now the mutual inclinations are smaller, so that there are few high-energy particles. This shows that
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a fragmentation occurring in planes nos. 3 and 4 is the least prone to damage the whole constellation. Nonetheless, in the third and fourth panel from the top, planes 4 and 5 experience fragment ¯uxes
Fig. 4. The ¯ux of fragments vs time produced by the simulated impact break-up of an IRIDIUM satellite. The seven panels show, from bottom to top, the ¯uxes at increasing impact energies E: 103 < 2E < 104 J; 104 < 2E < 105 J; 105 < 2E < 106 J; 106 < 2E < 107 J; 107 < 2E < 108 J; 108 < 2E < 109 J; 2E > 109 J: As in Fig. 2, the target satellite is located in a neighboring orbital plane, revolving in the same sense as the parent body of the fragments. The horizontal dotted lines represent the background ¯uxes in the same energy ranges, computed from the overall space debris population according to Kessler et al. (1996) (in the lower panels, these ¯uxes are out of scale on the higher side).
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higher than the backgrond, from about 1.5 years up to about 8.5 years after the initial event. Finally, in Fig. 8 we show the results of our simulation for another extreme situation, similar to that of Fig. 6. Now the plots refer to a fragmentation occurring in plane No. 6. Again we have counterand co-rotating planes close to the one where the break-up is assumed to occur, and this re¯ects in
relatively high ¯uxes in the third and fourth panels from the top. It is worth stressing that, in the energy range between about 107 and 108 J, corresponding to potentially disruptive projectiles of mass of the order of 1 kg, the collision probability stays at approximately twice the background level for several years, at least in the most dangerous counter-rotating case
Fig. 5. The same as in Fig. 4, but with the target satellite located in a plane revolving in the opposite sense with respect to the parent body.
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(Fig. 5). We recall that, at this altitude, the background impact hazard is already one of the highest in circum-terrestrial space (Pardini et al., 1998; Rossi and Farinella, 1992), such that the situation is already critical in the sense that collisions produce new debris faster than air drag can remove it (Kessler, 1991; Rossi et al., 1998b). Our results imply that, after the initial
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break-up, there will be a probability of the order of 10% that a second one will follow within ®ve years, and eventually this may trigger a chain-reaction eect with a characteristic time scale of about one century, much less than the current estimates with the general debris population (some 300±500 yr) (Cordelli et al., 1998).
Fig. 6. Flux of fragments vs time, on all the constellation planes, produced by a simulated impact break-up on plane No. 1 (see Fig. 1). Each curve represents the ¯ux aecting one of the six constellation planes. The seven panels are the same as in Fig. 4.
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4. Conclusions Our results show that for IRIDIUM-like, high-inclination satellite constellations, the self-generated collision hazard is far from negligible, and lasts at levels higher than, or comparable to, the background debris risk for a time span of several years. During this long ``transient'' interval, the orbital planes of the fragments
are not yet randomized, so there may be shorter periods of enhanced risk (corresponding to lower I angles and higher values of U and p for numerous ``streams'' of fragments) or even impact hazard spikes due to individual massive fragments which stay temporarily in intersecting, nearly coplanar and tangent orbits with respect to a potential target satellite. This long-lasting period of increased impact hazard
Fig. 7. The same as Fig. 6, but for a break-up event occurring in plane No. 3.
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is apparent, in particular, at energy levels comparable to the threshold for catastrophic break-up (we have veri®ed that this result is independent of the particular mass/velocity distribution models for fragments that we have adopted Ð very similar results are derived from the alternative NASA/JSC EVOLVE break-up model of Reynolds (1991)). We have found that while these conclusions hold for any relative geometry
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between the fragment swarm and the potential target, the hazard is particularly acute in the case of the ``counter-rotating'' geometry, due to the higher impact velocity and collision probability in this case. While satellite constellations represent just a speci®c example of the problems which are going to be faced by LEO satellites in the next decades due to the debris population, their peculiar dynamical architecture makes
Fig. 8. The same as Fig. 6, but for a break-up event occurring in plane No. 6.
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them particularly vulnerable to the consequences of an initial chance break-up event.
Acknowledgements A.R. contributed to this paper in the framework of the Cooperation Agreement (1997±2001) between the CNUCE Institute of the National Research Council (CNR) and the Italian Space Agency (ASI).
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