EVOLUTION OF SPACECRAFT ORBITAL MOTION DUE TO HYPERVELOCITY IMPACTS WITH DEBRIS AND METEOROIDS
Luc Sagnieres` and Inna Sharf Department of Mechanical Engineering, McGill University, 817 Sherbrooke W., Montreal, QC, H3A 0C3, Canada, Email: [email protected]
ABSTRACT From a somewhat different perspective, the impacts be- tween the small debris flux and a large object, be it a functional or defunct spacecraft, will lead to a transfer A framework is developed to include the effect of bom- of momentum from the impactor to the target, thus mod- bardment from space debris and meteoroids on the orbital ifying its orbital parameters. Moreover, at such high ve- motion of active and inoperative satellites by considering locities, the particles ejected during crater formation pro- the transfer of momentum from impactors as a stochastic vide an additional momentum transfer, an effect known process known as a compound Poisson process. The dif- as momentum enhancement [12]. While many of the en- ferential equation for orbital motion is set up as a stochas- vironmental factors influencing an object’s orbit in space tic differential equation and solved in a Monte Carlo sim- have been extensively studied, the effect of bombardment ulation. The orbital propagation is coupled to a determin- of debris and meteoroids on spacecraft orbit propagation, istic attitude propagation model in order to implement the on the other hand, is a research area that is still in its in- additional effect of ejecta momentum. The framework is fancy. The work proposed here follows the methodology applied to four spacecraft and two analytical test cases described in [14] for propagating the attitude motion of are presented in order to obtain a confirmation of the va- tumbling spacecraft undergoing these hypervelocity im- lidity of the stochastic model. The resulting probability pacts, however, applying that approach to orbital motion. distribution functions for the change in semi-major axis after a ten-year propagation period show that for the pre- This method enables the inclusion of hypervelocity im- sented cases, a systematic decrease is present, although pacts into spacecraft orbit propagation models by consid- small compared to observations and estimated decay due ering the transfer of linear momentum from collisions as to other effects. a stochastic jump process known as a compound Pois- son process [14]. The differential equation for orbital motion then becomes a stochastic differential equation Key words: hypervelocity impacts, orbit propagation, (SDE) and is solved in a Monte Carlo simulation by em- stochastic differential equation. ploying independent sets of randomly generated colli- sions [14]. These collisions can be obtained using impact fluxes converted into probability density functions (PDF), 1. INTRODUCTION the former obtained from the European Space Agency’s (ESA) Meteoroid and Space Debris Terrestrial Environ- ment Reference (MASTER) model describing the debris Collisions between small orbital debris and micromete- and meteoroid population around Earth [5]. Furthermore, oroids and active and inoperative satellites are typically the stochastic orbit propagation model can be coupled to a considered in the context of the increasing threat that deterministic attitude propagation model in order to ade- they present to space operations and missions [8]. The quately implement the additional effect of ejecta momen- current estimates of the space debris population in or- tum which is dependent on the orientation of the space- bit are 29,000 above 10 cm, 670,000 above 1 cm, and craft. The momentum contribution from ejecta can be more than 170 million above 1 mm [4]. Furthermore, ev- calculated for every collision by considering the velocity, ery day approximately 100 tons of dust and sand-sized mass and direction distribution of the ejecting particles meteoroids enter Earth’s atmosphere [9]. When collid- from a model developed for ESA defining the character- ing with orbiting spacecraft, these hypervelocity impacts, istics of such ejecta [12]. occurring at speeds up to 60 km per second, can have disastrous consequences because of the structural dam- First, a review of the differential equations governing age they may cause and electromagnetic interference they spacecraft dynamics will be presented in Section 2. Sec- may have with spacecraft systems [3]. When impacting ond, the SDEs for orbital motion will be discussed in large debris, this bombardment is responsible for gener- Section 3 along with the numerical integration method ating more small debris in orbit, further aggravating the used to solve them. In order to assess the importance space debris problem. of these collisions on orbit propagation, the developed
Proc. 7th European Conference on Space Debris, Darmstadt, Germany, 18–21 April 2017, published by the ESA Space Debris Office Ed. T. Flohrer & F. Schmitz, (http://spacedebris2017.sdo.esoc.esa.int, June 2017) model is then applied to several pieces of orbital ob- 3. INCLUDING HYPERVELOCITY IMPACTS IN jects in Section 4: the defunct European environmen- ORBITAL MOTION tal satellite Envisat, representing a potential active de- bris removal (ADR) target in low-Earth orbit (LEO); a high area-to-mass ratio (HAMR) object modeled after As hypervelocity impacts are random, they will have to multi-layer insulation debris in a highly elliptic orbit with be considered as a stochastic process and the differen- apogee at geostationary altitude, chosen to represent an tial equation for orbital motion will have to be set up as upper bound when the effect of the collisions is strongest; an SDE. A stochastic process is one that describes the one of the operational LAGEOS spherical satellites in evolution of a system’s random variables over time, and, a medium Earth orbit (MEO), satellites having been the unlike deterministic processes, can expand in multiple, cause of some debate due to unexplained semi-major axis or infinitely many, paths [11]. Including stochastic pro- decay; and lastly to the Japenese satellite Ajisai, also an cesses in differential equations is useful when wanting operational spherical satellite undergoing an orbital de- to describe uncertainties or random variations in any dy- cay due to aerodynamic drag. Finally, in Section 5, two namical system. Solving SDEs requires the use of numer- analytical test cases are presented to provide additional ical integration techniques that have been a subject of in- confidence in the validity of the propagation model. terest in the last few decades as most integration methods used for solving ordinary differential equations perform poorly when applied to SDEs [11]. 2. SPACECRAFT DYNAMICS 3.1. Collisions as a compound Poisson process Understanding the dynamics of satellites in orbit requires analysis of the six degrees of freedom governing the or- As explained in [14], the momentum transfer due to hy- bit and attitude of the spacecraft: three for translation pervelocity impacts can be represented as a compound and three for orientation. Three vector differential equa- Poisson process if a PDF describing the impacts can be tions describe the evolution of the corresponding vari- determined, and if the time between collisions can be ables. First is the dynamics equation for orbital motion shown to follow an exponential distribution. PDFs for in an Earth-centered inertial (ECI) coordinate frame [2]: the mass and velocity of the impactors can be obtained µ X by transforming the velocity and mass impact fluxes of ¨r(t) = − r(t) + a (t, r(t)) (1) r(t)3 j the MASTER-2009 model for a certain input orbit. A 2D j PDF for direction can also be obtained in an analogous where r is the position as a function of time t, r = krk, manner from the directional impact flux as a function of µ is the Earth’s gravitational parameter, and ai represents azimuth and elevation angles. These PDFs can then be the non-gravitational accelerations, and assuming a per- combined to define the linear momentum of an impactor. fectly uniform spherical Earth, ignoring Earth’s oblate- ness and third-body interactions. The time between collisions can be assumed to follow an exponential distribution due to the analogy between ki- The second differential equation, the dynamics equation netic gas theory and objects sweeping the space environ- for attitude motion, relates the evolution of angular veloc- ment filled with other debris and meteoroids [5]. There- ity, ω, to the sum of the external torques about the center fore, the PDF for time between collisions can be obtained of mass of the body, τ j [6]: from the total expected number of collisions as outputted by MASTER-2009. The four PDFs obtained then define × X Iω˙ (t) + ω(t) Iω(t) = τ j(t, q(t)) (2) the impactors that a spacecraft in the specified orbit will j collide with. where I is the matrix representation of the inertia ten- From Eq. (1), the differential equation of motion can be sor of the rigid body in the centroidal body-fixed frame set up in differential form: T T and q = q0 qv is the attitude parametrization, cho- sen here to be quaternions. µ X d˙r(t) = − r(t) + a (t, r(t)) dt (5) r(t)3 j Finally, the third differential equation is the kinematic j equation for orientation: Considering the momentum of the satellite, defined as 1 ˙q(t) = ΩΩ(ω)q(t) (3) p = M ˙r with M the mass of the spacecraft, Eq. (5) can 2 be rewritten as: where, expressed in terms of the body-referenced compo- µ X nents of ω: dp(t) = M − r(t) + a (t, r(t)) dt (6) r(t)3 j 0 −ωx −ωy −ωz j ω 0 ω −ω Ω = x z y (4) ωy −ωz 0 ωx Including the term for the momentum from hypervelocity ωz ωy −ωx 0 impacts as a function of the compound Poisson process, Yt, Eq. (6) is transformed into an SDE: Knowing the position and velocity of the satellite in its orbit at the current time step, the relative velocity in the inertial frame, v , can be obtained from the associated µ X rel dp(t) = M − r(t) + a (t, r(t)) dt+dY rotation matrix, C (t): r(t)3 j t IO j v = C (t )vO (11) (7) rel IO n+1 rel As shown in [14], hypervelocity impacts have a minimal From this, the relative momentum vector of the impactor effect on the evolution of angular velocity and a deter- in the inertial frame can be determined, and used in ministic approach to attitude propagation can therefore Eq. (9): be applied. Therefore, Eq. (7), along with Eqs. (2) and ∆Y = mv (12) (3) define the dynamics of any satellite orbiting Earth un- tn+1 rel dergoing bombardment of debris and meteoroids. 3.4. Momentum exchange and the jump condition 3.2. Solving the stochastic differential equation The derivation of Eq. (9) starts by looking at the mo- mentum exchange between the impactor and the target The numerical integration method used to solve the at the moment of impact t in the inertial frame, where stochastic jump process in Eq. (7) requires a jump- n+1 the subscript i here represents the impactor, and the su- adapted time discretization method, where the time dis- perscript ()− represents the time right before the impact. cretization is constructed to include the jump times on Thus, using conservation of momentum over the duration top of the regular time steps [11]. Once the jump times of impact, we have: are known, generated randomly by the method presented in [14], solving Eq. (7) then reduces to: first using any ∆pt = −∆pi,t (13) numerical integration method to propagate the position n+1 n+1 vector without considering collisions, i.e., integrating for- or equivalently: ward the following first-order ODE: − − pt = p + p − pi,t (14) n+1 tn+1 i,tn+1 n+1 dp(t) µ X = M − r(t) + a (t, r(t)) (8) In terms of velocity, where M is the mass of the target dt r(t)3 j j and m is the mass of the impactor: Mv = Mv− + mv− − mv (15) Second, if time step tn+1 is a jump time, the estimate of tn+1 tn+1 i,tn+1 i,tn+1 momentum obtained from Eq. (8) at the next time step, − pt , is updated using the following jump condition: The velocities after impact are equal, assuming the im- n+1 pactor is absorbed by the target: − pt = p + ∆Yt (9) n+1 tn+1 n+1 Mv = Mv− + mv− − mv (16) tn+1 tn+1 i,tn+1 tn+1 − where ∆Ytn+1 = Ytn+1 − Yt is the jump size of the n+1 The velocity of the impactor right before impact can be compound Poisson process at jump time tn+1, describing the relative linear momentum of the impactor, and calcu- written in terms of the velocity of the target and the rela- lated as shown below. Derivation of the jump condition tive velocity of the impactor: and the reason for the use of relative linear momentum, − − v = v + vrel (17) in contrast to absolute linear momentum, can be found in i,tn+1 tn+1 Sections 3.3 and 3.4. Therefore:
− − Mvtn+1 = Mvtn+1 +mvtn+1 +mvrel −mvtn+1 (18) 3.3. Generating independent collisions
and solving for vtn+1 : Following the methodology outlined in [14], one can ob- − (M + m)vt = (M + m)v + mvrel (19) tain random values for impactor mass m, relative impact n+1 tn+1 velocity vrel, elevation angle θ, and azimuth φ from the Simplifying for M m gives: obtained PDFs using inverse transform sampling. A vec- − tor for the relative impactor velocity in the orbital ref- Mvtn+1 = Mvtn+1 + mvrel (20) erence frame can then be computed by switching from spherical coordinates to Cartesian coordinates (x in the or in terms of momentum: spacecraft ram direction; z pointing towards Earth; y − pt = p + mv (21) completing the right hand rule) using: n+1 tn+1 rel
"−vrel cos θ cos φ# This is the jump condition used for propagating the mo- O mentum of the target during a collision as seen in Eq. (9) vrel = −vrel cos θ sin φ (10) vrel sin θ coupled with Eq. (12). 3.5. Ejecta Momentum and Coupling to Attitude 4.1. Envisat Propagation Envisat was a European Earth-observing environmental As an impactor collides with the target, a large amount satellite launched by ESA on March 1, 2002. It was sent of particulates in the vicinity of the location of impact in a Sun synchronous polar orbit at an altitude of 790 brakes off from the target’s surface and is ejected out- km. The purpose of the satellite was to monitor climate wards [12]. These particles have an associated momen- change and the Earth’s environmental resources. The ex- tum and therefore increase the effect of the momentum pected mission lifetime was 5 years but it lasted until transfer from the impactor to the target. A model was April 8, 2012, when ESA lost contact with the satellite for developed primarily for ESA to describe the ejecta pop- unknown causes, and then announced its end of mission ulation that is created during such a hypervelocity im- on May 9, 2012. To this day, Envisat remains in orbit and pact [12]. It postulates that three ejection processes oc- is one of the largest space debris objects currently being cur: small particles thrown off at grazing angles and high tracked by the United States Space Surveillance Network velocities, known as jetting; cone fragments, which are (SSN). Due to its large size and busy polar orbit, Envisat small and fast particles that are ejected at a constant el- has been of much interest to the community as a potential evation angle in a cone around the location of impact; ADR target. and spall products, large fragments ejected normal to the impact surface at low velocities [12]. This model de- Envisat is assumed to be spinning at a rate of 2.67◦ s−1 scribes the velocity, size, and direction distributions for and weighs approximately 7828 kg with a total surface each ejection process as a function of the impactor’s char- area of 332 m2 [7, 1]. The orbit parameters used are a acteristics and from these, a vector for the linear momen- semi-major axis (SMA) of 7136 km, an eccentricity of tum contribution of the ejecta can be calculated [14]. 0.0001 and an inclination of 98.3◦. Using its total sur- face area and current orbit as input to MASTER-2009, Implementing this effect in the orbit propagation of large a total of 5.77 × 105 impacts per year, or about one space debris requires knowledge of the target’s orienta- per minute, can be expected, with 58% coming from tion at the moment of impact. Therefore, the attitude of micrometeoroids. Figs. 2a and 2b show the PDFs for the target needs to be propagated in parallel. The equa- the change in SMA throughout the ten-year simulations tions for rotational motion and translational motion can for the case with and without momentum enhancement, be propagated separately and the coupling is only per- where a yearly orbital decay of 0.96 m and 0.74 m are formed at the times of impact when information about seen, respectively. Furthermore, the widths of the PDFs attitude is used in the calculation of the momentum con- increase with time, as expected, and are larger for the tribution from ejecta. case with momentum enhancement, similarly to what was obtained in [14] for attitude motion. However, for such a large spacecraft at a relatively low orbit, aerodynamic 4. CASE STUDIES drag will have a much larger influence on the orbital de- cay of the satellite, calculated to be up to 2 km per year depending on solar activity. Similar PDFs were obtained To understand the influence of hypervelocity impacts on for the orbit eccentricity, inclination, right ascension of orbital motion, the differential equations of motion were the ascending node and argument of perigee, but changes propagated for ten years considering only hypervelocity in them are minute and deemed negligible. impacts and the Earth’s gravitational force. The method was applied to four cases, Envisat, a HAMR object, LA- GEOS and Ajisai, and each ten-year run was repeated 4.2. High Area-to-Mass Ratio Object 10,000 times for different randomly generated sets of col- lisions including the effect of ejecta momentum. A solu- tion without momentum enhancement was also computed The same procedure was repeated using the properties of for Envisat in order to compare with our analytical test a HAMR object. A flat two-sided 1 × 1 m2 plate with a case in Section 5. As Envisat can be considered to be mass of 0.1 kg was employed, therefore having an area- spin-stabilized, a constant spin rate is assumed and exter- to-mass ratio of 20, simulating a piece of MLI debris [15]. T ◦ −1 nal torques can therefore be neglected. The HAMR ob- An initial angular velocity of ω0 = [1, 1, 1] s was ject is randomly tumbling, so excluding external torques used to simulate tumbling. The orbit used to calculate will also not provide a bias in the results. Finally, LA- the impact fluxes from MASTER-2009 has a SMA of GEOS and Ajisai are spherical satellites and the changes 24,802 km, an eccentricity of 0.7, and an inclination of in attitude are not important. The impact fluxes used 15◦ [15]. The total number of expected collisions for that are assumed to be constant throughout the propagation. orbit was found to be close to 870 per year per m2, with The flowchart outlining the complete methodology is pre- about 93% coming from meteoroids. 1739 collisions are sented in Fig. 1. The inputs include the orbital parame- therefore expected to occur per year, or one every 5 hours. ters and the initial conditions of the differential equations. The output includes the probability distributions for the For such an object, the effect of hypervelocity impacts orbital parameters. The fraction of the algorithm which will be much greater, similarly to what was seen in is in the Monte Carlo simulation is delineated with a box. [14]. Indeed, Fig. 3 shows distributions with much larger Figure 1. Flowchart of the Algorithm for Coupled Orbit-Attitude Propagation from Hypervelocity Impacts
0.016
) 0.014 -1 12 0.012 a) )
-1 10 0.01
0.008 8 0.006
6 0.004
Probability Distribution Function (m 0.002 4
0 -500 -400 -300 -200 -100 0 100 200 300 400 500 2 Semi-major Axis Change (m) Probability Distribution Function (m
0 -12 -10 -8 -6 -4 -2 0 Figure 3. SMA PDF Evolution for the HAMR Object 25
) b) -1 20
15 widths (hundreds of meters compared to < 1 m for En- visat), and a decrease in the distribution center of about 10 8 m per year. Similar large variations in the distributions of other orbital elements can also be seen. Due to the 5 small change in SMA and the large distribution widths, a
Probability Distribution Function (m non-negligible possibility of the orbit being raised exists. 0 The widths of the distributions are related to the spread -12 -10 -8 -6 -4 -2 0 Semi Major Axis Change (m) in directionality of the impacts: the wide spread is due to the fact that most impacts in this high-altitude orbit occur with meteoroids and these come from every direc- Figure 2. SMA PDF Evolution for Envisat (a) with Mo- tion. Because impacts are not mainly coming from the mentum Enhancement and (b) without ram direction, as is the case for Envisat, some of the one- year propagations end up with more collisions causing an increase in the SMA. 35 8 ) ) 7 -1
-1 30
6 25 5 20 4
15 3
10 2
Probability Distribution Function (m 1 5 Probability Distribution Function (m 0 0 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 Semi-major Axis Change (m) Semi-major Axis Change (m)
Figure 5. SMA PDF Evolution for Ajisai Figure 4. SMA PDF Evolution for LAGEOS
to be caused by aerodynamic drag. These values were 4.3. LAGEOS compared to the outputs of the stochastic model in Fig. 5, which shows the evolution of the PDF for the variation The LAGEOS satellites are spherical satellites used for in the SMA throughout a ten-year period. A decrease of providing a laser ranging benchmark for geodynami- approximately 29 cm per year is observed, which is less cal studies. LAGEOS-1 is an aluminum sphere with a than 1% of the observed decay due to aerodynamic drag brass core with a radius of 30 cm orbiting at a SMA of and is therefore negligible in comparison. 12274 km, an inclination of 109.9◦ and eccentricity of 0.0039. It was launched in 1976 and has been the cause From the results of the four case studies performed above, of some debate after observations determined it experi- a number of consistent and expected observations can be ences an orbital decay of approximately 1 mm per day made. First, there is a widening of the PDFs with time. [13]. Because of its high altitude, aerodynamic drag is Second, all PDFs indicate that the orbits are much more negligible and some possible decay mechanisms include likely to be lowered, although in the case of the HAMR the Yarkovsky thermal drag and charged particle drag, as object there is a non-negligible probability of the orbit be- well as the effect of the Earth’s inertial induction [13]. ing raised. This is due to the combination of a relatively We attempt here to establish whether hypervelocity im- small decrease in the expected SMA and wide probabil- pacts with micrometeoroids and small debris could play ity distributions. The widths of the distributions seem to a role in the observed decrease of the spacecraft altitude. be related to the directionality of the impacts, and there- fore the proportion of impacts coming from meteoroids: Fig. 4 shows the output of the stochastic model, with a larger percentage of meteoroid impacts leads to wider the yearly PDFs for the change in SMA. Approximately distributions. Third, all PDFs have a characteristic shape 5.1 × 103 collisions per year are expected, 90% of them sloping more shallowly and broadly towards negative val- from micrometeoroids. A yearly decrease in the distri- ues, again giving a clear bias towards orbit lowering. Fi- bution centers of approximately 8 mm is seen. This rep- nally, when comparing to other sources of orbit pertur- resents only about 2% of the orbital decay observations bations, hypervelocity impacts contribute little to the ob- for the satellite, showing that this effect is considerably served or expected decay values. smaller than the other effects suggested above. Once again, a small probability of orbit raising exists due to the high proportion of impacts from meteoroids, thus leading 5. ANALYTICAL TEST CASES to wide distributions.
Additional confidence in the validity of the orbit propa- 4.4. Ajisai gation model is given from two analytical test cases: one with a hypothetical spacecraft undergoing simple colli- sions, and the other with Envisat and its collisions accord- The last case that was looked at is the Japenese satellite ing to MASTER-2009. The validation process involves Ajisai launched in 1986. It is a 685 kg hollow sphere with comparing two solutions for each test case: on the one a diameter of 2.15 m. It is in a nearly circular orbit with hand, a simulation of bombardment of impactors lead- a SMA of 7865 km and an inclination of 50◦. From the ing to orbital decay; and on the other hand, a Hohmann MASTER-2009 fluxes, 7.8 × 104 collisions are expected transfer from the same initial orbit to the resulting de- to occur every year, with 75% from meteoroids. It was re- cayed orbit obtained by two hypothetical large impacts. ported in [10] that the satellite underwent a decay in SMA The validation involves comparing the total mass of the of 42.6 m and 94.6 m during three-year periods of low so- impactors in the simulation to the sum of the impactors lar activity and high solar activity, respectively, assumed in the equivalent analytical Hohmann transfer. 5.1. Hohmann Transfer from Hypervelocity Impacts This simulation showed a decay in the semi-major axis of 0.1582 m after one year. The sum of the impactor masses experienced during the one-year simulation is: From its definition, a Hohmann transfer from a circular orbit at (rA, vA) to a lower orbit at (rB, vB), with: X −12 mj = 365.25 × 60 × 24 × 10 µ j (30) vk = , k ∈ {A, B} (22) rk = 5.2596 × 10−7 kg where µ is Earth’s gravitational parameter, is a two- Substituting in Eqs. (23) through (29) the corresponding impulse maneuver using an elliptical transfer orbit with −1 values for M = 100 kg, vrel = 15 km s , rA = 7371 km eccentricity: −3 and rB = 7371 − (0.1582 × 10 ) km, we obtain:
rA − rB −7 e = (23) m1 = m2 = 2.6298 × 10 kg (31) rA + rB so that: The velocity changes to go from the outer circular orbit to the Hohmann transfer orbit, and from the Hohmann m + m = 5.2597 × 10−7 kg (32) transfer orbit to the inner circular orbit, are, respectively: 1 2 Comparing the results of Eqs. (30) and (32) gives cre- ∆vA = vA − vA H (24) dence to the validity of our stochastic model. ∆vB = vB − vBH where v is the velocity of the spacecraft at the point AH 5.3. Envisat that joins the outer orbit and the transfer orbit and vBH is the velocity of the spacecraft at the intersection of the inner orbit and the transfer orbit, calculated as: The second, and more general, test case involves the results of the case study presented in Section 4.1. As p (1 + e)rkµ shown in Fig. 2b, the decrease in SMA witnessed by vkH = , k ∈ {A, B} (25) rk Envisat after one year due to collisions only, without in- cluding the effect of ejecta momentum, is approximately By considering the two impulses leading to ∆vA and 0.736 m. From the MASTER-2009 impact fluxes, the ∆vB as caused by collisions with large masses, m1 and expected value for the impact velocity is calculated as −1 m2, impacting at some relative velocity, vrel, one can ap- 16.1449 km s and the median sum of the impactor ply conservation of momentum over the two impulses as: masses from each of the 10,000 runs is approximately 3.00 × 10−4 kg.
MvA + m1(vA − vrel) = (M + m1)vAH (26) Following the same methodology as in the previous sec- MvBH + m2(vBH − vrel) = (M + m2)vB (27) tion, one can calculate the masses of the two impactors needed to perform the corresponding Hohmann transfer. −1 Solving for m and m gives: In this case, M = 7828 kg, vrel = 16.1449 km s , 1 2 −3 rA = 7136 km, and rB = 7136 − (0.736 × 10 ) km. M∆vA From Eqs. (23)-(29), the two impact masses are therefore m1 = (28) found to be: −∆vA − vrel −5 M∆v m1 = m2 = 9.3436 × 10 kg (33) m = B (29) 2 −∆v − v B rel so that: The validation exercise is now applied to the two afore- −4 mentioned test cases, by comparing the sum of m1 and m1 + m2 = 1.8687 × 10 kg (34) m2 to the sum of the debris and meteoroid flux through- out the orbital propagation time-frame, between the cor- Comparing the result in Eq. (34) to the value of 3 × responding two orbits. 10−4 kg, as predicted by the Monte Carlo simulations, shows a significant difference. This is to be expected, as the directionality of the impactors, included in the 5.2. Hypothetical Spacecraft stochastic orbital propagation but not in the analytical Hohmann transfer, where the two collisions are assumed to be head-on, will produce a smaller effect in the reduc- The first test case involves a M = 100 kg satellite in a tion of the SMA. Indeed, from the MASTER-2009 direc- circular orbit at an initial altitude of 1000 km. A propaga- tion flux, approximately 60% of the collisions occur with tion was computed constraining the general formulation ±45◦ elevation and azimuth. in Section 3 to allow only the bombardment from head- on collisions every 60 s, with impactors having a mass However, it is possible to take this directionality spread of 10−12 kg and with an impact velocity of 15 km s−1. into account analytically by looking at the spacecraft’s derived PDF for direction. A reduction factor can be ob- Sciences and Engineering Research Council of Canada tained by integrating the direction-dependent flux over (NSERC). Computations were made on the supercom- the entire sphere of directions: puter Guillimin from McGill University, managed by Calcul Quebec´ and Compute Canada. Its operation is π π R R 2 funded by the Canada Foundation for Innovation (CFI), − −π − π F (θ, φ) cos θ cos φ dθ dφ π2 ´ R π R 2 ministere` de l’Economie, de la Science et de l’Innovation D = − π F (θ, φ) cos θ sin φ dθ dφ (35) −π − 2 du Quebec´ (MESI) and the Fonds de recherche du π R π R 2 − π F (θ, φ) sin θ dθ dφ Quebec´ - Nature et technologies (FRQ-NT). −π − 2 where the x-direction is the spacecraft ram direction, the z-direction points towards Earth, and the y-direction REFERENCES completes the right hand rule. Applying this to Envisat, we obtain: 1. Bastida Virgili B., Lemmens S., Krag H., (2014). In- vestigation on Envisat attitude motion, e.Deorbit Work- "−0.6001# shop, The Netherlands. D = −0.0073 (36) 2. Bate R. R., Mueller D. D., White J. E., (1971). Fun- 0.0986 damentals of astrodynamics, Courier Corporation. As one can see, the integrated flux is still mostly in the 3. Drolshagen G., (2008). Impact effects from small size incoming ram direction due to the symmetry of the di- meteoroids and space debris, Adv. Space Res., 41, 1123- rectional flux. The reduction factor obtained is there- 1131. fore kDk = 0.6082, which, when applied to the sum of 4. European Space Agency, (2017). Clean Space, the impactor masses from the Monte Carlo simulations