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, !, 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 I!_ (t) + !(t) I!(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- 0 1 sen here to be quaternions. µ X d_r(t) = − r(t) + a (t; r(t)) dt (5) @ r(t)3 j A Finally, the third differential equation is the kinematic j equation for orientation: Considering the momentum of the satellite, defined as 1 _q(t) = ΩΩ(!)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- 0 1 µ X nents of !: dp(t) = M − r(t) + a (t; r(t)) dt (6) @ r(t)3 j A 2 0 −!x −!y −!z3 j ! 0 ! −! Ω = 6 x z y7 (4) 4!y −!z 0 !x 5 Including the term for the momentum from hypervelocity !z !y −!x 0 impacts as a function of the compound Poisson process, Yt, Eq.