Long-term dynamics of high area-to-mass ratio objects in high-earth orbit

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1 Available online at ScienceDirect Advances in Sace Research 52 (2013) Long-term dynamics of high area-to-mass ratio objects in high-earth orbit Aaron J. Rosengren, Daniel J. Scheeres Deartment of Aerosace Engineering Sciences, University of Colorado at Boulder, Boulder, CO 80309, USA Received 19 February 2013; received in revised form 23 July 2013; acceted 24 July 2013 Available online 1 August 2013 Abstract The dynamics of high area-to-mass ratio (HAMR) objects has been studied extensively since the discovery of this debris oulation in near GEO orbits. A sound understanding of their nature, orbital evolution, and ossible origin is critical for sace situational awareness. In this aer, a new averaged formulation of HAMR object orbit evolution that accounts for solar radiation ressure, Earth oblateness, and lunisolar erturbations is exlored. The first-order averaged model, exlicitly given in terms of the Milankovitch orbital elements, is several hundred times faster to numerically integrate than the non-averaged counterart, and rovides a very accurate descrition of the long-term behavior. This model is derived and resented along with comarisons with exlicit long-term numerical integrations of HAMR objects in GEO. The dynamical configuration of the Earth Moon Sun system was found to have a significant resonance effect with HAMR objects leading to comlex evolutionary behavior. The roerties of this resonant oulation may serve as imortant constraints for models of HAMR debris origin and evolution. A systematic structure associated with their distribution in inclination and ascending node hase sace is identified. Given that these objects are difficult to target and correlate, this has many imlications for the sace surveillance community and will allow observers to imlement better search strategies for this class of debris. Ó 2013 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: High area-to-mass ratio objects; Sace debris; Long-term dynamical evolution; Averaging; Orbit erturbations; Milankovitch orbital elements 1. Introduction The motion of high area-to-mass ratio (HAMR) objects in high-earth orbits has been studied extensively since the discovery of this debris by Schildknecht and colleagues (ca. 2004). The HAMR debris oulation is thought to have origins in the geostationary (GEO) region, and many of these objects are uncharacterized with aarent area-tomass ratios of u to 30 meters squared er kilogram (Schildknecht et al., 2004; Liou and Weaver, 2005; Schildknecht, 2007). The orbits of HAMR objects are highly erturbed Corresonding author. Address: 431 UCB ECEE 166, Boulder, CO , USA. Tel.: addresses: aaron.rosengren@colorado.edu (A.J. Rosengren), scheeres@colorado.edu (D.J. Scheeres). from the combined effect of solar radiation ressure (SRP), anomalies of the Earth gravitational field, and third-body gravitational interactions induced by the Sun and the Moon (Chao, 2006; Valk et al., 2008; Anselmo and Pardini, 2010; Scheeres et al., 2011; Rosengren and Scheeres, 2011). The evolution of individual orbits of HAMR debris over a considerable time interval, taking into account both short-eriod and long-eriod terms, can be calculated by numerical integration of the recise set of differential equations. This rocess leads to the construction of high-recision ehemerides and the investigation of the emirical evolution of such objects, but not necessarily to general insight into the dynamics of the roblem. Numerical comutations of this tye have revealed that the amlitudes of the short-term fluctuations in the osculating orbital elements are, in general, quite small when comared with the long-term variations. For this /$36.00 Ó 2013 COSPAR. Published by Elsevier Ltd. All rights reserved. htt://dx.doi.org/ /j.asr

2 1546 A.J. Rosengren, D.J. Scheeres / Advances in Sace Research 52 (2013) reason, and the necessity of considering the orbital behavior of a large oulation of HAMR debris over many thousands of revolutions, it seems reasonable to investigate the equations that govern the long-term evolution of orbits. Such equations can be derived by the theory of secular erturbation. The theory of secular erturbation, also referred to as the theory of averaging, was introduced by Clairaut, Lagrange, and Lalace over two centuries ago. In the classical Lalace Lagrange theory, as in that of Gauss, Delaunay, Halhen, Poincaré-Lindstedt, von Zeiel, Brouwer, and the Lie-transformation of Hori and Derit, the goal is to eliminate the short-eriod erturbation terms and derive the equations that cature the secular evolution of the system (Sanders et al., 2007). When the equations of motion do not have a canonical form, short-eriod terms can be eliminated systematically by means of the of the Krylov Bogoliubov Mitroolski method of averaging. This method, develoed by Kyrloff and Bogoliuboff (1947) in the analysis of nonlinear oscillations, and generalized by Bogoliubov and Mitroolski (1961), was first alied to roblems in celestial mechanics and satellite theory in the early 1960s (Musen, 1960; Musen, 1961; Struble, 1961; Allan, 1962; Lidov, 1962). Alication of this method to astronomical roblems rovides a direct means of obtaining the first-order secular equations, and leads to equation which do not deend on exansions in either eccentricity or inclination. The averaged equations can be numerically integrated, with significantly reduced comutational requirements, and often reveal the essential characteristics of the exact solution in a more satisfactory way than a numerical solution of the non-averaged equations. Liou and Weaver (2005) used PROP3D, a fast orbit roagator based on the averaging rincile that was develoed for NASA s debris evolutionary models, to investigate the HAMR debris roblem. PROP3D accounts for the erturbations from Earth gravity u to the fourth zonal harmonic, low-order lunisolar gravitational interactions, and SRP with consideration of the Earth shadow effect. Through comarisons with a high-fidelity orbit integrator, they showed that HAMR objects in GEO are dominated by major erturbations, not those of higher order. Chao (2006) erformed long-term studies of the orbital evolution of GEO objects with high area-to-mass ratios through analytically averaged equations. Chao investigated the secular effects of SRP on the eccentricity and argument of erigee by averaging over the object s orbital eriod. The long-term motion of the orbit inclination and longitude of the ascending node were studied though doublyaveraged equations for the lunisolar attraction, ignoring higher-order terms in eccentricity, and singly-averaged equations for the SRP erturbation. The secular equations of motion were written in terms of the classical orbital elements, which leads to the resence of small numerical divisors the eccentricity and the sine of the inclination. We resent a non-singular model of first-order averaging, exlicitly given in terms of the Milankovitch vectorial elements, which accounts for solar radiation ressure, the oblateness of the Earth s gravitational figure, and lunisolar erturbations. The secular equations hold rigorously for all Kelerian orbits with nonzero angular momentum; they are free of the singularities associated with zero eccentricity and vanishing line of nodes. This aer is organized into the following discussions. We first resent the environment and force models for each erturbation, and discuss any underlying assumtions and aroximations. We then review the Milankovitch elements and give their erturbation equations in Lagrangian form, assuming the erturbing accelerations are exressible as gradients of a disturbing function. After deriving the first-order averaged equations for each erturbing force, we demonstrate their validity by comaring numerical integrations of them to integrations of the exact Newtonian equations. The extent to which the qualitative roerties of the orbit ersist with increasing area-to-mass is investigated. We then show how the geometry of the Earth Moon Sun system, and in articular the regression of the lunar node, causes a resonance effect with a articular class of HAMR objects, leading to comlex evolutionary behavior. The roerties of this resonant oulation may serve as imortant constraints for models of HAMR debris origin and evolution. We then launch a range of HAMR objects, released in geostationary orbit (equatorial, circular-synchronous) with area-to-mass ratios from 0 u to 40 m 2 /kg, uniformly distributed in lunar node, and redict the satial distribution of the oulation. We identify a systematic structure associated with their distribution in inclination and ascending node hase sace, which is similar to that of the uncontrolled GEO satellites. Finally, we discuss the imlications of these results for observing camaigns and our future research directions. 2. Environment and force models 2.1. The Earth Moon Sun system Any account of motion in the Earth Moon Sun systems has to start with a descrition of the dynamical configuration of this three-body roblem. Perozzi et al. (1991), using eclise records, the JPL ehemeris, and results from numerical integration of the three-body roblem, showed that the mean geometry of the Earth Moon Sun system reeats itself closely after a eriod of time equal in length to the classical eclise rediction cycle known as the Saros. That is, this dynamical system is moving in a nearly eriodic orbit. Saros means reetition, and indicates a eriod of 223 synodic months ( 6585:3213 days), after which the Sun has returned to the same lace it occuied with resect to the nodes of the Moon s orbit when the cycle began. While the motion of the Earth around the Sun can, over time sans of interest, be considered Kelerian, the Moon is incessantly subject to solar erturbations

3 A.J. Rosengren, D.J. Scheeres / Advances in Sace Research 52 (2013) resulting in eriodic and secular variations of its orbital elements (Roy, 2005). The semi-major axis, eccentricity, and inclination are subject to eriodic variations about their mean values of 384,400 km, , and , resectively. The line of ases advances, making one revolution in about 8.85 years. The node of the Moon s orbital lane (its intersection with the eclitic) regresses in the eclitic lane with a sidereal eriod of days (about years); the node moving westward on the eclitic at a rate of roughly 1 in 18.9 days. The regression of the lunar nodes is intrinsically related with the Saros cycle. This comlexity of lunar motion must be taken into account for long-term studies of HAMR object dynamics. To describe the motion of the Earth about the Sun, we define a heliocentric orbit frame, ð^e e ; ^E e? ; ^H e Þ, in which ^E e is the unit vector ointing to the orbit erihelion, ^E e? is the unit vector in the heliocentric lane of motion and normal to ^E e, and the cross roduct of these two vectors defines the orbit normal, secified as ^He, about which the Earth revolves. 1 With this formulation, the varying osition vector between the Earth and the Sun is secified as d e ¼ d e^de and slit into a magnitude d e and a direction ^d e, both of which are functions of the Earth true anomaly f e d e ¼ a eð1 e 2 e Þ ; 1 þ e e cos f e ð1þ ^d e ¼ cos f e ^Ee þ sin f e ^Ee? ; ð2þ where a e and e e are the semi-major axis and eccentricity of the Earth s heliocentric orbit, resectively. The Moon s actual motion is very comlex; high-recision lunar ehemerides are available through the Jet Proulsion Laboratory, which account for the relativistic n- body equations of motion for the oint-mass Sun, Moon, lanets, and major asteroids, erturbations on the orbit of the Earth Moon barycenter from the interaction of the oint-mass Sun with the figure, solid-body tides of both the Earth and Moon, and observations of lunar laser ranging (Folkner et al., 2009). Our urose in this aer is to adot the simlest ossible exressions useful for studying the long-term evolution of HAMR debris orbits. These exressions must reveal the qualitative regularities of motion, and they must rovide, with a certain degree of accuracy, a way of obtaining quantitative redictions of long-term changes. To that end, we assume that the Moon is on an osculating ellitical orbit in which the lunar node recesses clockwise in the eclitic lane with a eriod of about years. Note that we neglect the rotation of the lunar erigee as well as the eriodic variations in the Moon s semi-major axis, eccentricity, and inclination. We define a geocentric orbit frame, ð^e m ; ^E m? ; ^H m Þ, where ^E m is the unit vector ointing to the orbit erigee, ^E m? ¼ ^H m ^E m, and ^H m is the Moon s angular momentum 1 These are the orientation-defining integrals of the two-body roblem, and can be secified using the classical orbital elements relative to an inertial frame (Scheeres, 2012a). unit vector, about which the Moon revolves. These vectors are resolved using the Moon s mean eclitic orbital elements in which X m ðtþ ¼X m0 þ _ X m ðt t 0 Þ, where _X m ¼ 2=P saros and P saros is the sidereal eriod of nodal regression in seconds. The osition vector from the Earth to the Moon is then be secified as d m ¼ d m^dm, where d m and ^d m are given by Eqs. 1 and 2, resectively, using the Moon s orbit arameters Solar radiation ressure Solar radiation ressure is the largest non-gravitational erturbative force to affect the motion of HAMR objects in high-earth orbits, causing extreme variations in their orbital arameters over short time eriods. Tyical analysis of long-term orbit dynamics models the SRP acceleration using the cannonball model, which treats the object as a shere with constant otical roerties (Valk et al., 2008; Anselmo and Pardini, 2010; Scheeres et al., 2011). The total momentum transfer from the incident solar hotons is modeled as insolation lus reflection, and the force generated is indeendent of the body s attitude. Any force comonent normal to the object-sun line that results from an asherical shae or nonuniformly reflecting surface is thereby neglected. Then the net acceleration will act in the direction directly away from this line and have the general form (Scheeres, 2012a) ðd s rþ a sr ¼ ð1þqþða=mþp U ð3þ j d s rj 3 ¼ b ðd s rþ j d s rj ; ð4þ 3 where q is the reflectance, A=m is the aroriate cross-sectional area-to-mass ratio in m 2 /kg, P U is the solar radiation constant and is aroximately equal to kg km 3 =s 2 =m 2, and b ¼ð1þqÞðA=mÞP U. The vector from the Earth to the Sun is given by d s ¼ d e, and the osition vector of the orbiter relative to the Earth is r. This solar radiation ressure model can be rewritten as a disturbing function 1 R sr ¼ b j d s r j ; ð5þ where a sr sr =@r. If the object is close to the Earth, or r d s, the disturbing function can be further simlified by exanding 1= j d s r j and keeing the first term that contains the osition vector r: R sr ¼ b d d 3 s r; ð6þ s with the gradient giving a solar radiation ressure acceleration indeendent of the object s osition relative to the Earth: a sr ¼ b d 2 s ^d s : ð7þ

4 1548 A.J. Rosengren, D.J. Scheeres / Advances in Sace Research 52 (2013) The effects of the Earth s shadow are not taken into account; and as we are only considering distant Earth orbits, we also neglect Earth-albedo radiation ressure. For a given semi-major axis, a, reflectivity, and A=m value, we define the SRP erturbation angle (first defined by Mignard and Hénon (1984)) as tan K ¼ 3b rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 ll s a e ð1 e 2 e Þ ; ð8þ where l and l s are the gravitational arameters of the Earth and the Sun, resectively. We note that as the SRP erturbation becomes strong, K! =2; and as it becomes weak K! 0. The angle K can be used to rigorously characterize the strength of the SRP erturbation acting on a body as a function of its orbit, its non-gravitational arameter, and the orbit of the Earth about the Sun. As it combines these quantities into a single arameter, which comletely defines the long-term SRP-only solution (qq.v., Richter and Keller, 1995; Scheeres et al., 2011; Scheeres, 2012b), we find it efficacious to use as the fundamental defining characteristic of HAMR objects. Although the cannonball model catures the general nature of SRP, it does not rovide a recise rediction of how an individual object will evolve. However, this simle model is commonly used in the roagation of HAMR debris orbits since there is no method to incororate a hysically realistic SRP model with a lack of a riori information (i.e., object geometry, attitude behavior, surface roerties, thermal characteristics, etc.). Even though the cannonball model may not realistically reresent the SRP force acting on these objects, the fact that this model gives rise to many comlex dynamical behaviors necessitates a deeer understanding of this basic model before exloring these more comlex models. Thus, in the current aer, we will focus on the cannonball model as that will allow a direct comarison with earlier analyses of HAMR debris orbit dynamics (cf. Liou and Weaver, 2005; Chao, 2006; Valk et al., 2008; Anselmo and Pardini, 2010) Earth mass distribution We consider the effects of the C 20 and C 22 terms of the harmonic exansion of Earth s gravitational otential, which account for the olar and equatorial flattening of the Earth s figure. Inclusion of these redominant harmonics is sufficient to cature the main effects of nonshericity in the Earth s mass distribution at high-altitude orbits. The standard way to reresent the disturbing function of the second degree and order gravity field erturbation is using a body-fixed frame with latitude angle d measured from the equatorial lane and longitude angle k measured in the equator from the axis of minimum moment of inertia (Scheeres, 2012a) R 2 ¼ lc sin 2 3lC 22 d þ cos 2 d cos 2k; 2r 3 r 3 ð9þ where C 20 ¼ J 2 R 2 e is the dimensional oblateness gravity field coefficient, R e is the mean equatorial radius of the Earth, and C 22 is the dimensional elliticity gravity field coefficient. We can state the disturbing function in a general vector exression R 2 ¼ lc h i ð^r ^Þ 2 þ 3lC h i 22 ð^r ^sþ 2 ð^r ^qþ 2 ; 2r 3 r 3 ð10þ where we assume that the unit vectors ^; ^q, and ^s are aligned with the Earth s maximum, intermediate, and minimum axes of inertia. The erturbing acceleration is then a 2 ¼ 3lC nh i o ð^r ^Þ 2 ^r þ 2ð^r ^Þ^ 2r 4 3lC n h i o 22 5 ð^r ^sþ 2 ð^r ^qþ 2 ^r 2½ð^r ^sþ^s ð^r ^qþ^q Š : r Lunisolar gravitational attraction ð11þ Also necessary to incororate in this analysis is the erturbation of the Sun and Moon s gravity on the motion of the HAMR object. These can be modeled as third-body erturbations, and their functional form can be simlified by erforming an aroriate exansion. Taking Earth as the center of our dynamical system, the erturbation acceleration from a body with gravitational arameter l is (Scheeres, 2012a) " # r d a ¼ l j r d j þ d ; ð12þ 3 j d j 3 where d is the osition vector of the erturbing body relative to Earth. For use in erturbation analysis it is convenient to recast this as a disturbing function " # 1 R ¼ l j r d j d r ; ð13þ j d j 3 where a =@r. As the orbiter s distance from the Earth is small comared to the distance between the Earth and the third body, or r=d 1, the disturbing function can be reresented as an infinite series using the Legendre exansion, resulting in Scheeres (2012a) R ¼ l " X 1 i # r r d P i;0 d r : ð14þ d d rd d 2 i¼0 Keeing only the first non-constant term, with the Legendre olynomial P 2;0 ðxþ ¼1=2ð3x 2 1Þ, yields R ¼ l h 3ðr ^d i 2d 3 Þ 2 r 2 : ð15þ Thus, to lowest order, the gravitational attraction of the Sun and the Moon can be reresented as a quadratic form, which is the fundamental aroximation made in the Hill roblem. Under this aroximation, the erturbing acceleration simlifies to

5 A.J. Rosengren, D.J. Scheeres / Advances in Sace Research 52 (2013) a ¼ l h 3ðr ^d i d 3 Þ^d r : ð16þ or momentum sace) (q.v., Battin, 1999). These elements have not been widely used in celestial mechanics in the ast thirty years, and have recently been reformulated and exounded on in Rosengren and Scheeres (submited for 2.5. Non-averaged Equations of Motion ublication). The Milankovitch elements are articularly useful in Combining the above force models, we can define the finding the first-order long-eriod and secular variations equations of motion for a HAMR object in orbit about by means of the Krylov Bogoliubov Mitroolski method the oblate Earth disturbed by solar radiation ressure of averaging (qq.v., Musen, 1960; Musen, 1961; Allan, and lunisolar gravity. In an inertially fixed frame centered 1962). When the disturbing function is limited to its secular at the Earth, they can be stated in relative form art, in accordance with the method of averaging, the semimajor axis does not undergo any secular changes. Since the ; ð17þ semi-major axis is secularly invariable, the angular momentum vector can be scaled by la. For this vector, denoted UðrÞ ¼ l ffiffiffiffiffi r þr srðrþþr 2 ðrþþr s ðrþþr m ðrþ; ð18þ where R s and R m are third-body disturbing functions for the Sun and the Moon, resectively. Performing the artial derivatives, we can state the roblem in terms of the erturbation accelerations as r ¼ l r 3 r þ a sr þ a 2 þ a s þ a m : ð19þ 3. Averaged dynamics We now introduce the concet of averaging as this allows us to evaluate the secular effects of the erturbations on our system. For averaging to be valid, we assume that the erturbing forces are sufficiently small so that, over one orbital eriod, the deviations of the true trajectory from the Kelerian trajectory are relatively small. In this case, oscillations in the orbital elements will average out over reasonably small eriods. We refer the reader to Bogoliubov and Mitroolski (1961) for a theoretical discussion on the mathematical bases of averaging Milankovitch orbital elements In his monumental work on the astronomical theory of aleoclimates, Milutin Milankovitch ( ) reformulated the classical method of erturbation of elements using the two vectorial integrals of the unerturbed two-body roblem the angular momentum (areal) vector and the Lalace vector (Milankovitch, 1941). The vectorial integrals describe the satial orientation, geometrical shae, and size of the osculating Kelerian orbit, and, together with the sixth scalar integral that reresents the motion in time, constitutes a comlete set of orbital elements. Geometrically, the angular momentum vector, H, oints erendicular to the instantaneous orbit lane and the Lalace vector, b ¼ le, where e is the eccentricity vector, oints towards the instantaneous eriasis of the orbit. At a more fundamental level, the vector H reresents twice the areal velocity of the lanet, and hence exresses Keler s Second Law in vectorial form; the vector b can be interreted as the equation for the hodograh of Kelerian motion (i.e., the shae of Kelerian trajectories in velocity here as h, together with the eccentricity vector, e, the secular Milankovitch equations take a comact and symmetrical form. We can write these vectors in terms of the osition, r, and velocity, v, in dyadic notation 2 as h ¼ 1 ffiffiffiffiffi~r v; ð20þ la e ¼ 1 r ~v ~r v l jrj : ð21þ The first-order averaged equations in Lagrangian form can be stated as (Allan and Cook, 1964; Tremaine et al., 2009; Rosengren and Scheeres, submited for ublication) _h ¼ R h þ ; ð22þ _e ¼ h þ R ; ð23þ where the over bar indicates the averaged value and R ¼ Rð h; eþ= ffiffiffiffiffi la. We note that the artials of R are taken indeendently. The averaged disturbing function is defined as Z 2 Rð h; eþ ¼ 1 Rða; MÞ dm; ð24þ 2 0 where a is an arbitrary set of orbital elements excluding the mean anomaly, and Rð h; eþ is indeendent of the fast variable M. Eqs. 22 and 23 admit two integrals h e and h h þ e e; hysically meaningful solutions are restricted to the four-dimensional manifold on which h e ¼ 0and h h þ e e ¼ 1(Tremaine et al., 2009). Perturbation equations in terms of the Milankovitch elements in Gaussian form, that is, with the disturbing acceleration given exlicitly, were used after their incetion, most notably by Musen and Allan. Musen (1960) used these elements to study the effects of solar radiation ressure on the long-term orbit evolution of the Vanguard I satellite. Musen (1961) also develoed secular equations for third-body erturbations, which are valid for all eccen- 2 The notation ~a denotes the cross-roduct dyadic, defined such that ~a b ¼ a ~ b ¼ a b. See Rosengren and Scheeres, submited for ublication for a synosis of the basic notation and roerties of dyadics.

6 1550 A.J. Rosengren, D.J. Scheeres / Advances in Sace Research 52 (2013) tricity and all inclination. The third-body disturbing function was truncated to the lowest-order term in the Legendre exansion (i.e., Hill s aroximation), and averaged over the eriods of both the orbiter and the disturbing body. Allan (1962) extended Musen s work by incororating the arallactic term (third harmonic) into the disturbing function exansion. Because these equations are not widely known, and in order to draw attention to their alicability and elegance, we give here an outline of their derivations using the Lagrangian form of the secular Milankovitch equations. We also derive the secular equations resulting from Earth s oblateness (J 2 ), and combine them with the other erturbations to give a comlete first-order averaged model for the evolution of HAMR debris in GEO Averaged SRP dynamics Using the Lagrange lanetary equations, the disturbing function can be averaged rior to alication in this system. Substituting Eq. 6 into Eq. 24, we find R sr ¼ 1 Z 2 R sr dm ð25þ 2 0 ¼ b ^d d 2 s r: ð26þ s Thus, we only need to comute the average of the osition vector, a classically known result 3 r ¼ 3 2 ae: ð27þ This leads to R sr ¼ 3 rffiffiffi a b ^d 2 l d 2 s e: ð28þ s Stated in this form, the scaled averaged disturbing function can be substituted into the secular Milankovitch equations, Eqs. 22 and 23, giving (cf. Richter and Keller, 1995) _h sr ¼ 3 rffiffiffi a b ~^d 2 l d 2 s e; ð29þ s _e sr ¼ 3 rffiffiffi a b ~^d 2 l d 2 s h: ð30þ s Richter and Keller (1995) and Scheeres (2012b) showed that the averaged SRP equations, Eqs. 29 and 30, can be solved in closed form, yielding an analytical solution for the secular variation in the scaled angular momentum and eccentricity vectors. The analytical solution is exressed in a frame that rotates with the lanet-sun line, and solutions are eriodic in f e = cos K, reeating every Earth true anomaly 2 cos K. Thus, over one heliocentric orbit the solution will advance 1= cos K times. As the erturbation grows large, and K aroaches =2, the solution 3 The bar ( ) oerator is omitted from the Milankovitch elements in what follows because there is no ambiguity; i.e., all variables are averaged variables. will reeat many times over one year. Conversely, as the erturbation grows small the solution will reeat only once every year. In a revious study (q.v., Scheeres et al., 2011), we alied the SRP-only averaged solution to the dynamics of HAMR objects in GEO and GPS orbit regimes having a variety of K values. One of the surrising asects of the theory is that the extremely simle, eriodic behavior that occurs relative to the Earth-Sun rotating frame becomes quite comlex and aeriodic in the Earth equatorial frame. The resence of the comlex oscillations in the inclination and eccentricity vector, with short-eriod and long-eriod terms, is just an artifact of transforming into the inertial frame Averaged J 2 dynamics Tesseral harmonics in the Earth s gravitational otential can also introduce long-term effects, esecially if the mean motion of the object is commensurable with the angular velocity of Earth s rotation. The most interesting case is the influence of the elliticity of Earth s equator on the motion of a geostationary object. This effect was studied by Lemaıtre et al. (2009) for HAMR objects in GEO sace who found that a resonance occurs from the C 22 dynamics, giving rise to chaotic behavior localized to a narrow range of semi-major axis. The averaged effect of this erturbation must be treated using a resonance theory. With mean motion averaging, our formulation will be unable to account for this subtle, and otentially imortant, asect of motion. Therefore, we only consider the averaged C 20 dynamics in our system. The averaged C 22 dynamics will be studied in future work. Performing the average over the second zonal harmonic disturbing function gives R 20 ¼ lc 20 2 " # 1 ^r^r 3^ ^ r3 r 3 ; ð31þ where the roduct of two unit vectors, ab, is called a dyad and, in column and row vector notation is equivalent to the outer roduct (i.e., ½aŠ½bŠ T ). From Musen (1961), we have 1 r ¼ 1 3 a 3 h ; 3 ^r^r r 3 ¼ 1 2a 3 h 3 h U ^h^h i ð32þ ; ð33þ where U is the identity dyadic and has the general roerty U a ¼ a U ¼ a. Consequently, R 20 ¼ nc h ð^ ^hþ i 2 ; ð34þ 4a 2 h 3 in which n ¼ ffiffiffiffiffiffiffiffiffi l=a 3 is the orbiter s mean motion. Substituting R 20 into Eqs. 22 and 23, the secular equations for the oblateness gravity field erturbation can be written as

7 A.J. Rosengren, D.J. Scheeres / Advances in Sace Research 52 (2013) _h 20 ¼ 3nC 20 2a 2 h ð^ hþ~^ h; 5 _e 20 ¼ 3nC ð^ 4a 2 h 5 hþ2 ~h þ 2ð^ hþ ~^ e: 2 h ð35þ ð36þ h _¼ ¼ 1 Z 2 _h dm ¼ 3l! 4 ^d ^d 5e ~e h 2n d 3 ð43þ! 3 ^d ^d ~ h5; ð44þ d Averaged third-body dynamics For the third-body erturbations, there are two timescales or frequencies over which the dynamical motion occurs. These timescales are defined by the orbital rate, n, and the motion of the erturbing body with its angular rate, N.IfN =n 1, it is accetable to hold N constant while averaging over n; in which case, the remaining system has a time-varying term associated with N Singly-averaged equations The Hill-aroximated third-body disturbing function, Eq. 15, can be averaged over the orbiter s unerturbed two-body motion about the Earth as R ¼ l 2d 3 h 3^d rr ^d r 2 i ; ð37þ in which we note (Scheeres, 2012a) r 2 ¼ a 2 1 þ 3 2 e2 ; ð38þ rr ¼ 1 2 a2 5ee hh þð1 e 2 ÞU : ð39þ Substituting Eqs. 38 and 39 into Eq. 37, and disregarding the irrelevant constant term, we have R ¼ 3l h i 5ð^d 4nd 3 eþ 2 ð^d hþ 2 2e 2 : ð40þ From the secular Milankovitch equations, Eqs. 22 and 23, the singly-averaged third-body equations can be written in the form (cf. Allan, 1962) _h ¼ 3l ^d 2nd 3 ð5ee hhþ ~^d ; ð41þ _e ¼ 3l h ^d 2nd 3 ð5eh heþ ~^d 2 ~ i h e : ð42þ Doubly-averaged equations If the erturbing body is assumed to be in an ellitic orbit, and sufficient distance between the two timescales exists, then another averaging may be erformed. Since averaging is a linear rocess, it can be erformed over the secular Milankovitch equations directly to yield e _¼ ¼ 1 Z 2 _e dm ¼ 3l! 4 ^d ^d 5e 2n ~ h h d 3! ^d ^d ~e þ d 3 ð45þ! ~ h e5; d 3 ð46þ where (=) denotes the double averaged value. The average of these quantities are given by Eqs. 32 and 33. Consequently, the doubly-averaged third-body dynamics for an ellitically orbiting disturbing body become (cf. Musen, 1961) h _¼ ¼ e _¼ ¼ 3l 4na 3 h3 3l 4na 3 h3 ^H ð5ee hhþ e^h ; ð47þ h ^H ð5eh he Þ e^h 2 ~ i h e : ð48þ where ^H is the angular momentum unit vector of the erturbing body Secular equations of motion The secular evolution of the Milankovitch orbital elements in the resence of SRP, J 2 and lunisolar erturbations can be stated as _h ¼ h _ sr þ h _ 20 þ h _ s þ h _ m ; _e ¼ _e sr þ _e 20 þ _e s þ _e m ; ð49þ ð50þ where the over bar has been droed, the SRP dynamics are given by Eqs. 29 and 30, and the C 20 dynamics are given by Eqs. 35 and 36. The lunisolar dynamics can either be reresented by the singly-averaged equations, Eqs. 41 and 42, or the doubly-averaged equations, Eqs. 47 and 48. In this formulation, the motion of the disturbing bodies can either be sulied from theory, i.e., the two-body solution, or can be rovided by an ehemeris. Combining all of these erturbations leads to a highlynonlinear system, which does not aear integrable, but such simlifications hel to understand some qualitative features of the system. Although the exact averaged solution is resumably inaccessible, the exressions given in Eqs. 49 and 50 are several hundred times faster to numerically integrate than their non-averaged Newtonian counterarts. With these results it is ossible to redict accurately the long term orbital behavior of HAMR objects, given the initial values of the orbital elements and the initial geometry of the Earth Moon Sun system; the latter being imortant for calculation of Moon-induced erturbations.

8 1552 A.J. Rosengren, D.J. Scheeres / Advances in Sace Research 52 (2013) Table 1 Maximum long-term inclination and minimum eriasis radius reached over all 360 trajectories for each SRP erturbation angle. The corresonding effective area-to-mass ratios are also listed. Recall that the Saros resonance becomes imortant between K ¼ 10:5 and K ¼ 15:5. K [ ] ð1 þ qþa=m [m 2 /kg] max i [ ] min r [R e ] Long-term dynamics of HAMR objects Numerical integration of the recise non-averaged equations of motion reresents the most accurate means of calculating the exact trajectory of an orbiting body in a given time interval. Anselmo and Pardini have made several numerical investigations of the HAMR debris roblem, maing out the dynamics of these objects over long timesans with all relevant erturbations included. Their most recent work resents a detailed analysis concerning the long-term evolution of HAMR debris in high-earth orbit subject to SRP with Earth s shadow effects, geootential harmonics u to degree and order eight, and third-body gravitational interactions induced by the Sun and the Moon (Anselmo and Pardini, 2010). Comarison of these solutions with the results obtained from the averaged formulas is a significantly reliable estimate of the accuracy of the aroximated equations. Such comarison ermits us to conclude about the alicability of the averaged equations for considering the evolution of HAMR debris orbits. We restrict our attention to objects released in geostationary orbit (a 42164:2 km) on 1950 January 01 12:00:00 UTC; therefore, e 0 0 and h 0 ^ (Earth s rotation ole). To understand the main characteristics of motion and to determine the extent to which the qualitative roerties of the orbit ersist with increasing area-to-mass, the dynamics were simulated using a reflectance value of 0.36, and area-to-mass ratios between 0 and 40 m 2 /kg. For a given semi-major axis, reflectivity, and A=m value, we comute the corresonding K angle (Eq. 8), which we use to characterize the evolutionary behavior of HAMR debris orbits. The SRP erturbation angles and corresonding effective area-to-mass ratios are shown in Table Newtonian non-averaged dynamics For the non-averaged dynamics, we use the high-accuracy Solar System ehemeris (DE421), rovided by JPL, to calculate the osition vectors of the Sun and the Moon (Folkner et al., 2009). The long-term orbit evolution of several HAMR objects, obtained using numerical integrations of the Newtonian equations of motion (Eq. 19), are shown in Fig. 1. The eccentricity and inclination evolution, over 100 years, shown in Figs. 1(a) and 1(b), closely match the results obtained by Anselmo and Pardini (2010). 4 For the object with K ¼ 12:60 (corresonding to ð1 þ qþa=m ¼ 20:4 m 2 /kg), the eccentricity undergoes an aroximately yearly oscillation with amlitude of about 0.4 and long-eriod modulations of 0:05. The inclination undergoes aroximately yearly oscillations that are suerimosed on the long-term drift, which has a varying maximum amlitude between 25 and 35 and a long-term oscillation eriod of about 22 years. As shown in Fig. 1(c), the evolution of the two-dimensional eccentricity vector, e½cos x cos X cos i sin x sin X; cos x sin X þ cos i sin x cos XŠ, is characterized by both a yearly and long-term regression; the latter exhibiting comlex evolutionary behavior. The orbit ole, ^h, recesses clockwise, having the same characteristics as the inclination oscillation (see Fig. 1(d)) Averaged dynamics Since our Newtonian non-averaged results comare well, both quantitatively and qualitatively, with those of Anselmo and Pardini (2010), 5 they can be used as a logical basis for assessing the validity of our averaged model. We are articularly interested in distinguishing between cause and effect and in identifying the recise origin of any erturbation exerienced by the HAMR object. To that end, we avoid using the recise JPL ehemeris, and instead assume two-body dynamics for the Sun and the Moon, for which the lunar node regresses in the eclitic lane with a sidereal eriod of 18:61 years (see Section 2.1). The evolution of several HAMR objects, obtained using numerical integrations of the singly-averaged equations of motion, are shown in Fig. 2. The eccentricity evolution in Fig. 2(a) is shown over a shorter timescale to emhasize the resective amlitudes of the yearly oscillations for each object. The aroximated averaged equations, using twobody dynamics and accounting for the regression of the Moon s node, gives us nearly identical lots at this level of resolution as the full Newtonian non-averaged simulations. We do not use any secial formalism in our integrations to reserve the constraints on h and e, 6 yet after 100 years, they are satisfied to over one art er billion. With the Newtonian non-averaged formulation, the various erturbations are all lumed together and we obtain no indication as to the form and nature of any of them. 4 Note that we used slightly different initial conditions and a different release eoch for our simulations; the latter will change the initial dynamical configuration of the Earth Moon Sun system. 5 In addition to our force model, they account for higher-order gravity field erturbations and Earth shadow effects in their numerical integrations. 6 Recall that e h ¼ 0 and e e þ h h ¼ 1.

9 A.J. Rosengren, D.J. Scheeres / Advances in Sace Research 52 (2013) Fig. 1. Long-term orbit evolution (100 years) in the Earth-equatorial frame for different values of the SRP erturbation angle, as redicted by the full nonaveraged equations of motion, Eq. 19, using Eq. 5 for SRP, Eq. 11 for C 20 and C 22, and Eq. 12 for lunisolar erturbations. The osition vectors of both the Sun and the Moon were comuted using the JPL ehemeris (DE421). However, our averaged formulation gives significant qualitative indications and allows us to understand many asect of HAMR debris motion. Concerning the eccentricity evolution, solar radiation ressure acting alone induces subyearly oscillations with eriod 2 cos K; the amlitude increasing with increasing K. Inclusion of the C 20 dynamics causes only slight changes in the short-term oscillations, but induces long-eriod small fluctuations in the maximum amlitudes. The dynamical couling between SRP and oblateness becomes more ronounced with increasing K (q.v., Rosengren and Scheeres, 2011). The addition of third-body erturbations, rimarily the attraction of the Moon, causes a slight increase in the short-term amlitudes, and gives rise to long-term aeriodic oscillations in the maximum amlitudes. Regarding the inclination evolution, solar radiation ressure accounts for the sub-yearly oscillations that ride on to of the longer-term secular drift, and the reduction in the long-term oscillation eriods with increasing K (see Fig. 3). The addition of Earth oblateness brings about a reduction in both the amlitude and eriod of the long-term oscillations. Inclusion of lunisolar erturbations causes a slight increase in the long-term amlitudes and a decrease in the long-term oscillation eriods, and for certain values of K most notably K ¼ 12:60 (corresonding to ð1 þ qþa=m ¼ 20:4 m 2 /kg) causes large fluctuations (eak-to-eak changes) in the maximum amlitudes. These fluctuations manifest themselves as a beating henomenon in the evolution of the two-dimensional angular momentum unit vector, ½sin i sin X; sin i cos XŠ. Note that for lower values of K, this comlex behavior is not observed, as shown in Fig. 2(b). The origin of this henomenon, which also aears in the numerical results of Anselmo and Pardini (2010), will be discussed in Section 4.3. The limitations and domain of validity of the doublyaveraged third-body equations (Eqs. 47 and 48), is discussed extensively in Rosengren and Scheeres (2012), and the relevant figures will be omitted here. Note that in all cases considered, the doubly-averaged equations redict the qualitative nature of the orbits; however, the angular momentum vector evolution becomes misaligned with the singly-averaged results after several decades. For larger SRP erturbation angles, corresonding to faster reces-

10 1554 A.J. Rosengren, D.J. Scheeres / Advances in Sace Research 52 (2013) Fig. 2. Long-term orbit evolution in the Earth-equatorial frame for different values of the SRP erturbation angle, as redicted by the singly-averaged equations of motion, Eqs. 49 and 50, using Eqs. 29 and 30 for SRP, Eqs. 35 and 36 for C 20, and Eqs. 41 and 42 for lunisolar erturbations. The osition vectors of both the Sun and Moon were comuted using two-body dynamics, accounting for lunar nodal regression. Fig. 3. Accumulation of the effects on the long-term inclination evolution, as each erturbing force is added to the system. sions of the angular momentum vector, this deviation becomes more ronounced. For K ¼ 12:60, the doublyaveraged equations are able to cature the comlex beating henomenon, but redict a faster recession causing a shift in the inclination evolution. In our derivation of the doubly-averaged equations, we assumed the Moon followed a Kelerian ellise about the Earth; however, as the Moon s node regresses in the eclitic lane, this assumtion actually violates the averaging rincile. As will become aarent in the following section, a resonance theory is needed in order to average out the Moon s motion for lunar third-body erturbations.

11 A.J. Rosengren, D.J. Scheeres / Advances in Sace Research 52 (2013) Saros secular resonance The eoch date determines the initial geometry of the Earth Moon Sun system, and thus the initial location of the lunar ascending node. We found that when the nodal rate of the erturbed system is commensurate with the nodal rate of the Moon (i.e., the Saros), the erturbations build u more effectively over long eriods to roduce significant resonant effects on the orbit (Rosengren and Scheeres, 2012). Such resonances, which occur for a class of HAMR objects that are not cleared out of orbit through their eccentricity growth, gives rise to strongly changing dynamics over longer time eriods. This resonant behavior exlains the long-term beating henomenon that occurs for K ¼ 12:60 (see Fig. 2(b)). Its nodal eriod in the equatorial frame is close enough to the Saros (sidereal eriod of lunar nodal regression) that there is a strong interaction between the lunar effects and the overall recession rate. Figs. 4 6 show the inclination and two-dimensional angular momentum vector evolution in the Earth equatorial frame, for several HAMR objects roagated using the same initial conditions, but varying the initial lunar node. Varying the initial location of the Moon s ascending node over 2 is, in a sense, equivalent to varying the release eoch within a Saros cycle. For K ¼ 13:81, the nodal eriod in the equatorial frame is aroximately equal to years, thereby inducing a 1 : 1 resonance with the Saros. The qualitative icture of the evolution changes drastically based on this angle, which is indicative of resonance. Figs. 5 and 6 show the evolution of objects with nodal rates either too slow or too fast to resonantly interact with the Saros. Note that for these objects, their orbits will change quantitatively based on the initial node angle, but the qualitative evolution remains the same. The aroximate range of SRP erturbation angles for which resonance can be imortant at GEO is between K ¼ 10:5 and K ¼ 15:5 (corresonding to m 2 /kg 6 ð1 þ qþa=m m 2 / kg). We refer the reader to Rosengren and Scheeres (2012) for more details on the Saros resonance henomenon Global behavior of HAMR debris The classical Lalace lane Almost fifty years have elased since satellites were first launched into geostationary orbit. The orbital dynamics of uncontrolled GEO satellites is governed by the oblateness of the Earth and third-body gravitational interactions Fig. 4. Long-term orbit evolution in the Earth-equatorial frame of an object with K ¼ 13:81, as a function of the initial lunar node. Note the significant resonant effect caused by the Saros.

12 1556 A.J. Rosengren, D.J. Scheeres / Advances in Sace Research 52 (2013) Fig. 5. Long-term orbit evolution in the Earth-equatorial frame of an object with K ¼ 8:47, as a function of the initial lunar node. Note that the Saros resonance is not observable. induced by the Sun and the Moon. By itself, Earth s oblateness causes the ole of the orbital lane to recess around the ole of Earth s equator, the rate of rotation being roortional to J 2 ðr=aþ 2 n (as can be seen from Eq. 35). Lunisolar erturbations will have a similar effect, but the recession will now take lace about the oles of the orbital lanes of the Moon and the Sun, resectively, at a rate roortional to n 2 =n (see Eq. 47). The motion of the orbit ole of the satellite is a combination of simultaneous recession about these three different axes, one of which, the ole of the Moon s orbit, regresses around the ole of the eclitic with a eriod of years. The classical Lalace lane (or Lalacian lane) is the mean reference lane about whose axis the satellite s orbit recesses. On the Lalace lace, the secular orbital evolution driven by the combined effects of these erturbations is zero, so that the orbits are frozen. Under the aroximation that the lunar orbit lies in the eclitic, we can defined the Lalace lane at GEO, which lies between the lane of the Earth s equator and that of the eclitic and asses through their intersection (i.e., the Vernal equinox), and which has an inclination of about 7:5 with resect to Earth s equator (Allan and Cook, 1964). The significance of the classical Lalace lane is that the ole of an orbit inclined at a small angle to it will regress around the ole of this lane at nearly constant rate and inclination. This imlies that the orbital lanes of uncontrolled GEO satellites evolve in a systematic way; that is, their inclinations and ascending nodes are highly correlated. Shown in Fig. 7 is the long-term evolution of the inclination and ascending node, in the Earth equatorial frame, of initially geostationary satellites (note that SRP causes only negligible effects for a tyical satellite). These inactive satellites recess at a nearly constant inclination about the ole of the Lalace lane with a eriod of about 54 years (q.v., Schildknecht, 2007). The maximum values of inclination occur at X ¼ 0 (i.e., the ascending node of the Sun s orbit) and the minimum values occur at X ¼ 180 (i.e., the descending node) The modified Lalace lane The natural question arises as to whether the inclination and ascending node for HAMR objects, which have areato-mass ratios hundreds or thousands of times greater than that of a tyical satellite and are thus strongly erturbed by solar radiation ressure, is also systematic (strongly correlated). It has been noted by Anselmo and Pardini (2010) that an increase in area-to-mass ratio leads to a faster and wider clockwise recession of the orbit ole. Tamayo et al. (2013) argues, in the context of circumlanetary dust

13 A.J. Rosengren, D.J. Scheeres / Advances in Sace Research 52 (2013) Fig. 6. Long-term orbit evolution in the Earth-equatorial frame of an object with K ¼ 16:59, as a function of the initial lunar node. Note that the Saros resonance is not observable. articles, that solar radiation ressure modifies the classical Lalace lane equilibrium. In fact, Allan and Cook (1967), in considering the ossibility of a geocentric contribution to the zodiacal light, show that solar radiation ressure acting alone causes a secular recession of the orbit around the ole of the eclitic. They find an aroximate solution Fig. 7. Scatter lot of the time-series, over 54 years, of inclination and ascending node of initially geostationary objects subject to gravitational erturbations, as redicted by the averaged model. The long-term motion of the orbital lane is shown for four different initial ositions of the lunar node, i.e., four different launch dates. to the modified Lalace lane and note that the orbital lane of a given dust article, for a given semi-major axis, will regress around this lane. However, the validity of their analysis is limited to dust articles that are only weakly erturbed by solar radiation ressure; that is, articles which have low values of effective area-to-mass ratio. To understand the satial distribution of the HAMR debris oulation, and to determine whether their ði; XÞ attern is systematic, we investigated a range of HAMR objects, released in geostationary orbit with area-to-mass ratios from 0 u to 40 m 2 /kg. We roagated over 80 different K values for 100 years, uniformly distributed in initial lunar node (360 different node values from 0 to 2), giving nearly 30,000 simulations. As evident from Figs. 4 6, the initial location of the lunar node is an imortant arameter as different behavior occurs deending on where the object is relative to the Moon. Moreover, the initial lunar node can be correlated with any eoch within a Saros cycle, and thus may shed insight into the source of this debris. Fig. 8 shows the time-series, over 100 years, of inclination and right ascension of the ascending node in the Earth-equatorial frame, for a range of SRP erturbation angles, for two trajectories selected out of the 360 different trajectories that are a function of the initial lunar node. That is, we track the statistics over all 360 initial lunar