68th International Astronautical Congress, Adelaide, Australia. Copyright c 2017 by the authors. All rights reserved. IAC{17{A6.4.3 END-OF-LIFE DISPOSAL OF GEOSYNCHRONOUS SATELLITES Ioannis Gkolias Politecnico di Milano, Italy, [email protected] Camilla Colombo Politecnico di Milano, Italy, [email protected] End-of-life disposal of spacecraft in the GEO region is required for the further exploitation of this particularly important orbital regime. The orbital dynamics around the geostationary ring can be exploited for designing graveyard orbits or looking for re-entry solutions. Here we present an end-of-life trajectory design method based on a detailed cartography of the orbital space. Given a post-mission orbit of a decommissioned satellite and the available fuel on board, efficient two-burn transfers are calculated for each reachable orbit on the grid. Furthermore, an analysis of cost (delta-v) versus stability of target orbit or re-entry time is performed by means of finding the Pareto optimal solutions for each case. I. Introduction space. We focus our analysis on the resonant dy- namics, due to the characteristic frequencies of lu- Spacecraft in Geostationary Orbits (GEO) repre- nisolar and solar radiation pressure perturbations and sent a fundamental aspect of space activity and pro- their coupling with the asymmetries in Earth's grav- vide valuable services to mankind. According to the itational potential. publicly available two-line element data-sets, approx- Furthermore, we use the computed maps to design imately 1200 total objects are catalogued at a semi- manoeuvres for fuel efficient transfers to stable grave- major axis around the geostationary value. Apart yard orbits. Given the available delta-v for a satellite, from the active spacecraft, this population includes the admissible region is defined and the disposal ma- also defunct satellites, rocket bodies and smaller noeuvre is optimised to target a stable region of the pieces of space debris. orbit domain, for which the interaction with the GEO Safety procedures for operational spacecraft in- active spacecraft population minimises. clude the selection of orbits with lower collision risk with debris as well as the implementation of colli- II. Overview of dynamical mapping sion avoidance manoeuvres. Moreover, space debris guidelines aim at limiting the creation of new debris The dynamics of the geostationary orbit shows dy- by the prevention of in-orbit explosions and the im- namical behaviours influenced by the Earths triaxial- plementation of end-of life disposal manoeuvres to ity, through the resonant longitude angle, by the luni- free the GEO protected regions. To this end, ESA solar perturbation through long term oscillation in applies specific requirements on space debris mitiga- eccentricity, anomaly of the perigee, inclination and tion for all its projects. Those requirements suggest right ascension of the ascending node. that, space systems operating in the GEO protected For low initial inclinations, as expected, no re- zone shall be disposed into a graveyard orbit with entry conditions are found around the geostationary the eccentricity less than 0.005 and a given minimum orbit region; for this reason it is useful study the total perigee altitude above the geostationary altitude. variation of eccentricity cover during the long-term In this paper, we characterise the dynamical struc- propagation, as this can be used as a measure of the ture and study the long-term stability of the circum- stability of the orbit, for choosing, e.g., an appropri- terrestrial space at the geostationary altitude, in- ate graveyard orbit. cluding highly-inclined Geosynchronous Orbits (i.e. For higher initial inclination and low eccentric- orbits of the Beidou constellation). Semi-analytical ity conditions the Lidov-Kozai mechanics is present techniques and numerical high-fidelity models are em- also in this orbit regime. This can be seen by the ployed for the long-term propagation of the natural eccentricity-perigee plot, where the anomaly of the dynamics. We produce maps representing the long- perigee is represented with respect to the Earth-Moon term stability of the orbits in the orbital element plane. At starting inclinations above 55 degrees, re- IAC{17{A6.4.3 Page 1 of 13 68th International Astronautical Congress, Adelaide, Australia. Copyright c 2017 by the authors. All rights reserved. entry via natural eccentricity growth is possible with a time span of around 120 years. Therefore due to the natural dynamics in the or- bital region about the geosynchronous altitude divide the phase space into two distinct regions. This anal- ysis will form the basis for the disposal manoeuvres in the next phase of this work. III. Disposal design The dynamical mapping results presented in pre- vious works represent a very large set of data. A grid in semi-axis, eccentricity, inclination, anomaly of the ascending node and anomaly of the perigee was de- fined to represent all the possible initial orbit con- ditions in the LEO to GEO environment. All these initial conditions where integrated over a period of 120 years, considering two starting dates represent- ing different initial orbit configuration of the Earth, Moon and Sun. As a measure of the orbit stability the maximum variation of eccentricity over the integration time was stored, together with the re-entry time for those con- dition that achieve re-entry before 120 years due to eccentricity growth driven by lunisolar and Earths oblateness perturbation. The result obtained will be now used for the calculation of the optimal impulsive ∆v to reach the best disposal orbit. We use the same grid in orbital elements used for the dynamical mapping analysis and we consider this grid representative of any possible spacecraft opera- tional orbit. For each point of this grid, representing one selected operational orbit, we need to define the best disposal option. This could be: • to transfer to a neighbourhood orbit which over the long term will result in a re-entry via eccen- tricity growth, • to transfer to a neighbourhood orbit which will be stable over the long term, • do not to perform any transfer and let the space- craft orbit naturally evolve towards re-entry or towards a stable graveyard orbit. The following section describes the method used for calculating the optimal graveyard orbit for a given initial condition and to choose among different dis- posal options. Let us first focus on one single initial Fig. 1: Reachable orbital element domain starting condition and explain how the optimal disposal for from an initial orbit with a0 = 42165 km, e0 = ◦ ◦ ◦ such condition can be determined. The method de- 0:01, i0 = 0:01 ,Ω0 = 0 and !0 = 0 . veloped for this work is composed of 3 steps. IAC{17{A6.4.3 Page 2 of 13 68th International Astronautical Congress, Adelaide, Australia. Copyright c 2017 by the authors. All rights reserved. III.i Reachable orbital element domain anomalies are fast variables with respect to the other Given a maximum available ∆vmax the reachable orbital elements, we can assume this variable as free space in orbital elements ∆a where a = [a; e; i; Ω;!] one, as was done in.3 Indeed, once the optimal true via an instantaneous impulsive manoeuvre is calcu- anomaly on the initial and target orbit are found, lated by means of Gausss equations written for finite the permanence time of the initial orbit to reach the differences.1 starting true anomaly for the transfer can be calcu- lated. Therefore, a grid in time of flight for the Lam- 2 2a v bert is defined with a step of ∆T oF in the domain ∆a = ∆vt µEarth [0; T oFmax]; given the five orbital elements on the 1 r initial orbit a and the target orbit a , the true ∆e = 2(e + cos f)∆v − sin f∆v 0 target v t a n anomaly on the two orbits are also determined via a r grid search with a step of ∆M. For each point in this ∆i = cos(f + !)∆vh h three dimensional grid, a Lambert arc is calculated r sin(f + !) from the initial to the target orbit and the total ∆v ∆Ω = ∆v h sin i h is calculated as 1 r ∆! = 2 sin f∆vt + 2e + cos f ∆vn ev a ∆v = jjvtransfer0 (f) − v0(f)jj r sin(f + !) cos i − ∆v h sin i h where the dependence on the true anomaly f is shown. where ∆vt, ∆vn and ∆vh is the finite change in the On the grid a Lambert arc is calculated for each velocity in the tangential, normal and out-of-plane point on the initial orbit to each point on the target direction, so that orbit and the optimal transfer (i.e. the one corre- sponding to the minimum ∆v) is stored. ∆vt = ∆vmax cos α cos δ ∆vn = ∆vmax cos α sin δ III.iii Pareto front ∆vh = ∆vmax sin α From a given starting orbit, the minimum ∆v to reach each one of the reachable target orbit is stored. where 0 ≤ α ≤ 2π and −π=2 ≤ δ ≤ π=2 are the At this point the next step is to select among them right ascension and the co-declination that describe the best disposal strategy between re-entry or grave- the orientation of the ∆v manoeuvre with respect to yard. Each target orbit corresponds to a different the t − n − h frame. long term evolution, which in some cases may lead to Note that, as shown in,2 Gauss' equation for finite re-entry within the 120 year time frame. To quickly differences introduce a numerical error, proportional characterise the long term evolution of each target to the magnitude of ∆vmax, due to the fact that an orbit two parameters are saved: instantaneous manoeuvre only changes the velocity and not the position of the spacecraft.