Semi-analytical theory of mean orbital motion for geosynchronous space debris under gravitational influence

This paper provides a hamiltonian formulation of the equations of motion of an artificial satellite or space debris orbiting the geostationary ring. This theory of order 1 has been formulated using canonical and non-singular elements for eccentricity and inclination. The analysis is based on an expansion in powers of the eccentricity and of the inclination. The theory accounts for the influence of the Earth gravity field expanded in spherical harmonics, paying a particular attention to the resonance occurring for geosynchronous objects. The luni-solar perturbations are also taken into account. We present the resonant motion and its main characteristics: equilibria, stability, fundamental frequencies and width of the resonant area by comparison with a basic analytical model. Finally, we show some results concerning the long term dynamics of a typical space debris under the influence of the gravitational field of the Earth and the luni-solar interactions.     

Space debris mitigation in geosynchronous orbit

In order to preserve the geosynchronous region, the Inter-Agency Space Debris Coordination Committee (IADC) proposed and endorsed a re-orbiting strategy for spacecraft at the end-of-life: they should be disposed above the synchronous altitude and passivated, to reduce the risk of inadvertent explosions. The recommended perigee altitude of the disposal orbit took into account all relevant perturbations and was a function of the expected perturbing acceleration induced by solar radiation pressure. It was intended to prevent any further interference with a properly defined geostationary protected region.     

Shape and motion estimate of LEO debris using light curves

We have derived a tri-axial ellipsoidal model of an LEO object, a Cosmos 2082 rocket body, including its rotational axis direction, rotation period, precession, and a compositional parameter, using only light curve data from an optical telescope. The brightness of the object was monitored for two days and least-squares fitting was used to determine these values. The derived axial ratios are 100:18:18, the coordinates of the rotational axis direction on the celestial sphere are R.A. = 305.8° and Dec. = 2.6°, and its observed average rotation period is 41 s. When precession is considered, its amplitude and precession period are 30.5° and 29.4 min. These results show that optical light curve data are sufficient to determine the shapes and the motions of LEO objects.     

Estimation of orbital parameters of broken-up objects from in-situ debris measurements

Even sub-millimeter-size debris could cause a fatal damage on a spacecraft. Such tiny debris cannot be followed up or tracked from the ground. Therefore, Kyushu University has initiated IDEA the project for In-situ Debris Environmental Awareness, which conducts in-situ measurements of sub-millimeter-size debris. One of the objectives is to estimate the location of on-orbit satellite fragmentations from in-situ measurements. The previous studies revealed that it is important to find out the right nodal precession rate to estimate the orbital parameters of a broken-up object properly. Therefore, this study derives a constraint equation that applies to the nodal precession rate of the broken-up object. This study also establishes an effective procedure to estimate properly the orbital parameters of a broken-up object with the constraint equation.     

Medium Earth Orbit dynamical survey and its use in passive debris removal

The Medium Earth Orbit (MEO) region hosts satellites for navigation, communication, and geodetic/space environmental science, among which are the Global Navigation Satellites Systems (GNSS). Safe and efficient removal of debris from MEO is problematic due to the high cost for maneuvers needed to directly reach the Earth (reentry orbits) and the relatively crowded GNSS neighborhood (graveyard orbits). Recent studies have highlighted the complicated secular dynamics in the MEO region, but also the possibility of exploiting these dynamics, for designing removal strategies. In this paper, we present our numerical exploration of the long-term dynamics in MEO, performed with the purpose of unveiling the set of reentry and graveyard solutions that could be reached with maneuvers of reasonable $\mathrm{\Delta }V$ cost. We simulated the dynamics over 120–200?years for an extended grid of millions of fictitious MEO satellites that covered all inclinations from 0 to 90°, using non-averaged equations of motion and a suitable dynamical model that accounted for the principal geopotential terms, 3rd-body perturbations and solar radiation pressure (SRP). We found a sizeable set of usable solutions with reentry times that exceed $～40$ years, mainly around three specific inclination values: 46°, 56°, and 68°; a result compatible with our understanding of MEO secular dynamics. For $\mathrm{\Delta }V?300$ m/s (i.e., achieved if you start from a typical GNSS orbit and target a disposal orbit with $e<0.3$), reentry times from GNSS altitudes exceed $～70$ years, while low-cost ($\mathrm{\Delta }V?5–35$ m/s) graveyard orbits, stable for at lest 200?years, are found for eccentricities up to $e\approx 0.018$. This investigation was carried out in the framework of the EC-funded “ReDSHIFT” project.     

Symplectic integration of space debris motion considering several Earth’s shadowing models

In this work, we present a symplectic integration scheme to numerically compute space debris motion. Such an integrator is particularly suitable to obtain reliable trajectories of objects lying on high orbits, especially geostationary ones. Indeed, it has already been demonstrated that such objects could stay there for hundreds of years. Our model takes into account the Earth’s gravitational potential, luni-solar and planetary gravitational perturbations and direct solar radiation pressure. Based on the analysis of the energy conservation and on a comparison with a high order non-symplectic integrator, we show that our algorithm allows us to use large time steps and keep accurate results. We also propose an innovative method to model Earth’s shadow crossings by means of a smooth shadow function. In the particular framework of symplectic integration, such a function needs to be included analytically in the equations of motion in order to prevent numerical drifts of the energy. For the sake of completeness, both cylindrical shadows and penumbra transitions models are considered. We show that both models are not equivalent and that big discrepancies actually appear between associated orbits, especially for high area-to-mass ratios.     

Uncertainty propagation for statistical impact prediction of space debris

Predictions of the impact time and location of space debris in a decaying trajectory are highly influenced by uncertainties. The traditional Monte Carlo (MC) method can be used to perform accurate statistical impact predictions, but requires a large computational effort. A method is investigated that directly propagates a Probability Density Function (PDF) in time, which has the potential to obtain more accurate results with less computational effort. The decaying trajectory of Delta-K rocket stages was used to test the methods using a six degrees-of-freedom state model. The PDF of the state of the body was propagated in time to obtain impact-time distributions. This Direct PDF Propagation (DPP) method results in a multi-dimensional scattered dataset of the PDF of the state, which is highly challenging to process. No accurate results could be obtained, because of the structure of the DPP data and the high dimensionality. Therefore, the DPP method is less suitable for practical uncontrolled entry problems and the traditional MC method remains superior. Additionally, the MC method was used with two improved uncertainty models to obtain impact-time distributions, which were validated using observations of true impacts. For one of the two uncertainty models, statistically more valid impact-time distributions were obtained than in previous research.     

Using parallel computing for the display and simulation of the space debris environment

Parallelism is becoming the leading paradigm in today’s computer architectures. In order to take full advantage of this development, new algorithms have to be specifically designed for parallel execution while many old ones have to be upgraded accordingly. One field in which parallel computing has been firmly established for many years is computer graphics. Calculating and displaying three-dimensional computer generated imagery in real time requires complex numerical operations to be performed at high speed on a large number of objects. Since most of these objects can be processed independently, parallel computing is applicable in this field. Modern graphics processing units (GPUs) have become capable of performing millions of matrix and vector operations per second on multiple objects simultaneously.     

An adaptive threshold method for improving astrometry of space debris CCD images

Optical survey is a main technique for observing space debris, and precisely measuring the positions of space debris is of great importance. Due to several factors, e.g. the angle object normal to the observer, the shape as well as the attitude of the object, the variations of observed characteristics for low earth orbital space debris are distinct. When we look at optical CCD images of observed objects, the size and brightness are varying, hence it’s difficult to decide the threshold during centroid measurement and precise astrometry. Traditionally the threshold is given empirically and constantly in data reduction, and obviously it’s not suitable for data reduction of space debris. Here we offer a solution to provide the threshold. Our method assumes that the PSF (point spread function) is Gaussian and estimates the signal flux by a directly two-dimensional Gaussian fit, then a cubic spline interpolation is performed to divide each initial pixel into several sub-pixels, at last the threshold is determined by the estimation of signal flux and the sub-pixels above threshold are separated to estimate the centroid. A trail observation of the fast spinning satellite Ajisai is made and the CCD frames are obtained to test our algorithm. The calibration precision of various threshold is obtained through the comparison between the observed equatorial position and the reference one, the latter are obtained from the precise ephemeris of the satellite. The results indicate that our method reduces the total errors of measurements, it works effectively in improving the centering precision of space debris images.     

A robotic telescope network for space debris identification and tracking

The two TAROT (Télescopes à Action Rapide pour les Objets Transitoires; Rapid Action Telescopes for Transient Objects) installations are fully robotic optical observatories with optimized observation scheduling, data processing and archiving. Zadko is a 1 m telescope in Western Australia. The fully robotisation of the Zadko telescope has just been completed; it is now included in the TAROT network. In this paper we provide an overview of this international network of robotic optical telescopes. We discuss the advantages of using the network to participate in a satellite and space debris tracking program. This network will access almost all geostationary belt objects, and provide the first real-time satellite positioning capability. The inclusion of the 1 m Zadko telescope into the network significantly extends the efficiency and sensitivity of the existing two telescope configuration.     

A new algorithm for optical observations of space debris with the TAROT telescopes

Since 2004, we observe satellites in the geostationary orbit with a network of robotic ground based fully automated telescopes called TAROT. One of them is located in France and the second at ESO, La Silla, Chile. The system processes the data in real time. Its wide field of view is useful for the discovery, the systematic survey and for the tracking of both catalogued and un-catalogued objects. We present a new source extraction algorithm based on morphological mathematic, which has been tested and is currently under implementation in the standard pipeline. Using this method, the observation strategy will correlate the measurements of the same object on successive images and give better detection rate and false alarm rate than the previous one. The overall efficiency and quality of the survey of the geostationary orbit has drastically improved and we can now detect satellites and debris in different orbits like Geostationary Transfer Orbit (GTO). Results obtained in real conditions with TAROT are presented.     

Semi-analytical investigations of high area-to-mass ratio geosynchronous space debris including Earth’s shadowing effects

This paper investigates the long-term perturbations of the orbits of geosynchronous space debris influenced by direct radiation pressure including the Earth’s shadowing effects. For this purpose, we propose an extension of our homemade semi-analytical theory [Valk, S., Lemaître, A., Deleflie, F. Semi-analytical theory of mean orbital motion for geosynchronous space debris under gravitational influence. Adv. Space Res., submitted for publication], based on the method developed by Aksnes [Aksnes, K. Short-period and long-period perturbations of a spherical satellite due to direct solar radiation. Celest. Mech. Dyn. Astron. 13, 89–104, 1976] and generalized into a more convenient non-singular formalism. The perturbations accounting for the direct radiation pressure with the Earth’s shadow are computed on a revolution-by-revolution basis, retaining the original osculating Hamiltonian disturbing function. In this framework, we compute the non-singular mean longitude at shadow entry and shadow exit at every orbital revolution in opposition to classical approaches where the singular eccentric anomalies at shadow entry and shadow exit are computed. This new algorithm is developed using non-singular variables. Consequently, it is particularly suitable for both near-circular and near-equatorial orbits as well as orbits which transit periodically around null eccentricities and null inclinations.The algorithm is tested by means of numerical integrations of the equations, averaged over the short periods, including radiation pressure, J₂, the combined Moon and Sun third body attraction as well as the long-term effects of the 1:1 resonance occurring for geosynchronous objects. As an extension of [Valk, S., Lemaître, A., Anselmo, L. Analytical and semi-analytical investigations of geosynchronous space debris with high area-to-mass ratios influenced by solar radiation pressure. Adv. Space Res., doi:10.1016/j.asr.2007.10.025, 2007b], we especially apply our analysis to space debris with area-to-mass as high as 20 m²/kg. This paper provides numerical and semi-analytical investigations leading to a deep understanding of the long-term evolution of the semi-major axis. Finally, these semi-analytical investigations are compared with accurate numerical integrations of the osculating equations of motion over time scales as high as 25 years.     

First laser measurements to space debris in Poland

The Borowiec Satellite Laser Ranging station (BORL 7811, Borowiec) being a part of the Space Research Centre of the Polish Academy of Sciences (SRC PAS) went through modernization in 2014–2015. One of the main tasks of the modernization was the installation of a high-energy laser module dedicated to space debris tracking. Surelite III by Continuum is a Nd:YAG pulse laser with 10?Hz repetition rate, a pulse width of 3–5?ns and a pulse energy of 450?mJ for green (532?nm). This new laser unit was integrated with the SLR system at Borowiec performing standard satellite tracking. In 2016 BORL 7811 participated actively to the observational campaigns related to the space debris targets from LEO region managed by the Space Debris Study Group (SDSG) of the International Laser Ranging Service (ILRS).Currently, Borowiec station regularly tracks 36 space debris from the LEO regime, including typical rocket bodies (Russian/Chinese) and cooperative targets like the inactive TOPEX/Poseidon, ENVISAT, OICETS and others. In this paper the first results of space debris laser measurements obtained by the Borowiec station in period August 2016 – January 2017 are presented. The results gained by the SRC PAS Borowiec station confirm the rotation of the defunct TOPEX/Poseidon satellite which spins with a period of approximately 10?s. The novelty of this work is the presentation of the sample results of the Chinese CZ-2C R/B target (NORAD catalogue number 31114) which is equipped (probably) with retroreflectors. Laser measurements to space debris is a very desirable topic for the next years, especially in the context of the Space Surveillance and Tracking (SST) activity. Some targets are very easy to track like defunct ENVISAT or TOPEX/Poseidon. On the other hand, there is a big population of different LEO targets with different orbital and physical parameters, which are challenging for laser ranging like small irregular debris and rocket boosters.     

Photon counting detector for space debris laser tracking and lunar laser ranging

We are reporting on a design, construction and performance of solid state photon counting detector package which has been designed for laser tracking of space debris. The detector has been optimized for top photon detection efficiency and detection delay stability. The active area of the commercially available avalanche photodiode manufactured on Si (SAP500 supplied by Laser Components, Inc.) is circular with a diameter of 500 μm. The newly designed control circuit enables to operate the detection sensor at a broad range of biases 5–50 V above its breakdown voltage of 125 V. This permits to select a right trade-off between photon detection efficiency, timing resolution and dark count rate. The photon detection efficiency exceeds 70% at the wavelength of 532 nm. This is the highest photon detection efficiency ever reported for such a device. The timing properties of the detector have been investigated in detail. The timing resolution is better than 80 ps r.m.s, the detection delay is stable within units of picoseconds over several hours of operation. The detection delay stability in a sense of time deviation of 800 fs has been achieved. The temperature change of the detection delay is 0.5 ps/K. The detector has been tested as an echo signal detector in laser tracking of space debris at the satellite laser station in Graz, Austria. Its application in lunar laser ranging is under consideration by several laser stations.     

On the practical exploitation of perturbative effects in low Earth orbit for space debris mitigation

This paper presents the results of a numerical evaluation of the natural lifetime reduction in low Earth orbit, due to dynamical perturbations. The study considers two values for the area-to-mass ratio, a nominal ratio which resembles a typical value of spacecraft in orbit today, and an enhanced ratio which covers the surface augmentation. The results were obtained with two orbit propagators, one of a semi-analytical nature and the second one using non-averaged equations of motion. The simulations for both propagators were set up similarly to allow comparison. They both use the solar radiation pressure and the secular terms of the geopotential ($J_2,J_4$ and $J_6$). The atmospheric drag was turned on and off in both propagators to alternatively study the eccentricity build up and the residual lifetime. The non-averaging case also covers a validation with the full 6?×?6 geopotential. The results confirm the findings in previous publications, that is, the possibility for de-orbiting from altitudes above the residual atmosphere if a solar sail is deployed at the end-of-life, due to the combined effect of solar radiation pressure and the oblateness of the Earth. At near polar inclinations, shadowing effects can be exploited to the same end. The results obtained with the full, non-averaging propagator revealed additional de-orbiting corridors associated with solar radiation pressure which were not found by previous work on space debris mitigation. The results of both tools are compared for specific initial conditions. For nominal values of area-to-mass ratio, instead, it is confirmed that this resonance effect is negligible.The paper then puts the findings in the perspective of the current satellite catalogue. It identifies space missions which are currently close to a resonance corridor and shows the orbit evolution within the resonances with a significantly shorter residual orbital lifetime. The paper finishes with a discussion on the exploitation of these effects with regards to the long-term simulation of the space debris environment and a flux and collision probability comparison.     

Global dynamics of high area-to-mass ratios GEO space debris by means of the MEGNO indicator

In this paper we provide an extensive analysis of the global dynamics of high-area-to-mass ratios geosynchronous (GEO) space debris, applying a recent technique developed by Cincotta and Simó [Cincotta, P.M., Simó, C.Simple tools to study global dynamics in non-axisymmetric galactic potentials–I. Astron. Astrophys. (147), 205–228, 2000.], Mean Exponential Growth factor of Nearby Orbits (MEGNO), which provides an efficient tool to investigate both regular and chaotic components of the phase space.     

International scientific optical network for space debris research

A joint team of researchers under the auspices of the Center for Space Debris Information Collection, Processing and Analysis of the Russian Academy of Sciences collaborates with 15 observatories around the world to perform observations of space debris. For this purpose, 14 telescopes were equipped with charge-coupled device (CCD) cameras, Global Positioning System (GPS) receivers, CCD frame processing and ephemeris computation software, with the support of the European and Russian grants. Many of the observation campaigns were carried out in collaboration with the Astronomical Institute of the University of Bern (AIUB) team operating at the Zimmerwald observatory and conducting research for the European Space Agency (ESA), using the Tenerife/Teide telescope for searching and tracking of unknown objects in the geostationary region (GEO). More than 130,000 measurements of space objects along a GEO arc of 340.9°, collected and processed at Space Debris Data Base in the Ballistic Center of the Keldysh Institute of Applied Mathematics (KIAM) in 2005–2006, allowed us to find 288 GEO objects that are absent in the public orbital databases and to determine their orbital elements. Methods of discovering and tracking small space debris fragments at high orbits were developed and tested. About 40 of 150 detected unknown objects of magnitudes 15–20.5 were tracked during many months. A series of dedicated 22-cm telescopes with large field of view for GEO survey tasks is in process of construction. 7 60-cm telescopes will be modernized in 2007.     

The bistatic radar capabilities of the Medicina radiotelescopes in space debris detection and tracking

An accurate measurement of the position and trajectory of the space debris fragments is of primary importance for the characterization of the orbital debris environment. The Medicina Radioastronomical Station is a radio observation facility that is here proposed as receiving part of a ground-based space surveillance system for detecting and tracking space debris at different orbital regions (from Low Earth Orbits up to Geostationary Earth Orbits). The proposed system consists of two bistatic radars formed by the existing Medicina receiving antennas coupled with appropriate transmitters. This paper focuses on the current features and future technical development of the receiving part of the observational setup. Outlines of possible transmitting systems will also be given together with the evaluation of the observation strategies achievable with the proposed facilities.     

Properties of the high area-to-mass ratio space debris population at high altitudes

In the framework of its space debris research activities ESA established an optical survey program to study the space debris environment at high altitudes, in particular in the geostationary ring and in the geostationary transfer orbit region. The Astronomical Institute of the University of Bern (AIUB) performs these surveys on behalf of ESA using ESA’s 1-m telescope in Tenerife. Regular observations were started in 1999 and are continued during about 120–140 nights per year. Results from these surveys revealed a substantial amount of space debris at high altitudes in the size range from 0.1 to 1 m. Several space debris populations with different dynamical properties were identified in the geostationary ring. During the searches for debris in the geostationary transfer orbit region a new population of objects in unexpected orbits, where no potential progenitors exist, was found. The orbital periods of these objects are clustered around one revolution per day; the eccentricities, however, are scattered between 0 and 0.6. By following-up some of these objects using the ESA telescope and AIUB’s 1-m telescope in Zimmerwald, Switzerland, it was possible to study the properties of this new population. One spectacular finding from monitoring the orbits over time spans of days to months is the fact that these objects must have extreme area-to-mass ratios, which are by several orders of magnitudes higher than for ‘normal-type’ debris. This in turn supports the hypothesis that the new population actually is debris generated in or near the geostationary ring and which is in orbits with periodically varying eccentricity and inclination due to perturbations by solar radiation pressure. In order to further study the nature of these debris, multi-color and temporal photometry (light curves) were acquired with the Zimmerwald telescope. The light curves show strong variations over short time intervals, including signals typical for specular reflections. Some objects exhibit distinct periodic variations with periods ranging from 10 to several 100 s. All this is indicative for objects with complicated shapes and some highly reflective surfaces.