Determination of the area and mass distribution of orbital debris fragments

An important factor in modeling the orbital debris environment is the loss rate of debris due to atmospheric drag and luni/solar perturbations. An accurate knowledge of the area-to-mass ratio of debris fragments is required for the calculation of the effect of atmospheric drag. In general, this factor is unknown and assumed values are used. However, this ratio can be calculated for fragments for which changes in the orbital elements due to atmospheric drag as a function of time are known. This is the inverse of the technique used to determine the atmospheric density from the decay of satellites with accurately known area-to-mass ratios. These kinds of propagation programs are routinely used in predicting the decay of an orbiting vehicle. In this work the area-to-mass ratio of about 2600 fragments arising from the breakup of 24 artificial satellites have been determined. An analysis of the data on about 200 objects (rocket bodies, scientific satellites, etc.) with known mass, size, and shape has also been made. The value of the radar cross-section (RCS), as measured by the Eglin radar operating at 70 cm wavelength, has been correlated to the effective area of these objects. The measurements of the area-to-mass ratio of these objects then provide a calibration of the actual to the calculated mass. It has been shown that the debris mean mass, m, is related to the mean effective area, A, by a power law relation, m = k A ^1.86. However, for a given effective area the mass distribution is very broad. Moreover, the cumulative mass distribution, N(>m), can be expressed as N(>m) = D(m + b), where D, b, and c are constants. The asymptotic slope, c, of low intensity explosions is on the average lower than the slope for high intensity explosions, but there is considerable spread of this slope in each class. Part of the flattening, as indicated by the finite value of the parameter, b, can be understood as arising out of the spread in the RCS values due to the tumbling motion of the fragments and effects related to the detectability of the fragment by the Eglin radar. It has been established that the mass in a given breakup calculated using this technique is in good agreement with the expected mass value. These results can be used in modeling the breakups of other artificial earth satellites and safety analysis.     

Radiation Pressure and the Poynting-Robertson Effect for Fluffy Dust Particles

A correct understanding of the dynamical effect of solar radiation exerted on fluffy dust particles can be achieved with assistance of a light scattering theory as well as the equation of motion. We reformulate the equation of motion so that the radiation pressure and the Poynting-Robertson effect on fluffy grains are given in both radial and nonradial directions from the center of the Sun. This allows numerical estimates of these radiation forces on fluffy dust aggregates in the framework of the discrete dipole approximation, in which the first term of the scattering coefficients in Mie theory determines the polarizability of homogeneous spheres forming the aggregates.The nonsphericity in shape turns out to play a key role in the dynamical evolution of dust particles, while its consequence depends on the rotation rate and axis of the grains. Unless a fluffy dust particle rapidly revolves on its randomly oriented axis, the nonradial radiation forces may prevent, apart from the orbital eccentricity and semimajor axis, the orbital inclination of the particle from being preserved in orbit around the Sun. However, a change in the inclination is most probably controlled by the Lorentz force as a consequence of the interaction between electric charges on the grains and the solar magnetic field. Although rapidly and randomly rotating grains spiral into the Sun under the Poynting-Robertson effect in spite of their shapes and structures, fluffy grains drift inward on time scales longer at submicrometer sizes and shorter at much larger sizes than spherical grains of the same sizes. Numerical calculations reveal that the dynamical lifetimes of fluffy particles are determined by the material composition of the grains rather than by their morphological structures and sizes. The Poynting-Robertson effect alone is nevertheless insufficient for giving a satisfactory estimate of lifetimes for fluffy dust grains since their large ratios of cross section to mass would reduce the lifetimes by enhancing the collisional probabilities. We also show that the radiation pressure on a dust particle varies with the orbital velocity of the particle but that this effect is negligibly small for dust grains in the Solar System.     

A continuum model for the orbit evolution of self-propelled ‘smart dust’ swarms

A continuity equation is developed to model the evolution of a swarm of self-propelled ‘smart dust’ devices in heliocentric orbit driven by solar radiation pressure. These devices are assumed to be MEMs-scale (micro-electromechanical systems) with a large area-to-mass ratio. For large numbers of devices it will be assumed that a continuum approximation can be used to model their orbit evolution. The families of closed-form solutions to the resulting swarm continuity equation then represent the evolution of the number density of devices as a function of both position and time from a set of initial data. Forcing terms are also considered which model swarm sources and sinks (device deposition and device failure). The closed-form solutions presented for the swarm number density provide insights into the behaviour of swarms of self-propelled ‘smart dust’ devices an can form the basis of more complex mission design methodologies.     

Hamiltonian Formulation and Perturbations for Dust Motion Around Cometary Nuclei

In this paper we analyze the dynamical behavior of large dust grains in the vicinity of a cometary nucleus. To this end we consider the gravitational field of the irregularly shaped body, as well as its electric and magnetic fields. Without considering the effect of gas friction and solar radiation, we find that there exist grains which are static relative to the cometary nucleus; the positions of these grains are the stable equilibria. There also exist grains in the stable periodic orbits close to the cometary nucleus. The grains in the stable equilibria or the stable periodic orbits won’t escape or impact on the surface of the cometary nucleus. The results are applicable for large charge dusts with small area-mass ratio which are near the cometary nucleus and far from the Solar. It is found that the resonant periodic orbit can be stable, and there exist stable non-resonant periodic orbits, stable resonant periodic orbits and unstable resonant periodic orbits in the potential field of cometary nuclei. The comet gravity force, solar gravity force, electric force, magnetic force, solar radiation pressure, as well as the gas drag force are all considered to analyze the order of magnitude of these forces acting on the grains with different parameters. Let the distance of the dust grain relative to the mass centre of the cometary nucleus, the charge and the mass of the dust grain vary, respectively, fix other parameters, we calculated the strengths of different forces. The motion of the dust grain depends on the area-mass ratio, the charge, and the distance relative to the comet’s mass center. For a large dust grain (> 1 mm) close to the cometary nucleus which has a small value of area-mass ratio, the comet gravity is the largest force acting on the dust grain. For a small dust grain (< 1 mm) close to the cometary nucleus with large value of area-mass ratio, both the solar radiation pressure and the comet gravity are two major forces. If the a small dust grain which is close to the cometary nucleus have the large value of charge, the magnetic force, the solar radiation pressure, and the electric force are all major forces. When the large dust grain is far away from the cometary nucleus, the solar gravity and solar radiation pressure are both major forces.     

Coupled orbit-attitude dynamics of high area-to-mass ratio (HAMR) objects: influence of solar radiation pressure, Earth’s shadow and the visibility in light curves

The orbital and attitude dynamics of uncontrolled Earth orbiting objects are perturbed by a variety of sources. In research, emphasis has been put on operational space vehicles. Operational satellites typically have a relatively compact shape, and hence, a low area-to-mass ratio (AMR), and are in most cases actively or passively attitude stabilized. This enables one to treat the orbit and attitude propagation as decoupled problems, and in many cases the attitude dynamics can be neglected completely. The situation is different for space debris objects, which are in an uncontrolled attitude state. Furthermore, the assumption that a steady-state attitude motion can be averaged over data reduction intervals may no longer be valid. Additionally, a subset of the debris objects have significantly high area-to-mass ratio (HAMR) values, resulting in highly perturbed orbits, e.g. by solar radiation pressure, even if a stable AMR value is assumed. Note, this assumption implies a steady-state attitude such that the average cross-sectional area exposed to the sun is close to constant. Time-varying solar radiation pressure accelerations due to attitude variations will result in un-modeled errors in the state propagation. This work investigates the evolution of the coupled attitude and orbit motion of HAMR objects. Standardized pieces of multilayer insulation (MLI) are simulated in a near geosynchronous orbits. It is assumed that the objects are rigid bodies and are in uncontrolled attitude states. The integrated effects of the Earth gravitational field and solar radiation pressure on the attitude motion are investigated. The light curves that represent the observed brightness variations over time in a specific viewing direction are extracted. A sensor model is utilized to generate light curves with visibility constraints and magnitude uncertainties as observed by a standard ground based telescope. The photometric models will be needed when combining photometric and astrometric observations for estimation of orbit and attitude dynamics of non-resolved space objects.     

Application of the extended Kalman filter in satellite prediction

The extended Kalman filter is used in this paper to process single-station laser ranging data over a few revolutions to improve the satellite orbit. The aim is to provide accurate short-term predictions of the satellite position. The dynamical model includes the perturbations due to the Earth's oblateness, air drag, solar radiation pressure and the gravitational attractions of the Sun and the Moon.The proposed method is tested with simulated and real LAGEOS data. The results show that the above aim is achievable. Moreover, the computing program based on the present method can be realized on mini-computers.     

Periodic and quasi-periodic motions of a solar sail close to SL 1 in the Earth–Sun system

Solar sails are a proposed form of spacecraft propulsion using large membrane mirrors to propel a satellite taking advantage of the solar radiation pressure. To model the dynamics of a solar sail we have considered the Earth–Sun Restricted Three Body Problem including the Solar radiation pressure (RTBPS). This model has a 2D surface of equilibrium points parametrised by the two angles that define the sail orientation. In this paper we study the non-linear dynamics close to an equilibrium point, with special interest in the bounded motion. We focus on the region of equilibria close to SL ₁, a collinear equilibrium point that lies between the Earth and the Sun when the sail is perpendicular to the Sun–sail direction. For different fixed sail orientations we find families of planar, vertical and Halo-type orbits. We have also computed the centre manifold around different equilibria and used it to describe the quasi-periodic motion around them. We also show how the geometry of the phase space varies with the sail orientation. These kind of studies can be very useful for future mission applications.     

The impact of Earth’s shadow on the long-term evolution of space debris

Direct solar radiation pressure and Earth’s shadow crossings are known to be responsible for short-term variations of space debris orbital elements, the higher the area-to-mass ratio the larger the perturbation. Nevertheless, existing studies have always been performed on periods of time shorter than 150 years. Considering longer time scales of the order of a 1000 years, this paper focuses on the long-term periodic evolution of space debris trajectories caused by successive Earth’s shadow crossings. Other perturbations as the geopotential and third-body gravitational attractions obviously play a role and compete with the one which is described in this paper. Symplectic numerical propagations and new (semi-)analytical models are developed to identify a frequency associated to shadow entry and exit eccentric anomalies. It is shown that Earth’s shadow is responsible for large deviations from the initial orbital elements, even on shorter period of times, and that this effect increases along with the area-to-mass ratio.     

Long-term density evolution through semi-analytical and differential algebra techniques

This paper introduces and combines for the first time two techniques to allow long-term density propagation in astrodynamics. First, we introduce an efficient method for the propagation of phase space densities based on differential algebra (DA) techniques. Second, this DA density propagator is used in combination with a DA implementation of the averaged orbital dynamics through semi-analytical methods. This approach combines the power of orbit averaging with the efficiency of DA techniques. While the DA-based method for the propagation of densities introduced in this paper is independent of the dynamical system under consideration, the particular combination of DA techniques with averaged equations of motion yields a fast and accurate technique to propagate large clouds of initial conditions and their associated probability density functions very efficiently for long time. This enables the study of the long-term behavior of particles subjected to the given dynamics. To demonstrate the effectiveness of the proposed approach, the evolution of a cloud of high area-to-mass objects in Medium Earth Orbit is reproduced considering the effects of solar radiation pressure, the Earth’s oblateness and luni-solar perturbations. The method can propagate 10,000 random fragments and their density for 1 year within a few seconds on a common desktop PC.     

Iron/silicate fractionation and the origin of Mercury

Compared with the other terrestrial planets, Mercury has anomalously low mass and high iron content. Equilibrium condensation and inhomogeneous accretional models are not compatible with these properties, unless the solar nebula's thermal structure and history meet stringent conditions. Also, such models predict a composition which does not allow a presently molten core. It appears that most of the solid matter which originally condensed in Mercury's zone has been removed. The planet's composition may be explained if the removal process was only slightly more effective for silicates than for iron. It is proposed that planetesimal orbits in the inner solar nebula decayed because of gas drag. This process is a natural consequence of the non-Keplerian rotation of a centrally condensed nebula. A simple quantitative model shows good agreement with the observed mass distribution of the terrestrial planets. The rate of orbital decay is slower for larger and/or denser bodies, because of their smaller area-to-mass ratios. With plausible assumptions as to planetesimal sizes and compositions, this process can produce fractionation of the sense required to produce an iron-rich planet. Cosmogonical implications are discussed.     

Dynamical lifetime survey of geostationary transfer orbits

In this paper, we study the long-term dynamical evolution of highly elliptical orbits in the medium-Earth orbit region around the Earth. The real population consists primarily of Geosynchronous Transfer Orbits (GTOs), launched at specific inclinations, Molniya-type satellites and related debris. We performed a suite of long-term numerical integrations (up to 200 years) within a realistic dynamical model, aimed primarily at recording the dynamical lifetime of such orbits (defined as the time needed for atmospheric reentry) and understanding its dependence on initial conditions and other parameters, such as the area-to-mass ratio (A / m). Our results are presented in the form of 2-D lifetime maps, for different values of inclination, A / m, and drag coefficient. We find that the majority of small debris (\(>70\%\), depending on the inclination) can naturally reenter within 25–90 years, but these numbers are significantly less optimistic for large debris (e.g., upper stages), with the notable exception of those launched from high latitude (Baikonur). We estimate the reentry probability and mean dynamical lifetime for different classes of GTOs and we find that both quantities depend primarily and strongly on initial perigee altitude. Atmospheric drag and higher A / m values extend the reentry zones, especially at low inclinations. For high inclinations, this dependence is weakened, as the primary mechanisms leading to reentry are overlapping lunisolar resonances. This study forms part of the EC-funded (H2020) “ReDSHIFT” project.     

Dust ring formation due to sublimation of dust grains drifting radially inward by the Poynting-Robertson drag: An analytical model

Dust particles exposed to the stellar radiation and wind drift radially inward by the Poynting-Robertson (P-R) drag and pile up at the zone where they begin to sublime substantially. The reason they pile up or form a ring is that their inward drifts due to the P-R drag are suppressed by stellar radiation pressure when the ratio of radiation pressure to stellar gravity on them increases during their sublimation phases. We present analytic solutions to the orbital and mass evolution of such subliming dust particles, and find their drift velocities at the pileup zone are almost independent of their initial semimajor axes and masses. We derive analytically an enhancement factor of the number density of the particles at the outer edge of the sublimation zone from the solutions. We show that the formula of the enhancement factor reproduces well numerical simulations in the previous studies. The enhancement factor for spherical dust particles of silicate and carbon extends from 3 to more than 20 at stellar luminosities L_?=0.8-500L_⊙, where L_⊙ is solar luminosity. Although the enhancement factor for fluffy dust particles is smaller than that for spherical particles, sublimating particles inevitably form a dust ring as long as their masses decrease faster than their surface areas during sublimation. The formulation is applicable to dust ring formation for arbitrary shape and material of dust in dust-debris disks as well as in the Solar System.     

Dynamics of Mars Trojans

In this paper we explore the dynamical stability of the Mars Trojan region applying mainly Laskar's Frequency Map Analysis. This method yields the chaotic diffusion rate of orbits and allows to determine the most stable regions. It also gives the frequencies which are responsible for the instability of orbits. The most stable regions are found for inclinations between about 15° and 30°. For inclinations smaller than 15°, we confirm, by applying a synthetic secular theory, that the secular resonances ν₃, ν₄, ν₁₃, ν₁₄ rapidly excite asteroid orbits within a few Myrs, or even faster. The asteroids are removed from the Trojan region after a close encounter with Mars. For large inclinations, the secular resonance ν₅ clears a small region around 30° while the Kozai resonance rapidly removes bodies for inclinations larger than 35°. The dynamical lifetimes of the three L5 Trojans, (5261) Eureka, 1998 VF31, 2001 DH47, and the only L4 Trojan 1999 UJ7 are determined by numerically integrating clouds of corresponding clones over the age of the Solar System. All four Trojans reside in the most stable region with smallest diffusion coefficients. Their dynamical half-lifetime is of the order of the age of the Solar System. The Yarkovsky force has little effect on the known Trojans but for bodies smaller than about 1-5 m the drag is strong enough to destabilize Trojans on a timescale shorter than 4.5 Gyr.     

Concentration and sorting of chondrules and CAIs in the late Solar Nebula

The high concentration and sorting of chondrules, sub-mm sized spherules found in undifferentiated meteorites, is one of the great unsolved mysteries in planetology. Here we present a unifying explanation for these phenomena based on the assumption that chondrules were present when the Solar Nebula was optically thin but had a significant amount of gas. An immediate consequence is that chondrules feel a force known as photophoresis. Photophoresis is based on a temperature gradient over the surface of a particle resulting from absorption of radiation and non-uniform interaction with its gaseous environment. In comparison to well-known forces originating from starlight, i.e. radiation pressure, Poynting-Robertson drag, or the Yarkovski effect, photophoresis can be stronger by many orders of magnitude in gaseous environments. In the application discussed here photophoresis concentrates chondrules and CAIs, which are both found in chondrites, in the region of the asteroid belt. Chondrules from any place in the Solar Nebula will be dragged to the asteroid belt region, while smaller dust particles and their aggregates will be removed from this region at the same time. This leads to a high relative concentration of chondrules, sorted with respect to their thermal conductivity, density, and size, for building chondrite parent bodies. Furthermore, photophoresis prevents CAIs from being lost to the Sun.     

Light pressure effect on the orbital evolution of objects moving in the neighborhood of low-order resonances

The hardware-software complex of the SBG telescope of the Kourovka astronomical observatory of the Ural Federal University is described. The area-to-mass ratio of high-orbit objects was estimated from the results of positional observations at the SBG telescope. Positions and sizes of resonance zones are determined from the numerical simulation results depending on the area-to-mass ratio in the case of low-order resonances (1 : 1, 1 : 2, and 1 : 3). Secular perturbations of semimajor axes of orbits, caused by the Poynting-Robertson effect, are estimated in the neighborhood of low-order resonance zones at different area-to-mass ratios.     

Radiation drag driven mass accretion in a clumpy interstellar medium: implications for the supermassive black hole-to-bulge relation

We quantitatively scrutinize the effects of the radiation drag arising from the radiation fields in a galactic bulge in order to examine the possibility that the radiation drag could be an effective mechanism to extract angular momentum in a spheroidal system like a bulge and allow plenty of gas to accrete on to the galactic centre. For this purpose, we numerically solve the relativistic radiation hydrodynamical equation coupled with accurate radiative transfer, and quantitatively assess the radiation drag efficiency. As a result, we find that in an optically thick regime the radiation drag efficiency is sensitively dependent on the density distributions of the interstellar medium (ISM). The efficiency drops according to     in an optically thick uniform ISM, where τ _T is the total optical depth of the dusty ISM , whereas the efficiency remains almost constant at a high level if the ISM is clumpy . Hence, if bulge formation begins with a star formation event in a clumpy ISM, the radiation drag will effectively work to remove the angular momentum and the accreted gas may form a supermassive black hole. As a natural consequence, this mechanism reproduces a putative linear relation between the mass of a supermassive black hole and the mass of a galactic bulge, although further detailed modelling for stellar evolution is required for a more precise prediction.     

Regional positioning using a low Earth orbit satellite constellation

Global and regional satellite navigation systems are constellations orbiting the Earth and transmitting radio signals for determining position and velocity of users around the globe. The state-of-the-art navigation satellite systems are located in medium Earth orbits and geosynchronous Earth orbits and are characterized by high launching, building and maintenance costs. For applications that require only regional coverage, the continuous and global coverage that existing systems provide may be unnecessary. Thus, a nano-satellites-based regional navigation satellite system in Low Earth Orbit (LEO), with significantly reduced launching, building and maintenance costs, can be considered. Thus, this paper is aimed at developing a LEO constellation optimization and design method, using genetic algorithms and gradient-based optimization. The preliminary results of this study include 268 LEO constellations, aimed at regional navigation in an approximately 1000 km \(\times \) 1000 km area centered at the geographic coordinates [30, 30] degrees. The constellations performance is examined using simulations, and the figures of merit include total coverage time, revisit time, and geometric dilution of precision (GDOP) percentiles. The GDOP is a quantity that determines the positioning solution accuracy and solely depends on the spatial geometry of the satellites. Whereas the optimization method takes into account only the Earth’s second zonal harmonic coefficient, the simulations include the Earth’s gravitational field with zonal and tesseral harmonics up to degree 10 and order 10, Solar radiation pressure, drag, and the lunisolar gravitational perturbation.     

The influence of radiation pressure and drag on Saturn's rings

The influence of radiation pressure and drag on the optical thickness of ring C is calculated as a function of the saturnocentric distance. The results are compared with Franklin and Cook's 1958 data. It seems probable that drag exerts more important effects than radiation pressure, at least in inner parts of ring C. The drag effect might also explain the existence of the Cassini division.     

Dynamical evolution of Saturn's F ring dust particles

Saturn's F ring has been the subject of study due to its peculiar structure and the proximity to two satellites, named Prometheus (interior) and Pandora (exterior to the ring), which cause perturbations to the ring particles. Early results from Voyager data have proposed that the ring is populated with centimetre- and micrometre-sized particles. The Cassini spacecraft also detected a less dense part in the ring with width of 700 km. Small particles suffer the effects of solar radiation. Burns et al. showed that due to effects of one component of the solar radiation, the Poynting–Robertson drag, a ring particle will decay in the direction of the planet in a time much shorter than the age of the Solar system. In this work, we have analysed a sample of dust particles (1, 3, 5 and 10 μm) under the effects of solar radiation, the Poynting–Robertson drag and the radiation pressure components and the gravitational effects of the satellites Prometheus and Pandora. In this case, the high increase of the eccentricity of the particles leads almost all of them to collide with the outer edge of the A ring. The inclusion of the oblateness of Saturn in this system significantly changes the outcome, since the large variation of the eccentricity is reduced by the oblateness effect. As a result, there is an increase in the lifetime of the particle in the envelope region. Our results show that even the small dust particles, which are very sensitive to the effects of solar radiation, have an orbital evolution similar to larger particles located in the F ring. The fate of all particles is a collision with Prometheus or Pandora in less than 30 years. On the other hand, collisions of these particles with moonlets/clumps present in the F ring could change this scenario.     

Chaotic motion of comets in near-parabolic orbit: Mapping approaches

There exist many comets with near-parabolic orbits in the Solar System. Among various theories proposed to explain their origin, the Oort cloud hypothesis seems to be the most reasonable (Oort, 1950). The theory assumes that there is a cometary cloud at a distance 10³ – 10⁵ AU from the Sun and that perturbing forces from planets or stars make orbits of some of these comets become of near-parabolic type. Concerning the evolution of these orbits under planetary perturbations, we can raise the question: Will they stay in the Solar System forever or will they escape from it? This is an attractive dynamical problem. If we go ahead by directly solving the dynamical differential equations, we may encounter the difficulty of long-time computation. For the orbits of these comets are near-parabolic and their periods are too long to study on their long-term evolution. With mapping approaches the difficulty will be overcome. In another aspect, the study of this model has special meaning for chaotic dynamics. We know that in the neighbourhood of any separatrix i.e. the trajectory with zero frequency of the unperturbed motion of an Hamiltonian system, some chaotic motions have to be expected. Actually, the simplest example of separatrix is the parabolic trajectory of the two body problem which separates the bounded and unbounded motion. From this point of view, the dynamical study on near-parabolic motion is very important. Petrosky's elegant but more abstract deduction gives a Kepler mapping which describes the dynamics of the cometary motion (Petrosky, 1988). In this paper we derive a similar mapping directly and discuss its dynamical characters.