Natural highways for end-of-life solutions in the LEO region

We present the main findings of a dynamical mapping performed in the Low Earth Orbit region. The results were obtained by propagating an extended grid of initial conditions, considering two different epochs and area-to-mass ratios, by means of a singly averaged numerical propagator. It turns out that dynamical resonances associated with high-degree geopotential harmonics, lunisolar perturbations and Solar radiation pressure can open natural deorbiting highways. For area-to-mass ratios typical of the orbiting intact objects, these corridors can be exploited only in combination with the action exerted by the atmospheric drag. For satellites equipped with an area augmentation device, we show the boundary of application of the drag, and where the Solar radiation pressure can be exploited.     

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.     

Implications of the Galilean satellites ice envelope explosions III. The origin of the Trojans and of some comets

Secondary explosions of the primary ice fragments ejected in the explosion of the electrolyzed massive ice envelopes of the Galilean satellites are capable of imparting velocities of up to ~5km s^–1 to the secondary fragments. As a result, the secondary fragments can enter the orbits of the irregular satellites (Agafonova and Drobyshevski, 1984b) and the Trojan libration orbits. In the latter case a perturbation velocity of V 0.3–2 km s^–1 is sufficient.The primary fragments ejected by the gravitational perturbations due to the Galilean satellites sunward from Jupiter's sphere of action move faster relative to the Sun than Jupiter does and therefore reach their first aphelion ahead of Jupiter in the neighborhood of L ₄. At the same time the fragments propelled from Jupiter's sphere of action beyond the planet's orbit approach it again in their perihelia behind Jupiter in the region of L ₅. The concentration of the fragments and, hence, the frequency of their collisions and explosions at L ₄ turn out to be much greater than those at L ₅. As a result, the number of the secondary fragments of diameter 15 km captured into libration orbits ahead of Jupiter can be as high as many hundreds and should exceed by more than a factor 3.5 that captured behind Jupiter.Since the icy mix of the fragments contains hydrocarbons and particulate material (silicates and the like), after ice sublimation from the surface layers the Trojans should reveal type C and RD spectra typical for Jupiter's irregular satellites, comet nuclei and other distant ice bodies of similar origin. Among the Trojans there cannot be rocky or metallic objects which are known to exist in the main asteroid belt.It is shown that a velocity perturbation of 150–200 m s^–1 resulting from a purely mechanical impact of two bodies may be sufficient to move collision fragments from the orbits of the Trojans to horseshoe-shaped trajectories with a subsequent transfer to the cometary orbits of Jupiter's family.     

Implications of the Galilean satellites ice envelope explosions II. The origin of the irregular satellites

The problem of the origin of the irregular satellites is solved readily in the context of a hypothesis involving explosion of the massive ice envelopes of the Galilean satellites saturated by electrolysis products. The thrown-off unexploded (primary) ice fragments of the outermost cold layers of the envelopes are also saturated by electrolysis products. In the course of explosive ejection their internal energy increases due to shock wave heating, as a result of which they will be able to detonate in subsequent sufficiently energetic collisions. The secondary fragments from new explosions may acquire additional velocity up to a few km s^–1 without breakup into small pieces.Gravitational perturbations by the parent satellites can eject the primary fragments moving near their orbits into the periphery of or beyond Jupiter's sphere of action. If such a fragment explodes in the outer zone of the sphere, then secondary fragments may become irregular satellites resulting in the so-called internal capture (the possibilities of capture considered earlier involved only bodies entering the sphere of action from outside).The mass of the primary fragment responsible for the inner (direct) group of Jupiter's irregular satellites is estimated as 10¹⁹ kg, and the additional velocity acquired by secondary fragments as 1.3 km s^–1; evaluation of the mass of the fragment responsible for the outer (retrograde) group yields 10¹⁸ kg, and that of the additional velocity of secondary fragments, 2 km s^–1.The ice envelopes of the Galilean and similar moonlike satellites should contain impurities corresponding to the composition of type C1 carbonaceous chondrites; therefore after sublimation of water ice the irregular satellites (just as type C asteroids, the Trojans and comets) exhibit spectro-photometric properties similar to those of C-type objects.     

An analytic satellite theory using gravity and a dynamic atmosphere

An analytical solution is given for the motion of an artifical Earth satellite under the combined influences of gravity and atmospheric drag. The gravitational effects of the zonal harmonicsJ ₂,J ₃, andJ ₄ are included, and the drag effects of any arbitrary dynamic atmosphere are included. By a dynamic atmosphere, we mean any of the modern empirical models which use various observed solar and geophysical parameters as inputs to produce a dynamically varying atmosphere model. The subtleties of using such an atmosphere model with an analytic theory are explored, and real world data is used to determine the optimum implementation. Performance is measured by predictions against real world satellites. As a point of reference, predictions against a special perturbations model are also given.     

Physical processes in Jupiter's ring: Clues to its origin by Jove!

The particles making up the Jovian ring may be debris which has been excavated by micrometeoroids from the surfaces of many unseen (R ? 1 km) parent bodies (or “mooms” as we will occasionally call them) residing in the ring. A distribution of particle sizes exists: large objects are sources for the small visible ring particles and also account for the absorption of charged particles noted by Pioneer; the small grains are generated by micrometeoroid impacts, by jostling collisions among different-sized particles, and by self-fracturing due to electrostatic stresses. The latter are most effective in removing surface asperities to thereby produce smooth and crudely equidimensional grains. The presence of intermediate-sized (radius of several to several hundred microns) objects is also expected; these particles will have a total area comparable to the area of the visible ring particles. The nominal size (?2 μm) of the visible particles derived from their forward-scattering characteristics is caused, at least in part, by a selection effect but may also reflect a fundamental grain size or the preferential generation of certain sizes along with the destruction of others. The tiny ring particles have short lifetimes (?10²?10³ years) limited by erosion due to sputtering and meteoroid impacts. Plasma drag significantly modifies orbits in ～10² years but Poynting-Robertson drag is not effective (T_PR ～ 10⁵ years) in removing debris. The ring width is influenced by the distribution of source satellites, by the initial ejection velocity off them, by electromagnetic scattering, and by solar radiation forces. In the absence of electromagnetic forces, debris will reimpact a mother satellite or collide with another particle in about 10 years. A relative drift between different-sized particles, caused by a lessened effective gravity due to the Lorentz force, will substantially shorten these times to less than a month. The ring thickness is determined by a balance between initial conditions (abetted perhaps by electromagnetic scattering) and collisional damping; existence of the “halo” over the diffuse disk compared to its relative absence over the bright ring indicates the presence of mooms in the bright ring but not in the faint disk. Small satellites (R ? 1 km) will not reaccumulate colliding dust grains whereas satellites having the size of J14 or J16 may be able to do so, depending upon their precise shape, size, density, and location. Visible ring structure could indicate separate source satellites. The particles in the faint inner disk are delivered from the bright ring by orbital evolution principally under plasma drag. The halo is comprised of small particles (～0.1 μm) partially drawn out of the faint disk by interactions with the tilted Jovian magnetic field.     

An estimate of the Comet Shoemaker-Levy 9 fragment sizes from the debris fields

The impacts of Comet Shoemaker-Levy 9 left spots on Jupiter with diameters on the order of tens of thousands of kilometers, which have the appearance of debris fields strewn upon the Jovian cloud tops. In this note we employ a measurement of the optical depth of the debris at the impact site of fragment G to estimate mass in the debris field and lower limits to the G fragment mass of 4×10¹² – 4×10¹³ g and diameter of 0.1 – 0.3 km.The masses and sizes of the fragments of Comet Shoemaker-Levy 9 are still uncertain, with estimated sizes ranging from 0.1 to 4 km. The size of the cometary body before breakup is believed to have been between 1 and 10 km. (Asphaug & Benz 1994; Solen 1994; Weaver et al. 1994, Scott & Melosh 1993). These estimates were based on pre-impact images of the cometary fragments. A complimentary technique is to use post-impact images of the spots left on Jupiter to infer the sizes and masses of the fragments.Structure in the underlying clouds is clearly visible through spots imaged by the Hubble Space Telescope, implying that the debris fields are relatively thin. Shortly after the G impact, A'Hearn and collaborators (paper in preparation) used the University of Maryland CCD System at the Perth Observatory to image Jupiter in a variety of bandpasses. While a complete reduction is still underway, a preliminary examination of the raw data shows that the spot at the impact site of fragment G, when near the central meridian roughly three hours after impact, had an average optical depth of roughly 0.05 in several bandpasses between 0.62 and 0.73µm. The measured diameter of the spot was approximately D = 15,000 km.In this note we do not present the data for optical depth, but rather we show that measurements of this type can be used to determine the mass of the solid particles in the clouds and thus to set limits on the mass of the impactor. We assume that the spot consisted of a thin layer of dust in the upper atmosphere. Assuming a one-particle layer covering a fraction of 0.05 of the spot area (a valid assumption for an optically thin cloud), the mass of matter in the spot is M = (0.05/4) dD², where and d are the particle density and diameter. Particle sizes are not directly measured. However, the particle diameters cannot be much less than 1 µm because the CCD observations when compared with HST ultraviolet images show that extinction is not strongly wavelength dependent at optical and near-uv wavelengths. Typical grain sizes in comets and in the zodiacal dust range from 1 to 10 µm. For particle densities of 0.5 g cm^–3 and assumed particle diameters in the range 1 – 10 µm, we find masses, M = 4×10¹² – 4×10¹³ g. Assuming an impactor density of 0.5 g cm^–3 (Asphaug & Benz 1994), the corresponding fragment diameters are 0.1 – 0.3 km. Larger sizes for the grains would increase the estimated mass.The observed debris may not be actual comet dust. Since temperatures in the fireball are estimated to be several thousand degrees, all the material in the fragment should have been vaporized (Sekanina et al 1995; Takata et al 1994; Zahnle & MacLow 1994). Therefore the debris material could consist of recondensed matter, perhaps organics, from the fireball. An impactor collides with roughly its own mass of atmospheric material before disruption, so the estimates for the impactor mass hold to order of magnitude even if the debris contains matter with contributions from originally atmospheric gases.The estimate of 0.1 – 0.3 km diameter for the G fragment is a lower limit because the object would also contain material, for example ices, that would not appear in the debris field. Furthermore, since the HST images show structure in the spots that is unresolved in the observations used here, the spot may not be optically thin at all points, but only on average, and this leads to our estimate being a lower limit for the mass of particles. As noted above, the particles are unlikely to be much less than 1 µm in size; particles much larger than 10µm would also imply a larger mass of particles. The derived fragment size is comparable to those estimated from pre-impact observations.     

A second order analytical atmospheric drag theory based on the TD88 thermospheric density model

A second order atmospheric drag theory based on the usage of TD88 model is constructed. It is developed to the second order in terms of TD88 small parameters K _n,j . The short periodic perturbations, of all orbital elements, are evaluated. The secular perturbations of the semi-major axis and of the eccentricity are obtained. The theory is applied to determine the lifetime of the satellites ROHINI (1980 62A), and to predict the lifetime of the microsatellite MIMOSA. The secular perturbations of the nodal longitude and of the argument of perigee due to the Earth’s gravity are taken into account up to the second order in Earth’s oblateness.     

Gas drag in primordial circumplanetary envelopes: A mechanism for satellite capture

Some natural satellites may have been captured due to the gas drag they experienced in passing through primordial circumplanetary nebulas. This paper models such an encounter and derives the testable parameters from the known properties of current solar system objects and Bodenheimer's (1977, Icarus 31) model of the earliest phases of Jupiter's evolution. We propose that the clusters of prograde and retrograde irregular satellites of Jupiter originated when two parent bodies were decelerated and fragmented as they passed through an extended primordial Jovian nebula. Fragmentation occured because the gas dynamic pressure exceeded the parent bodies' strengths. These events must have occurred only shortly before the primordial nebula experienced hydrodynamical collapse so that subsequently the fragments underwent only limited orbital evolution. Because self-gravity exceeded the relative drag force, the fragments initially remained together, only to be dispersed at a later time by a collision with a stray body. Predictions of this hypothesis, such as orbital distance of the irregular satellites and size of the parent bodies, are found to be consistent with the observed properties of Jupiter's irregular satellites. In addition nebular drag at a later time may have caused the inner three Galilean satellites to undergo a modest amount of orbital evolution, accounting for their present orbital resonance. Gas drag capture of Saturn's Phoebe and Iapetus and Neptune's Nereid and Triton may also be possible. Reasonable differences in properties could explain why these satellites, in contrast to the Jovian ones, did not fracture upon capture. The current irregular satellites represent only a tiny fraction of the bodies captured by primordial nebulas. The dominant fraction would have spiraled into the center of the nebula as a result of continued gas drag and thus offer one source for the heavy element cores of the outer planets. If one is willing to postulate the presence of a massive gaseous nebula around primordial Mars, then gas drag capture could account for the origin of the Martian moons. We hypothesize that a single parent body was captured in a region of the nebula where the gas velocity approached the Keplerian value, that it fragmented upon deceleration into at least two bodies, Phobos and Deimos, and that continued nebular drag led to the low eccentricity and inclination that characterize the satellites' current orbits. Following the dissipation of this nebula, the more massive Phobos tidally evolved to its current position.     

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.     

A CCD search for geosynchronous debris

A search for objects in geosynchronous Earth orbit was conducted using the Spacewatch Camera system. The telescope drive was off so that during integrations the stars were trailed while geostationary objects appeared as round images. The technique should detect geostationary objects to a limiting apparent visual magnitude of 19. A total sky area of 8.8 deg² was searched for geostationary objects; the total sky area monitored for geosynchronous debris passing through the field of view was 16.4 deg². Ten objects were found, three of which were observed on separate nights. Seven of these objects are probably geostationary satellites having apparent visual magnitudes brighter than 13.1. Three objects having magnitudes equal to or fainter than 13.7 showed motion in the north-south direction. The absence of fainter stationary objects suggests that a gap in debris size exists between satellites and particles having diameters in the millimeter range known to exist in geosynchronous orbit.     

Theory of the motion of an artificial Earth satellite

An improved analytical solution is obtained for the motion of an artificial Earth satellite under the combined influences of gravity and atmospheric drag. The gravitational model includes zonal harmonics throughJ ₄, and the atmospheric model assumes a nonrotating spherical power density function. The differential equations are developed through second order under the assumption that the second zonal harmonic and the drag coefficient are both first-order terms, while the remaining zonal harmonics are of second order.Canonical transformations and the method of averaging are used to obtain transformations of variables which significantly simplify the transformed differential equations. A solution for these transformed equations is found; and this solution, in conjunction with the transformations cited above, gives equations for computing the six osculating orbital elements which describe the orbital motion of the satellite. The solution is valid for all eccentricities greater than 0 and less than 0.1 and all inclinations not near 0^o or the critical inclination. Approximately ninety percent of the satellites currently in orbit satisfy all these restrictions.     

Orbit determination of space debris: admissible regions

The main problem in the orbit determination of the space debris population orbiting our planet is identifying which separate sets of data belong to the same physical object. The observations of a given object during a passage above an observing station are collectively called a Too Short Arc (TSA): data from a TSA cannot allow for a complete determination of an orbit. Therefore, we have to solve first the identification problem, finding two or more TSAs belonging to the same physical object and an orbit fitting all the observations. This problem is well known for the determination of orbits of asteroids: we shall show how to apply the methods developed for preliminary orbit determination of heliocentric objects to geocentric objects. We shall focus on the definition of an admissible region for space debris, both in the case of optical observations and radar observations; then we shall outline a strategy to perform a full orbit determination.     

Nearby young single black holes

We consider nearby young black holes formed after supernova explosions in close binaries whose secondary components are currently observed as the so-called runaway stars. Using data on runaway stars and making reasonable assumptions about the mechanisms of supernova explosion and binary breakup, we estimate the present positions of nearbyyoung black holes. For two objects, we obtained relatively small error regions (～50–100 deg²). The possibility of detecting these nearby young black holes is discussed.     

Deducing the age of the dense Venus atmosphere

We show how crater size-density counts may be used to help constrain the history of the Venus atmosphere, based on the predictions of simple but reasonable models for crater production, surface erosion, and the effects of atmospheric drag and breakup on incident meteors in the Venus atmosphere. If the atmosphere is old, we may also be able to determine the importance of breakup as a mechanism for destroying incident meteors in a dense fluid. In particular, if the atmosphere is young, the old (uneroded) surfaces will have crater densities upward of 10^?4 km^?2 and a ratio of small (4 km) craters to large (128 km) craters near 10³. If the atmosphere is old and the breakup mechanism is dominant, absolute crater densities on Venus surfaces will be diminished by several orders of magnitude relative to the young atmosphere case. If atmospheric drag is dominant and the atmosphere is old, the absolute crater density will be lowered by perhaps an order of magnitude relative to the young atmosphere case, and the ratio of small to large craters will be reduced to a value near 10^1.5 according to the models. The comparison of crater populations on young, as well as old, surfaces on Venus can help in distinguishing the young and old atmosphere scenarios, especially since the situation may be complicated by currently undetermined erosional and tectonic processes. Once a large fraction of Venus surface has been imaged at kilometer resolution, as the VOIR project promises to do, it could be possible to make an early determination of the age of the Venus atmosphere.     

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.     

Deciphering the origin of the regular satellites of gaseous giants - Iapetus: The Rosetta ice-moon

Ever since their discovery the regular satellites of Jupiter and Saturn have held out the promise of providing an independent set of observations with which to test theories of planet formation. Yet elucidating their origins has proven elusive. Here we show that Iapetus can serve to discriminate between satellite formation models. Its accretion history can be understood in terms of a two-component gaseous subnebula, with a relatively dense inner region, and an extended tail out to the location of the irregular satellites, as in the SEMM model of Mosqueira and Estrada (2003a,b) (Mosqueira, I., Estrada, P.R. [2003a]. Icarus 163, 198-231; Mosqueira, I., Estrada, P.R. [2003b]. Icarus 163, 232-255). Following giant planet formation, planetesimals in the feeding zone of Jupiter and Saturn become dynamically excited, and undergo a collisional cascade. Ablation and capture of planetesimal fragments crossing the gaseous circumplanetary disks delivers enough collisional rubble to account for the mass budgets of the regular satellites of Jupiter and Saturn. This process can result in rock/ice fractionation as long as the make up of the population of disk crossers is non-homogeneous, thus offering a natural explanation for the marked compositional differences between outer solar nebula objects and those that accreted in the subnebulae of the giant planets. For a given size, icy objects are easier to capture and to ablate, likely resulting in an overall enrichment of ice in the subnebula. Furthermore, capture and ablation of rocky fragments become inefficient far from the planet for two reasons: the gas surface density of the subnebula is taken to drop outside the centrifugal radius, and the velocity of interlopers decreases with distance from the planet. Thus, rocky objects crossing the outer disks of Jupiter and Saturn never reach a temperature high enough to ablate either due to melting or vaporization, and capture is also greatly diminished there. In contrast, icy objects crossing the outer disks of each planet ablate due to the melting and vaporization of water-ice. Consequently, our model leads to an enhancement of the ice content of Iapetus, and to a lesser degree those of Titan, Callisto and Ganymede, and accounts for the (non-stochastic) compositions of these large, low-porosity outer regular satellites of Jupiter and Saturn. For this to work, the primordial population of planetesimals in the Jupiter-Saturn region must be partially differentiated, so that the ensuing collisional cascade produces an icy population of ?1 m size fragments to be ablated during subnebula crossing. We argue this is likely because the first generation of solar nebula ∼10 km planetesimals in the Jupiter-Saturn region incorporated significant quantities of ²⁶Al. This is the first study successfully to provide a direct connection between nebula planetesimals and subnebulae mixtures with quantifiable and observable consequences for the bulk properties of the regular satellites of Jupiter and Saturn, and the only explanation presently available for Iapetus’ low density and ice-rich composition.     

On the process of accretion in the formation of the planets and comets

The physically reasonable assumption that the seed bodies which initiated the accretion of the individual asteroids, planets, and comets (subsequently these objects are collectively called planetoids) formed by stochastic processes requires a radius distribution function which is unique except for two scaling parameters: the total number of planetoids and their most probable radius. The former depends on the ease of formation of the seed bodies while the second is uniquely determined by the average pre-encounter velocity, V, of the accretable material relative to an individual planetoid. This theoretical radius function can be fit to the initial asteroid radius distribution which Anders (1965) derived from the present-day distribution by allowing for fragmentation collisions among the asteroids since their formation. Normalizing the theoretical function to this empirical distribution reveals that there were about 10² precollision asteroids and that V = (2?4) × 10^?2 km/sec which was presumably the turbulent velocity in the Solar Nebula. Knowing V we can determine the scale height of the dust in the Solar Nebula and consequently its space density. The density of accretable material determines the rate of accretion of the planetoids. From this we find, for example, that the Earth formed in about 8 × 10⁶ yr and it attained a maximum temperature through accretion of about 3 × 10³°K. From the total mass of the terrestrial planets and the theoretical radius function we find that about 2 × 10³ planetoids formed in the vicinity of the terrestrial planets. Except for the asteroids the smaller planetoids have since been accreted by the terrestrial planets. About 15% of the present mass of the terrestrial planets was accumulated by the secondary accretion of these smaller primary planetoids. There are far fewer primary planetoids than craters on the Moon or Mars. The craters were likely produced by the collisional breakup of a few primary planetoids with masses between one-tenth and one lunar mass. This deduction comes from comparing the collision cross sections of the planetoids in this mass range to that of the terrestrial planets. This comparison shows that two to three collisions leading to the breakup of four to six objects likely occurred among these objects before their accretion by the terrestrial planets. The number of these fragments is quite adequate to explain the lunar and Martin craters. Furthermore the mass spectrum of such fragments is a power-law distribution which results in a power-law distribution of crater radii of just the type observed on the Moon and Mars. Applying the same analysis to the planetoids which formed in the vicinity of the giant planets reveals that it is unlikely that any fragmentation collisions took place among them before they were accreted by these planets due to the integrated collision cross section of the giant planets being about three orders of magnitude greater than that of the terrestrial planets. We can thus anticipate a marked scarcity of impact craters on the satellites of these outer planets. This prediction can be tested by future space probes. Our knowledge of the radius function of the comets is consistent with their being primary planetoids. The primary difference between the radius function of the planetoids which formed in the inner part of the solar system and that of the comets results from the fact that the seed bodies which grew into the comets formed far more easily than those which grew into the asteroids and the terrestrial planets. Thus in the outer part of the Solar Nebula the principal solid material (water and ammonia snow) accreted into a huge (～10¹²⁺) number of relatively small objects (comets) while in the inner part of the nebula the solid material (hard-to-stick refractory substances) accumulated into only a few (～10³) large objects (asteroids and terrestrial planets). Uranus and Neptune presumably formed by the secondary accretion of the comets.     

Very low strengths of interplanetary meteoroids and small asteroids

Abstract– We have assembled data on 13 cases of meteorite falls with accurate tracking data on atmospheric passage. In all cases, we estimate the bulk strength of the object corresponding to its earliest observed or inferred fragmentation in the high atmosphere, and can compare these values with measured strengths of meteorites in the taxonomic class for that fall. In all 13 cases, the strength corresponding to earliest observed or inferred fragmentation is much less than the compressive or tensile strength reported for that class of stony meteorites. Bulk strengths upon atmospheric entry of these bodies are shown to be very low, 0.1 to approximately 1 MPa on first breakup, and maximal strength on breakup as 1–10 MPa corresponding to weak and “crumbly” objects, whereas measured average tensile strength of the similar meteorite classes is about 30 MPa. We find a more random relation between bulk sample strength and sample mass than is suggested by a commonly used empirical power law. We estimate bulk strengths on entry being characteristically of the order of 10^?1–10^?2 times the tensile strengths of recovered samples. We conclude that pre‐entry, meter‐scale interplanetary meteoroids are typically highly fractured or in some cases rubbly in texture, presumably as a result of their parent bodies’ collisional history, and can break up under stresses of a few megapascals. The weakness of some carbonaceous objects may result from very porous primordial accretional structures, more than fractures. These conclusions have implications for future asteroid missions, sample extraction, and asteroid hazard mitigation.