Orbital evolution of asteroids with the effects of mutual gravitational attraction

The effects of the mutual gravitational attraction between asteroids were analyzed by two N-body calculations, in which N=4,516 (the Sun, the nine planets, and 4,506 asteroids). In one calculation the gravity of the asteroids was taken into account, and in the other it was ignored. These calculations were carried out for a time period of about 100 years. The largest difference in the positions of the asteroids between these two calculations is about 10^–3 AU. For the orbital elements of the semimajor axis, the eccentricity, and the inclination, the largest differences were 9 × 10^–6 AU, 4 × 10^–6, and 5 × 10^–4 degrees, respectively. It was found that the distribution of the differences of the semimajor axis between the two calculations is quite similar to the Cauchy distribution.     

The Effect of Tidal Interaction with a Gas Disk on Formation of Terrestrial Planets

We have performed N-body simulation on final accretion stage of terrestrial planets, including the effect of damping of eccentricity and inclination caused by tidal interaction with a remnant gas disk. As a result of runway and oligarchic accretion, about 20 Mars-sized protoplanets would be formed in nearly circular orbits with orbital separation of several to ten Hill radius. The orbits of the protoplanets would be eventually destabilized by long-term mutual gravity and/or secular resonance of giant gaseous planets. The protoplanets would coalesce with each other to form terrestrial planets through the orbital crossing. Previous N-body simulations, however, showed that the final eccentricities of planets are around 0.1, which are about 10 times higher than the present eccentricities of Earth and Venus. The obtained high eccentricities are the remnant of orbital crossing. We included the effect of eccentricity damping caused by gravitational interaction with disk gas as a drag force (“gravitational drag”) and carried out N-body simulation of accretion of protoplanets. We start with 15 protoplanets with 0.2M⊕ and integrate the orbits for 10⁷ years, which is consistent with the observationally inferred disk lifetime (in some runs, we start with 30 protoplanets with 0.1M⊕). In most runs, the damping time scale, which is equivalent to the strength of the drag force, is kept constant throughout each run in order to clarify the effects of the damping. We found that the planets' final mass, spatial distribution, and eccentricities depend on the damping time scale. If the damping time scale for a 0.2M⊕ mass planet at 1 AU is longer than 10⁸ years, planets grow to Earth's size, but the final eccentricities are too high as in gas-free cases. If it is shorter than 10⁶ years, the eccentricities of the protoplanets cannot be pumped up, resulting in not enough orbital crossing to make Earth-sized planets. Small planets with low eccentricities are formed with small orbital separation. On the other hand, if it is between 10⁶ and 10⁸ years, which may correspond to a mostly depleted disk (0.01-0.1% of surface density of the minimum mass model), some protoplanets can grow to about the size of Earth and Venus, and the eccentricities of such surviving planets can be diminished within the disk lifetime. Furthermore, in innermost and outermost regions in the same system, we often find planets with smaller size and larger eccentricities too, which could be analogous to Mars and Mercury. This is partly because the gravitational drag is less effective for smaller mass planets, and partly due to the “edge effect,” which means the innermost and outermost planets tend to remain without collision. We also carried out several runs with time-dependent drag force according to depletion of a gas disk. In these runs, we used exponential decay model with e-folding time of 3×10⁶ years. The orbits of protoplanets are stablized by the eccentricity damping in the early time. When disk surface density decays to ?1% of the minimum mass disk model, the damping force is no longer strong enough to inhibit the increase of the eccentricity by distant perturbations among protoplanets so that the orbital crossing starts. In this disk decay model, a gas disk with 10⁻⁴-10⁻³ times the minimum mass model still remains after the orbital crossing and accretional events, which is enough to damp the eccentricities of the Earth-sized planets to the order of 0.01. Using these results, we discuss a possible scenario for the last stage of terrestrial planet formation.     

A Period Study of the Rs CVn-Type Binary ER Vulpeculae

Many available published times of light minima of the late-type binary system ER Vul have been compiled and analyzed using a new method proposed by Kalimeris et al. (1994). It was shown that the orbital period of the system oscillation with a period of about 30.6 years and an amplitude of 3.2×10^-6 days while it undergoes a constant period decrease of about dP /P=7.84× 10^-8 day / year. The prospective physical mechanisms that could have modulated the orbital period behaviour (periodic or non-periodic), have been studied. We found that a combination of a magnetic activity cycle mechanism and an enhanced stellar wind could explain satisfactorily the period change.     

The case for a wet,warm climate on early Mars

Theoretical arguments are presented in support of the idea that Mars possessed a dense CO₂ atmosphere and a wet, warm climate early in its history. Calculations with a one-dimensional radiative-convective climate model indicate that CO₂ pressures between 1 and 5 bars would have been required to keep the surface temperature above the freezing point of water early in the planet's history. The higher value corresponds to globally and orbitally averaged conditions and a 30% reduction in solar luminosity; the lower value corresponds to conditions at the equator during perihelion at times of high orbital eccentricity and the same reduced solar luminosity.The plausibility of such a CO₂ greenhouse is tested by formulating a simple model of the CO₂ geochemical cycle on early Mars. By appropriately scaling the rate of silicate weathering on present Earth, we estimate a weathering time constant of the order of several times 10⁷ years for early Mars. Thus, a dense atmosphere could have persisted for a geologically significant time period (∼10⁹years) only if atmospheric CO₂ was being continuously resupplied. The most likely mechanism by which this might have been accomplished is the thermal decomposition of carbonate rocks induced directly and indirectly (through burial) by intense, global-scale volcanism. For plausible values of the early heat flux, the recycling time constant is also of the order of several times 10⁷ years. The amount of CO₂ dissolved in standing bodies of water was probably small; thus, the total surficial CO₂ inventory required to maintain these conditions was approximately 2 to 10 bars. The amount of CO₂ in Mars' atmosphere would eventually have dwindled, and the climate cooled, as the planet's internal heat engine ran down. A test for this theory will be provided by spectroscopic searches for carbonates in Mars' crust.     

Mass effects in the problem of three bodies

This is a study of the dynamical behavior of three point masses moving under their mutual gravitational attraction in a plane. The initial positions and velocities are identical for all cases studied and only the masses of the participating bodies change in the series of numerical experiments. In this way the effect of the coupling terms in the differential equations of motion are investigated. The motion in all 125 cases begins with an interplay between the three bodies, followed by temporary ejections or by an eventual escape. The total mass of the system is kept constant while the massratios change from 1 to 5. The initial velocities being zero, the total energy is negative in all cases.Approximately 74% of the cases disintegrated (i.e. two bodies formed a binary and the third body escaped) in less than 140 time units, 47% in less than 50 time units and 10% ended in escape in less than 10 time units. Considering three stars with total mass 12M , initially placed at 3, 4 and 5 parsec distances (or three galaxies with mass 2.4×10¹² M , initially placed 30, 40 and 50 kpc apart), the unit of time (approximately the crossing time) becomes 1.5×10⁷ y (3.2×10⁷ y). The average time of disintegration was found to be of the order of 10⁹ y. The average semi-major axis of the binaries left behind after disintegration was 0.7 parsec and the average value of the eccentricity was 0.76. The effect of the masses on the escapes was established and it was found that the bodynot with the smallest mass escaped in 13% of the disintegrated cases. The cases which did not disintegrate in 150 time units were analyzed in detail and the time of their eventual escape was estimated.The numerical results are tabulated regarding escape time, ejection period, total energy, escape energy, terminal velocity, semi-major axis, and eccentricity.The evolution of triple systems is followed from interplays through ejections to escapes and the orbital parameters for the separation of these classes are estimated.     

Orbital evolution of asteroids with the effects of mutual gravitational attraction

The effects of the mutual gravitational attraction between asteroids were analyzed by two N-body calculations, in which N=4,516 (the Sun, the nine planets, and 4,506 asteroids). In one calculation the gravity of the asteroids was taken into account, and in the other it was ignored. These calculations were carried out for a time period of about 100 years. The largest difference in the positions of the asteroids between these two calculations is about 10^?3 AU. For the orbital elements of the semimajor axis, the eccentricity, and the inclination, the largest differences were 9 × 10^?6 AU, 4 × 10^?6, and 5 × 10^?4 degrees, respectively. It was found that the distribution of the differences of the semimajor axis between the two calculations is quite similar to the Cauchy distribution.     

Orbital period analysis of eclipsing dwarf novae: U Geminorum

A new orbital period analysis for U Geminorum is made by means of the standard O–C technique based on 187 times of light minima including the three newest CCD data from our observation. Although there are large scatter near 70,000 cycles in its O–C diagram, there is strong evidence (>99.9% confidence level) to show the secular increase of orbital period with a rate  s⁻¹. Using the physical parameters recently derived by Echevarría et al. (Astron. J. 134:262, 2007), the range of mass transfer rate for U Geminorum is estimated as from −3.5(5)×10⁻⁹ M _⊙ yr⁻¹ to −1.30(6)×10⁻⁸ M _⊙ yr⁻¹. Moreover, the data before 60,000 cycles shows the obvious quasi-period variations. The least square estimation of a ∼17.4 yr quasi-periodic variation superimposed on secular orbital period increase is derived. Considering the possibility that solar-type magnetic activity cycles in the secondary star of U Geminorum may produce the quasi-period variations of the orbital period, Applegate’s mechanism is discussed and the results indicate such mechanism has difficulty explaining the quasi-period variation for U Geminorum. Hence, we attempted to apply the light-travel time effect to interpret the quasi-period variation and found the perturbation of ∼17.4 yr quasi-period may result from a brown dwarf. If the orbital inclination is assumed as i∼15°, corresponding to the upper limit of mass of a brown dwarf, then its orbital radii is ∼7.7 AU.     

Earth albedo and the orbit of Lageos

The long-period perturbations in the orbit of Lageos satellite due to the earth's albedo have been found using a new analytical formalism. The earth is assumed to be a sphere whose surface diffusely reflects sunlight according to Lambert's law. Specular reflection is not considered. The formalism is based on spherical harmonics; it produces equations which hold regardless of whether the terminator is seen by the satellite or not. Specializing in the case of a realistic zonal albedo shows that Lageos' orbital semimajor axis changes periodically by only about a centimeter and the eccentricity by two parts in 10⁵. The longitude of the node increases secularly by about 6×10^–4 arc sec yr^–1. The effect considered here can explain neither the secular decay of 1.1 mm day^–1 in the semimajor axis nor the observed along-track variations in acceleration of order 2×10^–12 ms^–2.     

The enigma of the Uranian satellites' orbital eccentricities

Using recently published determinations of the diameters and orbital elements of the uranian satellites and assuming reasonable dissipation functions and rigidities for icy satellites, the eccentricity decay times for the satellites were calculated. For the inner three, decay times are on the order of 10⁷–10⁸ years, making it difficult to understand why these satellites still have their observed eccentricities. The three inner satellites have a near-commensurability in their mean motions that may be able to force their eccentricities at some time in the future, but cannot force them now. Several possible explanations exist: (1) The reported eccentricities are incorrect, and are in fact near-zero. (2) The reported mean motions are incorrect, and an exact commensurability exists. (3) The physical properties that we have assumed for the satellites are grossly in error (e.g., dissipation function Q is in reality very large). (4) The system is evolving very rapidly, perhaps from a previous state of higher eccentricity. Cases 1 and 2 are unlikely when one considers the quality of existing data. Case 3 would be more consistent with non-icy compositions. Cases 2 and 4 would imply some tidal heating of the satellites, particularly Ariel. A new lower bound of ～ 1.7 × 10⁴ on the Q of Uranus is calculated from the mass of Ariel and its proximity to Uranus.     

Five new and three improved mutual orbits of transneptunian binaries

We present three improved and five new mutual orbits of transneptunian binary systems (58534) Logos-Zoe, (66652) Borasisi-Pabu, (88611) Teharonhiawako-Sawiskera, (123509) 2000 WK₁₈₃, (149780) Altjira, 2001 QY₂₉₇, 2003 QW₁₁₁, and 2003 QY₉₀ based on Hubble Space Telescope and Keck II laser guide star adaptive optics observations. Combining the five new orbit solutions with 17 previously known orbits yields a sample of 22 mutual orbits for which the period P, semimajor axis a, and eccentricity e have been determined. These orbits have mutual periods ranging from 5 to over 800 days, semimajor axes ranging from 1600 to 37,000 km, eccentricities ranging from 0 to 0.8, and system masses ranging from 2 × 10¹⁷ to 2 × 10²² kg. Based on the relative brightnesses of primaries and secondaries, most of these systems consist of near equal-sized pairs, although a few of the most massive systems are more lopsided. The observed distribution of orbital properties suggests that the most loosely-bound transneptunian binary systems are only found on dynamically cold heliocentric orbits. Of the 22 known binary mutual orbits, orientation ambiguities are now resolved for 9, of which 7 are prograde and 2 are retrograde, consistent with a random distribution of orbital orientations, but not with models predicting a strong preference for retrograde orbits. To the extent that other perturbations are not dominant, the binary systems undergo Kozai oscillations of their eccentricities and inclinations with periods of the order of tens of thousands to millions of years, some with strikingly high amplitudes.     

Long‐term orbital period behaviors of the neglected Algol type binaries: CC Herculis and XZ Aquilae

Orbital period variations of two neglected Algol type binaries, CC Her and XZ Aql, are studied based on all available times of minima. In the case of CC Her, it is found that the O –C curve displays a tilted sinusoidal variation with an eccentricity of 0.54 ± 0.03 and a period of 52.4 ± 0.4 yr, which can be explained by the light‐time effect due to the presence of an unseen component. The course of the orbital period change in XZ Aql appears less reliable but its O –C curve can be represented by a periodic variation with a period of 36.7 ± 0.6 yr superimposed on an upward parabola. The parabolic variation indicates a secular period increase with a rate of dP /dt = 7.1 s per century. The corresponding conservative mass transfer from less massive component to the more massive one is about 3.26 × 10^–7 M_⊙ yr^–1. It is interesting to see that the O –C variation of CC Her displays no evidence (as upward parabola) on the mass transfer characteristic for Algols. The periodic change of the orbital period of XZ Aql, like CC Her, may be caused by the presence of the thirdbody. The lower limits of the masses of the hypothetical unseen components for CC Her and XZ Aql are found to be 2.69 M_⊙ and 0.47 M_⊙, respectively. The third body of CC Her should be detectable not only spectroscopically but also photoelectrically, if it exists. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new light‐time effect study of AR Aurigae

A period study of the young binary AR Aur based on the extensive series of published photoelectric/ccd minima times indicates the cyclic (O – C) variation for the system. This continuous oscillatory variation covers almost three cycles, about 6000 orbital periods, by the present observational data. It can be attributed to the light‐time effect due to a third body with a period of 23.68 ± 0.17 years in the system. The analysis yields a light‐time semi‐amplitude of 0.0084 ± 0.0002 day and an orbital eccentricity of 0.20 ± 0.04. Adopting the total mass of AR Aur, the mass of the third body assumed in the co‐planar orbit with the binary is M₃ = 0.54 ± 0.03 M_⊙ and the semimajor axis of its orbit is a₃ = 13.0 + 0.2 AU. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Negative diffusion in planetary rings with a nearby moon

This paper analyzes a process that has been observed in simulations of numerous systems where ring material is strongly perturbed by a nearby moon. If the ring particles can be imparted with a forced eccentricity on the order of 10⁻⁵ in a single pass by the moon, particle orbits are observed to move towards regions of higher density as a result of the organized collisions that occur in the dense peaks of the satellite wake. The width of the ring can decrease by as much as 90% if the forced eccentricity is greater than 3 × 10⁻⁵ and the unperturbed geometric optical depth is greater than 0.03. The fractional change in ring width is relatively insensitive to the particle size so long as the particle radius is much less than the product of the semimajor axis and the forced eccentricity. Including a power law particle size distribution with slope of −2.8 spanning a decade in particle radius reduces the fractional width change by about 10% compared to the uniform particle-size case. Adding gravitational interactions between ring particles only has a significant effect on ring confinement if the unperturbed geometric optical depth exceeds .03, but a 40% reduction in ring width is still achieved in a self-gravitating ring of geometric optical depth 0.3 if the forced eccentricity exceeds 3 × 10⁻⁵. This process does not require the material to be in resonance with the moon, nor does it have any minimum mass constraints because particle self-gravity is not required. The collisional damping of satellite wakes therefore provides a simple mechanism by which a single moon can reduce the radial extent of any ringlet that is close to it and has sufficient optical depth for collisions to be significant.     

Solar Polarity Dependence of Cosmic Ray Power Spectra Observed with Mawson Underground Muon Telescopes

Power spectral density (PSD) of cosmic rays has been calculated from hourly averaged counts observed by underground muon telescopes located at Mawson over the low-frequency range 2.7×10⁻⁷ – 1.4×10⁻⁴ Hz. The first two harmonics of the solar daily variation are well defined for even cycles (20 and 22) whereas only the first harmonic is defined in cycle 21. The amplitude of the diurnal variation is lower for even cycles than for the odd cycle. The spectral power of the odd cycle exceeds those of the even cycles. The spectra are flatter and have lower power when the interplanetary magnetic field (IMF) is directed away from the Sun above the current sheet (A>0) than when the IMF is directed toward the Sun above the current sheet (A<0). The spectra imply that heliospheric magnetic turbulence may be more variable on time scales of several years than previously suspected.     

Spin states and climates of eccentric exoplanets

The known extrasolar planets exhibit a wide range of orbital eccentricities e. This has a profound influence on their rotations and climates. Because of tides in their interiors, mostly solid exoplanets are expected eventually to despin to a state of spin-orbit resonance, where the orbital period is some integer or half-integer times the rotation period. The most important of these resonances is the synchronous state, where the planet's spin period exactly equals its orbital period (like Earth's Moon, and indeed most of the regular satellites in the Solar System). Such planets seem doomed to roast on one side and freeze on the other. However, synchronous planets rock back and forth by an angle of ∼2Arcsine with respect to the sub-stellar point. For e=0.055 (as for the Moon), this optical libration amounts to only ∼6°; but for a synchronous planet with e=0.50, for example, it would rise to ∼59°. This greatly expands the temperate “twilight zone” near the terminator and considerably improves the planet's prospects for habitability. For e?0.72389, the optical libration exceeds 90°; for such planets, the sector of permanent night vanishes, while the sunniest region splits in two. Furthermore, the synchronous state is not the only possible spin resonance. For example, Mercury (with e≈0.206) has an orbital period exactly 1.5 times its rotation period. A terrestrial exoplanet with e=0.40, say, is liable to have an orbital period of 2.0, 2.5, or 3.0 times its spin period. The corresponding insolation patterns are generally complicated, and all different from the synchronous state. Yet these non-synchronous resonances also protect certain longitudes from the worst extremes of temperature and solar radiation, and improve the planet's habitability, compared to non-resonant rotation. These results also have implications for the direct detectability of extrasolar planets, and the interpretation of their thermal emissions.     

Orbital Period Studies of the RS CVn-Type BinariesII. UV Piscium

Many available published times of light minima of the active binary system UV Psc have been collected and analyzed using a new method proposed by Kalimeris et al. (1994). Similar to what was seen in other RS CVn-type binaries, the orbital period of UV Psc oscillates with a period of about 61 years and an semi-amplitude of 0.21 ×10^-5 days. Two possible mechanisms (magnetic activity cycle mechanism and a light-time effect due to a hypothetical third body) that could modulated the orbital period behaviour are studied. We think that the cyclical period change in UV Psc can most probably be attributed to a magnetic activity cycle in the primary component. This revised version was published online in July 2006 with corrections to the Cover Date.     

Excitation of Orbital Inclinations of Asteroids during Depletion of a Protoplanetary Disk: Dependence on the Disk Configuration

The pumping up of orbital inclinations of asteroids caused by sweeping secular resonances associated with depletion of a protoplanetary disk is discussed, focusing on the dependence on the disk inclinations and surface density distribution. The asteroids have large mean inclinations that cannot be explained by present planetary perturbations alone. It has been suggested that the sweeping secular resonances caused by disk depletion are responsible for these high inclinations. Nagasawa et al. (2000, Astron. J.119, 1480-1497) showed that the inclinations of asteroids are pumped up if the disk is depleted in an inside-out manner on a time scale longer than 3×10⁵ years. Their assumed disk midplane is not on the invariant plane. However, it should be affected by the inclination of the disk plane. Here we investigate the dependence on the disk inclinations. We assume a disk depletion model in which the disk inside the jovian orbit has been removed and the residual outer disk is uniformly depleted. We calculate the locations of the secular resonances and the excitation magnitude of the inclinations with analytical methods. We found that the inclinations are pumped up to the observational level for a depletion time scale longer than 10⁶ years in the case of the disk plane that coincides with the invariant plane. The required time scale is longest (3×10⁶ years) if the disk plane coincides with the jovian orbital plane. However, it is still within the observationally inferred depletion time scale. We also studied dependence on a disk surface density gradient and found that the results do not change significantly as long as the inner disk depletion is faster than the outer disk one.     

Influence of fast interstellar gas flow on the dynamics of dust grains

The orbital evolution of a dust particle under the action of a fast interstellar gas flow is investigated. The secular time derivatives of Keplerian orbital elements and the radial, transversal, and normal components of the gas flow velocity vector at the pericentre of the particle’s orbit are derived. The secular time derivatives of the semi-major axis, eccentricity, and of the radial, transversal, and normal components of the gas flow velocity vector at the pericentre of the particle’s orbit constitute a system of equations that determines the evolution of the particle’s orbit in space with respect to the gas flow velocity vector. This system of differential equations can be easily solved analytically. From the solution of the system we found the evolution of the Keplerian orbital elements in the special case when the orbital elements are determined with respect to a plane perpendicular to the gas flow velocity vector. Transformation of the Keplerian orbital elements determined for this special case into orbital elements determined with respect to an arbitrary oriented plane is presented. The orbital elements of the dust particle change periodically with a constant oscillation period or remain constant. Planar, perpendicular and stationary solutions are discussed. The applicability of this solution in the Solar System is also investigated. We consider icy particles with radii from 1 to 10 μm. The presented solution is valid for these particles in orbits with semi-major axes from 200 to 3000 AU and eccentricities smaller than 0.8, approximately. The oscillation periods for these orbits range from 10⁵ to 2 × 10⁶ years, approximately.     

Orbital evolution measurement of the accreting millisecond X-ray pulsar SAX J1808.4-3658

We present results from a pulse timing analysis of the accretion-powered millisecond X-ray pulsar SAX J1808.4-3658 using X-ray data obtained during four outbursts of this source. Extensive observations were made with the proportional counter array of the Rossi X-ray Timing Explorer (RXTE) during the four outbursts that occurred in 1998, 2000, 2002 and 2005. Instead of measuring the arrival times of individual pulses or the pulse arrival time delay measurement that is commonly used to determine the orbital parameters of binary pulsars, we have determined the orbital ephemeris during each observation by optimizing the pulse detection against a range of trial ephemeris values. The source exhibits a significant pulse shape variability during the outbursts. The technique used by us does not depend on the pulse profile evolution, and is therefore, different from the standard pulse timing analysis. Using 27 measurements of orbital ephemerides during the four outbursts spread over more than 7 years and more than 31,000 binary orbits, we have derived an accurate value of the orbital period of 7249.156862(5) s (MJD = 50915) and detected an orbital period derivative of (3.14 ± 0.21) × 10⁻¹² s s⁻¹. We have included a table of the 27 mid-eclipse time measurements of this source that will be valuable for further studies of the orbital evolution of the source, especially with ASTROSAT. We point out that the measured rate of orbital period evolution is considerably faster than the most commonly discussed mechanisms of orbital period evolution like mass transfer, mass loss from the companion star and gravitational wave radiation. The present time scale of orbital period change, 73 Myr is therefore likely to be a transient high value of period evolution and similar measurements during subsequent outbursts of SAX J1808.4-3658 will help us to resolve this.     

On the Inclination Distribution of the Jovian Irregular Satellites

Irregular satellites—moons that occupy large orbits of significant eccentricity e and/or inclination I—circle each of the giant planets. The irregulars often extend close to the orbital stability limit, about 1/3-1/2 of the way to the edge of their planet's Hill sphere. The distant, elongated, and inclined orbits suggest capture, which presumably would give a random distribution of inclinations. Yet, no known irregulars have inclinations (relative to the ecliptic) between 47 and 141°.This paper shows that many high-I orbits are unstable due to secular solar perturbations. High-inclination orbits suffer appreciable periodic changes in eccentricity; large eccentricities can either drive particles with ∼70°<I<110° deep into the realm of the regular satellites (where collisions and scatterings are likely to remove them from planetocentric orbits on a timescale of 10⁷-10⁹ years) or expel them from the Hill sphere of the planet.By carrying out long-term (10⁹ years) orbital integrations for a variety of hypothetical satellites, we demonstrate that solar and planetary perturbations, by causing particles to strike (or to escape) their planet, considerably broaden this zone of avoidance. It grows to at least 55°<I<130° for orbits whose pericenters freely oscillate from 0 to 360°, while particles whose pericenters are locked at ±90° (Kozai mechanism) can remain for longer times.We estimate that the stable phase space (over 10 Myr) for satellites trapped in the Kozai resonance contains ∼10% of all stable orbits, suggesting the possible existence of a family of undiscovered objects at higher inclinations than those currently known.