Orbital evolution of the Galilean satellites: Capture into resonance

C.F. Yoder's scenario 1979 for the capture into resonance of the first three Galilean satellites is reexamined. A more refined dynamical model for the resonance and for the tidal effects is proposed and analyzed. The results agree qualitatively with those of Yoder but differ numerically by 10 to 20%.     

Second order adiabatic invariants associated with the two-body problem with slowly varying mass

An adiabatic invariant is characterized by the property that its derivative is small and oscillatory. Therefore, assuming that such a quantity is constant does not lead to a cumulative error as t. In this paper, using action and angle variables, adiabatic invariants to 0(1) and 0() are found for the two-body problem with a slowly varying gravitational constant.     

Capture of grains into resonances through Poynting-Robertson drag

We review here some relevant problems connected to the evolution of circumstellar dust grains, subjected to Poynting-Robertson (PR) drag, and perturbed by first-order resonances with a planet on a circular orbit. We show that only outer mean motion resonances are able to counteract the damping effect of PR drag. However, the high orbital eccentricities reached by the particle lead to orbit crossings with the planet. This is a serious difficulty for a permanent trapping to be achieved. In any case, we show that the time spent in the resonance is long enough for statistical effects (accumulation at the resonant radius) to be significant. We underline some difficulties associated with this problem, namely, the non-adiabaticity of motion in the resonance phase space and the existence of close encounters with the planet at high eccentricities.     

Capture probability in the 3:1 mean motion resonance with Jupiter: an application to the Vesta family

We study the capture and crossing probabilities in the 3:1 mean motion resonance with Jupiter for a small asteroid that migrates from the inner to the middle Main Belt under the action of the Yarkovsky effect. We use an algebraic mapping of the averaged planar restricted three-body problem based on the symplectic mapping of Hadjidemetriou (Celest Mech Dyn Astron 56:563–599, 1993), adding the secular variations of the orbit of Jupiter and non-symplectic terms to simulate the migration. We found that, for fast migration rates, the captures occur at discrete windows of initial eccentricities whose specific locations depend on the initial resonant angles, indicating that the capture phenomenon is not probabilistic. For slow migration rates, these windows become narrower and start to accumulate at low eccentricities, generating a region of mutual overlap where the capture probability tends to 100 %, in agreement with the theoretical predictions for the adiabatic regime. Our simulations allow us to predict the capture probabilities in both the adiabatic and non-adiabatic cases, in good agreement with results of Gomes (Celest Mech Dyn Astron 61:97–113, 1995) and Quillen (Mon Not RAS 365:1367–1382, 2006). We apply our model to the case of the Vesta asteroid family in the same context as Roig et al. (Icarus 194:125–136, 2008), and found results indicating that the high capture probability of Vesta family members into the 3:1 mean motion resonance is basically governed by the eccentricity of Jupiter and its secular variations.     

Capture of comets from the Oort cloud into Halley-type and Jupiter-family orbits

This paper analyzes the capture of comets into Halley-type and Jupiter-family orbits from the nearparabolic flux of the Oort cloud. Two types of capture into Halley-type orbits are found. The first type is the evolution of near-parabolic orbits into short-period orbits (with heliocentric orbital periods P < 200 years) as a result of close encounters with giant planets. This process is followed by a very slow drift of cometary orbits into the inner part of the Solar System. Only those comets may pass from short-period orbits into Halley-type and Jupiter-family orbits, which move in orbits with perihelion distances q < 13 au. In the second type of capture, the perihelion distances of cometary orbits become rather small (< 1.5 au) during the first stage of dynamic evolution under the action of perturbations from the Galaxy, and then their semimajor axes decrease as a result of diffusion. The capture takes place, on average, in 500 revolutions of the comet about the Sun, whereas in the first case, the comet is captured, on average, after 12500 revolutions. The region of initial orbital perihelion distances q > 4 au is found to be at least as important a source of Halley-type comets as the region of perihelion distances q < 4 au. More than half of the Halley-type comets are captured from the nearly parabolic flux with q > 4 au. The analysis of the dynamic evolution of objects moving in short-period orbits shows that the distribution of Centaurs orbits agrees well with the observed distribution corrected for observational selection effects. Hence, the hypothesis associating the origin of Centaurs with the Edgeworth-Kuiper belt and the trans-Neptunian region exclusively should be rejected.     

Reducing the probability of capture into resonance

A migrating planet can capture planetesimals into mean motion resonances. However, resonant trapping can be prevented when the drift or migration rate is sufficiently high. Using a simple Hamiltonian system for first- and second-order resonances, we explore how the capture probability depends on the order of the resonance, drift rate and initial particle eccentricity. We present scaling factors as a function of the planet mass and resonance strength to estimate the planetary migration rate above which the capture probability drops to less than half. Applying our framework to multiple extrasolar planetary systems that have two planets locked in resonance, we estimate lower limits for the outer planet's migration rate, allowing resonance capture of the inner planet.
Mean motion resonances are comprised of multiple resonant subterms. We find that the corotation subterm can reduce the probability of capture when the planet eccentricity is above a critical value. We present factors that can be used to estimate this critical planet eccentricity. Applying our framework to the migration of Neptune, we find that Neptune's eccentricity is near the critical value that would make its 2 : 1 resonance fail to capture twotinos. The capture probability is affected by the separation between resonant subterms and so is also a function of the precession rates of the longitudes of periapse of both planet and particle near resonance.     

An extension of the Planck galaxy cluster catalogue

We present a catalogue of galaxy clusters detected in the Planck all-sky Compton parameter maps and identified using data from the WISE and SDSS surveys. The catalogue comprises about 3000 clusters in the SDSS fields. We expect the completeness of this catalogue to be high for clusters with masses larger than M ₅₀₀ ≈ 3 × 10¹⁴ M _⊙, located at redshifts z < 0.7. At redshifts above z ≈ 0.4, the catalogue contains approximately an order of magnitude more clusters than the 2nd Planck Catalogue of Sunyaev-Zeldovich sources in the same fields of the sky. This catalogue can be used for identification of massive galaxy clusters in future large cluster surveys, such as the SRG/eROSITA all-sky X-ray survey.     

The restricted problem: An extension of Breakwell-Perko's matching theory

This paper extends Breakwell and Perko's ‘first order’ matching theory (1965, 1966) to a more general matching theory which is applicable to a wider class of second species solutions. In a first stage, the matching theory is elaborated on the basis of new assumptions on the orders of magnitude of the small parameters. In a second stage, we construct a matching theory which takes into account general assumptions which include our assumptions and Breakwell-Perko's.     

Wisdom system: Dynamics in the adiabatic approximation

A simple approximate model of the asteroid dynamics near the 3:1 mean–motion resonance with Jupiter can be described by a Hamiltonian system with two degrees of freedom. The phase variables of this system evolve at different rates and can be subdivided into the ‘fast’ and ‘slow’ ones. Using the averaging technique, wisdom obtained the evolutionary equations which allow to study the long-term behavior of the slow variables. The dynamic system described by the averaged equations will be called the ‘Wisdom system’ below. The investigation of the, wisdom system properties allows us to present detailed classification of the slow variables’ evolution paths. The validity of the averaged equations is closely connected with the conservation of the approximate integral (adiabatic invariant) possessed by the original system. Qualitative changes in the behavior of the fast variables cause the violations of the adiabatic invariance. As a result the adiabatic chaos phenomenon takes place. Our analysis reveals numerous stable periodic trajectories in the region of the adiabatic chaos.     

An extension of Newton's apsidal precession theorem

Newton's apsidal precession theorem in Proposition 45 of Book I of the 'Principia' has great mathematical, physical, astronomical and historical interest. The lunar theory and the precession of the perihelion of the planet Mercury are but two examples of the applications of this theorem. We have examined the precession of orbits under varying force laws as measured by the apsidal angle θ( N , e ), where N is the index for the centripetal force law, for varying eccentricity e . The paper derives a general function for the apsidal angle, dependent only on e and N as the potential is spherically symmetric. Further, we explore approximate ways of the solution of this equation, in the neighbourhood of   N = 2  which happens to be the case of greatest historical interest. Exact solutions are derived where they are possible. The first derivatives  ∂θ/∂ N   and  ∂θ/∂ h   [where h ( N , e ) is the angular momentum] are analytically expressed in the neighbourhood of   N = 2  (case of the inverse square law). The value of  ∂θ/∂ N   is computed numerically as well for  1 ≤ N < 3  . The resulting integrals are interesting improper integrals with singularities at both limits. Some of the integrals, especially for   N = 2  , can be given in closed form in terms of generalized hypergeometric functions which are reducible in terms of algebraic and logarithmic functions. No evidence was found for isolated cases of zero precession as e was increased. The   N = 1  case of the logarithmic potential is also briefly discussed in view of its interest for the dynamics of eccentric orbits and its relevance to realistic galaxy models. The possibility of apsidal precession was also examined for a few cases of high-eccentricity asteroids and extrasolar planets. We find that these systems may provide interesting new laboratories for studies of gravity.     

Capture into first-order resonances and long-term stability of pairs of equal-mass planets

Massive planets form within the lifetime of protoplanetary disks, and therefore, they are subject to orbital migration due to planet–disk interactions. When the first planet reaches the inner edge of the disk, its migration stops and consequently the second planet ends up locked in resonance with the first one. We detail how the resonant trapping works comparing semi-analytical formulae and numerical simulations. We restrict to the case of two equal-mass coplanar planets trapped in first-order resonances, but the method can be easily generalized. We first describe the family of resonant stable equilibrium points (zero-amplitude libration orbits) using series expansions up to different orders in eccentricity as well as a non-expanded Hamiltonian. Then we show that during convergent migration the planets evolve along these families of equilibrium points. Eccentricity damping from the disk leads to a final equilibrium configuration that we predict precisely analytically. The fact that observed multi-exoplanetary systems are rarely seen in resonances suggests that in most cases the resonant configurations achieved by migration become unstable after the removal of the protoplanetary disk. Here we probe the stability of the resonances as a function of planetary mass. For this purpose, we fictitiously increase the masses of resonant planets, adiabatically maintaining the low-amplitude libration regime until instability occurs. We discuss two hypotheses for the instability, that of a low-order secondary resonance of the libration frequency with a fast synodic frequency of the system, and that of minimal approach distance between planets. We show that secondary resonances do not seem to impact resonant systems at low amplitude of libration. Resonant systems are more stable than non-resonant ones for a given minimal distance at close encounters, but we show that the latter nevertheless play the decisive role in the destabilization of resonant pairs. We show evidence that as the planetary mass increases and the minimal distance between planets gets smaller in terms of mutual Hill radius, the region of stability around the resonance center shrinks, until the equilibrium point itself becomes unstable.     

An extension of Szebehely's problem to holonomic systems

We study the problem of the determination of the potential function of forces generating a given family of orbits in the n-dimensional configuration space of the representative point of a holonomic system. We obtain first-order partial differential equations to solve the problem and we discuss some particular examples.

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Sunto Si studia il problema della determinazione del potenziale delle forze atte a far descrivere al punto rappresentativo di un sistema olonomo ad n gradi di libertà una traiettoria appartenente ad una assegnata famiglia di curve. Si scrivono le equazioni differenziali alle derivate parziali del primo ordine che reggono il problema e si considerano alcuni esempi particolari.

     

An analysis of the linear,adiabatic oscillations of a star in terms of potential fields

The linear adiabatic oscillations of a spherically symmetric star are analyzed in terms of potential fields. It is found that all displacement fields ϱξ can be described as either spheroidal or toroidal fields.     

On the adiabatic pulsations of uniformly rotating bodies

The problem of the oscillations and stability of compressible Maclaurin spheroids is reconsidered, on the basis of the third-order virial equations, in an arbitrarily rotating frame of reference. In contrast with the work of Kochhar and Trehan (1974), it is found that the frequencies evaluated in a rotating frame and those evaluated in an inertial frame are related to one another in a very simple way. Numerical calculations made for a wide range of the adiabatic exponent further clarify the effect of compressibility on the natural frequencies.     

An observational test for the origin of the Titan-Hyperion orbital resonance

If Hyperion's radius is near the upper limit of recent estimates, and tidal dissipation in Hyperion is reasonably well represented by a frequency-independent Q ? 2–300, finding Hyperion rotating in the 3:2 spin-orbit resonance like Mercury would imply a primordial origin for the Titan-Hyperion 4:3 orbital resonance. Independent of this test, observation of Hyperion's spin rate will place an upper bound on the average tidal effective Q for the satellite as a function of its assumed initial angular velocity.     

An analytical averaged model of orbital resonance

After having examined the characteristics of some models used at present in the study of orbital resonance, we propose an analytical averaged model, which offers certain convenience in the study of both the characteristics and the evolution of resonance.     

Capture of planetesimals by the giant planets

We investigate the possibility of gravitational capture of planetesimals as temporary or permanent satellites of Uranus and Neptune during the process of planetary growth. The capture mechanism is based in the enhancement of the Hill's sphere of action not only due to the mass acquired by the planet, but also by the variation of the planet-Sun distance as a consequence of the scattering of planetesimals by the planets of the outer solar system. Our calculations indicate that satellite capture was very important, specially during the first stages of the accretion process, contributing in a significant way to the planetary growth.