Mutual Gravitational Perturbations Research Articles

Towards a consistent model of the hot quadruple system HD 93206 = QZ Carinæ

Aims. HD 93206 is a massive early-type stellar system composed of components resolved by direct imaging (Ab, Ad, B, C, D) and a compact subsystem (Aa1, Aa2, Ac1, Ac2). Its geometry was already determined on the basis of extensive photometric, spectroscopic, and interferometric observations. However, the fundamental absolute parameters are still not known precisely enough. Methods. We use an advanced N-body model to account for all mutual gravitational perturbations among the four close components, and all observational data types, including astrometry, radial velocities, eclipse timing variations, squared visibilities, closure phases, triple products, normalized spectra, and spectral energy distribution (SED). The model has 38 free parameters, grouped into three sets of orbital elements, component masses, and their basic radiative properties (T, log g, vrot). Results. We revised the fundamental parameters of QZ Car as follows. For a model with the nominal extinction coefficient RV ≡ AV/E(B − V) = 3.1, the best-fit masses are m1 = 26.1 MS, m2 = 32.3 MS, m3 = 70.3 MS, and m4 = 8.8 MS, with uncertainties of the order of 2 MS, and the system distance d = (2800 ± 100) pc. In an alternative model, where we increased the weights of the radial velocity (RV) and transit timing variation (VTT) observations and relaxed the SED constraints, because extinction can be anomalous with RV ∼ 3.4, the distance is smaller: d = (2450 ± 100) pc. This corresponds to the distance of the Collinder 228 cluster. Independently, this is confirmed by dereddening the SED, which is only then consistent with the early-type classification (O9.7Ib for Aa1, O8III for Ac1). Future modelling should also account for an accretion disk around the Ac2 component.

Primordial migration of co-orbital satellites as a mechanism for the horseshoe orbit of Janus –Epimetheus

We analyze the orbital motion of two natural satellites initially in co-orbital configuration with a third (guiding) satellite embedded into a circumplanetary gas disc and undergoing tidal interactions with the central planet. By solving the exact equations of motion, including the dissipative effects and the mutual gravitational perturbations, we investigate the configuration of the system soon after the guiding satellite is disrupted when crossing the Roche limit with the central planet during its orbital decay. The application is done for the system composed of Saturn and the two small co-orbital satellites Janus and Epimetheus. We perform a large number of numerical simulations, varying the mass of the hypothetical guiding satellite, Saturn's tidal quality factor and the initial configuration and the masses of the small satellites. Analyzing the orbital configurations of Janus and Epimetheus soon after the disruption of the guiding satellite, we obtain that the mutual horseshoe co-orbital orbits appear as a natural outcome of the numerical simulations.

FOUR CENTURIES OF OBSERVATIONS OF THE GALILEAN SATELLITES OF JUPITER: INCREASING THE ASTROMETRIC ACCURACY

The main satellites of Jupiter, named Galilean after their discovery by Galileo Galilei, are among the most studied celestial objects. The dynamics of their motions represent one of the most complex challenges in the Solar System but the most interesting, including all the dynamical problems of a gravitational system. The modeling of their motions is difficult because of their size (Ganymede has a size similar to Mars or Mercury) and mutual gravitational perturbations, because of the flatness of Jupiter, the presence of Saturn and the Sun and strong tidal effects between them and the planet Jupiter. However, a good knowledge of their dynamics may help us understand their physical nature (their internal structure influences their motions), their formation and their evolution. For these purposes, accurate astrometric observations are essential to determine the physical parameters of their dynamics. Our purpose in this paper is to explore the history of the progress made in these studies during the last four centuries and the value of using old data in present-day research.

Precession of the orbital nodes of Jupiter and Saturn triggered by the mutual perturbation: A model of two rings

The problem of the precession of the orbital planes of Jupiter and Saturn under the influence of mutual gravitational perturbations was formulated and solved using a simple dynamical model. Using the Gauss method, the planetary orbits are modeled by material circular rings, intersecting along the diameter at a small angle α. The planet masses, semimajor axes and inclination angles of orbits correspond to the rings. What is new is that each ring has an angular momentum equal to the orbital angular momentum of the planet. Contrary to popular belief, it was proved that the orbital resonance 5: 2 does not preclude the use of the ring model. Moreover, the period of averaging of the disturbing force (T ≈ 1332 yr) proves to be appreciably greater than a conventionally used period (≈900 yr). The mutual potential energy of rings and the torque of gravitational forces between the rings were calculated. We compiled and solved the system of differential equations for the spatial motion of rings. It was established that a perturbing torque causes the precession and simultaneous rotation of the orbital planes of Jupiter and Saturn. Moreover, the opposite orbit nodes on the Laplace plane coincide and perform a secular movement in retrograde direction with the same velocity of 25.6″/yr and the period TJ = TS ≈ 50687 yr. These results are close to those obtained in the general theory (25.93″/yr), which confirms the adequacy of the developed model. It was found that the vectors of the angular velocity of orbital rings move counterclockwise over circular cones and describe circles on the celestial sphere with radii β1 ≈ 0.8403504° (Saturn) and β2 ≈ 0.3409296° (Jupiter) around the point which is located at an angular distance of 1.647607° from the ecliptic pole.

THE STABILITY AND DYNAMICS OF PLANETS IN TIGHT BINARY SYSTEMS

Planets have been observed in tight binary systems with separations less than 20 AU. A likely formation scenario for such systems involves a dynamical capture, after which high relative inclinations are likely and may lead to Kozai oscillations. We numerically investigate the fate of an initially coplanar double-planet system in a class of binaries with separation ranging between $12 - 20 $ AU. Dynamical integrations of representative four-body systems are performed, each including a hot Jupiter and a second planet on a wider orbit. We find that, although such systems can remain stable at low relative inclinations ($\lesssim 40^\circ$), high relative inclinations are likely to lead to instabilities. This can be avoided if the planets are placed in a \emph{Kozai-stable zone} within which mutual gravitational perturbations can suppress the Kozai mechanism. We investigate the possibility of inducing Kozai oscillations in the inner orbit by a weak coupling mechanism between the planets in which the coplanarity is broken due to a differential nodal precession. Propagating perturbations from the stellar companion through a planetary system in this manner can have dramatic effects on the dynamical evolution of planetary systems, especially in tight binaries and can offer a reasonable explanation for eccentricity trends among planets observed in binary systems. We find that inducing such oscillations into the orbit of a hot Jupiter is more likely in tight binaries and an upper limit can be set on the binary separation above which these oscillations are not observed.

Determination of the masses of planetary satellites from their mutual gravitational perturbations

We analyze the possibility of determining the masses of outer planetary satellites from their mutual gravitational perturbations via ground-based observations. Such a technique has been applied in (Emelyanov, 2005b) to determine the mass of the Jovian satellite Himalia. In this paper, we use the least-squares method to compute the errors of satellite masses inferred from simulated observations. We analyze several of the most suitable variants of groups of outer satellites of planets with maximum mutual attraction. We found that the mass of the Satumian satellite Phoebe (S9) can be refined by continuing observations of the satellite S25 Mundilfari until 2027. We show that the masses of other known outer planetary satellites cannot be determined from ground-based observations.

Yarkovsky detection opportunities: II. Binary systems

We consider the possibility of detecting the Yarkovsky orbital perturbation acting on binary systems among the near-Earth asteroids. This task is significantly more difficult than for solitary asteroids because the Yarkovsky force affects both the heliocentric orbit of the system's center of mass and the relative orbit of the two components. Nevertheless, we argue these are sufficiently well decoupled so that the major Yarkovsky perturbation is in the simpler heliocentric motion and is observable with the current means of radar astrometry. Over the long term, the Yarkovsky perturbation in the relative motion of the two components is also detectable for the best observed systems. However, here we consider a simplified version of the problem by ignoring mutual non-spherical gravitational perturbations between the two asteroids. With the orbital plane constant in space and the components' rotation poles fixed (and assumed perpendicular to the orbital plane), we do not examine the coupling between Yarkovsky and gravitational effects. While radar observations remain an essential element of Yarkovsky detections, lightcurve observations, with their ability to track occultation and eclipse phenomena, are also very important in the case of binaries. The nearest possible future detection of the Yarkovsky effect for a binary system occurs for (66063) 1998 RO 1 in September 2006. Farther out, even more statistically significant detections are possible for several other systems including 2000 DP 107, (66391) 1999 KW 4 and 1996 FG 3.

On the Origin of the Eccentricities of Extrasolar Planets

We develop a phenomenological theory that aims to account for the origin of the large eccentricities of extrasolar planets and that of the small eccentricities in the solar system, the preference for apsidal alignment in non-resonant multiplanet systems, and the origin of the similarities in the eccentricity distribution of extra-solar planets and that of spectroscopic binary stars. We show that if a physical process is weakly dependent on the local dynamics of the companion and imparts a small relative acceleration to the star-companion system, the eccentricity of the companion's orbit is excited to large values depending on the direction and duration of acceleration. A natural candidate for such processes are asymmetric stellar jets and star-disk winds. When the acceleration originates from a precessing jet, large eccentricities can be excited by the resonance of the jet's precession frequency with the induced acceleration's excitation frequency even for nearly perpendicular jets. Precession also reduces the eccentricity amplitude far inside the resonance radius. The acceleration's strength is best constrained in multiplanet systems because of the companions' mutual gravitational perturbations, while the acceleration's duration is bounded by the condition that the residual velocity imparted to the star is smaller than the stellar velocity dispersion in the Galaxy. In the outer parts of the star-companion system where the acceleration excitation time is comparable to or smaller than the orbital period, significant radial migration takes place which may have important consequences for the dynamics of the minor body populations in the solar system. The theory is illustrated with the $\upsilon$ Andromedae binary system.

Masses and Orbital Inclinations of Planets in the PSR B1257+12 System

We present measurements of the true masses and orbital inclinations of the two Earth-mass planets in the PSR B1257+12 system, based on the analysis of their mutual gravitational perturbations detectable as microsecond variations of the arrival times of radio pulses from the pulsar. The 6.2-millisecond pulsar, PSR B1257+12, has been regularly timed with the Arecibo telescope since late 1990. Assuming the standard pulsar mass of 1.4 M_Sun, the derived masses of planets B and C are 4.3 +/- 0.2 M_Earth and 3.9 +/- 0.2 M_Earth, respectively. The corresponding orbital inclinations of 53.4 and 47.3 deg (or 127 and 133 deg) imply that the two orbits are almost coplanar. This result, together with the known near 3:2 resonance between the orbits of the two planets, strongly supports the hypothesis of a disk origin of the PSR B1257+12 planetary system. The system's long-term stability is guaranteed by the low, Earth-like masses of planets B and C.

Mutual gravitational perturbations between planetesimals within protoplanetary disks

The result of several simulations of a planetesimal swarm orbiting a protostar are examined. It is shown that variations in both protoplanetary masses and inclinations can affect the behavior of these bodies, perhaps even causing ejection from the system or orbits that show large and/or sudden changes in their orbital parameters. Such effects are the result of mutual gravitational scattering between the planetesmials, which can cause motions ranging from nearly quiescent to wildly chaotic.

From Dust to Terrestrial Planets — Introduction

Terrestrial planets are accreted in a disk orbiting a central star. The basic challenge of their formation consists of assembling micron-sized or smaller dust grains to bodies with over 104 km in diameter. This formation process, ultimately based on collisions, occurs in three very different physical regimes depending upon the size of the bodies present: 1) Early on, micron- to mm-sized dust grains, chondrules and chondrites are strongly coupled to the gas. 2) As they grow larger, gravity increases the collisional cross section allowing runaway growth to occur. 3) After this runaway phase stops from exhaustion of matter in the immediate surroundings of the protoplanets, further growth occurs on timescales typical of mutual gravitational perturbations. The emphasis of this book is on the timescales corresponding to these formation phases as well as the characteristic chemical and isotopic composition of the bodies involved.

Large Scattered Planetesimals and the Excitation of the Small Body Belts

We study the dynamical excitation that large planetesimals, scattered either by Neptune or Jupiter, could have provided to the primordial Edgeworth–Kuiper belt and the asteroid belt. Using both a refined Monte Carlo approach and direct numerical integration, we show that the Monte Carlo method is useful only to give qualitative insight into the resulting excitation, but cannot be trusted from a quantitative viewpoint. According to our direct integrations, Neptune-scattered planetesimals of mass from a few tenths to one Earth mass could have ejected most of the bodies from the primordial Edgeworth–Kuiper belt, thus explaining the large mass deficiency of the present belt up to about 50 AU. The remaining bodies are left on orbits with eccentricity and inclination comparable to those observed. This dynamical excitation is not restricted to the inner part of the belt but may extend to 100 AU. We also show that Pluto has too small a mass to destabilize the motion of other bodies in the 2:3 mean motion resonance with Neptune. The same mechanism involving Jupiter-scattered planetesimals of about one Earth mass can excite the outer asteroid belt, hence depleting it of most of its primordial mass. However, this fails to excite the inner belt. In the case where the planetesimals are isolated by mutual gravitational perturbations on long-lived main-belt-like orbits, safe from encounters with Jupiter, the resulting asteroid belt is very similar to the currently observed one, in terms of mass deficiency, excitation in eccentricity and inclination, and radial mixing. Pallas-like bodies are also obtained. However, the decoupling of planetesimals from Jupiter on well-behaved orbits is rather improbable (2% of our simulations), and the resulting asteroid belt is very critically dependent on the mass of the scattered planetesimals and their residence time in the belt.

Orbital evolution of the terrestrial planets as a result of close encounters and collisions with planet-crossing asteroids

In this paper, we investigate the dynamical evolution of the terrestrial planets as a result of close encounters and collisions with planet-crossing asteroids. We also pay attention to the importance of time dependence in the flux rate of asteroids to planet-crossing orbits. Our results shown that the orbits of the terrestrial planets have been affected by this mechanism, although mutual gravitational perturbations among the terrestrial planets and by Jupiter dominate the variations in the orbital elements.

An observational test for the existence of a planetary system orbiting PSR1257 + 12

FOLLOWING the first report1 of an object of planetary mass orbiting a pulsar, Wolszczan and Frail2 have now reported the even more surprising discovery of two planet-size companions in orbit around the nearby millisecond pulsar PSR1257+12. The orbital periods of the two planets are about 98 days and 67 days, very close to a 3:2 ratio. Here we point out that, because of this near commensurability, the mutual gravitational perturbations of the two planets should produce not only small secular changes, but also larger periodic changes in their orbital elements. In particular, we find that changes in the eccentricities and orbital periods should become measurable within a few years. Such a measurement would help determine the three masses in the system and the inclinations of the orbits. More importantly, a detection of these changes, if they accord with the theoretical predictions presented here, would provide irrefutable confirmation that the periodic residuals observed by Wolszczan and Frail are indeed caused by orbiting planets, rather than some other effect. For the single planet-size object previously reported1 around the pulsar PS R1829–10, there is no dynamical test analogous to the one proposed here to confirm the planetary interpretation.

Stirring of a planetesimal swarm: The role of distant encounters

Planetesimals have random velocities (eccentric and inclined orbits) induced by their mutual gravitational perturbations. The viscous stirring algorithm of Stewart and Wetherill (1988, Icarus 74, 542–553) includes only scattering by close encounters of bodies in crossing orbits. A completetreatment should also include the effect of more distant bodies in noncrossing orbits. This paper presents expressions for the stirring rate due to distant encounters, derived with three-body formalism. The stirring rate has the same mass dependence as that for close encounters; the relative importance of the two mechanisms depends on the Safronov number θ. For small θ, close encounters dominate the stirring rate, but distant encounters may dominate for θ ≳ 10 2. Perturbations by a planetary embryo in scenarios involving “runaway growth” may affect the evolution of planetesimals in noncrossing orbits.

Numerical experiment applicable to planet formation

A three-dimensional numerical model has been developed with the goal of studying limited dynamical problems relevant to the latest stage of planet growth in the accretion theory. A small number of large protoplanets (~ moon-size) of different masses, moving around the Sun, are considered. The dynamical evolution and growth of the population is studied under mutual gravitational perturbations, accretion and collisional fragmentation processes. Gravitational encounters are treated exactly by numerical integration of the N-body problem. Outcomes of collisional fragmentation are modeled according to the results of Greenberg et al. (1978).

Numerical experiment applicable to the latest stage of planet growth

A three-dimensional numerical model was developed with the goal of studying limited dynamical problems relevant to the latest stage of planet growth in the accretion theory. A small number of large protoplanets (∼ Moon size) of different masses, moving around the Sun, are considered. The dynamical evolution and growth of the population is studied under mutual gravitational perturbations, accretion, and collisional fragmentation processes. Gravitational encounters are treated exactly by numerical integration of the N-body problem. Outcomes of collisional fragmentation are modeled according to the results of R. Greenberg et al. (1978, Icarus, 35, 1–26). In the present work, we consider 25 protoplanets with uniform mass distribution in the range 2 × 10 25−4 × 10 26 g on heliocentric orbits in the Earth zone. These bodies are initially confined to a small volume of space to permit gravitational perturbations by close approaches and collisions within a finite length of integration time. The dynamical evolution of the swarm is followed for four different sets of initial ranges in semimajor axis, eccentricity, and inclination: Δa=0.01, 0.02, 0.04, 0.08 AU; Δe= 0.005, 0.01, 0.02, 0.04; Δi=0°3, 0°6, 1°2, 2°4. Among other results, it is found that average eccentricities and inclinations evolve toward a steady state such that i ⋍ 1 2 , e ; it is also found that, whatever the initial conditions, the population evolves toward a quasi-equilibrium relative velocity distribution corresponding to a Safronov parameter value θ⋍10. Moreover, the growth process of the growing planet presents very similar behavior in the four cases considered, except for the time scale of evolution, which increases with the initial range of orbital elements. Earlier works of this kind have been presented by L.P. Cox and J.S. Lewis (1980, Icarus, 44, 706–721) and by G.N. Wetherill (1980b, In Geol. Soc. Canad. Spec. Publ., p. 20), although a number of differences exist between the three approaches.

FORMATION OF PLANETARY SYSTEMS

Stars are observed to be forming within cold regions of the galaxy called molecular clouds. A clump of gas and dust within a molecular cloud can collapse into a rotationally- supported disk orbiting the pre s sure -supported star. Such a disk has the same initial elemental composition as the growing star. At sufficient distances from the central star, it is cool enough for approx. 1 - 2% of this material to be in solid form, either remnant interstellar grains or condensates formed within the disk. During the infall stage, the disk is very active and probably highly turbulent. When the infall slows substantially or stops, the disk becomes more quiescent. The dust grows from micron-sized dust to kilometer-sized planetesimals by physical collisions, possibly aided by collective gravity. The dynamics of larger solid bodies within protoplanetary disks are better characterized. The primary perturbations on the orbits of kilometer-sized and larger planetesimals in protoplanetary disks are mutual gravitational interactions and physical collisions. These interactions lead to accretion (and in some cases erosion and fragmentation) of planetesimals. The most massive planets have the largest gravitationally-enhanced collision cross-sections, and accrete almost everything with which they collide. The size distribution of solid bodies becomes quite skewed, with a few large bodies growing much faster than the rest of the swarm, until they have accumulated most of the small bodies within their gravitational reach. Slower growth continues (at least for solid, earth-like planets) as the eccentricities of planetary embryos are pumped up by long-range mutual gravitational perturbations. As planetary masses increase, they become more efficient at stirring random velocities of neighboring bodies. If sufficiently massive and dense planets exist far enough from the star, they can eject material into interstellar space. In most models, giant planets begin their growth like terrestrial planets, but if they become massive enough before the protoplanetary disk dissipates, then they are able to accumulate substantial amounts of gas. Giant planet growth times predicted by current models are similar to estimates of the lifetime of the gaseous protoplanetary disk, so we don't know whether or not giant planets form in most protoplanetary disks. However, planets that become massive while a substantial amount of gas remains in the disk may migrate into the star as a consequence of their gravitational interactions with the disk.