New orbital elements for Moon and planets

The results of a simultaneous solution for the orbital elements of Moon and planets are given and their derivation is discussed. A modern Cowell integrator is used for orbit computations, and least-squares fits are made to some 40000 optical observations taken since 1913. The model includes relativistic terms, the leading zonal harmonics of Earth and Moon, the precession of the lunar equator, and the tidal couple between Earth and Moon. The tidal term in the Moon's mean longitude is found to be –19±4 per century squared. The solution also yields an extrapolation of the atomic time scale back to 1912.5. At that time, the difference between atomic and ephemeris time is about 6±2 s. Lunar declinations observed by the Washington transit circles, after receiving limb corrections and thus with respect to the center of Watts' reference sphere, are smaller than computed values by 0.33±0.01. It is found that solar oblateness cannot quite be determined with optical data covering about 50 yr, butJ ₂ is unlikely to be much larger than 10^–5. The advance of Mercury's perihelion is verified to within our resolution of 2 per century to match that predicted by Einstein.The solution presented here is believed to be the only simultaneous improvement of the orbits of Moon and planets. This simultaneity is found to be an essential feature in separating the Moon's mean motion, the lunar tidal deceleration, and the corrections to the Earth rotation rate. It is now possible to refer all astronomical events of the past 60 yr to a time with uniform rate, namely the atomic clock system. Considering the long baseline, this model should facilitate the prediction of fast variables, such as the lunar longitude, with considerably increased confidence. The planetary orbital elements compete with efforts of similar scope and accuracy at the Massachusetts Institute of Technology and the Jet Propulsion Laboratory.     

Relationship of comets and planets

We analyze our earlier data on the numerical integration of the equations of motion for 274 short-period comets (with the period P<200 yr) on a time interval of 6000 yr. As many as 54 comets had no close approaches to planets, 13 comets passed through the Saturnian sphere of action, and one comet passed through the Uranian sphere of action. The orbital elements of these 68 comets changed by no more than ±3 percent in a space of 6000 yr. As many as 206 comets passed close to Jupiter. We confirm Everhart’s conclusion that Jupiter can capture long-period comets with q = 4–6 AU and i < 9° into short-period orbits. We show that nearly parabolic comets cross the solar system mainly in the zone of terrestrial planets. No relationship of nearly parabolic comets and terrestrial planets was found for the epoch of the latest apparition of comets. Guliev’s conjecture about two trans-Plutonian planets is based on the illusory excess of cometary nodes at large heliocentric distances. The existence of cometary nodes at the solar system periphery turns out to be a solely geometrical effect.     

Evolutionary tracks of the terrestrial planets

On the basis of the model proposed by Matsui and Abe, we will show that two major factors — distance from the Sun and the efficiency of retention of accretional energy — control the early evolution of the terrestrial planets. A diagram of accretional energy versus the optical depth of a proto-atmosphere provides a means to follow the evolutionary track of surface temperature of the terrestrial planets and an explanation for why the third planet in our solar system is an aqua-planet.     

A model of the galactic tidal interaction with the oort comet cloud

We consider a model of the in situ Oort cloud which is isotropic with a random distrihution of perihelia directions and angular momenta. The energy distribution adopted has a continuous range of values appropriate for long-period (>200 yr) comets. Only the tidal torque of the Galaxy is included as a perturbation of comet orbits and it is approximated to be that due to a quasi-steady state distribution of matter with disk-like symmetry. The time evolution of all orbital elements can be analytically obtained for this case. In particular, the change in the perihelion distance per orbit and its dependence on other orbital elements is readily found. We further make the assumption that a comet whose perihelion distance was beyond 15 AU during its last passage through the Solar System would have orbit parameters that are essentially unchanged by planetary perturbations. Conversely, if the prior passage was inside 15 AU we assume that planetary perturbations would have removed the comet from the in situ energy distribution accessible by the galactic tide. Comets which had their perihelia changed from beyond 15 AU to within 5 AU in a single orbit are taken to be observable. We are able to track the evolution of 10⁶ comets as they are made observable by the galactic tidal touque. Detailed results are obtained for the predicted distribution of new (0 < 1/ < 10^–4 AU^–1) comets. Further, correlations between orbital elements can be studied. We present predictions of observed distributions and compare them with the random in situ results as well as with the actual observed distributions of class I comets. The predictions are in reasonable agreement with actual observations and, in many cases, are significantly different from random when perihelia directions are separated into galactic northern and southern hemispheres. However the well-known asymmetry in the north-south populations of perihelia remains to be explained. Such an asymmetry is consistent with the dominance of tidal torques today if a major stochastic event produced it in the past since tidal torques are unable to cause the migration of perihelia across the latitude barriers ±26°.6 in the disk model.     

Test for planetary influences on solar activity

A method due to Schuster is used to test the hypothesis that solar activity is influenced by tides raised in the Sun's atmosphere by planets. We calculate the distribution in longtitude of over 1000 flares occurring in a 61/2 yr segment of solar cycle 19, referring the longitude system in turn to the orbital positions of Jupiter and Venus. The resulting distributions show no evidence for a tidal effect.     

Accretion of terrestrial planets from oligarchs in a turbulent disk

We have investigated the final accretion stage of terrestrial planets from Mars-mass protoplanets that formed through oligarchic growth in a disk comparable to the minimum mass solar nebula (MMSN), through N-body simulation including random torques exerted by disk turbulence due to Magneto-Rotational Instability. For the torques, we used the semi-analytical formula developed by Laughlin et al. [Laughlin, G., Steinacker, A., Adams, F.C., 2004. Astrophys. J. 608, 489-496]. The damping of orbital eccentricities (in all runs) and type-I migration (in some runs) due to the tidal interactions with disk gas is also included. Without any effect of disk gas, Earth-mass planets are formed in terrestrial planet regions in a disk comparable to MMSN but with too large orbital eccentricities to be consistent with the present eccentricities of Earth and Venus in our Solar System. With the eccentricity damping caused by the tidal interaction with a remnant gas disk, Earth-mass planets with eccentricities consistent with those of Earth and Venus are formed in a limited range of disk gas surface density (∼10⁻⁴ times MMSN). However, in this case, on average, too many (?6) planets remain in terrestrial planet regions, because the damping leads to isolation between the planets. We have carried out a series of N-body simulations including the random torques with different disk surface density and strength of turbulence. We found that the orbital eccentricities pumped up by the turbulent torques and associated random walks in semimajor axes tend to delay isolation of planets, resulting in more coagulation of planets. The eccentricities are still damped after planets become isolated. As a result, the number of final planets decreases with increase in strength of the turbulence, while Earth-mass planets with small eccentricities are still formed. In the case of relatively strong turbulence, the number of final planets are 4-5 at 0.5-2 AU, which is more consistent with Solar System, for relatively wide range of disk gas surface density (∼¹⁰⁻⁴-¹⁰⁻² times MMSN).     

The mid-term and long-term solar quasi-periodic cycles and the possible relationship with planetary motions

This work investigates the solar quasi-periodic cycles with multi-timescales and the possible relationships with planetary motions. The solar cycles are derived from long-term observations of the relative sunspot number and microwave emission at frequency of 2.80 GHz. A series of solar quasi-periodic cycles with multi-timescales are registered. These cycles can be classified into three classes: (1) the strong PLC (PLC is defined as the solar cycle with a period very close to the ones of some planetary motions, named as planetary-like cycle) which is related strongly with planetary motions, including nine periodic modes with relatively short period (P<12 yr), and related to the motions of the inner planets and of Jupiter; (2) the weak PLC, which is related weakly to planetary motions, including two periodic modes with relatively long period (P>12 yr), and possibly related to the motions of outer planets; (3) the non-PLC, for which so far there has been found no clear evidence to show the relationship with any planetary motions. Among the planets, Jupiter plays a key role in most periodic modes due to its sidereal motion or spring tidal motions associated with other planets. Among planetary motions, the spring tidal motion of the inner planets and of Jupiter dominates the formation of most PLCs. The relationships between multi-timescale solar periodic modes and the planetary motions will help us to understand the essential nature and prediction of solar activities.     

Consequences of the tidal slowing of Mercury

Mercury, currently rotating very slowly, probably rotated faster in the past. If Mercury's rotation period had been near 8 hours initially, similar to that of most solar system bodies today, it would have been flattened by a few percent. As Mercury was slowed by solar tides, craters which were circular when they were emplaced would have been distorted by the same few percent. Substantial surface stresses, well above the fracture stress, would have been produced unless stress relief occurred; these stresses should have produced tensional fractures near the poles and two intersecting sets of shear planes in equatorial regions. Satellite orbits about the slowly spinning Mercury have been shown to collapse onto its surface: the impact craters resulting from these hypothetical lost satellites should be elongated along the orbit paths, which probably lie near the equator. However, none of these features has been found on the Mariner 10 images. They may be obscured by the effects of tidal heating that should cause an overall internal temperature increase of about 100°K although the increase would be substantially more in certain regions. Radial tides, sometimes called push-pull tides, are important at the present time because Mercury's large orbital eccentricity causes the planet to undergo significant tidal flexing each orbital period; the contemporary tidal heating due to this mechanism is estimated at more than 10¹⁶ erg/sec.     

Tidal evolution in close binary systems

The aim of the present paper will be to give a mathematical outline of the theory of tidal evolution in close binary systems of secularly constant total momentum — an evolution activated by viscous friction of dynamical tides raised by the two components on each other. The first section contains a general outline of the problem; and in Section 2 we shall establish the basic expressions for the energy and momenta of close binaries consisting of components of arbitrary internal structure. In Section 3 we shall investigate the maximum and minimum values of the energy (kinetic and potential) which such systems can attain for given amount of total momentum; while in Section 4 we shall compare these results with the actual facts encountered in binaries with components whose internal structure (and, therefore, rotational momenta) are known to us from evidence furnished by the observed rates of apsidal advance.The results show that all such systems — be these of detached or semi-detached type — disclose that more than 99% of their total momenta are stored in the orbital momentum. The sum of the rotational momenta of the constituent components amounts to less than a percent of the total — a situation characteristic of a state close to the minimum energy for given total momentum. This appears, moreover, to be true not only of the systems with both components on the Main Sequence, but also of those possessing evolved components in contact with their Roche limits.Under such conditions, a synchronism between rotation and revolution (characteristic of both extreme states of maximum and minimum energy) is not only possible, but appears to have been actually approached — if not attained — in the majority of cases. In other words, it would appear that — in at least a large majority of known cases — the existing close binaries have already attained orbits of maximum distension consistent with their momenta; and tidal evolution alone can no longer increase the present separations of the components to any appreciable extent.The virtual absence, in the sky, of binary systems intermediate between the stages of maximum and minimum energy for given momentum leads us to conjecture that the process of dynamical evolution activated by viscous tides may enroll on a time-scale which is relatively short in comparison with their total age — even for systems like Y Cygni or AG Persei, whose total age can scarcely exceed 10⁷ yr. A secular increase of the semi-major axes of relative orbits is dynamically coupled with a corresponding variation in the velocity of axial rotation of both components through the tidal lag arising from the viscosity of stellar material. The differential equations of so coupled a system are given in Section 5; but their solution still constitutes a task for the future.The Lunar Science Institute Contribution No. 90. The Lunar Science Institute is operated by the Universities Space Research Association under Contract No. NSR 09-051-001 with the National Aeronautics and Space Administration.     

Tidal evolution and the Pluto-Charon system

We analyze the system formed by Pluto and its satellite Charon from the point of view of the theory of tidal evolution. The singular feature of the system, i.e. the configuration of complete synchronism which has been suggested by the available data, is found to represent the stable end-product of the evolution. The time needed for the synchronization is shown to be less than the age of the solar system, provided that Pluto's tidal dissipation function is smaller than 10⁴–10⁵. Moreover, the initial orbital radius of the system could not be largerthan two or three times the present radius, so that Charon has been always a close satellite.Finally, we discuss Lyttleton's hypothesis that Pluto is an escaped satellite of Neptune, suggesting that a possible mechanism of Pluto's ejection could be connected with a retrograde capture of Triton by Neptune or with the subsequent tidal evolution of Triton's orbit.     

Effect of the rotation and tidal dissipation history of stars on the evolution of close-in planets

Since 20 years, a large population of close-in planets orbiting various classes of low-mass stars (from M-type to A-type stars) has been discovered. In such systems, the dissipation of the kinetic energy of tidal flows in the host star may modify its rotational evolution and shape the orbital architecture of the surrounding planetary system. In this context, recent observational and theoretical works demonstrated that the amplitude of this dissipation can vary over several orders of magnitude as a function of stellar mass, age and rotation. In addition, stellar spin-up occurring during the Pre-Main-Sequence (PMS) phase because of the contraction of stars and their spin-down because of the torque applied by magnetized stellar winds strongly impact angular momentum exchanges within star–planet systems. Therefore, it is now necessary to take into account the structural and rotational evolution of stars when studying the orbital evolution of close-in planets. At the same time, the presence of planets may modify the rotational dynamics of the host stars and as a consequence their evolution, magnetic activity and mixing. In this work, we present the first study of the dynamics of close-in planets of various masses orbiting low-mass stars (from \(0.6~M_\odot \) to \(1.2~M_\odot \)) where we compute the simultaneous evolution of the star’s structure, rotation and tidal dissipation in its external convective envelope. We demonstrate that tidal friction due to the stellar dynamical tide, i.e. tidal inertial waves excited in the convection zone, can be larger by several orders of magnitude than the one of the equilibrium tide currently used in Celestial Mechanics, especially during the PMS phase. Moreover, because of this stronger tidal friction in the star, the orbital migration of the planet is now more pronounced and depends more on the stellar mass, rotation and age. This would very weakly affect the planets in the habitable zone because they are located at orbital distances such that stellar tide-induced migration happens on very long timescales. We also demonstrate that the rotational evolution of host stars is only weakly affected by the presence of planets except for massive companions.     

Dynamics,interrelations and evolution of the systems of asteroids and comets

The dynamics of larger interplanetary bodies is reviewed, with emphasis on evolutionary problems, interrelations, and open questions. Observational biases distinguishing the sample of known objects from the whole population are briefly discussed. A schematic division of the solar system into zones with different regimes of motion, and a rough taxonomy of orbit types are attempted. The role of individual major planets in controlling the dynamical evolution of interplanetary objects, in particular by stabilizing resonances and destabilizing close encounters, is compared. There are significant evolution asymmetries due to boundary conditions and preferential evolutionary paths; individual major displacements in the phase space of orbital elements conserve the Tisserand invariant with respect to the planet responsible, thus favouring certain evolutionary sequences against others. Very limited lifetimes of some orbit types imply a long-term balance between source and sink, and require a continuous supply of objects from other types of orbits. In this respect, the ultimate fate of extinct comets is of particular interest. Under very specific conditions, nongravitational effects of mass loss can result in stabilization of a formerly unstable orbit. Since the dividing line between the two basic interplanetary populations distinguished by origin and composition — the asteroids and the comets — is essentially that between stable and unstable motion, orbital data can be used to specify which of the known asteroid-like objects may be devolatilized cometary nuclei.Paper presented at the European Workshop on Planetary Sciences, organised by the Laboratorio di Astrofisica Spaziale di Frascati, and held between April 23–27, 1979, at the Accademia Nazionale del Lincei in Rome, Italy.     

On the evolutionary stage of stars in the Draco dwarf Galaxy

An analysis of the colour-magnitude diagram of stars in Draco dwarf Galaxy is performed on the basis of Baade and Swope observational results. Some evidence is found for the occurrence of a clump in the luminosity distribution of Red Giants stars. By combining this characteristic with the morphology of Horizontal Branch one obtains — in the frame of present knowledge of star-evolution —a suggestion for accepting high helium content (Y-0.30) with a relatively small age (1–2×10⁹ yr) for the evolving stars.Some correlations among lifetimes of evolutionaries stages are derived.

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Sommario Sulla base dei risultati osservativi di Baade e Swope viene eseguita un'analisi del diagramma colore-magnitudine delle stelle della Galassia Nana in Draco. Viene così evidenziata la possibile esistenza di un picco nella distribuzione in luminosità delle Giganti Rosse. Combinando questa caratteristica con l'osservato andamento del Ramo Orizzontale si mostra come — nell'ambito delle attuali conoscenze sull'evoluzione stellare — un simile comportamento conduca a postulare un alto contenuto di elio (Y-0.30) e una età relativamente bassa (1–2×10⁹ anni) per le stelle del sistema.Sono infine ricavate alcune correlazioni, tra le vite medie delle principali fasi evolutive.

     

Cometary collisions and the dark material on Iapetus

Iapetus (S8) is unique in our solar system in that the albedo of its leading hemisphere is only 0.05 while that of the trailing side is 0.5. Several existing hypotheses are examined and found inadequate. Photometric studies of the dark side are compared to comet nuclei and class D asteroids. It is hypothesized that in the last 10⁶–10⁸ yrs the leading side suffered a high-velocity collision with a cometary body of mass 10¹³–10¹⁵ kg and traveling at a speed of 20 km s^–1. About 5–16% of the excavated material was ejected into space, where the vaporized ices dissipated while the dark carbonaceous/silicate material was reaccreted on the leading side. The collision, although not sufficient to break Iapetus' tidal lock, resulted in a period of oscillation of about 5 yr. Until tidal friction reasserted a lock, the oscillation gave rise to the longitude effect, viz., the observed fact that the dark material covers more than 220 of longitude but only 110 of latitude.     

On the relation between the solar activity cycle and the solar tidal force induced by the planets

The cross-correlation coefficient (t) of the solar tidal force induced by the planets f(x + t) with the sunspot number g(x) during a period of 44 years is about -0.7 when t is about -2 years. This fact will be useful for predicting solar activity. The solar tidal force was calculated from 1928 to 1971 for every degree on the equatorial plane and every time every planet moves one degree. As the solar tidal force, we used the moving annual average by months of the square of the vertical tidal force on the sun, and as the sunspot number we used the Zürich mean annual sunspot number.     

Evolution of low-mass close binaries and the problem of orbital period gap in cataclysmic variables

Evolution of close binary composed of a white dwarf primary and a Main-Sequence secondary has been calculated. Angular-momentum loss via gravitational radiation and magnetic stellar wind have been taken into account. We have found that magnetic stellar wind with a rate greater than (10^–10–10^–9)M yr^–1 is able to drive the evolution with mass exchange. If the time-scale of switch-off of wind when the primary becomes fully convective is not longer than 10⁶ yr, mass exchange interrupts due to a contraction of the secondary and the system becomes unobservable. Mass exchange resumes when components approach one another due to loss of momentum via gravitational radiation. The location and width of the thus-arising gap in the orbital periods are comparable to those observed.     

Metallicity of Sun-like G-stars that have Exoplanets

By considering the physical and orbital characteristics of G type stars and their exoplanets, we examine the association between stellar mass and its metallicity that follows a power law. Similar relationship is also obtained in case of single and multiplanetary stellar systems suggesting that, \(\hbox {Sun}^{\prime }\)s present mass is about 1% higher than the estimated value for its metallicity. Further, for all the stellar systems with exoplanets, association between the planetary mass and the stellar metallicity is investigated, that suggests planetary mass is independent of stellar metallicity. Interestingly, in case of multiplanetary systems, planetary mass is linearly dependent on the stellar absolute metallicity, that suggests, metal rich stars produce massive (\(\ge \)1 Jupiter mass) planets compared to metal poor stars. This study also suggests that there is a solar system planetary missing mass of \({\sim }\)0.8 Jupiter mass. It is argued that probably 80% of missing mass is accreted onto the Sun and about 20% of missing mass might have been blown off to the outer solar system (beyond the present Kuiper belt) during early history of solar system formation. We find that, in case of single planetary systems, planetary mass is independent of stellar metallicity with an implication of their non-origin in the host star’s protoplanetary disk and probably are captured from the space. Final investigation of dependency of the orbital distances of planets on the host stars metallicity reveals that inward migration of planets is dominant in case of single planetary systems supporting the result that most of the planets in single planetary systems are captured from the space.     

Secular effects of tidal friction on the planet-satellite systems of the solar system

We separate the tidal evolution of a planet-satellite system with zero eccentricity in two phases:phase 1—from the formation of the system to satellite's corotation (satellite's corotation means that its spin angular velocity equals the orbital angular velocity);phase 2—after satellite's corotation.We study the planet-satellite system during phase 1 with Darwin's graphical method and obtain an upper limit to satellite'sQ which discloses whether or not it is corotating. Moreover we obtain some qualitative information about the future evolution of the corotating satellites.The present work does not give any new result for the Earth-Moon case and for the Neptune-Triton case.     

Lunar paleotides and the origin of the Earth-Moon system

A new method for determining the early history of the Earth-Moon system is described. Called the study of lunar paleotides, it describes a method for explaining features of the remnant lunar gravity field, and the generation of the lunar mascons. A method for the determination of Earth-Moon distances compared with the radiometric ages of the maria is developed. It is shown that the Moon underwent strong anomalous gravitational tidal forces, for a durationt<10⁶yr, prior to the formation of the mascon surfaces. As these tidal forces had not been present at the time of the formation of the Moon, this shows that the Moon could not have been formed in orbit about the Earth.There are tides in the affairs of men which, taken at the flood, lead on to fortune... William Shakespeare 1564–1616     

Do planetary encounters reset surfaces of near Earth asteroids?

Processes such as the solar wind sputtering and micrometeorite impacts can modify optical properties of surfaces of airless bodies. This explains why spectra of the main belt asteroids, exposed to these ‘space weathering’ processes over eons, do not match the laboratory spectra of ordinary chondrite (OC) meteorites. In contrast, an important fraction of Near Earth Asteroids (NEAs), defined as Q-types in the asteroid taxonomy, display spectral attributes that are a good match to OCs. Here we study the possibility that the Q-type NEAs underwent recent encounters with the terrestrial planets and that the tidal gravity (or other effects) during these encounters exposed fresh OC material on the surface (thus giving it the Q-type spectral properties). We used numerical integrations to determine the statistics of encounters of NEAs to planets. The results were used to calculate the fraction and orbital distribution of Q-type asteroids expected in the model as a function of the space weathering timescale, t_sw (see main text for definition), and maximum distance, r^∗, at which planetary encounters can reset the surface. We found that t_sw ∼ 10⁶ yr (at 1 AU) and r^∗ ∼ 5R_pl, where R_pl is the planetary radius, best fit the data. Values t_sw < 10⁵ yr would require that r^∗ > 20R_pl, which is probably implausible because these very distant encounters should be irrelevant. Also, the fraction of Q-type NEAs would be probably much larger than the one observed if t_sw > 10⁷ yr. We found that t_sw ∝ q², where q is the perihelion distance, expected if the solar wind sputtering controls t_sw, provides a better match to the orbital distribution of Q-type NEAs than models with fixed t_sw. We also discuss how the Earth magnetosphere and radiation effects such as YORP can influence the spectral properties of NEAs.