Angular momentum and tidal evolution

Five satellites of Neptune orbit under the synchronous zone. In this sense the Neptune's system is similar to that of Uranus (nine satellites) and differs from Jupiter (two) and Saturn (zero). The basic parameters describing the angular momentum within the Neptune's system and of its tidal evolution are estimated. The main character of the tidal dynamics is due to the retrograde Triton. The total tidal decrease in the spin angular momentum of Neptune is compared with those of Uranus, Jupiter and Saturn.     

Tidal and rotational distortions in figures of Pluto and Charon

The supposition is, the tidal and rotational distortions should be fully responsible for the Pluto's and Charon's figure parameters. The mean polar and equatorial flattenings have been estimated about 10^–3, the second sectorial Stokes parameters about 9 × 10^–5, the differences between equatorial principal moments of inertia about 6 × 10³⁰ kg m² (Pluto) and 2 × 10²⁹ kg m² (Charon).     

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.     

On the accelerations of the Moon and Sun,the constant of gravitation,and the origin of mountains

In a previous paper Lyttleton (1976) has shown that the apparent secular accelerations of the Sun and Moon, as given by de Sitter, can be largely explained if the Earth is contracting at the rate required by the phase-change hypothesis for the nature of the core. More reliable values for these accelerations have since become available which warrant a redetermination of the various effects concerned on the basis of constantG, and this is first carried out in the present paper. The lunar tidal couple, which is the same whetherG is changing or not, is found to be (4.74±0.38)×10²³ cgs, about three-quarters that yielded by the de Sitter values, while within the theory the Moon would take correspondingly longer to reach close proximity to the Earth at about 1.5×10⁹ years ago.The more accurate values of the accelerations enable examination to be made of the effects that a decreasingG would have, and it is shown that a valueG/G=–3×10^–11 yr^–1 can be weakly satisfied compared with the close agreement found on the basis of constantG, while a value as large numerically asG/G=–6×10^–11 yr^–1 seems to be definitely ruled out. On the iron-core model, an intrinsic positive component of acceleration of the angular velocity cannot be reconciled at all with the secular accelerations even for constantG, and far less so ifG is decreasing at a rate suggested by any recent cosmological theory.ItG=0, the amount of contraction available for mountain-building would correspond to a reduction of surface area of about 49×10⁶ km² and a volume to be redistributed of 160×10⁹ km³ if the time of collapse were 2.5×10⁹ years ago. For earlier times, the values are only slightly reduced. IfG/G=–3×10^–11 yr^–1, the corresponding values are 44×10⁶ km² and 138×10⁹ km³ for collapse at –2.5×10⁹ yr, and not importantly smaller at 38×10⁶ km² and 122×10⁹ km³ for collapse at –4.5×10⁹ yr. Any of these values would suffice to account in order of magnitude for all the eras of mountain-building. An intense brief period of mountain-building on an immense scale would result from the Ramsey-collapse at whatever time past it may have occurred.     

A simple two-layer model for the Uranus satellites

A two-layer model of a satellite interior with a rocky core with a density 3–3.4 g cm^-3 and with a H₂O mantle with a density 0.94–1.2 g cm^-3 is applied for the icy satellites. The case of Mimas is discussed separately. A comparison of the results with these obtained for more complicated models as applied for Jupiter and Saturn icy satellites has been carried out. This comparison shows that the two-layer model offers a reasonable approximation and, therefore, it can be applied for the satellites of Uranus. We obtained the dimensionless core radii 0.55–0.74, 0.45–0.68, 0.59–0.67, 0.55–0.65, and dimensionless core masses 0.42–0.72, 0.26–0.63, 0.47–0.61, 0.41–0.57, for Ariel, Umbriel, Titania, and Oberon, respectively.Institute of Geophysics of Warsaw University, Warszawa, Poland.     

Formation of the regular satellites of giant planets in an extended gaseous nebula II: satellite migration and survival

For a satellite to survive in the disk the time scale of satellite migration must be longer than the time scale for gas dissipation. For large satellites (∼1000 km) migration is dominated by the gas tidal torque. We consider the possibility that the redistribution of gas in the disk due to the tidal torque of a satellite with mass larger than the inviscid critical mass causes the satellite to stall and open a gap (W.R. Ward, 1997, Icarus 26, 261-281). We adapt the inviscid critical mass criterion to include gas drag, and m-dependent nonlocal deposition of angular momentum. We find that such a model holds promise of explaining the survival of satellites in the subnebula, the mass versus distance relationship apparent in the saturnian and uranian satellite systems, the concentration of mass in Titan, and the observation that the satellites of Jupiter get rockier closer to the planet whereas those of Saturn become increasingly icy. It is also possible that either weak turbulence (close to the planet) or gap-opening satellite tidal torque removes gas on a similar time scale (10⁴-10⁵ years) as the orbital decay time of midsized (200-700 km) regular satellites forming in the inner disk (inside the centrifugal radius (I. Mosqueira and P.R. Estrada, 2003, Icarus, this issue)). We argue that Saturn’s satellite system bridges the gap between those of Jupiter and Uranus by combining the formation of a Galilean-sized satellite in a gas optically thick subnebula with a strong temperature gradient, and the formation of smaller satellites, closer to the planet, in a disk with gas optical depth ?1, and a weak temperature gradient.Using an optically thick inner disk (given gaseous opacity), and an extended, quiescent, optically thin outer disk, we show that there are regions of the disk of small net tidal torque (even zero) where satellites (Iapetus-sized or larger) may stall far from the planet. For our model these outer regions of small net tidal torque correspond roughly to the locations of Callisto and Iapetus. Though the precise location depends on the (unknown) size of the transition region between the inner and outer disks, the result that Saturn’s is found much farther out (at ∼3rcS, where rcS is Saturn’s centrifugal radius) than Jupiter’s (at ∼ 2rcJ, where rcJ is Jupiter’s centrifugal radius) is mostly due to Saturn’s less massive outer disk and larger Hill radius. However, despite the large separation between Ganymede and Callisto and Titan and Iapetus, the long formation and migration time scales for Callisto and Iapetus (I. Mosqueira and P.R. Estrada, 2003, Icarus, this issue) makes it possible (depending on the details of the damping of acoustic waves) that the tidal torque of Ganymede and Titan clears the gas disk out to their location, thus stranding Callisto and Iapetus far from the planet. Either way, our model provides an explanation for the presence of regular satellites outside the centrifugal radii of Jupiter and Saturn, and the absence of such a satellite for Uranus.     

Astronomical evidence concerning non-gravitational forces in the Earth-Moon system

The most conspicuous effects of non-gravitational forces in the Earth-Moon system are the accelerations of the Earth's spin and of the Moon's mean angular velocity. Evidence indicates that the present acceleration of the Moon is between –20 and –52 s of arc per century per century and that the present average acceleration of the Earth is between –5 and –23 parts in 10⁹ per century. Over the past 2000 yr, the average for the Moon has been about –42 s per century per century and for the Earth has been about –28 parts in 10⁹ per century; these values are probably correct within 10%. Evidence that does not involve any assumptions about the present values shows strongly that there was a square wave in the accelerations that lasted from about 700–1300, and that the accelerations were different by a factor of perhaps 5 during the time of this wave from what they were at neighboring times.An effect that seems to be changing the obliquity of the ecliptic has been reported in recent literature, on the basis of data obtained within the past century. The effect amounts to about 1/4 s of arc per century if it is real. Older data are not accurate enough to give information about an effect this small.There are no satisfactory explanations of the accelerations. Existing theories of tidal friction are quite inadequate.Paper presented at the AAAS Symposium on the Early History of the Earth and Moon in Philadelphia on 28 December 1971.     

Tidal rigidity of phobos

The Martian satellites Phobos and Deimos move along nearly circular coplanar, stable orbits and have created surfaces older than ～ 10⁹ years. The accretion hypothesis suggests that their primordial orbits were also very regular. However, tides raised on Mars and Phobos can substantially alter the semimajor axis a of Phobos' orbit over time. The effect of the Martian tidal torque alone on Phobos' orbit implies that the primordial e was ～0.1 to 0.2 about 4.6 × 10⁹ years ago if the present observed e = 0.015 is naively interpreted as a tidally damped remnant. Significant tidal friction in Phobos reduces the time scale for Phobos to achieve a crossing orbit with Deimos to less than 10⁹ years and permits the primodial e to approach unity. The consequences of orbital intersections cannot easily be resolved by assuming either a catastrophic origin for both satellites (namely, that both are fragments of a common parent body fractured by an impact) or that they were captured sequentially by Mars. Either hypothesis is difficult to accept, given that Deimos' orbit, which is only slightly affected by tides, is now so regular. An alternative scenario is proposed in this paper in which the observed e of Phobos results from several gravitational resonance excitations within the last 10⁹ years, assuming tidal friction in Phobos has had only a small effect on its orbit. In facr, both the primordial e and the inclination i may have been much smaller than presently observed. The constraints imposed on tidal friction in Phobos by both the apparent age of Phobos' surface (> 10⁹yrs) and the above scenario can be satisfied only of μQ > 10¹²dynes/cm². Since the Q factor is ～10², the rigidity μ > 10¹⁰dynes/cm². Thus Phobos should have substantial internal strength.     

Peculiarities of Uranus’s satellite system

The absence of Uranus’s equatorial satellites in the region of approximately equal influence of its oblateness and solar perturbations is explained in terms of an improved physical model. This model is more complete than the previously studied case of an integrable averaged problem. The model improvement stems from the fact that the inclination of Uranus’s equator to the ecliptic differs by 90° and that the orbital evolution of Uranus due to secular planetary perturbations is taken into account. The lifetime of Uranus’s hypothetical satellites in orbits with semimajor axes 1.3–7 million km can be estimated by numerically integrating the evolution equations to be ～10⁴ yr. This is the time scale on which the evolution of the orbits leads to their intersection with the orbits of inner satellites.     

A note on the total mass of comets in the solar system

The assumption that the very low albedo determined for Halley's comet is typical of all short period comets, taken together with the assumption that the average sizes of long and short period comets are approximately equal, leads to an increase in the total mass of comets in the solar system by almost two orders of magnitude. If gravitational ejection from the Uranus - Neptune zone during the later phases of planet formation is indeed responsible for the classical Oort cloud between 10⁴–10¹⁵ AU, then the mass of comets in this transplanetary region during cosmogonie times has to exceed the combined masses of Uranus and Neptune by over an order of magnitude. Furthermore, if the recent arguments for as many as 10¹⁴ comets in an inner Oort cloud between ~40– 10⁴AU are valid, then the total mass of comets in the solar system approaches 2% of a solar mass.     

Stability of the solar system

If the solar system is considered as a mechanical clockwork consisting of its present members which attract each other as mass-points, the extent of its present approach to secular stability (i.e., the state of minimum potential energy) — manifested by the existence of a number of nearcommensurabilities of the present orbital periods, not only of the planets, but also of their satellites —could not have been attained in a time-span of 4.6×10⁹ yr of its age by gravitational perturbations alone.The existence of such commensurabilities — striking in many instances— could then be understood only on the assumption that either (a) the solar system was actually born with the present 2-, 3- and 4-term couplings between the orbital period of the planets already built-in from the outset (which is improbable on any known grounds); or (b) that these couplings — in particular, the 25 Jupiter-Saturn commensurability — have arisen as a result of tidal interaction between proto-planetary globes of much larger dimensions than these planets possess today. For the present dimensions and mutual distances of these planets, their tidal interaction in 10⁹ yr would exert but negligible effects; and during that time neither their masses, nor the scale of the solar system underwent any essential change.Therefore, a hypothesis is proposed that the situation now obtaining had its origin in the early days of the formation of the solar system, when the planetary globes — in particular, those of Jupiter and Saturn (now in the terminal stage of Kelvin contraction) — were very much larger than they are today; and when, as a result, the tidal coupling between them operated at a much higher rate than at the present time.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.     

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.     

Dynamical history of coplanar two-satellite systems

One of the possible early states of the Earth-Moon system was a system of several large satellites around the Earth. The dynamical evolution of coplanar three-body systems is studied; a planet (Earth) and two massive satellites (proto-moons) with geocentric orbits of slightly different radii. Such configurations may arise in multiple satellite systems receding from a planet due to tidal friction. The numerical integration of the equations of motion shows that initially circular Keplerian orbits are soon transformed into disturbed elliptic orbits which are intersecting. The life-time of such a coplanar system between two probable physical collisions of satellites is roughly from one day to one year for satellite systems with radii less than 20R⊕, and may reach 100 yr for three-dimensional systems. This time-scale is short in comparison with the duration of the removal of satellites due to tides raised on the planet, which is estimated as 10⁶–10⁸ yr for the same orbital dimensions. Therefore, the life-time of a system of several proto-moons is mainly determined by their tidal interactions with the Earth. For conditions which we have considered, the most probable result of the evolution was coalescence of satellites as the consequence of the collisions.     

The shapes and surface features of Prometheus and Pandora

Topographic models of Saturn's F-Ring shepherd satellites Prometheus and Pandora were derived from the shapes of limbs and terminators in Voyager images, modified locally to accommodate large craters and ridges. The models are presented here in tabular and graphic form, including the first published maps of the satellites. The shape of Prometheus is approximated by a triaxial ellipsoid with axes of 145, 85 and 60 km. The volume is estimated to be 3.9 ± 1.0 × 10⁵ km³, significantly smaller than previous estimates. A system of prominent ridges and valleys cross the north polar region. Prometheus appears to be less heavily cratered than the other small satellites near the edge of the rings, though this may be an artifact of the low resolution of available images. Pandora is approximated by a triaxial ellipsoid with axes of 114, 84 and 62 km. The volume is estimated to be 3.1 ± 1.0 × 10⁵ km³. Its surface appears to be very heavily cratered.     

The cosmic-ray 1.68-year variation: A clue to understand the nature of the solar cycle?

Using the maximum entropy method (MEM), the cosmic-ray power spectral density in the frequency range 3 × 10^–9–2 × 10^–7 Hz has been estimated for the period 1947–1990. Cosmic-ray intensity data were integrated from the ion chamber at Huancayo and the neutron monitor at Deep River, following the method of Nagashima and Morishita (1980). The estimated spectrum shows power-law dependence (f ^–1.62), with several peaks superimposed. Periodicities of the different peaks are identified and related to solar activity phenomena; most of them were reported in the past. Once the 11-yr variation is eliminated, the most prominent feature in the spectrum is a variation, not reported before, with a period of 1.68 yr (604.8 d). This peak is correlated with fluctuations of similar periodicities found in the southern coronal hole area and in large active regions. The importance that this variation may have to elucidate the solar magnetic flux emergence and the activity cycle is discussed.Deceased 10 April, 1995.     

Possible effects of internal rotation in low-mass stars

The influence of internal rotation on the evolution of a 0.85M star is investigated by the construction of model sequences. Rotation is treated by a simple one-dimensional approximation. The calculations assume solid-body rotation on the zero-age Main Sequence, followed by conservation of angular momentum in shells. The 4 cases considered have the initial angular velocities 0,2×10^–4, 6×10^–4, and 8×10^–4/sec. All cases but the last are followed to helium ignition. Compared with the non-rotating case, the rotating models are older at Main-Sequence turnoff, develop fast-spinning central regions on the red-giant branch, and ignite helium at higher surface luminosities and at larger helium-core masses. The increases in the last two quantities are roughly proportional to the square of the initial angular velocity.The 6×10^–4 case is followed through the helium core flash to the zero-age horizontal branch. Under the assumption of spherical symmetry, the non-central ignition of helium leads to a sequence of flashes of decreasing amplitude occurring progressively closer to the center. The flashes are weaker than those encountered in previous studies and do not produce mixing.     

Coronal Magnetography of Solar Active Region 8365 with the SSRT and NoRH Radio Heliographs

Microwave maps of solar active region NOAA 8365 are used to derive the coronal magnetograms of this region. The technique is based on the fact that the circular polarization of a radio source is modified when microwaves pass through the coronal magnetic field transverse to the line of sight. The observations were taken with the Siberian Solar Radio Telescope (SSRT) on October 21 – 23 and with the Nobeyama Radio Heliograph (NoRH) on October 22 – 24, 1998. The known theory of wave mode coupling in quasi-transverse (QT) region is employed to evaluate the coronal magnetograms in the range of 10 – 30 G at the wavelength 5.2 cm and 50 – 110 G at 1.76 cm, taking the product of electron density and the scale of coronal field divergence to be constant of 10¹⁸ cm^–2. The height of the QT-region is estimated from the force-free field extrapolations as 6.2 × 10⁹ cm for the 20 G and 2.3 × 10⁹ cm for 85 G levels. We find that on large spatial scale, the coronal magnetograms derived from the radio observations show similarity with the magnetic fields extrapolated from the photosphere.     

A balloon-borne imaging telescope for high-energy astrophysics

We describe an imaging telescope for observations of celestial sources in the energy range between 30 keV and 1.8 MeV onboard stratospheric balloons. The detector is a 41 cm diameter, 5 cm thick NaI(Tl) crystal coupled to 19 photomultipliers in an Anger camera configuration. It is surrounded by a plastic scintillator 15 cm thick on the sides, 0.2 cm thick at the top and 20 cm thick at the bottom. The imaging device is based upon a 19 × 19 element square MURA (Modified Uniformly Redundant Array) coded mask mounted in an one-piece mask-antimask configuration. The detector's spatial resolution is about 10 mm at 100 keV. This is the first experiment to use such a mask pattern and configuration for astrophysical purposes. The expected 3 sensitivity for an on-axis source observed for 10⁴ s at a residual atmosphere of 3.5 g cm^–2 is 1.44 × 10^–5 photons cm^–2 s^–1 keV^–1 at 100 keV and 1.00 × 10^–6 photons cm^–2 s^–1 keV^–1 at 1 MeV. The angular resolution is approximately 14 arcminutes over a 13°field of view. The instrument is mounted in an automatic platform with a capability for pointing and stabilization in both azimuth and elevation axis with 2 arcmin accuracy.Presented at the 2nd UN/ESA Workshop, held in Bogotá, Colombia, 9-13 November, 1992.     

A period study of V450 Herculis

A detailed period study of the eclipsing binary system V450 Her has been presented. A new period (P= _. ^d 12724) has been given. The period changes in different portions of the O-C diagram, based on new period, have been estimated. The total period change ranges from 3.28×10^–6 d to 7.06×10^–5 d, which is appreciable.     

RS CVn-binary RW com: A possible three-body system

A first detailed period study of the eclipsing RS CVn-binary system RW Com is presented. A new period (P=0^d.2373455) based on 223 minima is given. The O–C diagrams of RW Com have been presented for the first time. Types of ten minima have been corrected judging the period trend. Period changes in different portions of the O–C diagram (Figure 2) have been estimated. The total change in period (P/P) ranges from 5.5×10^–7 to 6.4×10^–6. Thus, P ranges from 1.3×10^–7 d to 1.5×10^–6 d. Numerous minima are available in the time interval 1967 to 1986. This part of the O–C diagram (Figure 2) shows a sinusoidal variation, thus, it is suspected that RW Com could be a three-body system. The period of variation due to third body appears to be nearly 16 years.