Effect of 3rd-degree gravity harmonics and Earth perturbations on lunar artificial satellite orbits

In a previous work we studied the effects of (I) the J ₂ and C ₂₂ terms of the lunar potential and (II) the rotation of the primary on the critical inclination orbits of artificial satellites. Here, we show that, when 3rd-degree gravity harmonics are taken into account, the long-term orbital behavior and stability are strongly affected, especially for a non-rotating central body, where chaotic or collision orbits dominate the phase space. In the rotating case these phenomena are strongly weakened and the motion is mostly regular. When the averaged effect of the Earth’s perturbation is added, chaotic regions appear again for some inclination ranges. These are more important for higher values of semi-major axes. We compute the main families of periodic orbits, which are shown to emanate from the ‘frozen eccentricity’ and ‘critical inclination’ solutions of the axisymmetric problem (‘J ₂ + J ₃’). Although the geometrical properties of the orbits are not preserved, we find that the variations in e, I and g can be quite small, so that they can be of practical importance to mission planning.     

Report of the International Astronomical Union Division I Working Group on Precession and the Ecliptic

The IAU Working Group on Precession and the Equinox looked at several solutions for replacing the precession part of the IAU 2000A precession–nutation model, which is not consistent with dynamical theory. These comparisons show that the (Capitaine et al., Astron. Astrophys., 412, 2003a) precession theory, P03, is both consistent with dynamical theory and the solution most compatible with the IAU 2000A nutation model. Thus, the working group recommends the adoption of the P03 precession theory for use with the IAU 2000A nutation. The two greatest sources of uncertainty in the precession theory are the rate of change of the Earth’s dynamical flattening, ΔJ₂, and the precession rates (i.e. the constants of integration used in deriving the precession). The combined uncertainties limit the accuracy in the precession theory to approximately 2 mas cent⁻². Given that there are difficulties with the traditional angles used to parameterize the precession, z_A, ζ_A, and θ_A, the working group has decided that the choice of parameters should be left to the user. We provide a consistent set of parameters that may be used with either the traditional rotation matrix, or those rotation matrices described in (Capitaine et al., Astron. Astrophys., 412, 2003a) and (Fukushima Astron. J., 126, 2003). We recommend that the ecliptic pole be explicitly defined by the mean orbital angular momentum vector of the Earth–Moon barycenter in the Barycentric Celestial Reference System (BCRS), and explicitly state that this definition is being used to avoid confusion with previous definitions of the ecliptic. Finally, we recommend that the terms precession of the equator and precession of the ecliptic replace the terms lunisolar precession and planetary precession, respectively.     

The vector approach to the problem of physical libration of the Moon: the linearized problem

A flexible and informative vector approach to the problem of physical libration of the rigid Moon has been developed in which three Euler differential equations are supplemented by 12 kinematic ones. A linearized system of equations can be split into an even and odd systems with respect to the reflection in the plane of the lunar equator, and rotational oscillations of the Moon are presented by superposition of librations in longitude and latitude. The former is described by three equations and consists of unrestricted oscillations with a period of T ₁ = 2.878 Julian years (amplitude of 1.855″) and forced oscillations with periods of T ₂ = 27.201 days (15.304″), one stellar year (0.008″), half a year (0.115″), and the third of a year (0.0003″) (five harmonics altogether). A zero frequency solution has also been obtained. The effect of the Sun on these oscillations is two orders of magnitude less than that of the Earth. The libration in latitude is presented by five equations and, at pertrubations from the Earth, is described by two harmonics of unrestricted oscillations (T ₅ ≈ 74.180 Julian years, T ₆ ≈ 27.347 days) and one harmonic of forced oscillations (T ₃ = 27.212 days). The motion of the true pole is presented by the same harmonics, with the maximum deviation from the Cassini pole being 45.3″. The fifth (zero) frequency yields a stationary solution with a conic precession of the rotation axis (previously unknown). The third Cassini law has been proved. The amplitudes of unrestricted oscillations have been determined from comparison with observations. For the ratio $ \frac{{\sin I}} {{\sin \left( {I + i} \right)}} \approx 0.2311 $ \frac{{\sin I}} {{\sin \left( {I + i} \right)}} \approx 0.2311 , the theory gives 0.2319, which confirms the adequacy of the approach. Some statements of the previous theory are revised. Poinsot’s method is shown to be irrelevant in describing librations of the Moon. The Moon does not have free (Euler) oscillations; it has oscillations with a period of T ₅ ≈ 74.180 Julian years rather than T ≈ 148.167 Julian years.     

Analytical Reference Functions <Emphasis Type="Italic">F</Emphasis>(λ) for the Sun's Limb Darkening and Its Absolute Continuum Intensities (λλ 300 to 1100 m)

It is shown (1) that the coefficients A_i of the limb darkening functions I(μ)/I_center = P₅ (μ) = ∑A_i μⁱ (i = 0... 5; μ = cos ϑ), which had been published by Neckel and Labs (Solar Phys. 153, 91, 1994), can well be approximated by analytical functions of wavelength λ, and (2) that at first sight purely formal extrapolation of the functions P₅(μ) to the very limb (μ = 0.0) is not meaningless: in combination with absolute intensities for the disk center these functions yield ‘limb intensities’ which all correspond to almost the same ‘limb temperature’, T_limb≈4746 K. Together these results lead to ‘reference functions’ which can quickly yield rather reliable values of the Sun's continuum intensities, for any values of μ and λ.     

Numerical theory of rotation of the deformable Earth with the two-layer fluid core. Part 1: Mathematical model

Improved differential equations of the rotation of the deformable Earth with the two-layer fluid core are developed. The equations describe both the precession-nutational motion and the axial rotation (i.e. variations of the Universal Time UT). Poincaré’s method of modeling the dynamical effects of the fluid core, and Sasao’s approach for calculating the tidal interaction between the core and mantle in terms of the dynamical Love number are generalized for the case of the two-layer fluid core. Some important perturbations ignored in the currently adopted theory of the Earth’s rotation are considered. In particular, these are the perturbing torques induced by redistribution of the density within the Earth due to the tidal deformations of the Earth and its core (including the effects of the dissipative cross interaction of the lunar tides with the Sun and the solar tides with the Moon). Perturbations of this kind could not be accounted for in the adopted Nutation IAU 2000, in which the tidal variations of the moments of inertia of the mantle and core are the only body tide effects taken into consideration. The equations explicitly depend on the three tidal phase lags δ, δ _c, δ _i responsible for dissipation of energy in the Earth as a whole, and in its external and inner cores, respectively. Apart from the tidal effects, the differential equations account for the non-tidal interaction between the mantle and external core near their boundary. The equations are presented in a simple close form suitable for numerical integration. Such integration has been carried out with subsequent fitting the constructed numerical theory to the VLBI-based Celestial Pole positions and variations of UT for the time span 1984–2005. Details of the fitting are given in the second part of this work presented as a separate paper (Krasinsky and Vasilyev 2006) hereafter referred to as Paper 2. The resulting Weighted Root Mean Square (WRMS) errors of the residuals dθ, sin θd for the angles of nutation θ and precession are 0.136 mas and 0.129 mas, respectively. They are significantly less than the corresponding values 0.172 and 0.165 mas for IAU 2000 theory. The WRMS error of the UT residuals is 18 ms.     

Photometric and Colorimetric Observations of Asteroid 423 Diotima and Their Analysis

Series of photometric and colorimetric observations of the Main-Belt asteroid 423 Diotima during its five oppositions were obtained at the Crimean Astrophysical Observatory. It was concluded, based on the results of a frequency analysis of the V-band photometry obtained in 1990, that the asteroid is a binary system: the rotation period of the primary component is equal to 4_. ^h56, and the period of rotation and the orbital period of the satellite are equal to 14_. ^h90. An analysis of simultaneous BV and BVR observations made in 1993 and 1998–1999 yielded a rotation period of 4_. ^h54 ± 0_. ^h01 for the primary component. An analysis of the sets of V-band observations of the asteroid made from 1982 through 2000 allowed us to find the period of forced precession, which was equal to 113^d (or 226^d). It was suggested that the axis of the primary component of the binary asteroid precesses and the large amplitude of brightness variations (about 1 ^m ) is due to its lenticular shape.     

Determining the orientation and spin period of TOPEX/Poseidon satellite by a photometric method

We present the results of photometric observations of the TOPEX/Poseidon satellite performed during 2008–2016. The satellite become space debris after a failure in January, 2006, in a low Earth orbit. In the Laboratory of Space Research of Uzhhorod National University 73 light curves of the spacecraft were obtained. Standardization of photometric light curves is briefly explained. We have calculated the color indices of reflecting surfaces and the spin rate change. The general tendency of the latter is described by an exponential decay function. The satellite spin periods based on 126 light curves (including 53 light curves from the MMT-9 project operating since 2014) were taken into account. In 2016 the period of its own rotation reached its minimum of 10.6 s.A method to derive the direction of the spin axis of an artificial satellite and the angles of the light scattered by its surface has been developed in the Laboratory of Space Research of Uzhhorod National University. We briefly describe the “Orientation” program used for these purposes. The orientation of the TOPEX/Poseidon satellite in mid-2016 is given. The angle of precession β = 45°–50° and period of precession P _pr = 141.5 s have been defined. The reasons for the identified nature of the satellite’s own rotation have been found. They amount to the perturbation caused by a deviation of the Earth gravity field from a central-symmetric shape and the presence of moving parts on the satellite.     

Eccentricity Generation in Hierarchical Triple Systems with Non-coplanar and Initially Circular Orbits

In a previous paper, we developed a technique for estimating the inner eccentricity in coplanar hierarchical triple systems on initially circular orbits, with comparable masses and with well-separated components, based on an expansion of the rate of change of the Runge-Lenz vector. Now, the same technique is extended to non-coplanar orbits. However, it can only be applied to systems with I ₀ < 39.23° or I ₀ > 140.77°, where I is the inclination of the two orbits, because of complications arising from the so-called ‘Kozai effect’. The theoretical model is tested against results from numerical integrations of the full equations of motion. This revised version was published online in July 2006 with corrections to the Cover Date.     

Neutron star sequences and the starquake glitch model for the Crab and the Vela pulsars

We construct for the first time, the sequences of stable neutron star (NS) models capable of explaining simultaneously, the glitch healing parameters, Q, of both the pulsars, the Crab (Q≥0.7) and the Vela (Q≤0.2), on the basis of starquake mechanism of glitch generation, whereas the conventional NS models cannot give such consistent explanation. Furthermore, our models also yield an upper bound on NS masses similar to those obtained in the literature for a variety of modern equations of state (EOSs) compatible with causality and dynamical stability. If the lower limit of the observational constraint of (i) Q≥0.7 for the Crab pulsar and (ii) the recent value of the moment of inertia for the Crab pulsar (evaluated on the basis of time-dependent acceleration model of the Crab Nebula), I _Crab,45≥1.93 (where I ₄₅=I/10⁴⁵ g cm²), both are imposed together on our models, the models yield the value of matching density, E _b =9.584×10¹⁴ g cm⁻³ at the core-envelope boundary. This value of matching density yields a model-independent upper bound on neutron star masses, M _max≤2.22M _⊙, and the strong lower bounds on surface redshift z _R ≃0.6232 and mass M≃2.11M _⊙ for the Crab (Q≃0.7) and the strong upper bound on surface redshift z _R ≃0.2016, mass M≃0.982M _⊙ and the moment of inertia I _Vela,45≃0.587 for the Vela (Q≃0.2) pulsar. However, for the observational constraint of the ‘central’ weighted mean value Q≈0.72, and I _Crab,45>1.93, for the Crab pulsar, the minimum surface redshift and mass of the Crab pulsar are slightly increased to the values z _R ≃0.655 and M≃2.149M _⊙ respectively, whereas corresponding to the ‘central’ weighted mean value Q≈0.12 for the Vela pulsar, the maximum surface redshift, mass and the moment of inertia for the Vela pulsar are slightly decreased to the values z _R ≃0.1645, M≃0.828M _⊙ and I _Vela,45≃0.459 respectively. These results set an upper and lower bound on the energy of a gravitationally redshifted electron-positron annihilation line in the range of about 0.309–0.315 MeV from the Crab and in the range of about 0.425–0.439 MeV from the Vela pulsar.     

Periodic solutions for resonance 4/3: Application to the construction of an intermediary orbit for Hyperion’s motion

The motion of Hyperion is an almost perfect application of second kind and second genius orbit, according to Poincaré’s classification. In order to construct such an orbit, we suppose that Titan’s motion is an elliptical one and that the observed frequencies are such that 4n _H−3n _T+3n _ω=0, where n _H, n _T are the mean motions of Hyperion and Titan, n _ω is the rate of rotation of Hyperion’s pericenter. We admit that the observed motion of Hyperion is a periodic motion such as . Then, .N _H, N _T, k∈ N ⁺. With that hypothesis we show that Hyperion’s orbit tends to a particular periodic solution among the periodic solutions of the Keplerian problem, when Titan’s mass tends to zero. The condition of periodicity allows us to construct this orbit which represents the real motion with a very good approximation. This revised version was published online in July 2006 with corrections to the Cover Date.     

A magnetic mechanism for halting inward protoplanet migration: I. Necessary conditions and angular momentum transfer timescales

A magnetic torque associated with the magnetic field linking a giant, gaseous protoplanet to its host pre-main-sequence star can halt inward protoplanet migration. This torque results from a toroidal magnetic field generated from the star’s poloidal (dipole) field by the twisting differential motion between the star’s rotation and the protoplanet’s revolution. Outside the corotation radius, where a protoplanet orbits slower than its host star spins, this torque transfers angular momentum from the star to the protoplanet, halting inward migration. Necessary conditions for angular momentum transfer include the requirement that the Alfvén speed v _A in the region magnetically linking a protoplanet to its host star exceeds the protoplanet’s orbital speed v _K . In addition, the timescale for Ohmic dissipation τ _D must exceed the protoplanet’s orbital period P to ensure that the protoplanet is magnetically coupled to its host star. For a Jupiter-mass protoplanet orbiting a solar-mass pre-main-sequence star, v _A >v _K and τ _D >P only when the migrating protoplanet approaches within about 0.1 AU of its host star, primarily because of the rapid drop in the strength of the magnetic field with increasing distance from the central star. Because of this restricted reach, inwardly migrating gaseous protoplanets can be expected to “pile up” very close to their central stars, as is indeed observed for extrasolar planets. The characteristic timescale required for a magnetic torque to transfer angular momentum outward from a more rapidly spinning central star to a magnetically coupled protoplanet is found to be comparable to planet-forming disk lifetimes and protoplanet migration timescales.     

An absolute clock of the cosmos?

In 1968–2005 different observers (mainly, one of the authors—V.M. Lyuty) performed numerous measurements of luminosity of the nucleus of the Seyfert galaxy NGC 4151. It is shown that (a) luminosity of the object pulsated over 38 years with a period of 160.0106(7) min coinciding, within the error limits, with the well-known period P ₀ = 160.0101(2) min of the enigmatic “solar” pulsations, and (b) when registering oscillations of luminosity of NGC 4151 nucleus with the P ₀ period, time moments of observations must be reduced to the earth instead of the sun, i.e., to the reference frame of the observer. The coherent P ₀ oscillation is characterized, therefore, by invariability of both frequency and phase with respect to redshift z and the earth’s orbital motion, respectively. From these results it, thus, follows that the coherent P ₀ oscillation seems to be of a true cosmological origin. The P ₀ period itself might represent a course of the “cosmic clock” related to the existence of an absolute time of the Universe in Newton’s comprehension.     

Empirical models of pressure and density in Saturn's interior: Implications for the helium concentration, its depth dependence, and Saturn's precession rate

We present ‘empirical’ models (pressure vs. density) of Saturn's interior constrained by the gravitational coefficients J₂, J₄, and J₆ for different assumed rotation rates of the planet. The empirical pressure-density profile is interpreted in terms of a hydrogen and helium physical equation of state to deduce the hydrogen to helium ratio in Saturn and to constrain the depth dependence of helium and heavy element abundances. The planet's internal structure (pressure vs. density) and composition are found to be insensitive to the assumed rotation rate for periods between 10h:32m:35s and 10h:41m:35s. We find that helium is depleted in the upper envelope, while in the high pressure region (P?1 Mbar) either the helium abundance or the concentration of heavier elements is significantly enhanced. Taking the ratio of hydrogen to helium in Saturn to be solar, we find that the maximum mass of heavy elements in Saturn's interior ranges from ∼6 to 20 M_⊕. The empirical models of Saturn's interior yield a moment of inertia factor varying from 0.22271 to 0.22599 for rotation periods between 10h:32m:35s and 10h:41m:35s, respectively. A long-term precession rate of about ^0.754″ yr⁻¹ is found to be consistent with the derived moment of inertia values and assumed rotation rates over the entire range of investigated rotation rates. This suggests that the long-term precession period of Saturn is somewhat shorter than the generally assumed value of 1.77×¹⁰⁶ years inferred from modeling and observations.     

Magnetic field structure and accretion regimes of the asynchronous polar by Cam

Series of photometric CCD observations of the asynchronous polar BY Cam in a low accretion state (R = 14^m–16^m) were made on the K-380 telescope at the Crimean Astrophysical Observatory (CrAO) over 100 hours in the course of 31 nights during 2004–2005. A period of P ₁ = 0.137120±0.000002 days was found for the variations in the brightness, along with less significant periods of P ₂ = 0.139759±0.000003 and P₃ = 0.138428±0.000002 days, where P₂ and P₃ are obviously the orbital and rotation periods, while the dominant period P₁ is the sideband period. A modulation in the brightness and an amplitude of 0.137 days in the oscillations at the orbital-rotational beat period (synodic cycle) of 14.568±0.003 day are found for the first time. The profile of the modulation period is four humped. This indicates that the magnetic field has a quadrupole component, which shows up well during the low brightness state. Accretion takes place simultaneously into two or three accretion zones, but at different rates. The times of the times of maxima for the main accretion zone vary with the phase of the beat period. Three types of variation of this sort are distinguished: linear, discontinuous, and chaotic, which indicate changes in the accretion regimes. At synodic phases 0.25 and 0.78 the bulk of the stream switches by 180°, and at phase 0.55, by ∼75°. At phases of 0.25–0.55 and 0.55–0.78, the O-C shift with a period of 0.1384 days, which can be explained by a retrograde shift of the main accretion zone relative to the magnetic pole and/or a change in the angle between the field lines and the surface of the white dwarf owing to the asynchronous rotation. For phases of 0.78–1.25 the motion of the accretion zone is quite chaotic. It is found that synchronization of the components occurs at a rate of less than dP_rot/P_rot∼10⁻⁹ day/day. __________ Translated from Astrofizika, Vol. 49, No. 1, pp. 121–137 (February 2006).     

Long-term evolution of orbits about a precessing oblate planet. 2. The case of variable precession

We continue the study undertaken in Efroimsky [Celest. Mech. Dyn. Astron. 91, 75–108 (2005a)] where we explored the influence of spin-axis variations of an oblate planet on satellite orbits. Near-equatorial satellites had long been believed to keep up with the oblate primary’s equator in the cause of its spin-axis variations. As demonstrated by Efroimsky and Goldreich [Astron. Astrophys. 415, 1187–1199 (2004)], this opinion had stemmed from an inexact interpretation of a correct result by Goldreich [Astron. J. 70, 5–9 (1965)]. Although Goldreich [Astron. J. 70, 5–9 (1965)] mentioned that his result (preservation of the initial inclination, up to small oscillations about the moving equatorial plane) was obtained for non-osculating inclination, his admonition had been persistently ignored for forty years. It was explained in Efroimsky and Goldreich [Astron. Astrophys. 415, 1187–1199 (2004)] that the equator precession influences the osculating inclination of a satellite orbit already in the first order over the perturbation caused by a transition from an inertial to an equatorial coordinate system. It was later shown in Efroimsky [Celest. Mech. Dyn. Astron. 91, 75–108 (2005a)] that the secular part of the inclination is affected only in the second order. This fact, anticipated by Goldreich [Astron. J. 70, 5–9 (1965)], remains valid for a constant rate of the precession. It turns out that non-uniform variations of the planetary spin state generate changes in the osculating elements, that are linear in , where is the planetary equator’s total precession rate that includes the equinoctial precession, nutation, the Chandler wobble, and the polar wander. We work out a formalism which will help us to determine if these factors cause a drift of a satellite orbit away from the evolving planetary equator.By “precession,” in its most general sense, we mean any change of the direction of the spin axis of the planet—from its long-term variations down to nutations down to the Chandler wobble and polar wander.     

Classical and relativistic node precessional effects in WASP-33b and perspectives for detecting them

WASP-33 is a fast rotating, main sequence star which hosts a hot Jupiter moving along a retrograde and almost polar orbit with semi-major axis a=0.02 au and eccentricity provisionally set to e=0. The quadrupole mass moment J₂^*J_{2}^{\star} and the proper angular momentum S _⋆ of the star are 1900 and 400 times, respectively, larger than those of the Sun. Thus, huge classical and general relativistic non-Keplerian orbital effects should take place in such a system. In particular, the large inclination Ψ_⋆ of the orbit of WASP-33b to the star’s equator allows to consider the node precession [(W)\dot]\dot{\Omega} and the related time variation dt _d /dt of the transit duration t _d . The WASP-33b node rate due to J₂^*J_{2}^{\star} is 9×10⁹ times larger than the same effect for Mercury induced by the Sun’s oblateness, while the general relativistic gravitomagnetic node precession is 3×10⁵ times larger than the Lense-Thirring effect for Mercury due to the Sun’s rotation. We also consider the effect of the centrifugal oblateness of the planet itself and of a putative distant third body X. The magnitudes of the induced time change in the transit duration are of the order of 3×10⁻⁶,2×10⁻⁷,8×10⁻⁹ for J₂^*J_{2}^{\star}, the planet’s rotational oblateness and general relativity, respectively. A yet undiscovered planet X with the mass of Jupiter orbiting at more than 1 au would induce a transit duration variation of less than 4×10⁻⁹. A conservative evaluation of the accuracy in measuring dt _d /dt over 10 yr points towards ≈10⁻⁸. The analysis presented here will be applicable also to other exoplanets with similar features if and when they will ne discovered.     

The Diagnosis of Complex Rotation in the Lightcurve of 4179 Toutatis and Potential Applications to Other Asteroids and Bare Cometary Nuclei

The analysis of radar observations of the asteroid 4179 Toutatis by Hudson and Ostro (1995, Science270, 84-86) yielded a complex spin state. We revisit the visible lightcurve data on Toutatis (Spencer et al. 1995, Icarus117, 71-89) to explore the feasibility of using a rotational lightcurve to recover the signature of an excited spin state. For this, we apply Fourier transform and CLEAN algorithm (WindowCLEAN). WindowCLEAN yields clear and precise frequency signatures associated with the precession of the long axis about the total angular momentum vector and a combination of this precession and rotation about the long axis. For a long-axis mode state, our periodicities for Toutatis yield a mean long-axis precession period, P_φ, of 7.38 days and a rotation period around the long axis, P_ψ, of 5.38 days, which compare well with the respective periods of 7.42 and 5.37 days derived by Ostro et al. (1999, Icarus137, 122-139) and represent an independent confirmation of these values. We explain why the dramatic change in the Earth-Toutatis-Sun geometry during the time that the lightcurve was obtained has little effect on the final results obtained. Using the Toutatis example as a guide, we discuss the capabilities as well as the limitations on deriving information about complex spin states from asteroidal lightcurves.     

Exoplanet's Figure and Its Interior

Along with the development of the observing technology, the observation and study on the exoplanets’ oblateness and apsidal precession have achieved significant progress. The oblateness of an exoplanet is determined by its interior density profile and rotation period. Between its Love number k₂ and core size exists obviously a negative correlation. So oblateness and k₂ can well constrain its interior structure. Starting from the Lane-Emden equation, the planet models based on different polytropic indices are built. Then the flattening factors are obtained by solving the Wavre's integro-differential equation. The result shows that the smaller the polytropic index, the faster the rotation, and the larger the oblateness. We have selected 469 exoplanets, which have simultaneously the observed or estimated values of radius, mass, and orbit period from the NASA (National Aeronautics and Space Administration) Exoplanet Archive, and calculated their flattening factors under the two assumptions: tidal locking and fixed rotation period of 10.55 hours. The result shows that the flattening factors are too small to be detected under the tidal locking assumption, and that 28% of exoplanets have the flattening factors larger than 0.1 under the fixed rotation period of 10.55 hours. The Love numbers under the different polytropic models are solved by the Zharkov's approach, and the relation between k₂ and core size is discussed.     

Shear-free homogeneous cosmological model with heat flux

We present the condition of vanishing shear in a spatially homogeneous spacetime in terms of the Ricci rotation co-efficients corresponding to an orthonormal tetrad (ν ^α. _A η ^α) (whereν ^α is the unit vector along the time axis and _A η ^α are the three independent reciprocal group vectors). Assuming that the velocity vector can be expanded in the direction ofν ^α and any one of the _A η ^α’s it is shown that shear-free motion is possible only in case of some special Bianchi types, and these cases are studied assuming the velocity vector to be geodetic and that there may be a nonvanishing heat flux term.     

Dynamical fine structure of a quiescent filament

A quiescent filament was observed near the center of the disk (N5, W5) with the MSDP spectrograph of the 50 cm refractor of the Pic-du-Midi Observatory on June 17, 1986. We focus our study on the statistical moments of the Dopplershift,V ₁, and the intensity,I ₁, at the center of a chord of the Hα profile (±0.256 Å), versus the minimum intensityI ₀. We use a statistical model simulating a numbern _max of threads (of optical thicknessτ ₀ and source functionS ₀), seen over the chromosphere. The threadsj along the same line-of-sighti are identical except for the velocityv _j (gaussian distributionv ₀,σ _v). We search for the best fit between the observed and simulated quantities:V ₁,σ (V ₁),I ₁,σ(I ₁), and the histogram of theI ₀ values over the field of view. A good fit is obtained with: (a) threads characterized byτ ₀ = 0.2,S ₀ = 0.06 (unit of the continuum at disk center), mean upward velocityv ₀ = 1.7 km s⁻¹ and gaussian-type velocity distributionσ _v = 3.5 km s⁻¹. Other possible values ofτ ₀ andσ _v are discussed; (b) underlying chromosphere deduced from observed quiet Sun (outside the filament) by modifying the chromospheric velocities: additional mean upward velocity 0.7 km s⁻¹, standard deviation reduced by a factorF _c ∼ 0.7. The results are discussed in connection with the values deduced from prominence observations.