On the status of the rotation of Venus

Owing to its extremely slow rotation, Venus must be regarded as a triaxial body with differences of all three principal moments of inertia comparable in magnitude, thus rendering it a body essentially different from a rapidly rotating planet. The dynamical problem then arises of how such a body, with a rotation-period comparable with its orbital period, would be affected by couples exerted upon it by the gravitational action of the Sun. Equations for the rotatory motion are set up in a form suitable for numerical solution by machine-calculations, but the problem so presented can be adequately investigated only for a hypothetical planet with far larger differences of principal moments than could hold for Venus. Results obtained on this limited basis nevertheless suggest that for the actual planet the direction of the rotation axis may move almost randomly between the two hemispheres defined by the orbital plane and thus that the present direction near the south celestial pole of the orbit may be only a temporary situation. Order-of-magnitude considerations based on the equations of motion suggest that a time-scale of order 10⁷ to 10⁸ yr may on average be required for large changes in direction of the rotation axis to take place.     

On the free rotation of a rigid body

It is well known that the equations governing the motion of a freely-rotating rigid body possess an exact analytical solution, involving Jacobi's elliptic functions. Andoyer (1923) and Deprit (1967) have shown that the problem may be very usefully reduced to a one-degree-of-freedom Hamiltonian system. When two of the body's principal moments of inertia are very nearly equal, the Hamiltonian system has the same form as the Ideal Resonance Problem. In earlier publications (Jupp, 1969, 1972, 1973), the author has constructed formal power-series solutions of the latter problem.In this article, the general solution of the Ideal Resonance Problem is employed to formulate a second-order formal series solution of the problem of a freely-rotating rigid body which has two of its principal moments of inertia differing by a small quantity. This solution is firstly expressed in terms of the mean elements, and then in terms of the initial conditions. The latter solution is global in nature being applicable over the whole phase plane. It is demonstrated that the exact solution and the second-order formal series solution, written in terms of the initial conditions, differ by terms of at most third order in the small parameter, over the whole domain of possible motions. This serves as an important check on the general results published in the earlier articles.     

Analytical extension of lunar libration tables

Tables of lunar physical libration defining the analytical dependence upon the parameters of the lunar gravitational field are presented. The tables are obtained on the framework of the main problem in lunar libration by integration of the Hamilton equations reduced to the harmonic oscillator equations.The variables of physical libration have been obtained in the form of Poisson series. The distinguishing feature of the tables is that these series are the analytical extension of semianalytical solution computed for a number of dynamical parameters LURE2.A comparison with the Eckhardt's solution is briefly presented. The previously revealed disagreement of the mean inclination of lunar equator to ecliptic with that in Eckhardt's solution 500 has been maintained.     

On the solar rotation and activity

The interaction between differential rotation and magnetic fields in the solar convection zone was recently modelled by Brun (2004). One consequence of that model is that the Maxwell stresses can oppose the Reynolds stresses, and thus contribute to the transport of the angular momentum towards the solar poles, leading to a reduced differential rotation. So, when magnetic fields are weaker, a more pronounced differential rotation can be expected, yielding a higher rotation velocity at low latitudes taken on the average. This hypothesis is consistent with the behaviour of the solar rotation during the Maunder minimum. In this work we search for similar signatures of the relationship between the solar activity and rotation determined tracing sunspot groups and coronal bright points. We use the extended Greenwich data set (1878–1981) and a series of full-disc solar images taken at 28.4 nm with the EIT instrument on the SOHO spacecraft (1998–2000). We investigate the dependence of the solar rotation on the solar activity (described by the relative sunspot number) and the interplanetary magnetic field (calculated from the interdiurnal variability index). Possible rotational signatures of two weak solar activity cycles at the beginning of the 20th century (Gleissberg minimum) are discussed. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamical evolution of the rotation of Venus

By considering the torque of the bodily tides, the effect of the core-mantle viscous coupling and the torque of the atmospheric tides have been obtained by numerical calculation: the evolution of the spin angular velocity and the obliquity of the Venus are calculated numerically with the step-variable Runge-Kutta method of 7th order; and 7 sets of the probable Cytherean spin evolution have been obtained. It is indicated that the present spin state of Venus is the result of long-term evolution within the reasonable ranges of some disposable parameters. The early spin period is between 7 h to 2 d and the corresponding obliquity is about 90 ° ~ 100 °. The effects of the torques of body and atmospheric tides and the core-mantle viscous coupling of Venus on its spin angular velocity could nearly cancel out each other about a billion years ago. Therefore, Venus could have been captured in a spin-orbit resonant state by the gravitational torque of the Earth on the permanent deformation part of Venus; and this resonant state has lasted up to the present time.     

On the relation between activity and rotation in main-sequence stars

Es wird ein modifizierter Index für das chromosphärische Aktivitätsniveau bei Sternen, A'_Call, vorgeschlagen. Dieses wird auf Grund der in den H und K Ca II-Linienkernen gemessenen Strahlungsflußdichte berechnet, ist aber im Unterschied zu anderen bekannten Indizes frei vom Einfluß der Sternfarbe. Die Anwendung des neuen Indexes führt zu einer modifizierten Aktivität-Rotation-Beziehung bei Sternen im unteren Teil der Hauptreihe. Es ist nachgewiesen, daß im Koordinatensystem log A'_Call, log Ro (Ro ist die Rossby Zahl) F-M-Sterne einer allgemeinen Beziehung mit dem Parameter α = 1.6 gehorchen. Die Kurve weist keine Zeichen von Aktivitätssättigung bei Sternen mit niedrigen Rossby-Zahlen auf, aber das Aktivitätsniveau strebt gegen einen konstanten Wert bei langsam rotierenden Sternen mit dünnen Konvektionszonen.     

On the effect of the mantle elasticity on the earth's rotation

Starting from the Hamiltonian model for a solid Earth with an elastic mantle previously developped by the authors, analytical expressions are derived which give the nutation series corresponding to the plane perpendicular to the angular momentum vector, to the plane perpendicular to the rotational axis and to the equator of figure, as well as the series that give the polar motion. The effects of the different perturbations — solid Earth, centrifugal and tidal potentials — are calculated separately. The corrections due to the elasticity of the mantle, which mostly correspond to the Oppolzer terms, are calculated with an accuracy of 10^–6 arc sec., given that the intrinsic observational accuracy has reached 0.01 mas.     

On the rotation period of Capella

We present differential Hα and Hβ photometry of the very bright RS CVn‐binary α Aurigae (Capella)obtained with theVienna automatic photoelectric telescope in the years 1996 through 2000. Low‐level photometric variations of up to 0^m_.04 are detected in Hα. A multifrequency analysis suggests two real periods of 106 ± 3 days and 8.64 ± 0.09 days, that we interpret to be the rotation periods of the cool and the hot component of the Capella binary, respectively. These periods confirm that the hotter component of Capella rotates asynchronously, while the cooler component appears to be synchronized with the binary motion. The combined Hα data possibly contains an additional period of 80.4 days that we, however, believe is either spurious and was introduced due to seasonal amplitude variations or stems from a time‐variable circumbinary mass flow. The rotational periods result in stellar radii of 14.3 ± 4.6 R_⊙ and 8.5 ± 0.5 R_⊙ for the cool and hot component, respectively, and are in good agreement with previously published radii based on radiometric and interferometric techniques. The long‐period eclipsing binary Aurigae served as our check star, and we detected complex light variations outside of eclipse of up to 0^m_.15 in H α and 0^m_.20 in Hβ. Our frequency analysis suggests the existence of at least three significant periods of 132, 89, and 73 days. One of our comparison stars (HD 33167, F5V) was discovered to be a very‐low amplitude variable with a period of 2.6360 ± 0.0055 days.     

Possible dynamical evolution of the rotation of Venus since formation

The past evolution of the rotation of Venus has been studied by a numerical integration method using the hypothesis that only solar tidal torques and core-mantle coupling have been active since formation. It is found quite conceivable that Venus had originally a rotation similar to the other planets and has evolved in 4.5×10⁹ years from a rapid and direct rotation (12-hour spin period and nearly zero obliquity) to the present slow retrograde one.While the solid tidal torque may be quite efficient in despinning the planet, a thermally driven atmospheric tidal torque has the capability to drive the obliquity from 0° towards 180° and to stabilize the spin axis in the latter position. The effect of a liquid core is discussed and it is shown that core-mantle friction hastens the latter part of the evolution and makes even stronger the state of equilibrium at 180°. The model assumes a nearly stable balance between solid and atmospheric tides at the current rotation rate interpreting the present 243 day spin period as being very close to the limiting value.A large family of solutions allowing for the evolution, in a few billions years, of a rapid prograde rotation to the present state have been found. Noticeably different histories of evolution are observed when the initial conditions and the values of the physical parameters are slightly modified, but generally the principal trend is maintained.The proposed evolutionary explanation of the current rotation of Venus has led us to place constraints on the solid bodyQ and on the magnitude of the atmospheric tidal torque. While the constraints seem rather severe in the absence of core-mantle friction (aQ15 at the annual frequency is required, and a dominant diurnal thermal response in the atmosphere is needed), for a large range of values of the core's viscosity, the liquid core effect allows us to relax somewhat these constraints: a solid bodyQ of the order 40 can then be allowed. ThisQ value implies that a semi-diurnal ground pressure oscillation of 2 mb is needed in the atmosphere in order for a stable balance to occur between the solid and atmospheric tides at the current rotation rate. No model of atmospheric tides on Venus has been attempted in this study, however the value of 2 mb agrees well with that predicted by the model given in Dobrovolskis (1978).     

Theory of the rotation of the rigid earth

An analytical theory is developed for planes normal to the angular-momentum axis, to the figure axis, and to the rotational axis of the triaxial rigid Earth. One of the purposes of this paper is to determine the effect on nutation and precession of Eckertet al.'s improvement to Brown's tables of the Moon and to check Woolard's theory from a different point of view. The present theory is characterized by the use of Andoyer variables, a moving reference plane, and Hori's averaging perturbation method. A comparison with Woolard's results shows that (1) the maximum difference in nutation for the plane normal to the angular-momentum axis, calculated from the same constants as Woolard adopted, reaches 0.0017, (2) the discrepancy in Oppolzer terms is large compared with the discrepancy in nutation for the plane normal to the angular-momentum axis, and (3) the present theory does not include some of the secular terms that are incorporated in Woolard's theory and that have an effect on the establishment of a reference system. The nutation coefficients 0.0001 for the three above-mentioned planes are calculated by using the numerical values recommended at the working meeting of the International Astronomical Union held in Washington in September 1974. The effects on precession and nutation due to the higher geopotential (n3) are also investigated. Any future revision of the lunar theory will not alter the values of the coefficients of the nutational terms derived here.     

Spectral methods for analyzing the rotation of solar structures

The application of spectral analysis methods for studying the rotation of solar structures is considered. The time series characterizing the time variation of the solar He I 1083 nm emission in 5° latitude zones have been used. Three types of spectral analysis have been tested: the Welch method, the multitaper method, and the Schuster periodogram method. The first two methods have been chosen for the analysis of observing time intervals 26 and 3 years in length. The Schuster periodogram method is more suitable for the sliding spectral analysis in a 1-year-long temporal window with a shift by half a year. The chosen methods for analyzing the power spectra allow one to obtain the spectral densities, the powers of significant peaks in them and the corresponding periods, the total powers in the specified intervals of periods and to estimate the significance of the peaks found and the intervals in which the true periods corresponding to the peaks can be located.     

Quaternions and the rotation of a rigid body

The orientation of an arbitrary rigid body is specified in terms of a quaternion based upon a set of four Euler parameters. A corresponding set of four generalized angular momentum variables is derived (another quaternion) and then used to replace the usual three-component angular velocity vector to specify the rate by which the orientation of the body with respect to an inertial frame changes. The use of these two quaternions, coordinates and conjugate moments, naturally leads to a formulation of rigid-body rotational dynamics in terms of a system of eight coupled first-order differential equations involving the four Euler parameters and the four conjugate momenta. The equations are formally simple, easy to handle and free of singularities. Furthermore, integration is fast, since only arithmetic operations are involved.     

Mechanisms for the rigid rotation of coronal holes

We show that the rotation of coronal holes can be understood in terms of a current-free model of the coronal magnetic field, in which holes are the footpoint locations of open field lines. The coronal field is determined as a function of time by matching its radial component to the photospheric flux distribution, whose evolution is simulated including differential rotation, supergranular diffusion, and meridional flow. We find that ongoing field-line reconnection allows the holes to rotate quasi-rigidly with their outer-coronal extensions, until their boundaries become constrained by the neutral line of the photospheric field as it winds up to form stripes of alternating magnetic polarity. This wind-up may be significantly retarded by a strong axisymmetric field component which forces the neutral line to low latitudes; it is also gradually halted by the cross-latitudinal transport of flux via supergranular diffusion and a poleward bulk flow. We conclude that a strong axisymmetric field component is responsible for the prolonged rigid rotation of large meridional holes during the declining phase of the sunspot cycle, but that diffusion and flow determine the less rigid rotation observed near sunspot maximum, when the holes corotate with their confining polarity stripes.     

Atmospheric tides and the rotation of Venus II. Spin evolution

Tides in the atmosphere of Venus may help to stabilize its slow retrograde rotation. The frequency dependence of the body tides also affects its rotational stability. However, the obliquity is probably maintained near 180° by friction between the core and mantle of Venus. In any case, it appears most likely that Venus originated with an obliquity greater than 90°.     

A precise modeling of Eros 433 rotation

Thanks to the recent data obtained from the NEAR space probe, we calculate in this paper, with a precision never reached so far for an asteroid, the precession and the nutation of Eros 433. In a preliminary step, we show that Eros obliquity has a remarkable value of 89.0° which tends to align its figure axis along the orbital plane. This very specific obliquity has some consequences on the motion of the axis of figure: one is the very small amplitude of the precession in longitude, for which we get the value . Moreover, we calculate Eros nutation for the figure axis due to the Sun, after developing the perturbing potential at the 4th order of the eccentricity. We show that the figure axis undergoes very large oscillations in the direction perpendicular to Eros orbital plane, due to the nutation in obliquity. Peak to peak, these oscillations reach 55″, which is far larger than the amplitudes of the nutations of the Earth due to the Sun (of the order of 2″). Moreover, we give the analytical developments of Δψ and Δε, both for the axis of angular momentum, and the axis of figure.     

Modelling the differential rotation of F stars

We model stellar differential rotation based on the mean-field theory of fluid dynamics. DR is mainly driven by Reynolds stress, which is anisotropic and has a non-diffusive component because the Coriolis force affects the convection pattern. Likewise, the convective heat transport is not strictly radial but slightly tilted towards the rotation axis, causing the polar caps to be slightly warmer than the equator. This drives a flow opposite to that caused by differential rotation and so allows the system to avoid the Taylor-Proudman state. Our model reproduces the rotation pattern in the solar convection zone and allows predictions for other stars with outer convection zones. The surface shear turns out to depend mainly on the spectral type and only weakly on the rotation rate. We present results for stars of spectral type F which show signs of very strong differential rotation in some cases. Stars just below the mass limit for outer convection zones have shallow convection zones with short convective turnover times. We find solar-type rotation and meridional flow patterns at much shorter rotation periods and horizontal shear much larger than on the solar surface, in agreement with recent observations. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

New Trends in the Development of the Lunar Physical Libration Theory

A review of the modern state of the lunar libration theory is presented. A significant progress in the lunar investigation is achieved due to the simultaneous processing of results of the satellite Doppler tracing and of the lunar laser ranging. The data evidencing existence of a small iron core in the Moon are discussed. In this connection, the further development of the theory of rotation of the Moon presents the study of internal structure and dynamics of a lunar body. A model of a two-layer Moon can have a very advanced application to explain some observed phenomena and to be as a first approach in the modelling of internal processes determining the lunar rotation.     

On the reality of the Venus winds

An examination of the effect of assumptions in the interpretation of the Venera wind data is made as a rebuttal to the suggestion by A.T. Young that the 140 m/sec Venera 8 horizontal wind at 45 km may be either spurious or anomalous. The Venera measurements of wind speed along with the Mariner measurements of a lower region of strong turbulence are evidence for a wide band of variable high-speed retrograde horizontal winds which girdle Venus at the equator. In the prevalent interpretation of the Mariner 10 uv photographs, the region of the top of the visible cloud is characterized by variable high-speed retrograde horizontal winds which orbit Venus with an average period of 4 Earth days, and by many features indicating vertical convection. This interpretation, together with the possibility of atmospheric corotation due to frictional coupling, suggests that the Venera-Mariner band of winds at 45 km extends well beyond the top of the visible cloud, and that the upper region of strong turbulence detected by the Mariners may result in part from vertical convection currents carried along by high-speed horizontal winds. In an alternate interpretation of the Mariner 10 uv photographs Young suggests that the predominant motions may be traveling wavelike disturbances with a 4-day period rather than bulk motion of the atmosphere. For this case the upper region of strong turbulence is interpreted as due mostly to vertical wind shear resulting from a rapid decrease in wind speed within a relatively short distance above the Venera-Mariner band of high-speed winds.     

Revealing of periodicities in the variations of differential rotation of the Sun

It is presumed that a north-southern asymmetry of a solid-body rotation of large spots, depending on even and odd solar activity cycles (Gigolashvili and Khutsishvili 1 1989, 1990) may possibly be explained by the asymmetry of the appearance of large structures with strong magnetic fields in the corresponding hemispheres. Spectral analysis of the observational data shows the presence of cyclic variations of differential rotation of large- and middle-sized spots. Variations of differential rotation of small spots are either absent or overlapped by noise. It is also supposed that the discovered and most frequently realized component of the spectrum of solar differential rotation variations — a four-years periodicity — may be either a real phenomenon or the result of overlapping of multiple quasi bi-annual variations.     

Internal rotation of the Sun as inferred from GONG observations

Helioseismology is a direct and most informative method of studying the structure and dynamics of the Sun. Determining the internal differential rotation of the Sun requires that the frequencies of its eigentones be estimated with a high accuracy, which is possible only on the basis of continuous long-term observations. The longest quasi-continuous series of data have been obtained by the Global Oscillation Network Group (GONG). The parameters of each individual mode of solar acoustic oscillations with low spherical degrees l=0, 1, 2, 3, 4, 5, 6 are determined by using 1260-day-long series of GONG observations. The mean frequency splitting by rotation for the modes of each radial order n is calculated by using all possible combinations between the eigenfrequencies in multiplets. As a result, it has become possible to statistically estimate the splitting and its measurement errors for the modes of each radial order. The mean splitting for each given degree l=1–6 is presented under the assumption of its independence of oscillation frequency, which holds for the achieved accuracy. The frequencies and splittings for the modes with low spherical degrees l, together with the MDI group results for higher degrees l, are used to invert the radial profile of solar angular velocity. Using the SOLA method to solve the inverse problem of restoring the rotation profile has yielded solutions sensitive to the deepest stellar interiors. Our results indicate that the solar core rotates faster than the surface, and there may be a local minimum in angular velocity at its boundary.