Earth’s precession–nutation motion: the error analysis of the theories IAU 2000 and IAU 2006 applying the VLBI data of the years 1984–2006

The long-term systematic errors of the analytical theories IAU 2000 and IAU 2006 of the Earth’s precession–nutational motion are studied making use of the VLBI data of 1984–2007. Several independent methods give indubitable evidence of the significant quadratic error in the IAU 2000 residuals of the precessional angle while the adopted value of the secular decrease /cy of the Earth’s ellipticity e (derived from Satellite Laser Ranging data) should manifest itself in the residuals of as the negative quadratic trend . The problem with the precession of the IAU 2006 theory adopted as a new international standard and based on the precession model P03 (Capitaine et al., Astron Astrophys 432:355–367, 2005) appears to be even more serious because the above mentioned quadratic term has already been incorporated into the P03 precession. Our analysis of the VLBI data demonstrates that the quadratic trend of the IAU 2006 residuals does amount to the expected value (30.0 ± 3) mas/cy². It means, first, that the theoretical precession rate of IAU 2006 should be augmented by the large secular correction and, second, that the available VLBI data have potentiality of estimating the rate . And indeed, processing these data by the numerical theory ERA of the Earth’s rotation (Krasinsky, Celest Mech Dyn Astron 96:169–217, 2006, Krasinsky and Vasilyev, Celest Mech Dyn Astron 96:219–237, 2006) yields the estimate /cy statistically in accordance with the satellite-based . On the other hand, applying IAU 2000/2006 models, the positive value /cy is found which is incompatible with the SLR estimate and, evidently, has no physical meaning. The large and steadily increasing error of the precession motion of the IAU 2006 theory makes the task of replacing IAU 2006 by a more accurate model be most pressing.     

Tidal effects in the earth–moon system and the earth's rotation

Analytical expressions for tidal torques induced by a tide‐arising planet which perturbs rotation of a nonrigid body are derived. Corresponding expressions both for secular and periodic perturbations of the Euler's angles are given for the case of the earth's rotation. Centennial secular rates of the nutation angle θ and of the earth's angular velocity ω, as well as the centennial logarithmic decrement ν of the Chandler wobble are evaluated:  mas, . In the Universal Time (UT) a large out‐of‐phase (sine) dissipative term with the period 18.6 years and the amplitude 2.3 ms is found. Corrections to nutation coefficients, which presumably have not been taken into account in IAU theory, are given. This revised version was published online in July 2006 with corrections to the Cover Date.     

Completely Integrable Systems Connected with Lie Algebras

In a recent paper Ballersteros and Ragnisco (1998) have proposed a new method of constructing integrable Hamiltonian systems. A new class of integrable systems may be devised using the following sequence: , where A is a Lie algebra is a Lie–Poisson structure on R ₃, C is a Casimir for is a reduced Poisson bracket and (A, ▵) is a bialgebra. We study the relation between a Lie-Poisson stucture Λ and a reduced Poisson bracket , which is a key element in using the Lie algebra A to constructing this sequence. New examples of Lie algebras and their related integrable Hamiltonian systems are given. This revised version was published online in July 2006 with corrections to the Cover Date.     

A Model of the Dust Envelope of the Carbon Mira Star V CrB from Photometry,Infrared Spectroscopy,and Speckle Polarimetry

$$UBVJHKLM$$ photometry for the carbon Mira star V CrB are presented. The infrared observations were carried out in the time interval 1989–2018, while the $$U$$, $$B$$, and $$V$$ data were obtained in 2001–2014. The light and color curves are analyzed. The pulsation period of V CrB has been found to be $$355\overset{\textrm{d}}{.}2$$ in the infrared $$JHKLM$$ bands and $$352^{\textrm{d}}$$ for the optical $$BV$$ band. In the $$JHK$$ bands, apart from periodic pulsations, there are distinct sinusoidal variations in the average brightness level with a characteristic period of $${\sim}8300$$ days. Color–magnitude relationships have been revealed for the infrared and optical bands. The phase curves exhibit the wavelength dependence of the brightness variability amplitude. The light curves for various bands and colors are discussed. We have constructed the model of a spherically symmetric circumstellar dust envelope that allows the observed spectral energy distribution at both maximum and minimum light to be reproduced equally well (within the model assumptions) and is consistent with the observations of V CrB by differential speckle polarimetry. The model is characterized by the following parameters: the optical depth is $$\tau_{K}=0.33$$, the inner and outer radii of the envelope are 8 and 40 000 AU, respectively. The envelope contains spherical carbon dust grains ($$3/4$$ by mass) and silicon carbide dust grains. Dust grains with a radius of 0.5 $$\mu$$m account for $$90\%$$ of the envelope mass. The remaining $$10\%$$ of the mass is accounted for by finer dust with a grain radius of 0.1 $$\mu$$m. Based on the observational data, we have estimated the bolometric flux from V CrB: $$2.6\times 10^{-7}$$ and $$5.1\times 10^{-7}$$ erg cm$${}^{-2}$$ s$${}^{-1}$$ at minimum and maximum light, respectively. The effective temperature of the star is $$T_{\textrm{max}}=3000$$ K at maximum light and $$T_{\textrm{min}}=2400$$ K at minimum light.     

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.     

The motion of the line of apsides of ζ phoenicis

An estimate of the period of the rotation of the line of apsides of the double-star system Phe is obtained by representing the density function as a product of a normal Gaussian distribution and an associated Legendre polynomial .The asymptotic behaviour of this function coincides with the results obtained by Zeldovichet al. (1981).The period of motion of the line of apsides of Phe (about 63 years) obtained in this way comes close to the period determined by an empirical formula for of Batten (1973).     

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.     

An Algebraic Method to Compute the Critical Points of the Distance Function Between Two Keplerian Orbits

We describe an efficient algorithm to compute all the critical points of the distance function between two Keplerian orbits (either bounded or unbounded) with a common focus. The critical values of this function are important for different purposes, for example to evaluate the risk of collisions of asteroids or comets with the Solar system planets. Our algorithm is based on the algebraic elimination theory: through the computation of the resultant of two bivariate polynomials, we find a 16th degree univariate polynomial whose real roots give us one component of the critical points. We discuss also some degenerate cases and show several examples, involving the orbits of the known asteroids and comets.      

On the measure-preserving mappings with odd dimension

In this paper, using two methods: LCN'S (Lyapunov characteristic numbers) method and slice cutting method, we study numerically two mappings with odd dimension: $$T_1 :\left\{ {\begin{array}{*{20}c} {x_{n + 1} = x_n + z_n ,} \\ {y_{n + 1} = y_n + x_{n + 1} , (\bmod 2\pi )} \\ {z_{n + 1} = z_n + A\sin y_{n + 1} ,} \\ \end{array} } \right. T_2 :\left\{ {\begin{array}{*{20}c} {x_{n + 1} = x_n + y_n + B \sin z_n ,} \\ {y_{n + 1} = y_n + A \sin x_{n + 1} , (\bmod 2\pi ),} \\ {z_{n + 1} = z_n + B \sin y_{n + 1} ,} \\ \end{array} } \right.$$ whereA, B are parameters. For the mappingT ₁ the whole region is stochastic; however, we find two-dimensional invariant manifolds for the mappingT ₂.     

Ion cyclotron instability in the solar wind

An ion cyclotron instability, arising because of the relative drift between the beam and the main components of the proton distribution function in the solar wind at 1 AU, is studied. The instability is excited in a bounded range of wave numbers provided the relative drift exceeds a certain minimum value called instability threshold. For 1, the instability threshold is smaller than or equal to the threshold of magnetosonic and Alfvén instabilities. The growth rates are enhanced by increasing relative drift and ratio of beam to main proton number density and by decreasing the wave numbers.     

Motion Around The Triangular Equilibrium Points Of The Restricted Three-Body Problem Under Angular Velocity Variation

We study numerically the asymmetric periodic orbits which emanate from the triangular equilibrium points of the restricted three-body problem under the assumption that the angular velocity ω varies and for the Sun–Jupiter mass distribution. The symmetric periodic orbits emanating from the collinear Lagrangian point L ₃, which are related to them, are also examined. The analytic determination of the initial conditions of the long- and short-period Trojan families around the equilibrium points, is given. The corresponding families were examined, for a combination of the mass ratio and the angular velocity (case of equal eigenfrequencies), and also for the critical value ω = 2

Hα Emission and Dynamics in Groups of Galaxies

H_α luminosities of a sample of galaxies in nearby compactgroups are presented. Our purpose is to study the influence of thegroup environment on the star formation rates (SFRs) of the galaxies in thegroups, provided that the H_α luminosity is a good tracer of theSFR of disc galaxies. Measuring the global L /L _B of the groups – including early-type galaxies – we find that the average value of the H_α emission is not significantly different from thatmeasured for field galaxies, and that most of the groups that show thehighest level of L /L _B, with respect to a set of synthetic groups built out of field galaxies, show tidal features in at least one of their members. Finally, we have exploredthe relationship between the ratio L /L _B and severalrelevant dynamical parameters of the groups (velocity dispersion, crossingtime, radius and mass-to-luminosity ratio) and have found no clearcorrelation. This suggests that the exact dynamical state of a groupdoes not appear to control the SFR of the group as a whole. This revised version was published online in July 2006 with corrections to the Cover Date.     

On the influence of gradients in the angular velocity on the solar meridional motions

If fluctuations in the density are neglected, the large-scale, axisymmetric azimuthal momentum equation for the solar convection zone (SCZ) contains only the velocity correlations and where u are the turbulent convective velocities and the brackets denote a large-scale average. The angular velocity, , and meridional motions are expanded in Legendre polynomials and in these expansions only the two leading terms are retained (for example, where is the polar angle). Per hemisphere, the meridional circulation is, in consequence, the superposition of two flows, characterized by one, and two cells in latitude respectively. Two equations can be derived from the azimuthal momentum equation. The first one expresses the conservation of angular momentum and essentially determines the stream function of the one-cell flow in terms of : the convective motions feed angular momentum to the inner regions of the SCZ and in the steady state a meridional flow must be present to remove this angular momentum. The second equation contains also the integral indicative of a transport of angular momentum towards the equator.With the help of a formalism developed earlier we evaluate, for solid body rotation, the velocity correlations and for several values of an arbitrary parameter, D, left unspecified by the theory. The most striking result of these calculations is the increase of with D. Next we calculate the turbulent viscosity coefficients defined by whereC _ro ⁰ and C _o ⁰ are the velocity correlations for solid body rotation. In these calculations it was assumed that ₂ was a linear function of r. The arbitrary parameter D was chosen so that the meridional flow vanishes at the surface for the rotation laws specified below. The coefficients v _ro ⁱ and v _0o ⁱ that allow for the calculation of C _ro and C _0o for any specified rotation law (with the proviso that ₂ be linear) are the turbulent viscosity coefficients. These coefficients comply well with intuitive expectations: v _ro ¹ and –v _0o ³ are the largest in each group, and v _0o ³ is negative.The equations for the meridional flow were first solved with ₀ and ₂ two linear functions of r ( ₀ ¹ = – 2 × 10 ^–12 cm ^–1) and ( ₂ ¹ = – 6 × 10 ¹² cm ^–1). The corresponding angular velocity increases slightly inwards at the poles and decreases at the equator in broad agreement with heliosismic observations. The computed meridional motions are far too large ( 150m s^–1). Reasonable values for the meridional motions can only be obtained if _o (and in consequence ), increase sharply with depth below the surface. The calculated meridional motion at the surface consists of a weak equatorward flow for gq < 29° and of a stronger poleward flow for > 29°.In the Sun, the Taylor-Proudman balance (the Coriolis force is balanced by the pressure gradient), must be altered to include the buoyancy force. The consequences of this modification are far reaching: is not required, now, to be constant along cylinders. Instead, the latitudinal dependence of the superadiabatic gradient is determined by the rotation law. For the above rotation laws, the corresponding latitudinal variations of the convective flux are of the order of 7% in the lower SCZ.     

Nonlinear newtonian theory and the effect of time dilatation

The fact that the energy density ρ_g of a static spherically symmetric gravitational field acts as a source of gravity, gives us a harmonic function \(f\left( \varphi \right) = e^{\varphi /c^2 } \) , which is determined by the nonlinear differential equation $$\nabla ^2 \varphi = 4\pi k\rho _g = - \frac{1}{{c^2 }}\left( {\nabla \varphi } \right)^2 $$ Furthermore, we formulate the infinitesimal time-interval between a couple of events measured by two different inertial observers, one in a position with potential φ-i.e., dt _φ and the other in a position with potential φ=0-i.e., dt ₀, as $${\text{d}}t_\varphi = f{\text{d}}t_0 .$$ When the principle of equivalence is satisfied, we obtain the well-known effect of time dilatation.