Diurnal and Subdiurnal Terms in Rdan97 Series

In this paper, the series RDAN97 recently published (Roosbeek and Dehant, 1998) is completed by computing the diurnal and subdiurnal nutation terms. The method used is based on computing the torque induced by the external bodies on the rigid earth. The ephemerides used are analytical and based on celestial mechanics considerations. With a truncation level of 0.1 μas, 115 terms in longitude and 78 terms in obliquity have been computed. These terms correspond to the influence of the earth's geopotential coefficients c_2,2 and s_2,2, c_3,m and s_3,m (for the interaction between the earth and the moon and the sun), and c_4,m and s_4,m (for the interaction between the earth and the moon). A comparison with the recent theories REN‐2000 (Souchay and Kinoshita, 1996, 1997) and SMART97 (Bretagnon et al., 1997, 1998) shows that our series is at a very high precision, better than the most recent VLBI campaigns. This revised version was published online in July 2006 with corrections to the Cover Date.     

Does the proton-to-electron mass ratio μ = m p /m e vary in the course of cosmological evolution?

The possible cosmological variation of the proton-to-electron mass ratio μ = m _p /m _e was estimated by measuring the H₂ wavelengths in the high-resolution spectrum of the quasar Q 0347-382. Our analysis yielded an estimate for the possible deviation ofμ value in the past, 10 Gyr ago: for the unweighted valueΔ μ / μ = (3.0±2.4)×10^-5; for the weightedvalueΔ μ / μ = (5.02±1.82)×10^-5.Since the significance of the both results does not exceed3σ, further observations are needed to increase the statistical significance. In any case, this result may be considered as the most stringent estimate on an upper limit of a possible variation of μ (95% C.L.):|Δ μ / μ| < 8× 10^-5 .This value serves as an effective tool for selection of models determining a relation between possible cosmological deviations of the fine-structure constant α and the elementary particle masses (m_p, m_e, etc.). This revised version was published online in July 2006 with corrections to the Cover Date.     

Effects On the Nutation of the Non-Zonal Harmonics of Third Degree

In this paper, we calculate the coefficients of the nutation for a rigid Earth model due to the C _3m and S _3m (m ≠ 0) harmonics of the geopotential, starting from the Hamiltonian theory as developped by Kinoshita (1977). We show that these coefficients are far from being negligible as given the level of truncation of 0.1 μas which is necessary in the reconstruction of the tables of nutation, and also that their value is very close to that given by Bretagnon et al. (1997). This revised version was published online in July 2006 with corrections to the Cover Date.     

Robe’s problem: its extension to 2+2 bodies

In the problem of 2+2 bodies in the Robe’s setup, one of the primaries of mass m^*₁m^{*}_{1} is a rigid spherical shell filled with a homogeneous incompressible fluid of density ρ ₁. The second primary is a mass point m ₂ outside the shell. The third and the fourth bodies (of mass m ₃ and m ₄ respectively) are small solid spheres of density ρ ₃ and ρ ₄ respectively inside the shell, with the assumption that the mass and the radius of third and fourth body are infinitesimal. We assume m ₂ is describing a circle around m^*₁m^{*}_{1}. The masses m ₃ and m ₄ mutually attract each other, do not influence the motion of m^*₁m^{*}_{1} and m ₂ but are influenced by them. We also assume masses m ₃ and m ₄ are moving in the plane of motion of mass m ₂. In the paper, the equations of motion, equilibrium solutions, linear stability of m ₃ and m ₄ are analyzed. There are four collinear equilibrium solutions for the given system. The collinear equilibrium solutions are unstable for all values of the mass parameters μ,μ ₃,μ ₄. There exist an infinite number of non collinear equilibrium solutions each for m ₃ and m ₄, lying on circles of radii λ,λ′ respectively (if the densities of m ₃ and m ₄ are different) and the centre at the second primary. These solutions are also unstable for all values of the parameters μ,μ ₃,μ ₄, φ, φ′. Such a model may be useful to study the motion of submarines due to the attraction of earth and moon.     

RDAN97: An Analytical Development of Rigid Earth Nutation Series Using the Torque Approach

New series of rigid Earth nutations for the angular momemtum axis, the rotation axis and the figure axis, named RDAN97, are computed using the torque approach. Besides the classical J2 terms coming from the Moon and the Sun, we also consider several additional effects: terms coming from J3 and J4 in the case of the Moon, direct and indirect planetary effects, lunar inequality, J2 tilt, planetary‐tilt, effects of the precession and nutations on the nutations, secular variations of the amplitudes, effects due to the triaxiality of the Earth, new additional out‐of‐phase terms coming from second order effect and relativistic effects. Finally, we obtain rigid Earth nutation series of 1529 terms in longitude and 984 terms in obliquity with a truncation level of 0.1 μ (microarcsecond) and 8 significant digits. The value of the dynamical flattening used in this theory is HD=(C-A)/C=0.0032737674 computed from the initial value pa=50′.2877/yr for the precession rate. These new rigid Earth nutation series are then compared with the most recent models (Hartmann et al., 1998; Souchay and Kinoshita, 1996, 1997; Bretagnon et al., 1997, 1998. We also compute a benchmark series (RDNN97) from the numerical ephemerides DE403/LE403 (Standish et al., 1995) in order to test our model. The comparison between our model (RDAN97) and the benchmark series (RDNN97) shows a maximum difference, in the time domain, of 69 μas in longitude and 29 μas in obliquity. In the frequency domain, the maximum differences are 6 μas in longitude and 4 μ as in obliquity which is below the level of precision of the most recent observations (0.2 mas in time domain (temporal resolution of 1 day) and 0.02 mas in frequency domain). This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

Bifurcation of central configuration in the 2N+1 body problem

The bifurcation of central configuration in the Newtonian N-body problem for any odd number N ≥ 7 is shown. We study a special case where 2n particles of mass m on the vertices of two different coplanar and concentric regular n-gons (rosette configuration) and an additional particle of mass m₀ at the center are governed by the gravitational law he 2n+1 body problem. This system is of two degrees of freedom and permits only one mass parameter μ =m ₀/m. This parameter μ controls the bifurcation. If n≥ 3, namely any odd N ≥ 7, then the number of central configurations is three when μ ≥ μ _c , and one when μ ≥ μ _c . By combining the results of the preceding studies and our main theorem, explicit examples of bifurcating central configuration are obtained for N ≤ 13, for any odd N ∈ [15,943], and for any N ≥ 945.     

Effect of perturbation on the location of libration point in the robe restricted problem of three bodies

The effect of small perturbation in the Coriolis and centrifugal forces on the location of libration point in the ‘Robe (1977) restricted problem of three bodies’ has been studied. In this problem one body,m ₁, is a rigid spherical shell filled with an homogeneous incompressible fluid of densityϱ ₁. The second one,m ₂, is a mass point outside the shell andm ₃ is a small solid sphere of densityϱ ₃ supposed to be moving inside the shell subject to the attraction ofm ₂ and buoyancy force due to fluidϱ ₁. Here we assumem ₃ to be an infinitesimal mass and the orbit of the massm ₂ to be circular, and we also suppose the densitiesϱ ₁, andϱ ₃ to be equal. Then there exists an equilibrium point (−μ + (ɛ′μ)/(1 + 2μ), 0, 0).     

An optimal (m+3)[m+4] Runge Kutta algorithm

Recurrent power series methods are particularly applicable to problems in celestial mechanics since the Taylor coefficients may be expressed by recurrence relations. However, as the number of Taylor coefficients increases as is often necessary because of accuracy requirements, the computing time grows prohibitively large. In order to avoid this unfavorable situation, Dr E. Fehlberg introduced in 1960 Runge-Kutta methods that use the firstm Taylor coefficients obtained by recursive relations, or some other technique.Optimalm-fold Runge-Kutta methods are introduced. Embedded methods of order (m+3)[m+4] and (m+4)[m+5] are presented which have coefficients that produce minimum local truncation errors for the higher order pair of solutions of the method, as well as providing a near maximum absolute stability region. It is emphasized that the methods are formulated such that the higher order pair of solutions is to be utilized. These optimal methods are compared to the existingm-fold methods for several test problems. The numerical comparisons show that the optimal methods are more efficient. It is stressed that these optimal methods are particularly efficient whenm is small.     

Rosette Central Configurations, Degenerate Central Configurations and Bifurcations

In this paper we find a class of new degenerate central configurations and bifurcations in the Newtonian n-body problem. In particular we analyze the Rosette central configurations, namely a coplanar configuration where n particles of mass m₁ lie at the vertices of a regular n-gon, n particles of mass m₂ lie at the vertices of another n-gon concentric with the first, but rotated of an angle π /n, and an additional particle of mass m₀ lies at the center of mass of the system. This system admits two mass parameters μ = m₀/m₁ and ε = m₂/m₁. We show that, as μ varies, if n > 3, there is a degenerate central configuration and a bifurcation for every ε > 0, while if n = 3 there is a bifurcation only for some values of ε.     

On bridges B(pL, qS) around triangular libration points

When μ is smaller than Routh’s critical value μ ₁ = 0.03852 . . . , two planar Lyapunov families around triangular libration points exist, with the names of long and short period families. There are periodic families which we call bridges connecting these two Lyapunov families. With μ increasing from 0 to 1, how these bridges evolve was studied. The interval (0,1) was divided into six subintervals (0, μ ₅), (μ ₅, μ ₄), (μ ₄, μ ₃), (μ ₃, μ ₂), (μ ₂, μ ₁), (μ ₁, 1), and in each subinterval the families B(pL, qS) were studied, along with the families B(qS, qS′). Especially in the interval (μ ₂, μ ₁), the conclusion that the bridges B(qS, qS′) do not exist was obtained. Connections between the short period family and the bridges B(kS, (k + 1)S) were also studied. With these studies, the structure of the web of periodic families around triangular libration points was enriched.     

The Effect Of Zonal Tides On The DynamicalEllipticity Of The EarthAnd Its Influence On The Nutation

In this paper, the expressions of variations of the dynamical ellipticity and the principal moments of inertia due to the deformations produced by the zonal part of the tidal potential are obtained. Starting from these expressions, we have studied from equations related to Hamiltonian theory, their effects on the nutation and finally we have evaluated numerically such influences, with a level of truncation at 0.1 μas. Thus we have shown that some coefficients are quite large with respect to the usual accuracy of up-to-date observations. This revised version was published online in July 2006 with corrections to the Cover Date.     

Motion close to the Hopf Bifurcation Of the vertical family of periodic orbits of L4

The paper deals with different kinds of invariant motions (periodic orbits, 2D and 3D invariant tori and invariant manifolds of periodic orbits) in order to analyze the Hamiltonian direct Hopf bifurcation that takes place close to the Lyapunov vertical family of periodic orbits of the triangular equilibrium point L₄ in the 3D restricted three-body problem (RTBP) for the mass parameter, μ greater than (and close to) μ_R (Routh’s mass parameter). Consequences of such bifurcation, concerning the confinement of the motion close to the hyperbolic orbits and the 3D nearby tori are also described.     

Existence and stability of triangular points in the restricted three-body problem with numerical applications

In this paper, we prove that the locations of the triangular points and their linear stability are affected by the oblateness of the more massive primary in the planar circular restricted three-body problem, considering the effect of oblateness for J ₂ and J ₄. After that, we show that the triangular points are stable for 0<μ<μ _c and unstable when , where μ _c is the critical mass parameter which depends on the coefficients of oblateness. On the other hand, we produce some numerical values for the positions of the triangular points, μ and μ _c using planets systems in our solar system which emphasis that the range of stability will decrease; however this range sometimes is not affected by the existence of J ₄ for some planets systems as in Earth–Moon, Saturn–Phoebe and Uranus–Caliban systems.     

Calculation of effective optical depth of absorption line formation in homogeneous semi-infinite planetary atmosphere during anisotropic scattering

The algorithm for determining effective optical thickness of absorption line formation in a plane-parallel homogeneous planetary atmosphere is presented. The case of anisotropic scattering is considered. The results of numerical calculations of τ _e (μ₀) at the scattering angle γ = π for some values of the single scattering albedo λ and the parameter of the Heyney-Greenstein scattering indicatrix g are given. The refined equation for the function T ^m (−μ, μ₀) is presented.     

Collinear Equilibrium Solutions of Four-body Problem

We discuss the equilibrium solutions of four different types of collinear four-body problems having two pairs of equal masses. Two of these four-body models are symmetric about the center-of-mass while the other two are non-symmetric. We define two mass ratios as μ ₁ = m ₁/M _T and μ ₂ = m ₂/M _T, where m ₁ and m ₂ are the two unequal masses and M _T is the total mass of the system. We discuss the existence of continuous family of equilibrium solutions for all the four types of four-body problems.     

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.     

The Radio Recombination Line Intensities Of Atoms And Ions: Helium,Carbon And Oxygen

The radio recombination line intensities of heavy elements of helium, carbon and oxygen are calculated with accounting for dielectronic recombination. Dielectronic recombination rates are determined accurate to the second order of a perturbation theory and the rates are described as function of principal quantum number for helium-like atom or ion. Balance equations are solved for the departure coefficients from LTE b_n. The collision and spontaneous transition rates are accounted for the balance equations, in which non-equilibrium distribution source is dielectronic recombination. Non-equilibrium amplification coefficients are found as functions of a medium temperature, density and ion charge z = 1–3 for radio recombination lines. Optical depths are calculated for the heavy element low-frequency lines with the numbers 300 > n > 1200. For the chosen electronic temperatures and densities T_e = 0.8× 10⁴–10× 10⁴ K, N_e = 0.05–0.1 cm⁻³ the line optical depth is determined by the values τ_L∼ 0.1× 10⁻⁴–100× 10⁻⁴. Calculated for free-free transition rates, the optical depth is given by using the value τ_ff∼ 10⁻²τ_L.     

Note on the Generalized Hansen and Laplace Coefficients

Recently, Breiter et al. [Celest. Mech. Dyn. Astron., 2004, 88, 153–161] reported the computation of Hansen coefficients X _k ^{γ ,m} for non-integer values of γ. In fact, the Hansen coefficients are closely related to the Laplace b _s ^(m), and generalized Laplace coefficients b _s,r ^(m) [Laskar and Robutel, 1995, Celest. Mech. Dyn. Astron., 62, 193–217] that do not require s,r to be integers. In particular, the coefficients X ₀ ^{γ ,m} have very simple expressions in terms of the usual Laplace coefficients b _{γ +2} ^(m), and all their properties derive easily from the known properties of the Laplace coefficients.     

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.