New values of gravitational moments J 2 and J 4 deduced from helioseismology

By applying the theory of slowly rotating stars to the Sun, the solar quadrupole and octopole moments J ₂ and J ₄ were computed using a solar model obtained from CESAM stellar evolution code (Morel, 1997) combined with a recent model of solar differential rotation deduced from helioseismology (Corbard et al., 2002). This model takes into account a near-surface radial gradient of rotation which was inferred and quantified from MDI f-mode observations by Corbard and Thompson (2002). The effect of this observational near-surface gradient on the theoretical values of the surface parameters J ₂, J ₄ is investigated. The results show that the octopole moment J ₄ is much more sensitive than the quadrupole moment J ₂ to the subsurface radial gradient of rotation.     

A new estimate of the quadrupole moment of the sun

Recent solar observations at Pic du Midi are reported that yield a value of J ₂=(2.57 ± 2.36) x 10^–6 for the quadrupole moment of the Sun. These observations were conducted from July 1993 to July 1994 after several improvements of the scanning heliometer. This instrument operates by fast photoelectric scans of opposite limbs of the Sun quasi-simultaneously, which provides the distance between both inflection points of the limb profiles. Any number of solar diameters in any position angle can be measured within a time interval short enough to minimize the scattering of the observational parameters. Errors due to atmospheric deterioration are discussed. From our results, compared to previous values obtained by other authors, it can be concluded than an upper limit for J ₂ is probably 1.0 × 10^-5.     

Impact of the Quadrupole Moment of the Sun on the Dynamics of the Earth-Moon System

Range of values of the Sun's mass quadrupole moment of coefficient J₂ arising both from experimental and theoretical determinations enlarge across literature on two orders of magnitude, from around 10^-7 until to 10^-5. The accurate knowledge of the Moon's physical librations, for which the Lunar Laser Ranging data reach an outstanding precision level, prove to be appropriate to reduce the interval of J₂ values by giving an upper bound of J₂. A solar quadrupole moment as high as 1.1 10^-5 given either from the upper bounds of the error bars of the observations, or from the Roche's theory, is not compatible with the knowledge of the lunar librations accurately modeled and observed with the LLR experiment. The suitable values of J₂ have to be smaller than 3.0 10^-6. As a consequence, this upper bound of 3.0 10^-6 is accepted to study the impact of the Sun's quadrupole moment of mass on the dynamics of the Earth-Moon system. Such as effect (with J₂ = 5.5±1.3 × 10^-6) has been already tested in 1983 by Campbell & Moffat using analytical approximate equations, and thus for the orbits of Mercury, Venus, the Earth and Icarus. The approximate equations are no longer sufficient compared with present observational data and exact equations are required. As if to compute the effect on the lunar librations, we have used our BJV relativistic model of solar system integration including the spin-orbit coupled motion of the Moon. The model is solved by numerical integration. The BJV model stems from general relativity by using the DSX formalism for purposes of celestial mechanics when it is about to deal with a system of n extended, weakly self-gravitating, rotating and deformable bodies in mutual interactions. The resulting effects on the orbital elements of the Earth have been computed and plotted over 160 and 1600 years. The impact of the quadrupole moment of the Sun on the Earth's orbital motion is mainly characterized by variations of , , and . As a consequence, the Sun's quadrupole moment of mass could play a sensible role over long time periods of integration of solar system models. This revised version was published online in July 2006 with corrections to the Cover Date.     

Ion-Acoustic Wave Generation by Two Kinetic Alfvén Waves and Particle Heating

In this paper we have investigated the beat wave excitation of an ion-acoustic wave at the difference frequency of two kinetic (or shear) Alfvén waves propagating in a magnetized plasma with β<1 (β=8π n _e0 T _e/B ₀ ² , where n _e0 is the unperturbed electron number density, T _e is the electron temperature, and B ₀ is the external magnetic field). On account of the interaction between two kinetic Alfvén waves of frequencies ω ₁ and ω ₂, the ponderomotive force at the difference frequency ω ₁−ω ₂ leads to the generation of an ion-acoustic wave. Also because of the filamentation of the Alfvén waves, magnetic-field-aligned density dips are observed. In this paper we propose that the ion-acoustic wave generated by this mechanism may be one of the possible mechanisms for the heating and acceleration of solar wind particles.     

On the existence of J 2 invariant relative orbits from the dynamical system point of view

The present paper addresses the existence of J ₂ invariant relative orbits with arbitrary relative magnitude over the infinite time using the Routh reduction and Poincaré techniques in the J ₂ Hamiltonian problem. The current research also proposes a novel numerical searching approach for J ₂ invariant relative orbits from the dynamical system point of view. A new type of Poincaré mapping is defined from different central manifolds of the pseudo-circular orbits (parameterized by the Jacobi energy E, the polar component of momentum H _z and the measure of distance Δr between the fixed point and its central manifolds) to the nodal periods T _d and the drifts of longitude of the ascending node during one period (ΔΩ), which differs from Koon et al.’s (AIAA 2001) definition on central manifolds parameterized by the same fixed point. The Poincaré mapping is surjective because it compresses the three-dimensional variables into two-dimensional images, and the mapping degenerates into a bijective mapping in consideration of the fixed points. An iteration algorithm to the degenerated bijective mapping is proposed from the continuation procedure to perform the ergodic representation of E- and H _z -contour maps on the space of T _d –ΔΩ. For the surjective mapping with Δr ≠ 0, different pseudo-circular or elliptical orbits may share the same images. Hence, the inverse surjective mapping may achieve non-unique variables from a single image, which makes the generation of J ₂ invariant relative orbits possible. The pseudo-circular or elliptical orbits generated from the surjective mapping will be defined in different meridian planes. Hence, the critical contribution of the present paper is the assignment of J ₂ invariant relative orbits to different invariant parameters E and H _z depending on the E- and H _z -contour map, which will hold J ₂ invariant relative orbits for extended durations. To investigate the high-order nonlinearity neglected by previous studies, a formation configuration with a large magnitude of 500 km is successfully generated from the theory developed in the present work, which is beyond the scope of the linear conditions of J ₂ invariant relative orbits. Therefore, the existence of J ₂ invariant relative orbit with an arbitrary relative magnitude over the infinite time is achieved from the dynamical system point of view.     

On Parameters of the Earth-Like Model of Venus

A brief review on the history of constructing Earth-like models of the Venusian interior is presented. Available observational data are analyzed. An explanation for the anomalously large ratio of the quadrupole gravitational moment J₂ to the small parameter of Venus q is proposed. Thus, estimates were obtained for the precession constant and principal moments of inertia for the Earth-like model of Venus.

     

Three-dimensional time optimal double angular momentum reversal trajectory using solar sails

A new concept of three dimensional non-Keplerian trajectories with double angular momentum reversal is investigated with high performance solar sails. The main discussion of this paper is about such 3D solar inverse orbits with inner constraints. The problem is addressed in a time optimal control framework solved by an indirect method. Two typical solar inverse orbits have been achieved and presented in a 3D non-dimensional dynamic model in the Heliocentric Inertial Frame. Starting from the Earth orbit ecliptic plane, a sailcraft in the inverse orbit exhibits a butterfly shape trajectory. As such, the new orbits are symmetrical with respect to a plane which contains the Sun-perihelion line. The relation of the sail attitude angles between the two symmetrical parts of the orbits are used to reduce the simulation effort. The quasi-heliostationary property at its aphelia is demonstrated with variation of the orbital radius. Evolutions of the orbital velocity and optimal sail orientations are also outlined and discussed to benefit future design work. As is suited for space observation guaranteed by its butterfly shape, the inverse orbits are thoroughly studied in terms of the concerned parameters. The discussion of the parametric influence is ranked in order as perihelion distance r _E , required maximum position z _max, perihelion position z _f and the sail lightness number β. Suitable ranges of each parameter are adopted to illustrate the orbital variation trend. Through numerical simulations the features of such inverse orbits are further emphasized to provide an initial reference for future researchers.     

Dependence of the aerosol component of optical thickness and the relative concentration of methane on depth in atmospheres of giant planets

Based on the data on a spectral dependence of the geometric albedo of giant planet discs, we obtained depth variations in the optical thickness τ _a of the aerosol component and relative concentration γ of methane (Uranium, Neptune) lnτ _a = −0.720 + 1.507Δlnp (for −2.2085 ≤ lnp ≤ −1.0018), lnτ _a = +1.224 + 1.160Δlnp (for −1.0018 ≤ lnp ≤ −0.0595), lnτ _a = +2.318 + 0.192Δlnp (for −0.0595 ≤ lnp), γ = 0.0027 for Jupiter; lnτ _a = −0.846 + 1.598Δlnp (for −3.3619 ≤ lnp ≤ −2.0575), lnτ _a = +1.238 + 1.342Δlnp (for −2.0575 ≤ lnp ≤ −1.2074), lnτ _a = +2.379 + 0.722 (for −1.2074 ≤ lnp ≤ −0.6501), lnτ _a = +2.781 + 0.326Δlnp (for 0.6501 ≤ lnp), γ = 0.0027 for Saturn; lnτ _a = −2.694 + 0.087Δlnp (for +0.3685 ≤ lnp ≤ +1.2314), lnτ _a = −2.619 + 7.341Δlnp (for +1.2314 ≤ lnp ≤ +1.7556), lnτ _a = +1.229 + 0.956Δlnp (for +1.7556 ≤ lnp) for Uranium; lnτ _a = −1.861 + 1.248Δlnp (for +0.3204 ≤ lnp ≤ +0.9051), lnτ _a = −1.131 + 0.347Δlnp (for +0.9051 ≤ lnp) for Neptune; depth-averaged relative methane concentration lnγ = −9.982 + 2.676Δlnp(0.3584 ≤ lnp ≤ 1.5445); ln γ = −9.738 + 2.561Δlnp(0.3237 ≤ lnp ≤ 1.6156) and γ = 0.00382(lnp ≥ 1.6156); 0.00554(lnp ≥ 1.6156) for Uranium and Neptune, respectively (p is in bar).     

An estimation of the oblateness of Sun from the motion of Icarus

From astrometric observations of minor planet (1566) Icarus from 1949 to 1987 were made solutions for improved orbital elements of Icarus and the quadrupole moment of the Sun. The formal result was J₂ = -0.6±5.8 &d 10^–6. From this we can conclude that J ₂ is very probably less than 2 · 10^–-5.     

Bianchi type-I cosmological models for binary mixture of perfect fluid and dark energy

We consider a self consistent system of Bianchi Type-I cosmology and Binary Mixture of perfect fluid and dark energy. The perfect fluid is taken to be obeying equations of state p _PF =γρ _PF with γ∈[0,1]. The dark energy is considered to be obeying a quintessence-like equation of state where the dark energy obeys equation of state p _DE =ωρ _DE where ω∈[−1,0]. Exact solutions to the corresponding Einstein field equations are obtained. Some special cases are discussed and studied. Further more power law models and exponential models are investigated.     

Quasar correlation function and redshift space distortion from SDSS DR5 data

For z = 0.8–2.2 redshift interval, quasar pair correlation function parameters and β redshift space distortion parameter (connected to large-scale potential flows) values are estimated. We base them on the Main QSO Sample from SDSS Data Release 5. Standard correlation function form ξ(r) = (r ₀/r)^γ is used for comoving distances r = 2–50 Mpc between quasars. We fix the parameters of the cosmological model: Ω_Λ = 1 − Ω _M = 0.726 and H ₀ = 70.5 km/(s Mpc). We come to the best-fit parameter values of γ = 1.77 ± 0.20, r ₀ = 5.52 ± 0.95 Mpc/h for r in the range 2–30 Mpc, γ = 1.91 ± 0.11, r ₀ = 5.82 ± 0.61 Mpc for r in the range 2–50 Mpc. The mean β value is β = 0.43 ± 0.22.     

On the theory of the oblateness of the Sun

In this paper we first emphasize why it is important to know the successive zonal harmonics of the Sun's figure with high accuracy: mainly fundamental astrometry, helioseismology, planetary motions and relativistic effects. Then we briefly comment why the Sun appears oblate, going back to primitive definitions in order to underline some discrepancies in theories and to emphasize again the relevant hypotheses. We propose a new theoretical approach entirely based on an expansion in terms of Legendre's functions, including the differential rotation of the Sun at the surface. This permits linking the two first spherical harmonic coefficients (J ₂ and J ₄) with the geometric parameters that can be measured on the Sun (equatorial and polar radii). We emphasize the difficulties in inferring gravitational oblateness from visual measurements of the geometric oblateness, and more generally a dynamical flattening. Results are given for different observed rotational laws. It is shown that the surface oblateness is surely upper bounded by 11 milliarcsecond. As a consequence of the observed surface and sub-surface differential rotation laws, we deduce a measure of the two first gravitational harmonics, the quadrupole and the octopole moment of the Sun: J ₂=−(6.13±2.52)×10⁻⁷ if all observed data are taken into account, and respectively, J ₂=−(6.84±3.75)×10⁻⁷ if only sunspot data are considered, and J ₂=−(3.49±1.86)×10⁻⁷ in the case of helioseismic data alone. The value deduced from all available data for the octopole is: J ₄=(2.8±2.1)×10⁻¹². These values are compared to some others found in the literature. Supplementary material to this paper is available in electronic form at http://dx.doi.org/10.1023/A:1005238718479     

On the Aerodynamic Drag Force Acting on Interplanetary Coronal Mass Ejections

It is well known that the interaction of an interplanetary coronal mass ejection (ICME) with the solar wind leads to an equalisation of the ICME and solar wind velocities at 1 AU. This can be understood in terms of an aerodynamic drag force per unit mass of the form F _D/M=−(ρ_e AC _D/M)(V _i−V _e)∣V _i−V _e∣, where A and M are the ICME cross-section and sum of the mass and virtual mass, V _i and V _e the speed of the ICME and solar wind, ρ_e the solar wind density, C _D a dimensionless drag coefficient, and the inverse deceleration length γ=ρ_e A/M. The optimal radial parameterisation of γ and C _D beyond approximately 15 solar radii is calculated. Magnetohydrodynamic simulations show that for dense ICMEs, C _D varies slowly between the Sun and 1 AU, and is of order unity. When the ICME and solar wind densities are similar, C _D is larger (between 3 and 10), but remains approximately constant with radial distance. For tenuous ICMEs, the ICME and solar wind velocities equalise rapidly due to the very effective drag force. For ICMEs denser that the ambient solar wind, both approaches show that γ is approximately independent of radius, while for tenuous ICMEs, γ falls off linearly with distance. When the ICME density is similar to or less than that in the solar wind, inclusion of virtual mass effects is essential.     

A new outlook on the `Differential Theory' of the solar Quadrupole Moment and Oblateness

The modeling of the quadrupole moment J ₂ and of the oblateness , two key solar parameters, derives from the development in successive spherical harmonics of the gravitational potential. These harmonics are representative of the shape of the Sun, by studying the local distortion of the internal layers, under their distribution of mass and velocity. The first aim of this paper is to study, over the radius r and the colatitude , the structure of the internal layers of the Sun through a geometrical approach, considering J ₂ and under a differential form. The second aim is to determine their theoretical values, after integration over r and , taking the best available models of density and rotation into account constrained by helioseismic data. The novelty of our approach lies in the analysis of the profiles of the two above-mentioned solar parameters, under differential form, from the core to the surface. This analysis allows us to comply with the physical processes located in the transition regions, namely the tachocline and maybe a new subsurface layer which could be called the leptocline. The profiles of tildeJ ₂ show two sharp decreases, which are directly connected to the shear layers located at 0.7 R and beneath the surface. The profiles of tilde show five changes of curvature, which seem to be connected to solar processes, such as the matter circulation flows, seismic events or the storage of the magnetic field, phenomena taking place in the transition regions. These sets of profiles allow us to propose a configuration scenario composed of a double layer. Moreover, as a result of the theoretical determination of tildeJ ₂ and tilde, the values at the surface of the quadrupole moment and of the oblateness can be deduced, which are 1.60×10^–7 and 8.77×10^–6, respectively. As a result of an analysis of available data, we may admit J ₂=(2.0±0.4)×10^–7. The theoretical computations of J ₂ and at the surface will be confronted in the near future with the values measured in space by means of the Picard microsatellite. This mission should permit one to measure at the same time both the total solar irradiance and the latitudinal diameters in any position angle (after removing the passing spots or faculae at the limb).     

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.     

Statistical Comparison of Magnetic Clouds with Interplanetary Coronal Mass Ejections for Solar Cycle 23

We compare the number and characteristics of interplanetary coronal mass ejections (ICMEs) to those of magnetic clouds (MCs) by using in-situ solar wind plasma and magnetic field observations made at 1 AU during solar cycle 23. We found that ≈ 28% of ICMEs appear to contain MCs, since 103 magnetic clouds (MCs) occurred during 1995  – 2006, and 307 ICMEs occurred during 1996 – 2006. For the period between 1996 and 2006, 85 MCs are identified as part of ICMEs, and six MCs are not associated with ICMEs, which conflicts with the idea that MCs are usually a subset of ICMEs. It was also found that solar wind conditions inside MCs and ICMEs are usually similar, but the linear correlation between geomagnetic storm intensity (Dst _min) and relevant solar wind parameters is better for MCs than for ICMEs. The differences between average event duration (Δt) and average proton plasma β (〈β〉) are two of the major differences between MCs and ICMEs: i) the average duration of ICMEs (29.6 h) is 44% longer than for MCs (20.6 hours), and ii) the average of 〈β〉 is 0.01 for MCs and 0.24 for ICMEs. The difference between the definition of a MC and that for an ICME is one of the major reasons for these average characteristics being different (i.e., listed above as items i) and ii)), and it is the reason for the frequency of their occurrences being different.     

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.     

Relativistic gravitational deflection of light and its impact on the modeling accuracy for the Space Interferometry Mission

We study the impact of relativistic gravitational deflection of light on the accuracy of future Space Interferometry Mission (SIM). We estimate the deflection angles caused by the monopole, quadrupole and octupole components of gravitational fields for a number of celestial bodies in the solar system. We observe that, in many cases, the magnitude of the corresponding effects is significantly larger than the 1 μas accuracy expected from SIM. This fact argues for the development of a relativistic observational model for the mission that would account for the influence of both static and time-varying effects of gravity on light propagation. Results presented here are different from the ones obtained elsewhere by the fact that we specifically account for the differential nature of the future SIM astrometric measurements. We also obtain an estimate for the accuracy of possible determination of the Eddington’s parameter γ via SIM global astrometric campaign; we conclude that accuracy of ∼7 × 10⁻⁶ is achievable via measurements of deflection of light by solar gravity. The article was translated by the authors.     

A diffraction limit on the gravitational lens effect

The focussing of gravitational radiation by the interior and exterior gravitational field of a Newtonian gravitational lens is considered. A graphical method for determining the caustic structure of a Newtonian gravitational lens is presented and the caustic structure of a solar type gravitational lens is discussed. Estimates of the amplitude magnification in the caustic region indicate that waves with frequencies less than a critical cutoff frequency ω _c are not amplified significantly. For a lens of massM this cutoff frequency is ω _c ≈(10^-1πM)^-1; for the Sun ω _c ≈10⁴s^-1. Work supported in part by National Science Foundation Grant PHY78-05368.     

Dark energy and flatness from observational H(z)+WMAP constraint

We analyse the dark energy problem using observational H(z) data plus the curvature constraint given by WMAP. After a non-parametric statistical study covering the most probable range of Ω _m0 and H ₀ from different combination of data, we investigate the possibility of having the dark energy EoS parameter ω _x ≠−1. In order to keep strict flatness (1% of deviation from Ω=1), our results point out this is the case for 0.20≲Ω _m0≲0.23 and H ₀≈67 km/s/Mpc, with ω _x ≈−0.55. However, if we admit a 10% deviation from the flatness condition, ω _x may have any value in the range [−1.2,−0.5] for 0.20≲Ω _m0≲0.35 and 67≲H ₀≲74 km/s/Mpc.