Tidal pattern instabilities on multi-moon planets

The equilibrium tide-generating forces in the lunar orbital plane of a planet of radius R are calculated for the case of N moons of mass Mi orbiting the planet at instantaneous polar coordinates (Di, αi). For the case of a single moon, there are only two high tides. For the case of two moons, it is found that there can exist a critical lunar orbital distance at which two high tides become unstable with respect to formation of three high tides. Bifurcation diagrams are presented which depict how the angular positions of the high and low tides on the planet vary with the lunar distances and lunar separation angle. Tidal stability diagrams, which illustrate the stability regions for various tidal patterns as a function of lunar distances and lunar separation angle, are presented for various values of D₂/D₁ and M₂/M₁. Generally speaking, the aforementioned tidal instability, and hence the propensity for formation of three high tides on a two-moon planet, exists over a significant range of lunar distances and separation angles provided that M₂/M₁∼³(D₂/D₁). For the case of N>2 moons, the tidal stability diagram becomes more complex, revealing a diversity of potential tidal patterns.     

A refined model for the influence of ocean tides on UT1 and polar motion

Periodic changes in Universal Time and polar motion have been derived from a numerical ocean model for a number of small partial tides. Combined with the results reported earlier our model now includes the 34 most important constituents of degree 2 (measured by the amplitude of the tidal potential). The results show that the terms neglected in all existing models amount to 80 μs in UT1 and to 0.3 mas in polar motion.     

Researches in China on the Effects of Tides on the Earth Rotation

In this paper the tidal phenomena on the Earth are concisely specified, including solid tides, ocean tides and atmospheric tides due to the luni-solar tide-generating force, and the Earth pole tide due to the motion of the Earth's rotation axis (polar motion); as well as their effects on the Earth rotation. The outcomes of scientific researches of Chinese astronomers on these topics are described in some detail. These researches deal with the mechanisms responsible for tidal effects on the earth rotation, and on the measurements of the Earth rotation parameters. Finally, the effects discovered by Chinese researchers on the measurements of the period and change in period of pulsars are discussed. These effects are very small in magnitude but not negligible.     

“Stickiness” in mappings and dynamical systems

We present results of a study of the so-called “stickiness” regions where orbits in mappings and dynamical systems stay for very long times near an island and then escape to the surrounding chaotic region. First we investigated the standard map in the form x_i+1 = x_i+y_i+1 and y_i+1 = y_i+K/2π · sin(2πx_i) with a stochasticity parameter K = 5, where only two islands of regular motion survive. We checked now many consecutive points—for special initial conditions of the mapping—stay within a certain region around the island. For an orbit on an invariant curve all the points remain forever inside this region, but outside the “last invariant curve” this number changes significantly even for very small changes in the initial conditions. In our study we found out that there exist two regions of “sticky” orbits around the invariant curves: A small region I confined by Cantori with small holes and an extended region II is outside these cantori which has an interesting fractal character. Investigating also the Sitnikov-Problem where two equally massive primary bodies move on elliptical Keplerian orbits, and a third massless body oscillates through the barycentre of the two primaries perpendicularly to the plane of the primaries—a similar behaviour of the stickiness region was found. Although no clearly defined border between the two stickiness regions was found in the latter problem the fractal character of the outer region was confirmed.     

Variations of the solid Earth's center of mass due to oceanic tides

Periodic variations in the location of the center of mass of the solid Earth due to ocean tides are derived from the ocean tide model of Seiler (1991). The sum of nine partial tides leads to changes in the coordinates X, Y, Z within a range of 1cm.     

Simulations of the Polar Field Reversals during Cycle 22

The revised Mount Wilson synoptic magnetic data for the period September 1987 through March 1996 are used as the basis of numerical simulations of the evolution of both the northern and southern polar magnetic fields during the reversal and declining phases of cycle 22. The simulations are based on numerical solutions of the flux-transport equation which involve, as parameters, the maximum meridional flow speed, v ₀, and the supergranule diffusivity, . By matching characteristics of the observed and simulated fields, such as the observed reversal times, the evolution of the net flux above 60 °, and the migration of the polar crown, empirical values of these parameters, i.e., v ₀=11 m s^–1,=600 km² s^–1, may be determined. Further, the observed decrease in the mean net flux above 60 ° during the late declining phase of cycle 22 can be simulated only by increasing the diffusivity to 900 km² s^–1. However, direct observations of the supergranule velocities yield values of the diffusivity of order 200 km² s^–1, and we show that the inclusion of a pattern of emerging bipoles in the simulations can increase the diffusion of these fields and that, together with a more realistic value of the diffusivity, it is possible to reproduce qualitatively the features of the observed polar field reversals.     

Earth Tides and Lense–Thirring Effect

The general relativistic Lense—Thirring effect can be measured by inspecting a suitable combination of the orbital residuals of the nodes of LAGEOS and LAGEOS II and the perigee of LAGEOS II. The solid and ocean Earth tides affect the recovery of the parameter by means of which the gravitomagnetic signal is accounted for in the combined residuals. Thus an extensive analysis of the perturbations induced on these orbital elements by the solid and ocean Earth tides is carried out. It involves the l=2 terms for the solid tides and the l=2,3,4 terms for the ocean tides. The perigee of LAGEOS II turns out to be very sensitive to the l=3 part of the ocean tidal spectrum, contrary to the nodes of LAGEOS and LAGEOS II. The uncertainty in the solid tidal perturbations, mainly due to the Love number k ₂, ranges from 0.4% to 1.5%, while the ocean tides are uncertain at 5–15% level. The obtained results are used in order to check in a preliminary way which tidal constituents the Lense-Thirring shift is sensitive to. In particular it is tested if the semisecular 18.6-year zonal tide really does not affect the combined residuals. It turns out that, if modeled at the level of accuracy worked out in the paper, the l=2,4 m=0 and also, to a lesser extent, the l=3, m=0 tidal perturbations cancel out.This revised version was published online in October 2005 with corrections to the Cover Date.     

Primordial noble gases in “phase Q” in carbonaceous and ordinary chondrites studied by closed‐system stepped etching

Abstract— The HF/HCI‐resistant residues of the chondrites CM2 Cold Bokkeveld, CV3 (ox.) Grosnaja, CO3.4 Lancé, CO3.7 Isna, LL3.4 Chainpur, and H3.7 Dimmitt have been measured by closed‐system stepped etching (CSSE) in order to better characterise the noble gases in “phase Q”, a major carrier of primordial noble gases. All isotopic ratios in phase Q of the different meteorites are quite uniform, except for (²⁰Ne/²²Ne)_Q. As already suggested by precise earlier measurements (Schelhaas et al., 1990; Wieler et al., 1991, 1992), (²⁰Ne/²²Ne)_Q is the least uniform isotopic ratio of the Q noble gases. The data cluster ～10.1 for Cold Bokkeveld and Lancé and 10.7 for Chainpur, Grosnaja, and Dimmitt, respectively. No correlation of (²⁰Ne/²²Ne)_Q with the classification or the alteration history of the meteorites has been found. The Ar, Kr, and Xe isotopic ratios for all six samples are identical within their uncertainties and similar to earlier Q determinations as well as to Ar‐Xe in ureilites. Thus, an unknown process probably accounts for the alteration of the originally incorporated Ne‐Q. The noble gas elemental compositions provide evidence that Q consists of at least two carbonaceous carrier phases “Q1” and “Q2” with slightly distinct chemical properties. Ratios (Ar/Xe)_Q and (Kr/Xe)_Q reflect both thermal metamorphism and aqueous alteration. These parent‐body processes have led to larger depletions of Ar and Kr relative to Xe. In contrast, meteorites that suffered severe aqueous alteration, such as the CM chondrites, do not show depletions of He and Ne relative to Ar but rather the highest (He/Ar)_Q and (Ne/Ar)_Q ratios. This suggests that Q1 is less susceptible to aqueous alteration than Q₂. Both subphases may well have incorporated noble gases from the same reservoir, as indicated by the nearly constant, though very large, depletion of the lighter noble gases relative to solar abundances. However, the elemental ratios show that Q₁ and Q₂ must have acquired (or lost) noble gases in slightly different element proportions. Cold Bokkeveld suggests that Q₁ may be related to presolar graphite. Phases Q₁ and Q₂ might be related to the subphases that have been suggested by Gros and Anders (1977). The distribution of the ²⁰Ne/²²Ne ratios cannot be attributed to the carriers Q₁ and Q₂. The residues of Chainpur and Cold Bokkeveld contain significant amounts of Ne‐E(L), and the data confirm the suggestion of Huss (1997) that the ²²Ne‐E(L) content, and thus the presolar graphite abundances, are correlated with the metamorphic history of the meteorites.     

Druse clinopyroxene in D'Orbigny angritic meteorite studied by single‐crystal X‐ray diffraction,electron microprobe analysis,and Mössbauer spectroscopy

Abstract— The crystal structure of druse clinopyroxene from the D'Orbigny angrite, (Ca_0.944 Fe²⁺_0.042 Mg_0.010Mn_0.004) (Mg_0.469Fe²⁺_0.317Fe³⁺_0.035Al_0.125Cr_0.010Ti_0.044) (Si_1.742Al_0.258) O₆, a = 9.7684(2), b = 8.9124(2), c = 5.2859(1) Å, β = 105.903(1)°, V = 442.58 ų, space group C2/c, Z = 2, has been refined to an R₁ index of 1.92% using single‐crystal X‐ray diffraction data. The unit formula, calculated from electron microprobe analysis, and the refined site scattering values were used to assign site populations. The distribution of Fe²⁺and Mg over the M1 and M2 sites suggests a closure temperature of 1000 °C. Mössbauer spectroscopy measurements were done at room temperature on a single crystal and a powdered sample. The spectra are adequately fit by a Voigt‐based quadrupole‐splitting distribution model having two generalized sites, one for Fe²⁺with two Gaussian components and one for Fe³⁺with one Gaussian component. The two ferrous components are assigned to Fe²⁺at the M1 site, and arise from two different next‐nearest‐neighbor configurations of Ca and Fe cations at the M2 site: (3Ca,0Fe) and (2Ca,1Fe). The Fe³⁺/Fe_tot ratio determined by Mössbauer spectroscopy is in agreement with that calculated from the electron microprobe analysis. The results are discussed in connection with the redox and thermal history of D'Orbigny.     

Ocean tide perturbations in the orbits of GEOS 1 and GEOS 2

Douglaset al. (1973) have estimated tidal parameters from the orbits of GEOS 1 and GEOS 2. Their results, interpreted in terms of Love numbers, are rather dispersive due in part to their neglect of the ocean tides. The ocean tidal corrections are estimated in this paper, but although they do not explain all of the discrepancy they do emphasise the importance of these perturbations on the motion of close Earth satellites. The remaining discrepancies could result in part from the fact that part of the long period tidal perturbations have been absorbed by the zonal harmonics in the Earth's gravity field.     

Long-term spectroscopic observations of Mars using IRTF/CSHELL: Mapping of O2 dayglow, CO, and search for CH4

Long-term spectroscopic observations of the O₂ dayglow at 1.27 μm result in a map of the latitudinal and seasonal behavior of the dayglow intensity for the full martian year. The O₂ dayglow is a sensitive tracer of Mars' photochemistry, and this map reflects variations of Mars' photochemistry at low and middle latitudes. It may be used to test photochemical models. Long-term observations of the CO mixing ratio have been also combined into the seasonal-latitudinal map. Seasonal and latitudinal variations of the mixing ratios of CO and the other incondensable gases (N₂, Ar, O₂, and H₂) discovered in our previous work are caused by condensation and sublimation of CO₂ to and from the polar regions. They reflect dynamics of the atmosphere and polar processes. The observed map may be used to test global circulation models of the martian atmosphere. The observed global abundances of CO are in reasonable agreement with the predicted variations with the 11-year solar cycle. Despite the perfect observing conditions, methane has not been detected using the IRTF/CSHELL with a 3σ upper limit of 14 ppb. This upper limit does not rule out the value of 10 ppb observed using the Canada-France-Hawaii Telescope and the Mars Express Planetary Fourier Spectrometer.     

Motion of rigid bodies in a set of redundant variables

In this article we study the conditions for obtaining canonical transformationsy=f(x) of the phase space, wherey(y ₁,y ₂,...,y _2n ) andx(x ₁,x ₂,...,x _2m ) in such a way that the number of variables is increased. In particular, this study is applied to the rotational motion in functions of the Eulerian parameters (q ₀,q ₁,q ₂,q ₃) and their conjugate momenta (Q ₀,Q ₁,Q ₂,Q ₃) or in functions of complex variables (z ₁,z ₂,z ₃,z ₄) and their conjugate momenta (Z ₁,Z ₂,Z ₃,Z ₄) defined by means of the previous variables. Finally, our article include some properties on the rotational motion of a rigid body moving about a fixed point.     

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.     

Low‐mass visual binaries in the solar neighbourhood: The case of HD 141272

We search for stellar and substellar companions of young nearby stars to investigate stellar multiplicity and formation of stellar and substellar companions. We detect common proper‐motion companions of stars via multi‐epoch imaging. Their companionship is finally confirmed with photometry and spectroscopy. Here we report the discovery of a new co‐moving (13 σ) stellar companion ∼17.8 arcsec (350AU in projected separation) north of the nearby star HD141272 (21 pc).With EMMI/NTT optical spectroscopy we determined the spectral type of the companion to be M3±0.5V. The derived spectral type as well as the near infrared photometry of the companion are both fully consistent with a M_⊙ dwarf located at the distance of HD141272 (21 pc). Furthermore the photometry data rules out the pre‐main sequence status, since the system is consistent with the ZAMS of the Pleiades. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Instability of the 2+2 body problem

The equations of motion of the 2+2 body problem (two interacting particles in the gravitational field of two much more massive primaries m₁ and m₂ in circular keplerian orbit) have an integral analogous to the Jacobi integral of the circular 2+1 body problem. We show here that with 2+2 bodies this integral does not give rise to Hill stability, i.e. to confinement for all time in a portion of the configuration space not allowing for some close approaches to occur. This is because all the level manifolds are connected and all exchanges of bodies between the regions surroundingm ₁,m ₂ and infinity do not contradict the conservation of the integral. However, it is worth stressing that some of these exchanges are physically meaningless, because they involve either unlimited extraction of potential energy from the binary formed by the small bodies (without taking into account their physical size) or significant mutual perturbations between the small masses without close approach, a process requiring, for the Sun-Jupiter-two asteroids system, timescales longer than the age of the Solar System.     

国内天文地球动力学中的潮汐研究

简要说明了天文地球动力学范畴内所研究的潮汐现象，包括由日月引潮力引起的固体潮、海洋潮、大气潮和由于地球自转轴的极移引起的极潮，以及这些潮汐对地球自转和地球自转的测量产生的效应。重点阐述中国天文学界在这一领域里的研究成果。这些研究涉及潮汐影响地球自转的机制，也就是各种潮汐效应与极移、自转速率变化和章动的关系，包括构建这类关系的理论模型，分析潮汐对它们的影响，利用中国古代丰富的天象记录计算地球自转的长期减慢，计算弹性或滞弹地球的洛夫数，依据某一地球模型计算潮汐效应或章动序列等等。研究也涉及在测量地球自转参数的不同技术中各种潮汐效应对测量结果产生的影响及其改正，并涉及与潮汐有关的观测方法的优化和数据处理过程的改进。最后介绍了中国学者所发现的脉冲星的周期和周期变率测量中的潮汐效应，尽管它们的量级甚微，但不容忽视。     

Numerical integration of periodic orbits in the main problem of artificial satellite theory

We describe a collection of results obtained by numerical integration of orbits in the main problem of artificial satellite theory (theJ ₂ problem). The periodic orbits have been classified according to their stability and the Poincaré surfaces of section computed for different values ofJ ₂ andH (whereH is thez-component of angular momentum). The problem was scaled down to a fixed value (–1/2) of the energy constant. It is found that the pseudo-circular periodic solution plays a fundamental role. They are the equivalent of the Poincaré first-kind solutions in the three-body problem. The integration of the variational equations shows that these pseudo-circular solutions are stable, except in a very narrow band near the critical inclincation. This results in a sequence of bifurcations near the critical inclination, refining therefore some known results on the critical inclination, for instance by Izsak (1963), Jupp (1975, 1980) and Cushman (1983). We also verify that the double pitchfork bifurcation around the critical inclination exists for large values ofJ ₂, as large as |J ₂|=0.2. Other secondary (higher-order) bifurcations are also described. The equations of motion were integrated in rotating meridian coordinates.     

Tidal effects on the LAGEOS–LARES satellites and the LARASE program

The effects of solid and ocean tides have been computed on the right ascension of the ascending node of the two LAGEOS and LARES satellites and on the argument of pericenter of LAGEOS II. Their effects—together with the possible mis-modeling related to systematic errors in the estimate of the tidal coefficients, especially in the case of ocean tides—are quite important to be well established for the key role of the LAGEOS satellites, as well as of the newly LARES, in space geodesy and geophysics as well as in fundamental physics measurements. In the case of the measurement of the Lense–Thirring effect, the mis-modeling of long-period tides may mimic a secular effect on the cited orbital elements, thus producing a degradation in the measurement of the relativistic precession. A suitable combination of the orbital elements of the three satellites can help in avoiding the effects of the long-period tides of degree \(\ell =2\) (as for the Lunar solid tides with periods of 18.6 and 9.3 years) and \(\ell =4\), but other long-period tides, as the ocean \(K_1\) tide, which has the same periodicities of the right ascension of the ascending node \(\varOmega \) of the satellites, may strongly influence the measurement, especially if it is performed over a relatively short time span. These results are particularly important in the case of LARES, since they are new and because of the role that the orbit of LARES, and especially of its ascending node right ascension, will have in a new measurement of the Lense–Thirring effect by the joint analysis of its orbit with that of the two LAGEOS.     

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.     

Discovery of a companion at the brown dwarf limit to the solar‐type star Gliese 29

Gliese 29 is a 7 to 8 Gyr old, southern Population I turnoff star with a large proper motion of 1″/yr. Using recent direct imaging observations with the 0.8 m Infrared Imaging System (IRIS) of the Universitätssternwarte Bochum near Cerro Armazones in Chile, we demonstrate that the faint source 2MASS J00402651–5927168 at a projected angular separation ρ = 6.″35 is a common‐proper‐motion companion to Gl 29. Provided this source is not part of a further subsystem, the IRIS J ‐ and K_s‐band photometry either implies a spectral type of about L2, based on its absolute magnitude, or an approximate mass M_B ≃ 0.077 M_⊙, suggesting that it may even be a brown dwarf. Assuming a face‐on circular orbit this faint companion orbits Gl 29 in 1880 years. (© 2016 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)