Intrinsic nonlinearity and method of disturbed observations in inverse problems of celestial mechanics

Orbit determination from a small sample of observations over a very short observed orbital arc is a strongly nonlinear inverse problem. In such problems an evaluation of orbital uncertainty due to random observation errors is greatly complicated, since linear estimations conventionally used are no longer acceptable for describing the uncertainty even as a rough approximation. Nevertheless, if an inverse problem is weakly intrinsically nonlinear, then one can resort to the so-called method of disturbed observations (aka observational Monte Carlo). Previously, we showed that the weaker the intrinsic nonlinearity, the more efficient the method, i.e. the more accurate it enables one to simulate stochastically the orbital uncertainty, while it is strictly exact only when the problem is intrinsically linear. However, as we ascertained experimentally, its efficiency was found to be higher than that of other stochastic methods widely applied in practice. In the present paper we investigate the intrinsic nonlinearity in complicated inverse problems of Celestial Mechanics when orbits are determined from little informative samples of observations, which typically occurs for recently discovered asteroids. To inquire into the question, we introduce an index of intrinsic nonlinearity. In asteroid problems it evinces that the intrinsic nonlinearity can be strong enough to affect appreciably probabilistic estimates, especially at the very short observed orbital arcs that the asteroids travel on for about a hundredth of their orbital periods and less. As it is known from regression analysis, the source of intrinsic nonlinearity is the nonflatness of the estimation subspace specified by a dynamical model in the observation space. Our numerical results indicate that when determining asteroid orbits it is actually very slight. However, in the parametric space the effect of intrinsic nonlinearity is exaggerated mainly by the ill-conditioning of the inverse problem. Even so, as for the method of disturbed observations, we conclude that it practically should be still entirely acceptable to adequately describe the orbital uncertainty since, from a geometrical point of view, the efficiency of the method directly depends only on the nonflatness of the estimation subspace and it gets higher as the nonflatness decreases.     

A new method for the statistical simulation of the virtual values of parameters in inverse orbital dynamics problems

A Monte Carlo-type method for simulating virtual values of the parameters in inverse orbital dynamics problems for highly nonlinear cases is proposed. The method is based on imitating Fisher’s statistics employed to specify the confidence region, and is implemented by solving repeatedly nonlinear least-squares problems with various samples of simulated observations obtainable by suitable random variations.     

Turbulence and its parameterization in accretion discs

Accretion disc turbulence is investigated in the framework of the shearing box approximation. The turbulence is either driven by the magneto‐rotational instability or, in the non‐magnetic case, by an explicit and artificial forcing term in the momentum equation. Unlike the magnetic case, where most of the dissipation occurs in the disc corona, in the forced hydrodynamic case most of the dissipation occurs near the midplane. In the hydrodynamic case evidence is presented for the stochastic excitation of epicycles. When the vertical and radial epicyclic frequencies are different (modeling the properties around rotating black holes), the beat frequency between these two frequencies appear to show up as a peak in the temporal power spectrum in some cases. Finally, the full turbulent resistivity tensor is determined and it is found that, if the turbulence is driven by a forcing term, the signs of its off‐diagonal components are such that this effect would not be capable of dynamo action by the shear–current effect. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A bulk cloud parameterization in a Venus General Circulation Model

A condensing cloud parameterization is included in a super-rotating Venus General Circulation Model. A parameterization including condensation, evaporation and sedimentation of mono-modal sulfuric acid cloud particles is described. Saturation vapor pressure of sulfuric acid vapor is used to determine cloud formation through instantaneous condensation and destruction through evaporation, while pressure dependent viscosity of a carbon dioxide atmosphere is used to determine sedimentation rates assuming particles fall at their terminal Stokes velocity. Modifications are described to account for the large range of the Reynolds number seen in the Venus atmosphere.Two GCM experiments initialized with 10 ppm-equivalent of sulfuric acid are integrated for 30 Earth years and the results are discussed with reference to “Y” shaped cloud structures observed on Venus. The GCM is able to produce an analog of the “Y” shaped cloud structure through dynamical processes alone, with contributions from the mean westward wind, the equatorial Kelvin wave, and the mid-latitude/polar Mixed Rossby/Gravity waves. The cloud top height in the GCM decreases from equator to pole and latitudinal gradients of cloud top height are comparable to those observed by Pioneer Venus and Venus Express, and those produced in more complex microphysical models of the sulfur cycle on Venus. Differences between the modeled cloud structures and observations are described and dynamical explanations are suggested for the most prominent differences.     

Solution of the three-dimensional inverse problem

The three dimensional inverse problem for a material point of unit mass, moving in an autonomous conservative field, is solved. Given a two-parametric family of space curvesf(x, y, z)=c ₁,g(x, y, z)=c ₂, it is shown that, in general, no potentialU=U(x, y, z) exists which can give rise to this family. However, if the given functionsf(x, y, z) andg(x, y, z) satisfy certain conditions, the corresponding potentialU(x, y, z), as well as the total energyE=E(f, g) are determined uniquely, apart from a multiplicative and an additive constant.     

The parameterization of solid metal‐liquid metal partitioning of siderophile elements

Abstract— Many solar system processes involve a metallic liquid, and the composition of the metallic liquid, such as the liquid's concentrations of S, P, and C, will influence the partitioning of elements during such processes. We present a method for parameterizing solid metal‐liquid metal partition coefficients for siderophile (metal‐loving) elements as a function of the metallic liquid composition. Our parameterization method is based on an older theory of Jones and Malvin (1990), which stated that the metallic liquid is composed of metal and non‐metal‐bearing domains, and the domains are the dominant influence on the partitioning behavior. By revising the means by which the metal domains are calculated, our revised parameterization method is able to match experimental partitioning data from the Fe‐Ni‐S, Fe‐Ni‐P, Fe‐Ni‐S‐P, and Fe‐Ni‐C systems. Mathematical expressions were derived for the solid metal‐liquid metal partitioning of 13 siderophile elements. Elements that are chalcophile (S‐loving), P‐loving, or C‐loving prefer the non‐metal‐bearing domains in the metallic liquid and, consequently, aren't fit by the parameterization method presented here. Possible applications for our parameterization method include modeling the crystallization of iron meteorites, planetary differentiation, and the solidification of Earth's inner core.     

The double inverse polarity paradigm

A model is presented for the origin of inverse polarity magnetic fields in the perpendicular as well as in the axial direction of quiescent prominences. The model is based on the presence of a discrete coronal arcade structure where magnetic separating surfaces can be identified. On the crossing of these separating surfaces magnetic reconnection driven by photospheric shear and converging motions can create the observed field direction in quiescent prominences.Dedicated to Cornelis de Jager     

A classical derivation of inverse Compton radiation

A classical method is used to derive formulas describing inverse Compton radiation in vacuo. It is shown that several expressions, including the power radiated and the rate at which photons are scattered, can be calculated exactly for the case of isotropically distributed unpolarized waves scattered by particles with any energy.     

On the two-dimensional inverse problem of dynamics

The authors extend the deduction of the equations satisfied by the force fields from inertial to rotating frames, when the curves of a certain family are known to be solutions for the equations of motion. Then Drǎmbâ's equation is obtained as a consequence of this result. The works of Hadamard and Moiseev are proved to be closely related to the inverse problem of dynamics.     

Surface soil moisture parameterization of the VIC-2L model: Evaluation and modification

HAPEX-MOBILHY data, consisting of one year of hourly atmospheric forcing data at Caumont (SAMER No. 3, 43.68°N, 0.1°W) were used repeatedly to run the two-layer Variable Infiltration Capacity (VIC-2L) land-surface scheme until the model reached equilibrium in its water and energy balance. The equilibrium results are compared with one year of weekly soil moisture measurements at different depths, the estimated latent heat fluxes for 35 days of the intensive observation period (IOP), and the accumulated evaporation, runoff and drainage for the entire soya crop season. The latent heat flux comparisons show that VIC-2L tends to underestimate the evaporation due to the low soil moisture in its upper layer. The soil moisture comparison shows that the total soil water content is well simulated in general, but the soil water content in the top 0.5 m is underestimated, especially in May and June. These comparisons suggest that the lack of a mechanism for moving moisture from the lower to the upper soil layer in VIC-2L is the main cause for model error in the HAPEX-MOBILHY application. A modified version of VIC-2L, which has a new feature that allows diffusion of moisture between soil layers, and a 0.1 m thin layer on top of the previous upper layer, is described. In addition, the leaf area index (LAI) and the fraction vegetation cover are allowed to vary at each time step in a manner consistent with the rest PILPS-RICE Workshop, rather than being seasonally fixed. With these modifications, the VIC-2L simulations are re-evaluated. These changes are shown to resolve most of the structural deficiencies in the original version of the model. The sensitivity analysis of the new version of the model to the choices of soil depths and root distribution show that the evapotranspiration and soil moisture at the model equilibrium state are more sensitive to the root distribution than to the soil depth.     

Non-linear compton and inverse compton effect

A charged particle of arbitrary initial motion and position is subjected to an intense circularly polarized electromagnetic wave. The electromagnetic radiation which results is calculated exactly and the results are expressed in terms of the power observed per unit bandwidth per unit solid angle.Several special cases are considered in order to demonstrate the range of possibilities which this type of interaction may have in astrophisics and numerical examples are given for the Crab pulsar.     

Statistical masses of Abell clusters

The analysis of a homogeneous sample of 108 Abell clusters has led to an average peculiar velocity for the center of mass motion of these clusters of 610±750 km s^–1. From this result, an upper limit for the average mass of the Abell clusters of (1.6±2.4)×10¹⁵ M was obtained under the assumption that the peculiar motion is due to the excess of neighbours with respect to an uniform background. A lower limit of (2.42.9) x 10¹⁴ h ^-10.4 M was derived if one assumes that the peculiar velocity results from the mutual acceleration with the nearest neighbour.     

Statistical mechanics of Keplerian orbits

Theoretical predictions agree with computer simulations at least for those collisional systems in which the restitution coefficient is independent of impact velocity. An uncertainty principle for the orbits restricts the validity of the theory and its predictions. Discussion of the whole theory and of computer simulations shows that a velocity-dependent restitution coefficient provides the only astronomically interesting applications of the collisional processes. The Saturnian and Uranian ring systems correspond very well to theoretical expectations if the restitution coefficient is of this type.     

Statistical studies of pulsar glitches

Shemar & Lyne have previously presented observations and an analysis of 32 glitches and their subsequent relaxations observed in a total of 15 pulsars. These data are brought together in this paper with those published by other authors. We show quantitatively how glitch activity decreases linearly with decreasing rate of slow-down. As indicated previously from studies of the Vela pulsar, the analysis suggests that 1.7 per cent of the moment of inertia of a typical neutron star is normally contained in pinned superfluid which releases its excess angular momentum at the time of a glitch. There is a broad range of glitch amplitude and there is a strong indication that pulsars with large magnetic fields suffer many small glitches while others show a smaller number of large glitches. Transient effects following glitches are very marked in young pulsars and decrease linearly with decreasing rate of slow-down, suggesting that the amount of loosely pinned superfluid decreases with age. We suggest that the low braking index of the Vela and Crab pulsars cannot be caused by a decreasing moment of inertia and should be attributed to step increases in the effective magnetic moment of the neutron star at the glitches.     

Statistical theory of thermal instability

A new statistical approach is presented to study the thermal instability of an optically thin unmagnetized plasma. In the framework of this approach the time evolution of the mass distribution function over temperature φ( T ) is calculated. Function φ( T ) characterizes the statistical properties of the multiphase medium of arbitrarily spaced three-dimensional structure of arbitrary (small or large) temperature perturbations. We construct our theory under the isobarical condition ( P  = constant over space), which is satisfied in the short-wavelength limit of the perturbations. The developed theory is illustrated for the case of the thermal instability of a slowly expanding interstellar cloud (smooth scenario). Numerical solutions of equations of the statistical theory are constructed and compared with hydrodynamical solutions. The results of both approaches are identical in the short-wavelength range when the isobarity condition is satisfied. Also the limits of applicability of the statistical theory are estimated. The possible evolution of the initial spectrum of perturbations is discussed. The proposed theory and numerical models can be relevant to the formation of the two-phase medium in the ∼ 1 pc region around quasars. Then small warm ( T  ≃ 10⁴  K ) clouds are formed as the result of thermal instability in an expanded gas fragment, which is a product of either star–star or star–accretion disc collision.     

Statistical mechanics of keplerian orbits

A dispersion in particle sizes is incorporated into the theory of partially elastic collisions on Keplerian orbits. Very different particles can never co-exist for long periods, if their elasticity parameter depends of the kinetic energy in collisions. If a system initially consists of a mixture of sizes, two possibilities for its evolution exist. If the smaller particles initially dominate, then they fall rapidly to the central body without affecting the remaining system. If the larger particles dominate, then a ringshaped system is formed. This is one possible mechanism for the origin of Saturn's rings. A dust content is also able to create a gap like the Cassini division.     

On the atmospheric muon energy spectrum in the deep ocean and its parameterization

The energy spectrum of those muons in the deep ocean ( 1000 m) that have been produced in the terrestrial atmosphere and have penetrated sea-water is calculated by Monte Carlo simulation and presented in a simplified form for easy use.     

Statistical tests of chondrule sorting

Abstract– The variation in sizes of chondrules from one chondrite to the next is thought to be due to some sorting process in the early solar nebula. Hypotheses for the sorting process include chondrule sorting by mass and sorting by some aerodynamic mechanism; one such aerodynamic mechanism is the process of turbulent concentration (TC). We present the results of a series of statistical tests of chondrule data from several different chondrites. The data do not clearly distinguish between various options for the sorting parameter, but we find that the data are inconsistent with being drawn from lognormal or (three‐parameter) Weibull distributions in chondrule radius. We also find that all but one of the chondrule data sets tested are consistent with being drawn from the TC distribution.