A postulate leading to the Titius-Bode law

The revised Titius-Bode law (Balsano and Hughes, 1979) giving distances of planets from the Sun shows integers that recall the Bohr law in the early quantum theory of hydrogen atom. The author searchs for a formalism, similar to the Sommerfeld's one accounting for the planetary distribution of the solar system. It is shown that such a formalism might be started with the assumed relation $$\int_{r_0 }^{r_n } {U(r)dr = nk,} $$ whereU(r) stands for the gravitation potential created by the Sun at distancer,k=cte,r _o=cte,r _n=distance of then ^th planet andn=1, 2, 3... Although the inspiration of the present note came from the early quantum theory, it is emphasized that there is no connection between the above assumed equation and the Sommerfeld quantum rule, but a pure formal similarity. the true significance of that equation is still unknown. It is either a fortuitous coincidence leading to the T-B law or any possible unsuspected property of gravitation.     

On gravitational orbits and the virial theorem

We emphasize the sharp distinctions between different one-body gravitational trajectories made by the ratio of time averagesR(t)–E _kin/E _pot.R is calculated as a function of the eccentricity (e) and of the energy (E). Whent, independently ofe andE, R1/2 for closed orbits (this clearly illustrates the fulfillment of the virial theorem in classical mechanics); whereasR1, at any time, for open orbits.     

Modification of the NUT solution and its physical interpretation

Erevan State University. Translated from Astrofizika, Vol. 32, No. 3, pp. 465–469, May–June, 1990.     

One of the solutions for the Laplace equation and its physical interpretation

In this paper it is confirmed once more that there exists the general solution of Laplace's equation in ellipsoidal coordinates which satisfies the Stäckel theorem and which was derived earlier by M. Jarov-Jarovoi and S. J. Madden. The author interprets physically the general solution in real space as potentials of layers of charge and double layers in which the distribution of densities is defined by Green's formula.     

A physical interpretation of the 'red Sirius' anomaly

A hot-gas halo is predicted by chemodynamical models during the early evolution of spheroidal galaxies. Cold condensations, arising from thermal instabilities in the hot gas, are expected to be embedded in the hot halo. In the early phases of the galaxy ( t ≲1 Gyr), a strong X-ray and EUV emission is produced by the extended hot-gas distribution, ionizing the cold clouds. This self-irradiating two-phase halo model successfully explains several line ratios observed in QSO absorption-line systems, and reproduces the temperature distribution of Lyman α clouds.     

On the connection between the Nekhoroshev theorem and Arnold diffusion

The analytical techniques of the Nekhoroshev theorem are used to provide estimates on the coefficient of Arnold diffusion along a particular resonance in the Hamiltonian model of Froeschlé et al. (Science 289:2108–2110, 2000). A resonant normal form is constructed by a computer program and the size of its remainder ||R _opt || at the optimal order of normalization is calculated as a function of the small parameter ${\epsilon}$ . We find that the diffusion coefficient scales as ${D \propto ||R_{opt}||^3}$ , while the size of the optimal remainder scales as ${||R_{opt}|| \propto {\rm exp}(1/\epsilon^{0.21})}$ in the range ${10^{-4} \leq \epsilon \leq 10^{-2}}$ . A comparison is made with the numerical results of Lega et al. (Physica D 182:179–187, 2003) in the same model.     

On the symmetric difference quotient and its application to the correction of orbits

In this paper we consider the analytical foundations of numerical applications of the symmetric difference quotient for orbits correction. From these considerations it follows that the better results obtained in numerical calculations of values of derivatives at point, obtained by replacing the ordinary difference quotient with the symmetric difference quotient, have not been obtained fortuitandy.     

On the physical connection between quasars and neighboring galaxies

A statistical analysis of galaxies and the quasars forming pairs with them shows that at least a fairly large share of the quasars lie at cosmological distances and are not physical neighbors of nearby galaxies. Translated from Astrofizika, Vol. 41. No. 2, pp. 227–234, April-June, 1998.     

Fragmentation in molecular clouds and its connection to the IMF

We present an analysis of star-forming gas cores in a smooth particle hydrodynamics simulation of a giant molecular cloud. We identify cores using their deep potential wells. This yields a smoother distribution with clearer boundaries than density. Additionally, this gives an indication of future collapse, as bound potential cores (p-cores) represent the earliest stages of fragmentation in molecular clouds. We find that the mass function of the p-cores resembles the stellar initial mass function and the observed clump mass function, although p-core masses  (∼0.7 M_⊙)  are smaller than typical density clumps. The bound p-cores are generally subsonic, have internal substructure and are only quasi-spherical. We see no evidence of massive bound cores supported by turbulence. We trace the evolution of the p-cores forward in time, and investigate the connection between the original p-core mass and the stellar mass that formed from it. We find that there is a poor correlation, with considerable scatter suggesting accretion on to the core is dependent on more factors than just the initial core mass. During the accretion process the p-cores accrete from beyond the region first bound, highlighting the importance of the core environment to its subsequent evolution.     

On the symmetric difference quotient and its application to the correction of orbits (II). A numerical analysis

In this paper we present, among other things, the numerical analysis of behaviour of ordinary and symmetric difference quotients for some elementary functions, for rectangular coordinates (velocity components) in relation to orbital elements and initial values of coordinates (velocity components) in Keplerian motion.     

The Titius-Bode law and the possibility of recent large-scale evolution in the solar system

Although it is by no means clear that the Titius-Bode law of planetary distances is indeed a “law” (even though there are enticing indications), it is proposed that if one assumes that the law is a “law” and that the planets obey it, then this argues against recent large-scale evolution in the solar system. Put another way: one can believe in the Titius-Bode law or in recent large-scale evolution or in neither of them. But it appears difficult to believe in both of them.     

Observation of a type of sporadic solar mass ejection and its physical interpretation with numerical simulation

Seven mediated and small ejective events on the sun observed at Ganyu Observing Station of Purple Mountain Observatory in 2000 are investigated. It is found that they were not accompanied by brightening. Their lengths were in the range 1–2.5×10⁴ km, their widths, 3–5×10³ km, and their lifetimes, 3–7 minutes. They were produced at places of weak magnetic fields and far away from large sunspots. These ejections are interpreted by numerical simulation with 1-D hydrodynamic equations of flow along magnetic arcs. As demonstrated by the results, they are different from the spicules and surges simulated by Suematsu et al. and Shibata et al. They are not matter with photospheric or chromospheric densities pushed by shock waves or rebound shock waves toward the solar corona, rather, they are ejections formed by continuous matter flows after magnetic reconnection. After evolving for about 5 minutes, they can attain a stationary hydrodynamic state.     

A crater and its ejecta: An interpretation of Deep Impact

We apply recently updated scaling laws for impact cratering and ejecta to interpret observations of the Deep Impact event. An important question is whether the cratering event was gravity or strength-dominated; the answer gives important clues about the properties of the surface material of Tempel 1. Gravity scaling was assumed in pre-event calculations and has been asserted in initial studies of the mission results. Because the gravity field of Tempel 1 is extremely weak, a gravity-dominated event necessarily implies a surface with essentially zero strength. The conclusion of gravity scaling was based mainly on the interpretation that the impact ejecta plume remained attached to the comet during its evolution. We address that feature here, and conclude that even strength-dominated craters would result in a plume that appeared to remain attached to the surface. We then calculate the plume characteristics from scaling laws for a variety of material types, and for gravity and strength-dominated cases. We find that no model of cratering alone can match the reported observation of plume mass and brightness history. Instead, comet-like acceleration mechanisms such as expanding vapor clouds are required to move the ejected mass to the far field in a few-hour time frame. With such mechanisms, and to within the large uncertainties, either gravity or strength craters can provide the levels of estimated observed mass. Thus, the observations are unlikely to answer the questions about the mechanical nature of the Tempel 1 surface.     

A new tool for the study of periodic orbits

We present the generalization of the method of rational Fourier-series approximations for nonlinear ordinary differential equations in galactic dynamics. In previous papers we presented the formulae which describe simple families of periodic orbits with very good accuracy. In this Letter we generalize our formulae to describe periodic orbits of any resonancem:n.     

An approach to the radial distribution of planetary orbits

The problem of determination of the radial distribution of the planetary orbits is approached under the assumption that the average present radial sizes of the orbits were already determined when the protoplanetary cloud flattened by initial angular momentum aggregated into a set of concentric rings from which the planetary material was ultimately collected. The object of this argument is to derive a consistent stationary distribution of orbits so that the problem of the non-stationary formation of the orbital rings is not here considered. Under the flattening assumption the 3D Poisson equation is replaced by the 2D Helmholtz equation (inhomogeneous) which is solved by use of an averaging theorem generalization of the well-known averaging theorem for the homogeneous Helmholtz equation. Augmenting the ring potentials obtained by specializing the mass distribution in the disk by a solar potential term and a rotational potential, differentiation leads to a generalization of the Kepler 3D law suitable for the many-body problem of a solar system with circular orbits. In this way a system of transcendental equations involving Bessel functions of the first and second kind are obtained which must be satisfied by the orbital radii. Naturally the restriction to circular orbits represents only an approximation to the orbital determination problem, but considering that no arguments have previously been available for the determination even of circular orbits it would seem to represent an advance.     

A way to generate a model of secular perturbations for planetary orbits

The reciprocal distance between two material points that rotate around a central body in nonintersecting orbits is expanded and the results are presented. The expansion is obtained accurate to the tenth order with respect to small parameters: the eccentricities and sine of the orbital inclination angle. The result is the basis of the averaging operation of the perturbation function in the system of eight major planets in the solar system, and of the numerical integration of the averaged equations of motion. The averaged Hamiltonian contains the terms whose period of variation is greater than 200 years. Forty eight equation of first order are numerically integrated with increments of 100 years for two intervals from the beginning of the Christian era: 25 million years forward and 25 million years backward over time. To present the results of calculation, the website (URL: http://vadimchazov.narod.ru/secequat.htm) was developed, where the initial codes, executable program modules, the results of calculations presented in graphical form, text files with initial conditions, tables for expanding the reciprocal distance between two material points, and the tables with the results of expansion of the perturbation function for eight major planets of the solar system are presented.     

Upper air density derived from the orbits of Discoverer satellites and its variation with latitude

Observations on artificial satellites have been used to investigate how the air density at heights between 190 and 260 km varies with latitude The Discoverer series of satellites was used because the position of their perigees moved over the latitude range from 80°S to 80°N.

It is concluded that the air density at a fixed height is a function of latitude and is about 30 per cent smaller at the poles than at the equator. This result is applicable to a local time of 14^h in the years 1959–1960: it is different from that obtained by Groves who concluded that the density is independent of latitude.     

A high-energy solar flare burst complex and the physical properties of its source region

We discuss a solar flare microwave burst complex, which included a major structure consisting of some 13 spikes of 60 ms FWHM each, observed 21 May, 1984 at 90 GHz (3 mm). It was associated with a simultaneous very hard X-ray burst complex. We suggest that the individual spikes of both bursts were caused by the same electron population: the X-bursts by their bremsstrahlung, and the microwave bursts by their gyrosynchrotron emission. This latter conclusion is based on the evidence that the radio turnover frequency was 150 GHz. It follows that the emission sources were characterized by an electron density of about 10¹¹ cm^–3, a temperature of 5 × 10⁸ K and a magnetic field of about 1400–2000 G. They had a size of about 350 km; if the energy release is caused by reconnection the sources of primary instability could have been smaller and in the form of thin sheets with reconnection speed at a fraction of the Alfvén velocity and burst-like energy injections of 10²⁷ erg during about 50 ms each. The energized plasma knots lost their injection energy by saturated convective flux (collisionless conduction) in about 30 ms.     

A second-order solution to the two-point boundary value problem for rendezvous in eccentric orbits

A new second-order solution to the two-point boundary value problem for relative motion about orbital rendezvous in one orbit period is proposed. First, nonlinear differential equations to describe the relative motion between a chaser and a target are presented considering the second-order terms in the gravity. Then, by regarding the second-order terms as external accelerations, we establish second-order state transition equations. Moreover, the J2 perturbations effects can also be considered in the state transition equations. Last, the initial relative velocity to fulfill a rendezvous is determined by solving the state transition equations. Numerical simulations show that the new second-order state transition equations are accurate. The second-order solution to the two-point boundary value problem on eccentric orbits is valid even if the relative range is farther than 500 km.