On the stability of hierarchical four-body systems

Generalized Jacobian coordinates can be used to decompose anN-body dynamical system intoN-1 2-body systems coupled by perturbations. Hierarchical stability is defined as the property of preserving the hierarchical arrangement of these 2-body subsystems in such a way that orbit crossing is avoided. ForN=3 hierarchical stability can be ensured for an arbitrary span of time depending on the integralz=c ² h (angular momentum squared times energy): if it is smaller than a critical value, defined by theL ₂ collinear equilibrium configuration, then the three possible hierarchical arrangements correspond to three disconnected subsets of the invariant manifold in the phase space (and in the configuration space as well; see Milani and Nobili, 1983a). The same definitions can be extended, with the Jacobian formalism, to an arbitrary hierarchical arrangement ofN≥4 bodies, and the main confinement condition, the Easton inequality, can also be extended but it no longer provides separate regions of trapped motion, whatever is the value ofz for the wholeN-body system,N≥4. However, thez criterion of hierarchical stability applies to every 3-body subsystem, whosez ‘integral’ will of course vary in time because of the perturbations from the other bodies. In theN=4 case we decompose the system into two 3-body subsystems whosec ² h ‘integrals’,z ₂₃ andz ₃₄, att=0 are assumed to be smaller than the corresponding critical values \(\tilde z_{23} \) and \(\tilde z_{34} \) , so that both the subsystems are initially hierarchically stable. Then the hierarchical arrangement of the 4 bodies cannot be broken until eitherz ₂₃ orz ₃₄ is changed by an amount \(\tilde z_{ij} - z_{ij} \left( 0 \right)\) ; that is the whole system is hierarchically stable for a time spain not shorter than the minimum between \(\Delta t_{23} = {{\left( {\tilde z_{23} - z_{23} \left( 0 \right)} \right)} \mathord{\left/ {\vphantom {{\left( {\tilde z_{23} - z_{23} \left( 0 \right)} \right)} {\dot z_{23} }}} \right. \kern-0em} {\dot z_{23} }}\) and \(\Delta t_{34} = {{\left( {\tilde z_{34} - z_{34} \left( 0 \right)} \right)} \mathord{\left/ {\vphantom {{\left( {\tilde z_{34} - z_{34} \left( 0 \right)} \right)} {\dot z_{34} }}} \right. \kern-0em} {\dot z_{34} }}\) . To estimate how long is this stability time, two main steps are required. First the perturbing potentials have to be developed in series; the relevant small parameters are some combinations of mass ratios and length ratios, the? _ij of Roy and Walker. When an appropriate perturbation theory is based on the? _ij , the asymptotic expansions are much more rapidly decreasing than the usual expansions in powers of the mass ratios (as in the classical Lagrange perturbation theory) and can be extended also to cases such as lunar theory or double binaries. The second step is the computation of the time derivatives \(\dot z_{ij} \) (we limit ourselves to the planar case). To assess the long term behaviour of the system, we can neglect the short-periodic perturbations and discuss only the long-periodic and the secular perturbations. By using a Poisson bracket formalism, a generalization of Lagrange theorem for semimajor axes and a generalization of the classical first order theories for eccentricities and pericenters, we prove that thez _ij do not undergo any secular perturbation, because of the interaction with the other subsystem, at the first order in the? _ik . After the long-periodic perturbations have been accounted for, and apart from the small divisors problems that could arise both from ordinary and secular resonances, only the second order terms have to be considered in the computation of Δt ₂₃, Δt ₃₄. A full second order perturbative theory is beyond the scope of this paper; however an order-of-magnitude lower estimate of the Δt _ij can be obtained with the very pessimistic assumption that essentially all the second order terms affect in a secular way thez _ij . The same method could be applied also toN≥5 body systems. Since almost everyN-body system existing in nature is strongly hierarchical, the product of two? _ij is very small for almost all the real astronomical problems. As an example, the hierarchical stability of the 4-body system Sun, Mercury, Venus, and Jupiter is investigated; this system turns out to be stable for at least 110 million years. Although this hierarchical stability time is ～10 times less than the real age of the Solar System, taking into account that many pessimistic assumptions have been done we can conclude that the stability of the Solar System is no more a forbidden problem for Celestial Mechanics.     

An exact solution of transfer equations for interlocked multiplets

The transfer equations for non-coherent scattering arising from interlocking of principal lines without redistribution is exactly solved in a very simple way by Laplace tranform and Wiener-Hopf technique which are easily applied by the use of the new representation ofH-functions obtained recently by the author (1977). The emergent intensity in therth line is expressed in terms of anH-function and a Cauchy type integral admitting of closed form evaluation.     

On topological stability in the general three-body problem

The confining curves in the general three-body problem are studied; the role of the integralc ² h (angular momentum squared times energy) as bifurcation parameter is established in a very simple way by using symmetries and changes of scale. It is well known (Birkhoff, 1927) that the bifurcations of the level manifolds of the classical integrals occur at the Euler-Lagrange relative equilibrium configurations. For small values of the mass ratio ε=m ₃/m ₂ both the positions of the collinear equilibrium points and thec ² h integral are expanded in power series of ε. In this way the relationship is found between the confining curves resulting from thec ² h integral in the general problem, and the zero velocity curves given by the Jacobi integral in the corresponding restricted problem. For small values of ε the singular confining curves in the general and in the restricted problem are very similar, but they do not correspond to each other: the offset of the two bifurcation values is, in the usual, system of units of the restricted problem, about one half of the eccentricity squared of the orbits of the two larger bodies. This allows the definition of an approximate stability criterion, that applies to the systems with small ε, and quantifies the qualitatively well known destabilizing effect of the eccentricity of the binary on the third body. Because of this destabilizing effect the third body cannot be bounded by any topological criterion based on the classical integrals unless its mass is larger than a minimum value. As an example, the three-body systems formed by the Sun, Jupiter and one of the small planets Mercury, Mars, Pluto or anyone of the asteroids are found to be ‘unstable’, i.e. there is no way of proving, with the classical integrals, that they cannot cross the orbit of Jupiter. This can be reliably checked with the approximate stability criterion, that given for the most important three-body subsystems of the Solar System essentially the same information on ‘stability’ as the full computation of thec ² h integral and of the bifurcation values.     

Rosat and EUVE observations of B stars

Recent observations of the X-ray and EUV emission of non-supergiant B stars are summarized. As compared with O stars, the X-rays of most of the near-main-sequence B stars are soft, and the stars show a departure from theL _x = 10^?7 L _bol relation. Using line driven wind models to provide an estimate of the density distribution, it is concluded that a major fraction of the wind emission measure is hot, whereas in shocked wind theory less than 10 percent of the wind emission measure should be hot. The X-ray observations suggest that all of the B stars are X-ray emitters with a basal X-ray luminosity of about 10^?8.5 L _bol . A hard component dominates the X-ray emission of τ Sco, and possible causes are discussed. For the Be stars, the X-ray emission is that which is expected from a normal B-star wind coming from the poles, as in the Wind Compressed Disk (WCD) model of Be stars. None of the stars, including theβ Cep stars, show noticeable variability in their X-rays.EUVE observations of∈ CMa B2 II, find it to be the brightest object in the EUV sky at 500 to 700 Å. It shows a Lyman continuum flux that is a factor of 30 higher than line blanketed model atmospheres. The continuum is seen on both sides of the He I 504 Å edge, and the discrepancy with model atmospheres is even greater shortward of 504 Å. TheEUVE spectra show emission lines both from high stages of ionization ( Feix to Fexvi) and from low stages (Heii and Oiii). The Heii Lymanα results from recombination following X-ray photoionization in the wind, and the Oiii resonance line is found to be present because of the Bowen fluorescence mechanism. Thus, there is and interesting coupling between the wind production by the EUV photospheric emission, the production of X-ray and line EUV emission by winds, and the production of fluorescence by recombination in the wind; all of these processes are now observable in B stars.     

New photometric investigation of the Herbig Ae/Be star HD 52721, a close binary system: Evidence for the existence of large-scale azimuthal inhomogeneities

We present new results of our photometry for the Herbig Be star HD 52721 obtained from January 16 to March 25, 2013. A new data reduction technique is used. Using this technique, we have also reanalyzed the previous results of our photometry for this object pertaining to the period from March 7 to March 28, 2010. The Be star HD 52721 is known as an eclipsing variable with the period P = 1^d. 610. Two photometric minima observed during one period are a peculiarity of its photometric variability. They are separated in phase of the period P by 0.5 and differ from one another in depth by 0 ^m . 04. We have also detected additional minima observed at the phases of maximum brightness. We hypothesize that they can be associated with the existence of local azimuthal inhomogeneities rotating synchronously with the orbital motion of the binary component stars in the circumstellar envelope. When processing our CCD frames, we have applied an efficient CCD-frame rejection method that has allowed the accuracy of observations to be increased considerably. The CCD frames have been further processed using the Apex II software package, which is a universal software platform for astronomical image processing. We justify the need for additional photometric observations of HD 52721 in various color bands to confirm the hypothesis about the existence of azimuthal inhomogeneities in the program binary system and to analyze their physical properties.     

Reconstruction of objects by direct demodulation

High resolution reconstruction of complicated objects from incomplete and noisy data can be achieved by solving modulation equations iteratively under physical constraints. This direct demodulation method is a powerful technique for dealing with inverse problem in general case. Spectral and image restorations and computerized tomography are only particular cases of general demodulation. It is possible to reconstruct an object in higher dimensional space from observations by a simple lower dimensional instrument through direct demodulation. Our simulations show that wide field and high resolution images of space hard X-rays and soft-rays can be obtained by a collimated non-position-sensitive detector without coded aperture masks.     

The rotation of small asteroids

The addition of the unbiased sample of R. P. Binzel and J. D. Mulholland (Icarus56, 519–533) nearly triples the sample size of photoelectrically determined rotational parameters for main belt asteroids with estimated diameters (D) ≤30 km. Nonparametric stattistical tests which require no assumptions about the distributions or variances of the samples are used to examine rotational parameters for all D ≤ 30 km asteroids. A comparison of photoelectric and photographic results shows that the techniques have a highly significant difference in the range of detected frequencies. This difference does not allow photographic and photoelectric observations to be combined for meaningful statistical tests since a photographic bias toward smaller sample variances can induce statistical results that appear overly significant. Photographic observations also show a highly significant bias toward detecting asteroids with larger lightcurve amplitudes. The fit of a Maxwellian to the observed rotational frequency distribution can be rejected at a highly significant confidence level but the observed distribution can be acceptably fit by two Maxwellian distributions, which is consistent with the hypothesis that there are separate populations of slow and fast rotating asteroids. The frequency distributions of <15 km main belt asteroids and Earth and Mars crossers are not found to differ significantly. However, the larger mean lightcurve amplitude of the Earth and Mars crossing asteroids is found to be statistically significant. This latter result is interesting in view of the lack of any strong inverse amplitude versus diameter relation for small asteroids. No significant diameter dependence on rotational frequency is seen among only D ≤ 30 km asteroids. However, the inverse frequency versus diameter relation for D ≤ 120 km asteroids found by S. F. Dermott, A. W. Harris, and C. D. Murray (Icarus, in press) is found to be statistically significant using a linear least-squares analysis of photoelectric data only. No significant diameter dependence on rotational lightcurve amplitude is seen using linear least-squares analysis of photoelectric data for D≤30 and D≤90 km asteroids. However, a significant inverse amplitude versus diameter relation is found when this analysis is extended to D≤120 km asteroids. This finding may be consistent with the hypothesis of Dermott et al. that near 120 km there is a transition between primordial asteroids and their collisional fragments.     

The occurrence of high-order resonances and Kozai mechanism in the scattered disk

By means of numerical methods we explore the relevance of the high-order exterior mean motion resonances (MMR) with Neptune that a scattered disk object (SDO) can experience in its diffusion to the Oort cloud. Using a numerical method for estimate the strength of these resonances we show that high-eccentricity or high-inclination resonant orbits should have evident dynamical effects. We investigate the properties of the Kozai mechanism (KM) for non-resonant SDO's and the conditions that generate the KM inside a MMR associated with substantial changes in eccentricity and inclination. We found that the KM inside a MMR is typical for SDO's with Pluto-like or greater inclinations and is generated by the oscillation of ω inside the mixed (e,i) resonant terms of the disturbing function. A SDO diffusing to the Oort cloud should experience temporary captures in MMR, preferably of the type 1:N, and when evolving inside a MMR and experiencing the KM it can reach regions where the strength of the resonance drops and consequently there is a possibility of being decoupled from the resonance generating by this way a long-lived high-perihelion scattered disk object (HPSDO).     

The mode of creation of stars,stellar systems and the hubble system of galaxies

It is confirmed that the creation of stars in spiral (and perhaps also Irri) galaxies requires a physical parameter (X factor) additional to gas density. Consequently theX factor is an essential feature of stellar patterns and perhaps of stellar systems (spiral and other disk and spheroidal systems, globular clusters) and may be the key to the origin of the few, yet remarkably varied Hubble system of galaxies.

1. It is shown that theX factor is organized over the whole galaxy and is a function of azimuth φ as well as radiusr. Only a galaxy-wide force field seems capable of explaining such anX(r, φ) factor either magnetic or gravitational in origin.
2. If gravitational in origin, theX factor must be a shock wave, but a survey of observations in eight galaxies, including our own, shows no large-scale shocks associated with star creation. This provides further strong evidence against the density-shock theory of twin spiral arms.
3. It is confirmed that galaxies of different Hubble types did not evolve from one another, so that each protogalaxy must possess a genetic factor which predetermines its evolution, and in particular its stellar systems. Thus the protogalactic genetic factor may be identical with theX factor.
4. The case for a primordial magnetic field is strengthened, and it is shown that in our Galaxy and some others the field must be generally oblique to the disk. Such a field can explain theX(r, φ) factor in terms of a magneto-gravitational mechanism of gas clumping.
5. An earlier, hydromagnetic theory of the Hubble types and of radio galaxies is extended by including theX factor to explain the various stellar systems observed in spiral, elliptical, lenticular and irregular galaxies.

     

Statistical parameterization of inverse problems

Image restoration, computerized tomography, and other similar problems are considered as a unified class of stochastic inverse problems. The conventional approach to these problems that proceeds from some integral or functional equations suffers from three main shortcomings: (i) subjectivity, (ii) inability to account for the inner (radiational) noise, and (iii) inability to include the fundamental concept of the natural limit of solution accuracy. A general approach is developed, the Statistical Parameterization of Inverse Problems (SPIPR), that takes into account both the inner and external random noise and gives an explicit form of the above-mentioned natural limit. Applications of the SPIPR to various problems show that the maximum likelihood method as the concrete way to obtain an object estimate has practically limiting efficiency.Two new fields of applications of the SPIPR are outlined along with the image restoration problem: the elimination of blurring due to atmosphere turbulence and reconstruction of an object structure in the computerized tomography. The expressions for the main distribution function in all these problems are found. The corresponding real examples and model cases are considered as well.     

The problem of the critical inclination revisited

The behaviour of the argument of the pericentre is investigated for the orbit of an artificial satellite which is moving under the potential when the inclination of the orbit is close to thecritical value tan^?1 2. The theory is developed to first order and it is applicable to all values of the eccentricity, with the exception of those in the neighbourhood of zero and unity. Four principal types of behaviour are noted and these are illustrated in appropriate phase-plane diagrams. It is shown that the two types which exhibit double libration in the argument of the pericentre are restricted to a relatively small domain in the (a, e)-plane of possible motions. Moreover, it is demonstrated that for double libration to occur it is necessary, but not sufficient, that \(e > \sqrt 6/13\) . The ranges of values of the inclination for which libration of the pericentre is a possibility are given for the more important cases. The general results are applied to the specific case of artificial Earth satellites whose orbits are inclined to the equator at angles close to the value of the critical inclination.     

High temperature plasma inβ Lyrae,observed fromCopernicus

High-resolution ultraviolet spectrophotometry of the complex close binary systemβ Lyrae was performed with the Princeton Telescope Spectrometer onCopernicus. Observations were made at phases 0.0, 0.25, 0.5, and 0.75 with the resolution of 0.2 Å (far-ultraviolet) and 0.4 Å (mid-ultraviolet). The far-ultraviolet spectrum is completely dominated by emission lines indicating existence of high temperature plasma in this binary. The spectrum ofthis object is unlike any other object observed fromCopernicus. It is believed that this high temperature plasma results from dynamic mass transfer taking place in this binary. The current results are compared with the OAO-2 Wisconsin Experiment Package observations and other observational results. The possibility that the secondary component is a collapsed object is also discussed; theCopernicus observations are consistent with the hypothesis that the spectroscopically invisible secondary component is a black hole.     

Iron meteorite bulk densities determined via 3-D laser imaging

This study tested the feasibility of using 3-D laser imaging to measure the bulk density of iron meteorites. 3-D laser imaging is a technique in which a 3-D model of an object is built after aligning and merging individual detailed images of its surface. Assuming that the mass of the object is known, the volume of the model is calculated by software and an estimate of bulk density can be obtained by dividing mass by volume. The 3-D laser imaging technique was used to determine the density of 46 fragments from 11 different iron meteorites. The technique proved to be robust and was applied successfully to study samples ranging close to four orders of magnitude in mass (8 g to 156 kg) and exhibiting a variety of surface textures (e.g., cracks, regmaglypts), reflectivities (e.g., polished surfaces, fusion crust, rust), and morphologies (e.g., sharp angular edges, shrapnel tendrils). Three metrics were considered to estimate the error associated with density measurements: the range accuracy of the laser camera, image alignment error, and inter-operator variability during model building. Inter-operator variability was the largest source of error and was highest when assembling models of samples which either lacked distinctive features or were very complex in shape. Excluding two anomalous Zagora samples where silicate inclusions might have lowered density, the densities measured using 3-D laser imaging ranged from 6.98 to 7.93 g cm⁻³, consistent with previous studies. There is overlap between bulk density and iron meteorite class, and therefore bulk density cannot be used in isolation as a classification criterion. It is a good indicator, however, of weathering effects and of the potential presence of low-density inclusions.     

Tidal disruptions: II. A continuum theory for solid bodies with strength, with applications to the Solar System

We present a comprehensive theory for the breakup conditions for ellipsoidal homogeneous secondary bodies subjected to the tidal forces from a nearby larger primary: for materials ranging from purely fluid ones, to granular rubble-pile gravel-like ones, and to those with either cohesive or granular strength including cohesive rocks and metals. The theory includes but greatly extends the classical analyses given by Roche in 1847, which dealt only with fluids, and also our previous analysis [Holsapple, K.A., Michel, P., 2006. Icarus 183, 331-348], which dealt only with solid but non-cohesive bodies. The results here give the distance inside of which breakup must occur, for both a steadily orbiting satellite and for a passing or impacting object. For the fluid bodies there is a single specific shape (a “Roche Ellipsoid”) that can be in equilibrium at any given distance from a primary, and especially only one shape that can exist at the overall minimum distance (d/R)(ρ/ρp)^1/3=2.455, the classical well-known “Roche limit.” In contrast, solid bodies can exist at a given distance from a primary with a range of shapes. Here we give multiple plots of the minimum distances for various important combinations of body shape, spin, mass density, and the strength parameters characterized by an angle of friction and cohesive strength. Such results can be used in different ways. They can be used to estimate limits on strengths and mass densities for orbiting bodies at a known distance and shape. They can be used to determine breakup distances for passing bodies with an assumed strength and shape. They can be used to constraint physical properties such as bulk density of bodies with a known shape that were known to breakup at a given distance. A collection of approximately 40 satellites of the Solar System is used for comparison to the theory. About half of those bodies are closer than the Roche fluid limit and must have some cohesion and/or friction angle to exist at their present orbital distance. The required solid strength for those states is determined. Finally, we apply the theory to the break up of the SL9 comet at close approach with Jupiter. Our results make clear that the literature estimates of its bulk density depend markedly on unknown parameters such as shape, orientation and spin, and most importantly, material strength characterization.     

The Umov effect for single irregularly shaped particles with sizes comparable with wavelength

The Umov effect manifests itself as an inverse correlation between the linear polarization maximum of an object’s scattered light P_max and its geometric albedo A. This effect is observed for the Moon, Mercury and Mars, and there are data suggesting this effect is valid for asteroids. The Umov effect is due to the contribution of interparticle multiple scattering that increases albedo and decreases polarization. We here study if the Umov effect can be extended to the case of single irregularly shaped particles with sizes comparable with the wavelength. This, in particular, is important for cometary dust polarimetry. We show the Umov effect being valid for weakly absorbing irregular particles (Im(m) ? 0.02) almost through the entire range of size parameters x considered. Highly absorbing particles (Im(m) > 0.02) follow the Umov effect only if x exceeds 14. In the case of weakly absorbing particles, the inverse correlation is essentially non-linear, which is caused by the contribution of particles with small x. However, averaging over many different types of irregularly shaped particles could make it significantly more linear. The size averaging does not change qualitatively the diagram log(P_max)-log(A) for weakly absorbing particles. For single irregular particles whose sizes are comparable with wavelength, there is no reliable correlation between the slope of the polarization curve h near the inversion phase angle and geometric albedo A. Using the extended Umov Law, we estimate the geometric albedo of dust particles forming cometary circumnuclear haloes A = 0.1 − 0.2, which is a few times larger than the average geometric albedo over the entire comae. Note that, using the obtained values for A of cometary particles, one can derive their number density in circumnuclear haloes from photometric observations.     

Evidence forr-process elements in HR 8911

The results of a spectral analysis of some high-dispersion plates of the peculiar star HR 8911 ( Piscium) are reported, and the observational evidence for somer-only elements in the atmosphere of this object is discussed.Many uranium lines were observed, so that the presence of this element may be considered certain. The presence of uranium and of some elements with mass number around theA195 peak gives strong evidence for anr-process mechanism which originated the elemental peculiarities in this object. The implications about the origin of peculiar stars are discussed on the ground of an explosive origin of ther-process elements observed in this star. Finally, the uncommon element holmium seems very likely to be present in the atmosphere of this peculiar star.     

A Unified and Very Fast Way for Computing the Global Potential and Linear Force-Free Fields

We present a fast solver for computing potential and linear force-free fields (LFFF) above the full solar disk with a synoptic magnetic map as input. The global potential field and the LFFF are dealt with in a unified way by solving a three-dimensional Helmholtz equation in a spherical shell and a two-dimensional Poisson equation on the solar surface. The solver is based on a combination of the spectral method and the finite-difference scheme. In the longitudinal direction the equation is transformed into the Fourier spectral space, and the resulting two-dimensional equations in the r?C?? plane for the Fourier coefficients are solved by finite differencing. The solver shows an extremely fast computing speed, e.g., the computation for a magnetogram with a resolution of 180(??)×360(?) is completed in less than 2 s. Even on a high-resolution 600×1200 grid, the solution can be obtained within only about one minute on a single CPU. The solver can potentially be applied directly to the original resolution of observed magnetograms from SDO/HMI for routinely analyzing daily full-disk data.     

Solar system history as recorded in the saturnian ring structure

The paper is based on Holberg's analysis of the Voyager photographs in both reflected and transparent light, combined with occultation data of stars seen through the rings. Besides rapidly varying phenomena (spokes, braided ring, etc.), which according to Mendis are due to gravito-electromagnetic effects, the ring consists of abulk structure, a fine structure, and also ahyperfine structure, showing more than 10000 ringlets. The large number of ringlets can be explained by the Baxter-Thompson ‘negative diffusion’. This gives the ringlets a stability which makes it possible to interprete them as ‘fossils’, which originated at cosmogonic times. It is shown that thebulk structure can be explained by the combined ‘cosmogonic shadows’ of Mimas, the co-orbiting satellites, and the Shepherd satellites. This structure originated at the transition from the plasma phase to the planetesimal phase (which probably took place 4–5×10⁹ y ago). Further, Holberg has discovered that the shadows are not simple void region but exhibit a certain characteristic ‘signature’. This is not yet understood theoretically. Parts of thefine structure are explained by Holberg as resonances with the satellites. Parts are here interpreted as cosmogonic shadow effects. However, there are a number of ringlets which can neither be explained by cosmogonic nor by resonance effects. The most important conclusion is that an analysis of the ring data is likely to lead to areconstruction of the plasma-planetesimal transition with an accuracy of a few percent.     

Zero-population stars

The expected characteristics of stars with zero metal content are investigated for a large range of masses and in varying the original helium-content. It is shown that, in such an extreme case (Z=0), the Main Sequence locus is much less sensitive than expected to the He-content;Y=0 models are found to ignite 3α reactions in the Zero-Age Main Sequence at masses larger by ～ 5M _⊙ than in theY=0.2 case. A comparison with previous results discloses the M.S. slope to be a sensitive function of the metal content only. Evolutionary computations confirm that possible survivors of the zero population can develop a giant branch, akin to that of the well-known Population II stars.