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.     

Analysis of fossil pigments from algae and bacteria in meromictic Lake Fidler, Tasmania, and its application to lake management

Lake Fidler is an ectogenic meromictic lake with a monimolimnion maintained by periodic incursions of brackish water from the lower Gordon River estuary. A dam across the middle reaches of the Gordon River has restricted these incursions of brackish water and meromictic stability has rapidly declined. A palaeolimnological study was carried in order to assess the historical development of meromixis and the impact of the dam on the microbiological communities in the lake. Fossil pigments in a 17 m sediment core were analysed using reverse phase high performance liquid chromatography (rp-HPLC) and mass spectrometry (MS). In addition, taphonomic studies of pigment production, deposition and degradation in the water column and surface sediments were used to identify planktonic and benthic pigment degradation processes and constrain the stratigraphic interpretation. Results comparing the pigment composition of pelagic sediment traps and littoral surface sediments indicated that the core from the centre of the lake would permit a historical reconstruction of planktonic bacterial and algal communities. Marked increases in prokaryotic pigments ca 3500 yr B.P. suggested the possible colonisation of a chemocline by phototrophic bacteria. Further changes in chlorophyll: carotenoid ratios and changes in relative abundances of both chlorophyll a and bacteriochlorophyll c derivatives also indicated that a change in the depositional environment had occurred; possibly due to altered stratification or anoxia. From this we infer the onset of either intermittent or permanent meromixis. Further increases in prokaryotic pigment abundance suggested that the present state of permanent meromixis was firmly established by 2070 ±50 ¹⁴C yr B.P., and diatom analysis confirmed the development of a stable mixolimnion. High resolution studies of the top 10 cm of sediments measured pigments in mean concentrations of 15.1 ng g^-1 with a mean S.D. of only 2.78 indicating little change in pigment abundance since the construction of the dam. Thus, Lake Fidler still retains most of the features of meromixis. However, evidence from nearby Lake Morrison and Sulphide Pool has shown that any further declines in meromictic stability will cause a rapid reversion to holomixis. Palaeolimnological evidence from the early stages of meromictic development of Lake Fidler suggests that such reversion to holomixis may not permanently eliminate all the microbiological communities, and that, given time, they may return and prosper with re-establishment of a suitable chemocline. These studies will guide recommendations for a management strategy to prevent the further decay of meromixis in the Gordon River lakes.     

10-GHz Tenerife cosmic microwave background observations at 8° resolution and their analysis using a new maximum entropy method

The complete set of data from the Tenerife 10-GHz (8° FWHM) twin-horn, drift scan experiment is described. These data are affected by both long-term atmospheric baseline drifts and short-term noise. A new maximum entropy procedure, utilizing the time invariance and spatial continuity of the astronomical signal, is used to achieve a clean separation of these effects from the astronomical signal, and to deconvolve the effects of the beam-switching. We use a fully positive/negative algorithm to produce two-dimensional maps of the intrinsic sky fluctuations. Known discrete sources and Galactic features are identified in the deconvolved map. The data from the 10-GHz experiment, after baseline subtraction with MEM, are then analysed using conventional techniques, and new constraints on Galactic emission are made.     

Star-forming regions in blue compact dwarf galaxies

We present near-infrared 2.1-μm continuum and Brγ and 1–0S(1) line images of three blue compact dwarf galaxies: II Zw 40, NGC 5253 and He 2-10. Comparison of the morphologies of the emission-line regions and continuum together with the line ratios shows that in these cases the starbursts are typified by strongly peaked Brγ and weak diffuse H₂ emission, indicating the presence of one or more compact nuclei, and also tidal tails. We consider the possible origins of the H₂ emission, and conclude that shock excitation in cloud collisions, and hence dynamical processes such as interactions or mergers, is important. By comparing models of the equivalent width of Brγ to other age constraints from the literature we show that the star formation must have occurred not only recently but in a short-duration burst. The current centres of star formation in all three galaxies are of a similar age but, whereas the hotspots in He 2-10 are coeval, we find phase differences between those in NGC 5253. We compare the masses and sizes of the star formation sites to those of present-day globular clusters.     

The transformation of nesquehonite into hydromagnesite

Nesquehonite, synthesized in the laboratory, can be readily altered to hydromagnesite via an intermediate phase which is morphologically similar to hydromagnesite. This intermediate phase exhibits an X-ray pattern similar to that of the newly discovered mineral dypingite. The transformation is accompanied by a large loss of water, with resulting increase in magnesium. Chemical analysis indicates that dypingite occurs in a range of phases from slightly altered nesquehonite to hydromagnesite. The alteration of nesquehonite at 52°C is exceedingly rapid, explaining why hydromagnesite is the more common hydrate in nature. This formation of a new mineral via the aqueous phase also occurs in the solid state at temperatures in excess of 100°C, as indicated by differential thermal analysis and thermal gravimetric analysis.     

Heat deposition and retention in a solid planet growing by impacts

This paper considers the scaling of impact effects with impactor size and velocity (or planetary radius) and the retention of heat deposited by impacts in a solid planet (i.e., with no convective motions). Some previously used scalings are inconsistent with the general scaling rules of Holsapple and Schmidt (1982), and no study of impact heating has considered the full permissible range of scalings. A sample physical impact model which spans this range is presented. There are three length scales which control impact heat retention: the depth scales of heat deposition and impact stirring and the ratio κ/ν, where κ is the thermal diffusivity and ν is the upward velocity of the accreting surface. These are evaluated in the contexts of the general scaling rules and Safronov's (1972) distribution of impactor sizes. It is found that the effeciency of heat retention (i.e., the fraction of deposited heat which is retained) is independent of the planetary growth rate. It may be low at small planetary radii, but tends to level out around 3000 km radius to values of 40–70%. Combined with an assumed heat deposition effeciency of 20%, this gives melting at a radius between about 2000 and 3000 km in terrestrial planets.     

The angular distribution of cosmic rays at the boundary of the heliosphere

Measurements of the sidereal daily variation of the muon intensity at a depth of 60 m.w.e. have been carried out in London using telescopes inclined at 70° to the zenith for the period 1972 to the present. The direction of maximum sensitivity for these telescopes lies in the Earth's equatorial plane and the asymptotic directions of look at the boundary of the heliosphere have been determined by integrating the equation of motion of the primary particles in a model interplanetary magnetic field. In this way the measured sidereal variation can be related to the cosmic ray intensity distribution in interstellar space. It is shown that the observational data are consistent with an axially symmetric intensity distribution of the form ΔI = 0.09 (1 + cosα) % where ΔI is the direction from the mean intensity and α is measured from the direction of maximum intensity which lies at 1^Π = 250° b^Π = ?60°. The most likely interpretation of this result is that the axis of this distribution corresponds to the local direction of the interstellar magnetic field and that the cosmic rays have a bulk streaming motion of 65±15 km s^?1 along the field direction.     

Analysis of two-decade meteorological and air quality trends in Rome (Italy)

Theoretical and Applied Climatology - The extreme temperature events are a concern in recent years due to climate variability particularly in India as there is an increase in the temperature...     

Determination of Electron Flux Spectra in a Solar Flare with an Augmented Regularization Method: Application to Rhessi Data

Kontar et al. (2004) have shown how to recover mean source electron spectra in solar flares through a physical constraint regularization analysis of the bremsstrahlung photon spectra I() that they produce. They emphasize the use of non-square inversion techniques, and preconditioning combined with physical properties of the spectra to achieve the most meaningful solution to the problem. Higher-order regularization techniques may be used to generate forms with certain desirable properties (e.g., higher-order derivatives). They further note that such analysis may be used to infer properties of the electron energy spectra at energies well above the maximum photon energy observed. In this paper we apply these techniques to data from a solar flare observed by RHESSI on 26 February, 2002. Results using different orders of regularization are presented and compared for various time intervals. Clear evidence is presented for a change in the value of the high-energy cutoff in the mean source electron spectrum with time. We also show how the construction of the injected electron spectrum F₀(E₀) (assuming that Coulomb collisions in a cold target dominate the electron transport) is facilitated by the use of higher-order regularization methods.     

A tentative determination of methylmercury in seawater

A bioassay experiment using caged mussels has suggested a total concentration of methylmercury in the seawater of a polluted estuary of 0.06 ng dm^?3. The technique may provide a useful tool for the assessment of relative concentrations of methylmercury in seawater at different locations.