Dynamic discretization method for solving Kepler’s equation

Kepler’s equation needs to be solved many times for a variety of problems in Celestial Mechanics. Therefore, computing the solution to Kepler’s equation in an efficient manner is of great importance to that community. There are some historical and many modern methods that address this problem. Of the methods known to the authors, Fukushima’s discretization technique performs the best. By taking more of a system approach and combining the use of discretization with the standard computer science technique known as dynamic programming, we were able to achieve even better performance than Fukushima. We begin by defining Kepler’s equation for the elliptical case and describe existing solution methods. We then present our dynamic discretization method and show the results of a comparative analysis. This analysis will demonstrate that, for the conditions of our tests, dynamic discretization performs the best.     

The next generation stellar population synthesis library

Most of the analysis of absorption lines in galaxies is done in the Lick system, defined in the 1970's and 1980's by Faber, Burstein and collaborators. Working in this system is difficult, since it is based on low resolution spectra taken with a non-linear detector without flux-calibration. Due to intrinsic errors in the published indices the Lick system limits the accuracy with which one can analyse galaxy spectra. In the last 2 years we have observed a new stellar library to replace the Lick system. It consists of 1100 stars, covering a wide range in metallicity, abundance ratio, temperature and gravity, observed at a resolution of 2.0Å (9 for Lick) from 3500–7500Å, and calibrated in flux. With this library galaxy spectra can be analysed much more accurately than before, and in much more detail. Most important, the library now makes it possible to study absorption lines of galaxies at intermediate and high redshift, something which was impossible with the Lick system. With this library it will be possible for the first time to synthesise full Spectral energy distributions of galaxies in the optical based fully on observational spectra.     

Bimodality in low-luminosity E and S0 galaxies

Stellar population characteristics are presented for a sample of low-luminosity early-type galaxies (LLEs) in order to compare them with their more luminous counterparts. Long-slit spectra of a sample of 10 LLEs were taken with the ESO New Technology Telescope, selected for their low luminosities. Line strengths were measured on the Lick standard system. Lick indices for these LLEs were correlated with velocity dispersion (σ), alongside published data for a variety of Hubble types. The LLEs were found to fall below an extrapolation of the correlation for luminous ellipticals and were consistent with the locations of spiral bulges in plots of line strengths versus σ. Luminosity weighted average ages, metallicities and abundance ratios were estimated from  χ²  fitting of 19 Lick indices to predictions from simple stellar population models. The LLEs appear younger than luminous ellipticals and of comparable ages to spiral bulges. These LLEs show a bimodal metallicity distribution, consisting of a low-metallicity group (possibly misclassified dwarf spheroidal galaxies) and a high-metallicity group (similar to spiral bulges). Finally, they have low α-element to iron peak abundance ratios indicative of slow, extended star formation.     

The case for the Universe to be a quantum black hole

We present a necessary and sufficient condition for an object of any mass m to be a quantum black hole (q.b.h.): “The product of the cosmological constant Λ and the Planck’s constant ℏ, Λ and ℏ corresponding to the scale defined by this q.b.h., must be of order one in a certain universal system of units”. In this system the numerical values known for Λ are of order one in cosmology and about 10¹²² for Planck’s scale. Proving that in this system the value of the cosmological ℏ _c is of order one, while the value of ℏ for the Planck’s scale is about 10⁻¹²², both scales satisfy the condition to be a q.b.h., i.e. Λℏ≈1. In this sense the Universe is a q.b.h. We suggest that these objects, being q.b.h.’s, give us the linkage between thermodynamics, quantum mechanics, electromagnetism and general relativity, at least for the scale of a closed Universe and for the Planck’s scale. A mathematical transformation may refer these scales as corresponding to infinity (our universe) and zero (Planck’s universe), in a “scale relativity” sense.     

An optimized Hβ index for disentangling stellar population ages

We have defined a new Hβ absorption index definition,  Hβ_o  , which has been optimized as an age indicator for old and intermediate age stellar populations. Rather than using stellar spectra, we employed for this purpose a library of stellar population spectral energy distributions of different ages and metallicities at moderately high resolution.  Hβ_o  provides us with improved abilities for lifting the age–metallicity degeneracy affecting the standard Hβ Lick index definition. The new index, which has also been optimized against photon noise and velocity dispersion, is fully characterized with wavelength shift, spectrum shape, dust extinction and [α/Fe] abundance ratio effects.  Hβ_o  requires spectra of similar qualities as those commonly used for measuring the standard Hβ Lick index definition. Aiming at illustrating the use and capabilities of  Hβ_o  as an age indicator we apply it to Milky Way globular clusters and to a well selected sample of early-type galaxies covering a wide range in mass. The results shown here are particularly useful for applying this index and understand the involved uncertainties.     

Spectroscopic study of globular clusters in the halo of M31 with the Xinglong 2.16 m telescope

We present spectroscopic observations for 11 confirmed globular clusters(GCs) of M31 with the OMR spectrograph on the 2.16 m telescope at the Xinglong site of National Astronomical Observatories,Chinese Academy of Sciences.Nine of our sample clusters are located in the halo of M31 and the most remote one is out to a projected radius of 78.75 kpc from the galactic center.For all our sample clusters,we measured the Lick absorption-line indices and radial velocities.It is noted that most GCs in our sample are ...     

A review of Titan’s atmospheric phenomena

Saturn’s satellite Titan is a particularly interesting body in our solar system. It is the only satellite with a dense atmosphere, which is primarily made of nitrogen and methane. It harbours an intricate photochemistry, that populates the atmosphere with aerosols, but that should deplete irreversibly the methane. The observation that methane is not depleted led to the study of Titan’s methane cycle, starting with its atmospheric part. The features that inhabit Titan’s atmosphere can last for timescales varying from year to day. For instance, the reversal of the north–south asymmetry is linked to the 16-year seasonal cycle. Diurnal phenomena have also been observed, like a stratospheric haze enhancement or a possible tropospheric drizzle. Furthermore, clouds have been reported on Titan since 1993. From these first detections and up to now, with the recent inputs from the Cassini–Huygens mission, clouds have displayed a large range of shapes, altitudes, and natures, from the flocky tropospheric clouds at the south pole to the stratiform ones in the northern stratosphere. It is still difficult to compose a clear picture of the physical processes governing these phenomena, even though of lot of different means of observation (spectroscopy, imaging) are available now. We propose here an overview of the phenomena reported between 1993 and 2008 in the low atmosphere of Titan, with indications on the location, altitude, and their characteristics in order to give a perspective of our up-to-date understanding of Titan’s meteorological manifestations. We shall focus mainly on direct imaging observations, from both space- and ground-based facilities. All of these observations, published in more than 30 different refereed papers to date, allow us to build a precise chronology of Titan’s atmospheric changes (including the north–south asymmetry, diurnal and seasonal effects, etc). Since the interpretation is at an early stage, we only briefly mention some of the current theories regarding the features’ nature.     

Kinematics and stellar populations of 17 dwarf early-type galaxies

We present kinematics and stellar population properties of 17 dwarf early-type galaxies in the luminosity range -14 ≥ M _B ≥ -19. Our sample fills the gap between the intensively studied giant elliptical and Local Group dwarf spheroidal galaxies. The dwarf ellipticals of the present sample have constant velocity dispersion profiles within their effective radii and do not show significant rotation, hence are clearly anisotropic. The dwarf lenticulars, instead, rotate faster and are, at least partially, supported by rotation. From optical Lick absorption indices, we derive metallicities and element abundances. Combining our sample with literature data of the Local Group dwarf spheroidals and giant ellipticals, we find a surprisingly tight linear correlation between metallicity and luminosity over a wide range: -8 ≥ M _B ≥ -22. The α/Fe ratios of our dwarf ellipticals are significantly lower than the ones of giant elliptical galaxies, which is in agreement with spectroscopy of individual stars in Local Group dwarf spheroidals. Our results suggest the existence of a clear kinematic and stellar population dichotomy between dwarf and giant elliptical galaxies. This result is important for theories of galaxy formation, because it implies that present-day dwarf ellipticals are not the fossiled building blocks of giant ellipticals. This revised version was published online in August 2006 with corrections to the Cover Date.     

Fitting formulae for the effects of binary interactions on lick indices and colors of stellar populations

More than about 50% of stars are in binaries, but most stellar population studies take single star stellar population (ssSSP) models, which do not take binary interactions into account. In fact, the integrated peculiarities of ssSSPs are different from those of stellar populations with binary interactions (bsSSPs). Therefore, it is necessary to investigate the effects of binary interactions on the Lick indices and colors of populations in detail.We show some formulae for calculating the difference between the Lick indices and colors of bsSSPs, and those of ssSSPs. Twenty-five Lick indices and 12 colors are studied in this work. The results can be conveniently used for calculating the effects of binary interactions on stellar population studies and for adding the effects of binary interactions into present ssSSP models. The electronic data and fortran procedures in the paper can be obtained on request from the authors.     

The Stellar Populations of Early-Type Galaxies in the Fornax Cluster

We have measured central line strengths for a complete sample of early-type galaxies in the Fornax Cluster comprising 11 elliptical and 11 lenticular galaxies, more luminous than M _B=–17. We find that the centres of Fornax ellipticals follow the locus of galaxies of fixed age in Worthey's models and have metallicities varying roughly from half to 2.5 times solar. The centres of (relatively low luminosity) lenticular galaxies, however, exhibit a substantial spread to younger luminosity-weighted ages indicating amore extended star formation history. Our conclusions are based on two age/metallicity diagnostic diagrams in the Lick/IDS system comprising established indices such as [MgFe]and H_β as well as new and more sensitive indices, such as Fe3and H . The inferred difference in the age distribution between lenticular and elliptical galaxies is a robust conclusion as the models generate consistent relative ages using different age and metallicity indicators even though the absolute ages remain uncertain. The absolute age uncertainty is caused mainly by the effects of non-solar abundance ratios, which are not accounted for in the stellar population models. We find that Es are generally overabundant in magnesium and the most luminous galaxies show stronger overabundances. The luminosity-weighted stellar populations of young S0s are consistent with solar abundance ratios or a weak Mg under abundance. Two of the faintest lenticular galaxies in our sample have blue continua and extremely strong Balmer-line absorptions suggesting starbursts <2 Gyr ago. This revised version was published online in July 2006 with corrections to the Cover Date.     

The Effect of Method of Constructing the Mass Distribution on Single Stellar Populations

We use two methods of constructing the initial mass distribution, the traditional way and Monte Carlo simulation, to obtain integrated U - B, B - V, V-R and V-I colours and absorption-line indices denned by the Lick Observatory image dissector scanner (referred to as Lick/IDS), for instantaneous burst solarmetallicity single stellar populations with ages in the range 1-15 Gyr. We find that the evolutionary curves of all colours obtained by the traditional method are smoother than those by Monte Carlo simulation, that the U - B and B - V colours obtained by the two methods agree with one another, while the V - R and V - I colours by the traditional method are bluer than those by Monte Carlo simulation. A comparison of the Lick/IDS absorption-line indices shows that the variations in all the indices by the traditional method are smoother than that for the Monte Carlo simulation, and that all the indices except for TiO1 and TiO2 are consistent with those for the Monte Carlo simulation.     

Fast Procedure Solving Universal Kepler's Equation

We developed a procedure to solve a modification of the standard form of the universal Kepler’s equation, which is expressed as a nondimensional equation with respect to a nondimensional variable. After reducing the domain of the variable and the argument by using the symmetry and the periodicity of the equation, the method first separates the case where the solution is so small that it is given an inverted series. Second, it separates the cases where the elliptic, parabolic, or hyperbolic standard forms of Kepler’s equation are suitable. Here the separation is done by judging whether detouring these nonuniversal equations will cause a 1-bit loss of information to their nonuniversal solutions or not. Then the nonuniversal equations are solved by the author’s procedures to solve the elliptic Kepler’s equation (Fukushima, 1997a), Barker’s equation (Fukushima, 1998), and the hyperbolic Kepler’s equation (Fukushima, 1997b), respectively. And their nonuniversal solutions are transformed back to the solution of the universal equation. For the rest of the case, we obtain an approximate solution by solving roughly the approximated cubic equation as we did in solving Barker’s equation. Then the correction to the approximate solution is obtained by Halley’s method precisely. There the special function appeared in the universal equation is rewritten into a combination of similar special functions of small arguments, so that they are efficiently evaluated by their Taylor series. Numerical measurements showed that, in the case of Intel Pentium II processor, the new method is 10–25 times as fast as Shepperd’s method (Shepperd, 1985) and 7–13 times as fast as the standard Newton method. This revised version was published online in July 2006 with corrections to the Cover Date.     

A semi-analytic algorithm for constructing lower dimensional elliptic tori in planetary systems

We adapt the Kolmogorov’s normalization algorithm (which is the key element of the original proof scheme of the KAM theorem) to the construction of a suitable normal form related to an invariant elliptic torus. As a byproduct, our procedure can also provide some analytic expansions of the motions on elliptic tori. By extensively using algebraic manipulations on a computer, we explicitly apply our method to a planar four-body model not too different with respect to the real Sun–Jupiter–Saturn–Uranus system. The frequency analysis method allows us to check that our location of the initial conditions on an invariant elliptic torus is really accurate.     

On the change of the inner boundary of an optically thick accretion disk around white dwarfs using the dwarf nova SS Cyg as an example

We present the results of our studies of the aperiodic optical flux variability for SS Cyg, an accreting binary systemwith a white dwarf. The main set of observational data presented here was obtained with the ANDOR/iXon DU-888 photometer mounted on the RTT-150 telescope, which allowed a record (for CCD photometers) time resolution up to 8 ms to be achieved. The power spectra of the source’s flux variability have revealed that the aperiodic variability contains information about the inner boundary of the optically thick flow in the binary system. We show that the inner boundary of the optically thick accretion disk comes close to the white dwarf surface at the maximum of the source’s bolometric light curve, i.e., at the peak of the instantaneous accretion rate onto the white dwarf, while the optically thick accretion disk is truncated at distances 8.5 × 10⁹ cm ∼10R _WD in the low state. We suggest that the location of the inner boundary of the accretion disk in the binary can be traced by studying the parameters of the power spectra for accreting white dwarfs. In particular, this allows the mass of the accreting object to be estimated.     

Characteristics of the Re-Calculated <Emphasis Type="Italic">Yohkoh</Emphasis>/SXT Temperature Response

The temperature response functions of the Yohkoh/SXT are re-calculated based on the most recent elemental abundances and ionization balance available in the CHIANTI atomic database version 6.0.1. The new standard responses are calculated for three types of abundance; i.e., ‘coronal’, ‘hybrid’, and ‘photospheric’ abundances included in the CHIANTI database, and are available in SolarSoft since 2010. Comparison plots of the new and old response functions and filter ratios are available at the Yohkoh Legacy data Archive (). The three new responses generally peak at higher temperatures (at ≈ 10 MK) than the former standard response (at ≈ 5.6 MK) based on Mewe’s spectral model. The new responses with coronal and hybrid abundances have higher peak counts by a factor of 3 and 2, respectively, than those with the photospheric abundances and the former response based on Mewe’s model. The correction of the filter ratios depends on the type of filter and the range of the ratios to be used. In the significant cases, the new filter ratio produces 20 to 30% higher temperatures than the previous calibration. The choice of elemental abundance has a strong influence on the derived temperatures and emission measures, and often produces a variation significantly larger than the statistical and systematic errors considered so far.     

Gravitational Mesoscopic Constraints in Cosmological Dark Matter Halos

We present an analysis of the behaviour of the ‘coarse-grained’ (‘mesoscopic’) rank partitioning of the mean energy of collections of particles composing virialized dark matter halos in a Λ-CDM cosmological simulation. We find evidence that rank preservation depends on halo mass, in the sense that more massive halos show more rank preservation than less massive ones. We find that the most massive halos obey Arnold’s theorem (on the ordering of the characteristic frequencies of the system) more frequently than less massive halos. This method may be useful to evaluate the coarse-graining level (minimum number of particles per energy cell) necessary to reasonably measure signatures of ‘mesoscopic’ rank orderings in a gravitational system.     

Inclusion of binaries in evolutionary population synthesis

Using evolutionary population synthesis we present integrated colours, integrated spectral energy distributions and absorption-line indices defined by the Lick Observatory image dissector scanner (referred to as the Lick/IDS) system, for an extensive set of instantaneous-burst binary stellar populations with and without binary interactions. The ages of the populations are in the range 1–15 Gyr and the metallicities are in the range 0.0001–0.03. By comparing the results for populations with and without binary interactions we show that the inclusion of binary interactions makes the integrated U – B , B – V , V – R and R – I colours and all Lick/IDS spectral absorption indices (except for H_β) substantially smaller. In other words, binary evolution makes a population appear bluer. This effect raises the derived age and metallicity of the population.
We calculate several sets of additional solar-metallicity binary stellar populations to explore the influence of the binary evolution algorithm input parameters (the common-envelope ejection efficiency and the stellar wind mass-loss rate) on the resulting integrated colours. We also look at the dependence on the choice of distribution functions used to generate the initial binary population. The results show that variations in the choice of input model parameters and distributions can significantly affect the results. However, comparing the discrepancies that exist between the colours of various models, we find that the differences are less than those produced between the models with and those without binary interactions. Therefore it is very necessary to consider binary interactions in order to draw accurate conclusions from evolutionary population synthesis work.     

Wisdom system: Dynamics in the adiabatic approximation

A simple approximate model of the asteroid dynamics near the 3:1 mean–motion resonance with Jupiter can be described by a Hamiltonian system with two degrees of freedom. The phase variables of this system evolve at different rates and can be subdivided into the ‘fast’ and ‘slow’ ones. Using the averaging technique, wisdom obtained the evolutionary equations which allow to study the long-term behavior of the slow variables. The dynamic system described by the averaged equations will be called the ‘Wisdom system’ below. The investigation of the, wisdom system properties allows us to present detailed classification of the slow variables’ evolution paths. The validity of the averaged equations is closely connected with the conservation of the approximate integral (adiabatic invariant) possessed by the original system. Qualitative changes in the behavior of the fast variables cause the violations of the adiabatic invariance. As a result the adiabatic chaos phenomenon takes place. Our analysis reveals numerous stable periodic trajectories in the region of the adiabatic chaos.     

Habitable zones of host stars during the post-MS phase

A star will become brighter and brighter with stellar evolution, and the distance of its habitable zone will become larger and larger. Some planets outside the habitable zone of a host star during the main sequence phase may enter the habitable zone of the host star during other evolutionary phases. A terrestrial planet within the habitable zone of its host star is generally thought to be suitable for the existence of life. Furthermore, a rocky moon around a giant planet may be also suitable for life to survive, provided that the planet–moon system is within the habitable zone of its host star. Using Eggleton’s code and the boundary flux of the habitable zone, we calculate the habitable zone of our Solar system after the main sequence phase. It is found that Mars’ orbit and Jupiter’s orbit will enter the habitable zone of the Solar system during the subgiant branch phase and the red giant branch phase, respectively. And the orbit of Saturn will enter the habitable zone of Solar during the He-burning phase for about 137 million years. Life is unlikely at any time on Saturn, as it is a giant gaseous planet. However, Titan, the rocky moon of Saturn, may be suitable for biological evolution and become another Earth during that time. For low-mass stars, there are similar habitable zones during the He-burning phase as our Solar, because there are similar core masses and luminosities for these stars during that phase.     

Imaging the Earth’s Interior: the Angular Distribution of Terrestrial Neutrinos

Decays of radionuclides throughout the earth’s interior produce geothermal heat, but also are a source of antineutrinos; these geoneutrinos are now becoming observable in experiments such as KamLAND. The (angle-integrated) geoneutrino flux has been shown to provide a unique probe of geothermal heating due to decays, and an integral constraint on the distribution of radionuclides in the earth. In this paper, we calculate the angular distribution of geoneutrinos, which opens a window on the differential radial distribution of terrestrial radionuclides. We develop the general formalism for the neutrino angular distribution. We also present the inverse transformation which recovers the terrestrial radioisotope distribution given a measurement of the neutrino angular distribution. Thus, geoneutrinos not only allow a means to image the earth’s interior, but offer a direct measure of the radioactive earth, both revealing the earth’s inner structure as probed by radionuclides, and allowing a complete determination of the radioactive heat generation as a function of radius. Turning to specific models, we emphasize the very useful approximation in which the earth is modeled as a series of shells of uniform density. Using this multishell approximation, we present the geoneutrino angular distribution for the favored earth model which has been used to calculate the geoneutrino flux. In this model the neutrino generation is dominated by decays of potassium, uranium, and thorium in the earth’s mantle and crust; this leads to a very “peripheral” angular distribution, in which 2/3 of the neutrinos come from angles θ ≳ 60° away from the nadir. We note that a measurement of the neutrino intensity in peripheral directions leads to a strong lower limit to the central intensity. We briefly discuss the challenges facing experiments to measure the geoneutrino angular distribution. Currently available techniques using inverse beta decay of protons require a (for now) unfeasibly large number of events to recover with confidence the forward scattering signal from the background of subsequent elastic scatterings. Nevertheless, it is our hope that future large experiments, and/or more sensitive techniques, can resolve an image of the earth’s radioactive interior.