Comments on the application of power series solutions to problems in celestial mechanics

The author relates his experiences in utilizing the power series method to generate trajectories for orbital and sub-orbital vehicles and for then-body problem.     

Hamilton-like vectors for a class of Kepler problems with a force proportional to the velocity

The Kepler problem for the resistive force r/r ² is known to have a conserved vector which is the analogue to Hamilton's vector for the standar Kepler problem. In this note it is shown in a very elementary way that many similar force laws display the same property. The orbit equation can be obtained easily in such cases.     

Light curve solutions of the eccentric Kepler binaries KIC 4281895 and KIC 5115178 with tidally-induced humps

The paper presents our light curve solutions of the Kepler data of two eccentric, eclipsing binaries: KIC 4281895 and KIC 5115178. The derived orbital and stellar parameters reveal that their components are of G spectral type and undergo partial eclipses. We found tidally-induced light humps around the periastron phases of the two targets which are appearances of the recently discovered “heartbeat” phenomenon.     

Unified analytical solutions to two-body problems with drag

The two-body problem with a generalized Stokes drag is discussed. The drag force is proportional to the product of the velocity vector and the inverse square of the distance. The generalization consists of allowing two different proportionality constants for the radial and the transverse components of the force. Under the 'generalized Robertson transformation', the equation of the orbit takes the form of the Lommel equation and admits solutions in terms of Bessel and Lommel functions. The exact, analytical solutions for this type of drag reveal a paradoxical effect of increasing eccentricity for all trajectories. The Poynting–Robertson drag and Poynting–Plummer–Danby problems are discussed as particular cases of the general solution.     

Summary of solutions to the spherically symmetric accretion and stellar wind problems

The stellar wind and accretion problems described by the steady, spherically symmetric continuum equations incorporating thermal conduction and viscosity are studied. A summary of solutions, including some new solutions, is presented and the solution properties are briefly examined.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Light‐curve solutions of 20 ecipsing Kepler binaries,most of them with pronounced spot and flare activity

We carried out light curve solutions of the Kepler light curves of twenty detached eclipsing binaries with circular orbits and determined the orbital inclinations, temperatures. relative radii and luminosities of their components. We studied the quality of the solutions with respect to the adopted limb‐darkening law and its coefficients. The detailed tracing of the numerous and uninterrupted data of our targets gave us an unique possibility to detect and learn their spot and flare activity. We established that the out‐of‐eclipse variability of the most targets gradually changes from small‐amplitude two‐waved type to big‐amplitude one‐waved type and vice versa, i.e. their spot activity cycles pass through phase of two almost diametrically opposite spots and phase of big polar cool spot. We found that the low‐temperature targets show flare activity of UV Cet‐type with amplitudes of 0.002–0.22 mag and duration of up to several hours. (© 2016 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Determining the fundamental parameters of F and G supergiants

A technique for determining the effective temperature T _eff and the acceleration of gravity log g of F and G supergiants is discussed using four bright stars as examples, specifically two F supergiants, α Lep(F0 Ib) and π Sgr (F2 II), and two G supergiants, β Aqr (G0 Ib) and α Aqr (G2 Ib). In all four cases the parameter log g was derived from the high precision parallaxes recently obtained by van Leeuwen in a new reduction of data from Hipparcos. Because of this, the accuracy of the determinations of log g is much greater than before. Estimates of the parameter T _eff were checked using accurate values of T _eff obtained previously by the infrared flux method (IRFM). In the case of the early F supergiants, this method confirms the good accuracy of the T _eff values derived from the Balmer lines and the β-index. Measurements of the Balmer lines for the G supergiants are difficult because of strong blending, so the indices [c ₁] and β serve as indicators of T _eff . It is shown that the indices [c ₁] and β yield a systematic difference in the values of T _eff ; the IRFM confirms that deriving T _eff from the index [c ₁] is more accurate. Based on the values of T _eff and log g that have been found here, with the aid of the evolutionary tracks, we estimate the mass M and age t of each star. The Fe II lines, which are insensitive to departures from LTE, have been used to determine the microturbulence parameter V _t and the iron abundance. The latter is close to the solar iron abundance. Some problems concerning the chemical composition of these stars are discussed using the supergiant α Lep as an example. Translated from Astrofizika, Vol. 52, No. 2, pp. 237–257 (May 2009).     

Light curve solutions of the eclipsing Kepler binaries KIC 5080652, KIC 5285607, KIC 9236858 and KIC 11975363

We carried out light curve solutions of four detached binaries with circular orbits, observed by Kepler. As a result their orbital inclinations, temperatures and relative stellar radii were determined. We estimated also their global parameters on the base of the obtained solutions and empirical relation “temperature, luminosity” for MS stars. The out-of-eclipse light curves of KIC 5080652, KIC 9236858 and KIC 11975363 reveal a trend the bigger amplitudes to correspond to single-waved shape while the two-waved shape to be inherent to the smaller amplitudes. This type of variability was attributed to gradually transition between state with two almost opposite cool spots and state with bigger in size polar spot. We detected also several microflares of KIC 11975363 with amplitudes of 0.002–0.003 mag.     

Deciphering the pulsations of G 29-38 with optical time series spectroscopy

We present optical time series spectroscopy of the pulsating white dwarf star G 29-38 taken at the Very Large Telescope (VLT). By measuring the variations in brightness, Doppler shift and line shape of each spectrum, we explore the physics of pulsation and measure the spherical degree (ℓ) of each stellar pulsation mode. We measure the physical motion of the g modes correlated with the brightness variations for three of the eight pulsation modes in this data set. The varying line shape reveals the spherical degree of the pulsations, an important quantity for properly modelling the interior of the star with asteroseismology. Performing fits to the Hβ, Hγ and Hδ lines, we quantify the changing shape of the line and compare them to models and previous time series spectroscopy of G 29-38. These VLT data confirm several ℓ identifications and add four new values, including an additional ℓ= 2 and a possible ℓ= 4. In total, from both sets of spectroscopy of G 29-38, eleven modes now have known spherical degrees.     

The Lambert <Emphasis Type="Italic">W</Emphasis> function and solutions to Kepler’s equation

This paper presents a method for the truncation of infinite Fourier–Bessel representations for functions requiring a solution to Kepler’s equation. Use is made of the Lambert W function to solve for the desired index that bounds the remainder terms of the series, within the prescribed tolerance. The enforcement of a maximum on the number of Bessel functions is also useful in truncating the Bessel functions themselves, resulting in an analytical representation of the solution to a desired tolerance, without the use of infinite series.     

Transformations to extend the range of application of power series solutions of differential equations of motion

Let the solution of a differential equation, expanded in powers of the independent variablet, have radius of convergenceT. let , wheret=t(), be a new independent variable, and let the corresponding power series in have radius of convergenceS. Thent(S) will not in general be equal toT. Ift(S)>T, then the series in powers of may have advantages over those in powers oft. Mathematical consequences of this distinction have been appreciated since the time of Poincaré. In this note the practical applications of some transformations are investigated.     

Information from the Kinematics of F and G Stars in the Solar Neighborhood

We have calculated the orbital parameters for 90 stars in Chen et al. and updated the kinematic data for stars in Edvardsson et al. by using the accurate Hipparcos parallaxes and proper motions, and recalculated the \\\\\\\\\\\\-element abundances in Edvardsson et al. in a way consistent with Chen et al. The two sets of data are combined in a study of stellar populations and characteristics of F & G stars in the solar neighborhood. We confirm the result of Chen et al. that a distinguishable group of stars may belong to the thick disk rather than the thin disk. The ages for the stars are determined using the theoretical isochrones of VandenBerg et al. The age-metallicity relation is investigated for different subgroups according to distance from the sun and galactic orbital parameters. It is found that a mixing of stars with different orbital parameters significantly affect the age-metallicity relation for the disk. Stars with orbits confined to the solar circle all have metallicities [Fe/H] > -0.3 irresp     

Evolution of the Radio Luminosities of the Tycho and Kepler Supernovae Remnants

Radio emission of the historical supernovae remnants Tycho (SNR1572) and Kepler (SNR1604) and evolution of their luminosity are considered. Measurement data of secular luminosity decrease rate, obtained earlier by the authors, were corrected with account of variation in time of the flux density of the reference sources. As a result, it is found that the SNR1604 luminosity at 1667 MHz is weakening with an annual mean rate equal to (0.2 ± 0.07)%. The corresponding rate for SNR1572 is (0.47 ± 0.05)%. Since the radio luminosity evolution, as well as energy densities of magnetic field and relativistic electrons inside SNR1604 and SNR1572 are essentially different, these remnants should be considered as different types of supernovae. Bandiera classified SN1604 as type SNIb or SNII.     

Interpretation of the frequency distribution of isochrone ages of nearby F and G stars

The frequency distribution of isochrone ages of nearby F and G stars is used in a synthetic approach to the history of the star-formation rate (SFR). On the basis of stellar evolutionary sequences, age distributions are calculated for different assumptions on the SFR history. Models with a constant past SFR provide a good fit of the empirical age distributions, unless a feature at ages 4 Gyr. If real, a shoulder in the empirical frequency distribution of isochrone ages at 4–5 Gyr points toward an enhanced SFR during a more or less (1 to 6 Gyr) extended period which ceased with a peak about 4 Gyr ago. It is shown that models with such a nonmonotonous SFR history reproduce the empirical stellar age distribution and, moreover, the age-metallicity relation and the metallicity distribution of nearby stars. An era of enhanced star formation in this time interval is consistent with recently published interpretation of several age-dependent properties in different samples of nearby stars.     

Red supergiants in the LMC – III: luminous F and G stars

New BVRI observations for 40 and spectrophotometric measurements for 3 F to G LMC supergiant candidates (and 3 galactic F to G supergiants) are presented. The errors of the BVRI data are 0.01 to 0.03 mag in most cases. The wavelength range of the spectra is 3400 to 6400 Å, their resolution 10 Å. The mean error of the fluxes is 0.03 mag. Spectral indices measuring the strengths of the Hβ, Hγ, Hδ, NaD and CaII H+K lines, the CHα₀ and CNβ₀ bands, of the Balmer jump and the slope of the continuum redwards are discussed as measures of effective temperature and luminosity on the basis of galactic stars with accurate MK types and parallaxes. The Hγ line and the continuum gradient are very good temperature criteria, the CHα₀ band and especially the Balmer jump for luminosity. The luminosity classification given for F to G supergiant candidates in the LMC in the literature is often doubtful. 5 of the 3 stars observed spectrophotometrically turn out to be probably galactic foreground dwarfs on the basis both of the Balmer jump and the comparison of their flux distributions with synthetic ones based on the Kurucz model atmospheres. Surface gravities derived purely on the basis of flux distributions and such ones given by models of stellar evolution agree with each other for dwarfs and giants only. For supergiants the former are about 1.0 dex higher than the latter. As a consequence effective temperatures and metallicities given by these two methods deviate from each other for such stars, too. The intrinsic colours and temperatures of galactic and LMC supergiants do not differ. With absolute magnitudes up to -9.6 mag the upper luminosity limit in the LMC does not exceed that in the Galaxy, where Ia-0 supergiants have M_V of up to -9.5 mag. The metallicities of the supergiants show a rather large scatter. Nevertheless the mean metallicities of 0.0 ± 0.09 dex for the Galaxy and -0.6 ± 0.10 dex for the LMC agree well with other observations.     

Abundances and absolute stellar magnitudes for F and G supergiants of Magellanic Clouds

Using high-resolution spectra, we study 31 yellow supergiants of the Large and Small Magellanic Clouds by the method of atmosphere models. Abundance values of 20 chemical elements are determined. It is shown that ??-elements are in a slight excess and neutron-capture elements have an excess up to 0.60 dex. Approbation of a new technique for the determination of absolute stellar magnitudes of late-type supergiants is performed. The technique is based on the use of the spectroscopic criterion, namely, depth-line ratios for iron. Absolute stellar magnitudes of nine supergiants of the Large Magellanic Cloud are calculated using this technique. A value of the distance modulus of the Large Magellanic Cloud is estimated: m ? M = 18.4 ± 0.3 ^m .