Stabilization by manipulation of the Hamiltonian

Numerical integration of unstable differential equations should be avoided since a numerical error during thenth step produces erroneous initial values for the next step and thus deteriorates the subsequent integration in an unstable manner. A method is offered to stabilize the equations of motion corresponding to a given HamiltonianH by transformingH into a new HamiltonianH ^* which is equivalent to the Hamiltonian of a harmonic oscillator. In contrast to other methods of stabilization the realm of canonical mechanics is thus not abandoned. Perturbations are discussed and as examples the Keplerian motion and the motion of a gyroscope are presented.     

A new time element for a general time transformation

The paper introduces a new time element to be used with a general time transformation for satellite equations of motion. The purpose of this time element is to reduce the growth of the numerical errors with respect to the time integration. It is characteristic for the new time element , that the relation between the timet and the element does not depend on the independent variables.     

General time elements for the two body problem

When integrating a perturbed two-body problem, very often the propagation of the numerical error is reduced by using a new time variables defined by dt/ds=|q| ⁿ , (|q| is the radial distance,t the time). This paper introduces a time element for such transformations, i.e., a new variablet _n is defined so that dt _n/ds=1+ (perturbing terms) andt=F _n(t_n), whereF _n is a known function. The time element equation should be useful in reducing the error in the determination of the timet.F _n is given explicitly forn=1, 3/2, 2, 5/2 and 3, and a general expression is given for other values.The work was performed while the author was an NRC Senior Research Associate, Goddard Space Flight Center, Greenbelt, Md., U.S.A.     

On the Tetrad Formulation of the Equations of Motion in General Relativity

It is shown that the equation of motion Du _j/Ds = (e/mc ²)F _ji u ⁱ , a natural generalization to the curved spacetime of the Heaviside-Lorentz law of ponderomotive force, is equivalent to the metric independent and covariant Van Dantzig's equations of motion dx ^j _[jpi] = 0 or L _v p _i = 0, where p _i is the conjugate momentum 4-vector and v a vector determined by the condition p _i v ⁱ, only with respect to holonomic coordinates. With respect to an anholonomic system, the Heaviside-Lorentz equation is a particular case of the VD equations valid for a privileged class of anholonomic frames, those consisting of orthogonal unit vectors.     

Numerical stabilization of the differential equations of Keplerian motion

A stabilization of the classical equations of two-body motion is offered. It is characterized by the use of the regularizing independent variable (eccentric anomaly) and by the addition of a control-term to the differential equations. This method is related to the KS-theory (Stiefel, 1970) which performed for the first time a stabilization of the Kepler motion. But in contrast to the KS-theory our method does not transform the coordinates of the particle. As far as the theory of stability and the numerical experiments are concerned we restrict ourselves to thepure Kepler motion. But, of course, the stabilizing devices will also improve the accuracy of the computation of perturbed orbits. We list, therefore, also the equations of the perturbed motion.     

The photospheric velocity field of Procyon

BUSS observations of the profiles of two well observed spectral lines in the ultraviolet spectrum of CMi (Procyon; F5 IV–V) are analysed with a Fourier transform method in order to determine values of various parameters of the velocity field of the upper photosphere. We find a microturbulent line-of-sight velocity componentL = 0.9 ± 0.4 km s^–1, a macroturbulent velocity componentL _M = 5.3 ± 0.2 km s^–1, and a rotational velocity componentv _R sini=10.0±1.2 km s^–1. In these calculations a single-moded sinusoidal isotropic macroturbulent velocity function was assumed. The result appears to be sensitive to the assumed shape of the macroturbulence function: for an assumed Gaussian shape the observations can be described withv _R sini=4 km s^–1 andL _M = 11.6 ± 2.7 km s^–1. A comparison is made with other results and theoretical predictions.     

A proposed correction to the solar abundances of carbon and oxygen utilizing new and accurate theoretical forbidden transition probabilities

Previously published solar abundances of oxygen and carbon can be corrected to be logN(O) = 8.93 and logN(C) = 8.60 on the hydrogen log-scale when new accurate forbidden electric quadrupole transition probabilities A _Q(s^–1) are used. Such A _Q's, based on the new atomic structure and electron correlation theory, developed recently by Sinanolu and coworkers, are reported for the (¹ S ₀-¹ D ₂) lines of [C i], [N ii, [O i] and [O iii] and the (² P-² D) lines of [N i] and [O ii]. The available experimental values are also given for comparison.Work supported by Grant No. GP-29471 from the U.S. National Science Foundation.     

Al xii line ratios in the Sun

Recent calculations of electron impact excitation rates in He-like Alxii are used to derive the theoretical electron temperature and density sensitive emission line ratios G ( = (f + i)/r and R ( = f/i, where f, i, and r are the forbidden 1s ^{2 1} S – 1s2s ³ S, intercombination 1s ^{2 1} S – 1s2p ³ P and resonance 1s ^{2 1} S – 1s2p ¹ P transitions, respectively. These ratios are found to be significantly different from earlier calculations, and are in much better agreement with X-ray spectral data for two solar flares obtained with the SMM and P78-1 satellites.     

Note on aTheorem of Relativistic Hydrodynamics

A well known theorem of relativistic hydrodynamics states that the streamlines of an isentropic perfect fluid are the future-pointing timelike (FPT) curves extremizing the integral J = ∫ _S1 ^S2 fds, where f is the so-called index function and s the proper time on the world line of the fluid particle. The integral is taken over all possible FPT curves with regular representations xⁱ = xⁱ (s) joining the fixed end events E₁, E₂. The purpose of this note is to show that the streamlines of an adiabatic perfect fluid can likewise be regarded as extremizing curves of the functional J provided the class of admissible curves consists of those FPT curves satisfying the side condition uⁱ∂_iS = 0, uⁱ unit 4-velocity and S the specific proper entropy of the fluid, with the first end point fixed and the second being the end point variable. __________ Published in Astrofizika, Vol. 48, No. 4, pp. 641–647 (October–December, 2005).     

160-min pulsation of the Sun: New observational results

The 1974–1988 Crimean measurements of the solar line-of-sight velocity continue to show the presence of a statistically significant periodicity P ₁ = 160.009 (±) min with an average harmonic amplitude of about 21 cm s^–1. The period is supposed to be that of the global pulsation of the Sun but with a little-known physical mechanism of excitation.The new observations give some evidence for the existence of a second periodicity, P ₁ = 160.014 (±) min. It is hypothesized that the appearance of P ₁ might be a sidelobe mode (of the P ₀-oscillation) induced by rapid rotation of the central solar core.It is also noted that the spacing, in frequency, between P ₀ and P ₁, corresponds to a beat period of 10 ± 3 yr, which happens to be in good agreement with the average duration of the 11 yr cycle of the magnetic activity of the Sun. Accordingly, we suppose that the phase shift of the P ₀-mode between the 1974–1982 and 1986–1988 time intervals reflects a remarkable change of the general magnetic field of the Sun in the course of the 22 yr solar cycle.     

Determination of the mass-ratio distribution,I: Single-lined spectroscopic binary stars

For single-lined spectroscopic binary stars (SBI), the mass ratioq=M _sec/M _prim is calculated from the mass functionf(m), which is determined from observations. For statistical investigations of the mass-ratio distribution, the term sin³ i, that remains in the cubic equation from whichq is solved, has to be dealt with.This paper compares the common practise of taking an average value for sin³ i to a deconvolution scheme that takes into account the precise (expected) behaviour ofP (sin³ i) d sin³ i. The behaviour ofP(sin³ i) d sin³ i depends on how orbital planes of binary stars are oriented in space. For a random orientation of orbital planes,P _i(i) di=sini di. For the average value method, it is generally assumed thatP _i(i) di=(4/) sin² i di.For verification purposes, the deconvolution scheme is applied to an observed sample of double-lined spectroscopic binary stars (SBII), and to a synthetic sample of SBI systems, produced by a numerical model. In both cases, the scheme produces better results with the assumption thatP _i(i) di=(4/gp) sin² i di, rather than a purely random orientation of orbital planes. In the case of the synthetic sample of SBI systems, the deconvolution scheme does not produce better results than the method that assumes an average value for sin³ i.Application of the deconvolution method to the double-lined spectroscopic binary systems in theEighth Catalogue of the Orbital Elements of Spectroscopic Binary Stars, provides results which are compatible with the assumption that the orbital planes of these systems are oriented randomly in space.     

The Bar and Spiral Structure Legacy (BeSSeL) survey: Mapping the Milky Way with VLBI astrometry

Astrometric Very Long Baseline Interferometry (VLBI) observations of maser sources in the Milky Way are used to map the spiral structure of our galaxy and to determine fundamental parameters such as the rotation velocity (Θ₀) and curve and the distance to the Galactic center (R₀). Here, we present an update on our first results, implementing a recent change in the knowledge about the Solar motion. It seems unavoidable that the IAU recommended values for R₀ and Θ₀ need a substantial revision. In particular the combination of 8.5 kpc and 220 km s^–1 can be ruled out with high confidence. Combining the maser data with the distance to the Galactic center from stellar orbits and the proper motion of Sgr A* gives best values of R₀ = 8.3 ± 0.23 kpc and Θ₀ = 239 or 246±7 km s^–1, for Solar motions of V_⊙ = 12.23 and 5.25 km s^–1, respectively. Finally, we give an outlook to future observations in the Bar and Spiral Structure Legacy (BeSSeL) survey (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

V792 Her=HD 155638: A totally eclipsing RS CVn binary

Photometry of HD 155638=V792 Her has been analyzed to determine the elements of this totally eclipsing RS CVn binary. The light variation outside eclipse was found to have a period of 27^d.07±0^d.07, which is slightly different from the 27^d.5384±0^d.0045 orbital period. Analysis of the eclipses was achieved by a modification of the Russell-Merrill technique. With the aid of radial velocity measures, absolute elements were obtained for the hot and cool stars, respectively;R _h=2.58R ,R _c=12.28R ,M _h=1.40M ,M _c=1.46M ,i=80^o.61 and velocity semi-amplitudesK _c=48.36 km s^–1±0.79 km s^–1, andK _h=50.50 km s^–1±0.33 km s^–1. The apparent magnitudes areV _h=9 ^m .73 andV _c=8 ^m .48. The distance to HD 155638 was estimated to be 310 parsecs.     

Normal form,Lie-Poisson structure and reduction for the Henon-Heiles system

The reduced Henon-Heiles system is investigated as a Hamiltonian dynamical system obtained by applying the normalization of the HamiltonianH=1/2(p ₁ ² +p ₂ ² +q ₁ ² +q ₂ ² )+1/3q ₁ ³ –q ₁ q ₂ ² to fourth-degree terms. The related equations of motion are bi-Hamiltonian and possess the Lie-Poisson structure. Each Lie-Poisson structure possesses an associated Casimir function. When reduced to level sets of these functions, the equations of motion take various symplectic forms. The various reductions give different coordinate representations of the solutions. These coordinate representations are used to seek the simplest representation of the solutions.     

Galactic Sun's motion in the cold dark matter,MOdified Newtonian Dynamics and modified gravity scenarios

We numerically integrate the equations of motion of the Sun in Galactocentric Cartesian rectangular coordinates for –4.5 Gyr ≤ t ≤ 0 in Newtonian mechanics with two different models for the Cold Dark Matter (CDM) halo, in MOdified Newtonian Dynamics (MOND) and in MOdified Gravity (MOG) without resorting to CDM. The initial conditions used come from the latest kinematical determination of the 3D Sun's motion in the Milky Way (MW) by assuming for the rotation speed of the Local Standard of Rest (LSR) the recent value Θ₀ = 268 km s^–1 and the IAU recommended value Θ₀ = 220 km s^–1; the Sun is assumed located at 8.5 kpc from the Galactic Center (GC). For Θ₀ = 268 km s^–1 the birth of the Sun, 4.5 Gyr ago, would have occurred at large Galactocentric distances (12–27 kpc depending on the model used), while for Θ₀ = 220 km s^–1 it would have occurred at about 8.8–9.3 kpc for almost all the models used. The integrated trajectories are far from being circular, especially for Θ₀ = 268 km s^–1, and differ each other with the CDM models yielding the widest spatial extensions for the Sun's orbital path (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Galactic kinematics from OB3 stars with distances determined from interstellar Ca II lines

Based on data for 102 OB3 stars with known proper motions and radial velocities, we have tested the distances derived by Megier et al. from interstellar Ca II spectral lines. The internal reconciliation of the distance scales using the first derivative of the angular velocity of Galactic rotation Ω′₀ and the external reconciliation with Humphreys’s distance scale for OB associations refined by Mel’nik and Dambis show that the initial distances should be reduced by ≈20%. Given this correction, the heliocentric distances of these stars lie within the range 0.6–2.6 kpc. A kinematic analysis of these stars at a fixed Galactocentric distance of the Sun, R ₀ = 8 kpc, has allowed the following parameters to be determined: (1) the solar peculiar velocity components (u _⊙, v _⊙, ω _⊙) = (8.9, 10.3, 6.8) ± (0.6, 1.0, 0.4) km s⁻¹; (2) the Galactic rotation parameters Ω₀ = −31.5 ± 0.9 km s⁻¹ kpc⁻¹, Ω′₀ = +4.49 ± 0.12 km s⁻¹ kpc⁻², Ω″₀ = −1.05 ± 0.38 km s⁻¹ kpc⁻³ (the corresponding Oort constants are A = 17.9 ± 0.5 km s⁻¹ kpc⁻¹, B = −13.6 ± 1.0 km s⁻¹ kpc⁻¹ and the circular rotation velocity of the solar neighborhood is |V ₀| = 252 ± 14 km s⁻¹); (3) the spiral density wave parameters, namely: the perturbation amplitudes for the radial and azimuthal velocity components, respectively, f _R = −12.5±1.1 km s⁻¹ and f _ϑ = 2.0 ± 1.6 km s⁻¹; the pitch angle for the two-armed spiral pattern i = −5.3° ± 0.3°, with the wavelength of the spiral density wave at the solar distance being λ = 2.3 ± 0.2 kpc; the Sun’s phase in the spiral wave x _⊙ = −91° ± 4°.     

The theory of relativity and super-luminal-speeds

According to our Paper I (Cao Shenglin, 1988), it could be supposed that the physical space-time is a Finsler space-time, characterized by the metric ds ⁴=g _ijkf dx ⁱ dx ^j dx ^k dx ^f . If so, a new space-time transformation could be found by invariant ds ⁴ and the theory of relativity is discussed on this transformation.The project supported by National Natural Science Foundation of China.     

Concerning the derivation of the KS-transformation

The Kustaanheimo-Stiefel (KS) transformation is shown to follow naturally from the general solution of the two-body motion if half-arguments are introduced. Application to collision orbits and to the exact triangular solutions of Lagrange (vide E. Stiefel and G. Scheifele: 1971,Linear and Regular Celestial Mechanics, Springer, Berlin-Heidelberg-New York, p. 23–35).Notations x Position vector (x, y, z) - r=|x| Distance from the origin - 1/2h Energy constant or Kepler motion - c Angular momentum vector of Kepler motion - t physical time ()^·=d/dt () - new independent variable ()=d/d () Note by editor: This is the well-known Three-dimensional regularization, published in 1965 by P. Kustaanheimo and E. Stiefel, Perturbation Theory of Kepler Motion Based on Spinor Regularization,J. reine angewandte Mathematik 218, 204. The present article was written during Professor Volk's stay at the Zurich Technische Hochschule in 1972, when he also celebrated his 80th birthday.     

HD 1: The number‐one star in the sky

We present the first ever study of the bright star HD 1. The star was chosen arbitrarily just because of its outstanding Henry Draper number. Surprisingly, almost nothing is known about this bright 7.^m4 star. Our observations were performed as part of the commissioning of the robotic telescope facility STELLA and its fiber‐fed high‐resolution optical echelle spectrograph SES in the years 2007–2010. We found long‐term radial velocity variations with a full amplitude of 9 km s^–1 with an average velocity of –29.8 km s^–1 and suggest the star to be a hitherto unknown single‐lined spectroscopic binary. A preliminary orbit with a period of 6.2 years (2279±69 days) and an eccentricity of 0.50±0.01 is given. Its rms uncertainty is just 73 m s^–1. HD 1 appears to be a G9‐K0 giant of luminosity class IIIa with T_eff = 4850±100 K, logg = 2.0±0.2, L ≈ 155 L_⊙, a mass of 3.0±0.3 M_⊙, a radius of 17.7 R_⊙, and an age of ≈350 Myr. A relative abundance analysis led to a metallicity of [Fe/H] = –0.12 ± 0.09. The α ‐element silicon may indicate an overabundance of +0.13 though. The low strengths of some s‐process lines and a lower limit for the ¹²C/¹³C isotope ratio of ≥16 indicate that HD 1 is on the first ascend of the RGB. The absorption spectral lines appear rotationally broadened with a v sin i of 5.5±1.2 km s^–1 but no chromospheric activity is evident. We also present photometric monitoring BV (RI)_C data taken in parallel with STELLA. The star is likely a small‐amplitude (<10 mmag) photometric variable although no periodicity was found (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Candidate red clump giants in the PPMX catalogue

This paper presents the RCGP catalogue of more than 0.5 million candidate red clump stars with the limiting magnitude K _s = 9.5 ^m . These stars are selected from the PPMX catalogue as the most probable red clump members by analyzing the color-reduced proper motion diagrams built from the proper motions given in PPMX and J, K _s -photometry given in the 2MASS catalogue. Reddening of the selected stars is used to find extinction in the K _s -band and to consider it in the further analysis. The two-dimensional galactic rotation model generalized by Ogorodnikov is used to investigate the tangential velocity field of the selected red clump members, most of which are thin disk stars located within 1.5 kpc from the sun. The values of kinematic parameters and solar components are determined as a function of stellar heights above the galactic equatorial plane and their heliocentric distances.