Trigonometric series representations of the orbital inclination function

In this paper, new trigonometric series representations of the orbital inclination functionF _imp (i) in multiples of cosines or sines will be established for all possible values ofl, m, andp. For such representations, the literal analytical expressions and the recurrence formulae satisfied by their coefficients will be established. Moreover, an economic algorithm for the table formulation of these coefficients for the possible values ofl, m, andp is given. Finally, numerical examples of the representations forl=2(1)4;m=0(1)l;p=0(1)l are also included.     

Spectrum of the galactic magnetic fields

The magnetic fields observed in the galactic disc are generated by the differential rotation and the helical turbulent motions of interstellar gas. On the scalesl=2k ^–1 which lie in the intervall ₀<l<l _e (l ₀100 pc is the energy-range scale of the galactic turbulence), the spectral density of the kinetic energy of turbulence (k ^–5/3) exceeds the magnetic energy spectral density (k ^–1). The equipartition between magnetic and kinetic energies is reached atl=l _e 6 pc in the intercloud medium and is maintained down to the scalel=l _d 0.03 pc. In dense interstellar cloudsl _e is determined by the individual cloud size andl _d 0.1 pc.The internal turbulent velocities in Hi clouds (cloud size is assumed to be 10 pc) lie in the range from 1.8 to 5.6km s^–1, fitting well within the observed range of internal rms velocities. Dissipation of the interstellar MHD turbulence leads to creation of temperature fluctuations with amplitudes of 150 K and 65 K in dense clouds and intercloud medium, respectively. The small-scale fluctuations observed in the interstellar medium may arise from such perturbations due to the thermal instability, for instance. Dissipation of the MHD turbulence energy provides nearly half of the energy supply needed to maintain the thermal balance of the interstellar medium.     

Sensitivity of solar eigenfrequencies to the age of the sun

The increasing central concentration of the Sun with age modifies the acoustic eigenfrequencies. In particular, the frequency separation d _l =3(2l+3)^–1v _n,l –v _n–1,l+2 for modes with l + 1/2 n decreases as nuclear reactions augment the molecular-weight gradient in the energy-generating core. If, for example, the Sun were older than is generally believed, one might therefore expect d _l to be smaller than current theoretical predictions. On the other hand, to ensure that the luminosity is consistent with observations, the presumed initial hydrogen abundance would need to be enhanced, thereby reducing the resultant molecular-weight gradient. Thus there is some degree of cancellation of the two major factors that determine d _l .Various authors have either reported directly on the sensitivity of d _l , or have provided the information from which it can be calculated. We have added our own computations. There is broad agreement amongst the results: d _l diminishes with the presumed age of the Sun at the rate of about 1 Hz per Gy for l = 0; the magnitude of the rate appears to decline with increasing l.     

Observations of the Solar Limb Shape Distortions

Two new campaigns devoted to the observation of the solar limb distortions were made at the Pic-du-Midi Observatory, in September 2000 and September 2001, by means of the scanning heliometer. This apparatus can be used now routinely to accurately determine solar limb profiles (at two wavelengths), at any heliographic latitudes. Each measurement is made within 44 milliseconds (of time) which permits to record a limb profile together with the seeing. Scans are automatically rejected for seeing larger than 1.3 arc sec. Such conditions are essential to perform high-quality observations necessary to obtain the quadrupole term (l=2) in the polynomial expansion of the radius contour R() _{= constant} = R ₀ left(1+_l c _l P _l()right). Exceptional meteorological conditions in September 2001 (seeing of the order of 18 cm, for a 50 cm clear aperture of the refractor) enabled us to determine c ₂ and c ₄ (see Table I) with an accuracy of a few milli-arc-sec. Results indicate a distorted solar shape, the departures from a pure spherical body not exceeding 20 milli-arc-sec. We propose a model to interpret such results (the combination of a nearly uniform rotating core with a prolate solar tachocline and an oblate surface), which is briefly discussed. Our results are confronted to those obtained from space. We conclude that measurements of the quadrupole term from the ground are possible, but of high difficulty and can be obtained only during excellent weather conditions. The hexadecapole term should be only obtained from space. We show that an astrometric satellite would be required, whose mission would be also to accurately determine the solar rotation profiles (both surface and in depth) in order to unambiguously determine the inertia moments of the Sun through the J _n terms. Such values are also briefly discussed.     

The Relationship Between Solar Activity and the Large-Scale Axisymmetric Magnetic Field

To investigate the relationship between solar activity and the large-scale axisymmetric magnetic field of the Sun, we inferred from sunspot data over the period 1964–1985 a latitude–time distribution of magnetic field associated with active regions. This has been done allowing for both bipolar structure of the active regions and inclination of their axes to parallels of latitude, so the inferred magnetic field characterizes latitudinal separation of magnetic polarities which might be related to the large-scale magnetic field of the Sun according to the Babcock–Leighton model. The inferred magnetic field, A _z, is compared with the longitude-averaged (zonal) magnetic field of the Sun, B _z, derived from series of magnetograms obtained at Mount Wilson Observatory in the years 1964–1976, and at Kitt Peak National Observatory during the period from 1976 to 1985. The inferred magnetic field, A _z, exhibits a complex structure distribution of magnetic polarities with respect to latitude and time. Apart from concentration of the different polarity magnetic fields inside the high- and low-latitude portions of the sunspot belts, bipolar active regions produce an intensive, shorter-scale component of the magnetic field which varies on the time scale of about 2 years. Such a short-term variation of A _z reveals substantial correlation with the short-term component of B _z which has the form of the poleward-drifting streams of magnetic field. Most significant correlation takes place between the short-term variations of A _z occurring at latitudes below 20° and those of the large-scale magnetic fields occurring at middle latitudes of 40–50°. Moreover we analyze harmonic coefficients a _l and b _l obtained by expanding A _z and B _z into series in terms of the spherical harmonics. Power spectra of the time-dependent harmonic coefficients indicate that both A _z and B _z reveal a number of resonant modes which oscillate either with the 22-year period in the case of the anti-symmetric (odd-l) modes or with periods of about 2 years in the case of the symmetric (even-l) modes, but the resonant modes of A _z have significantly larger values of the spherical harmonic degree l (and, hence, smaller spatial scales) as compared to those of B _z. It is found that there is a close relationship between the harmonic coefficients b _l and a _m for which either m–l16 (even l=4,...,10) or m–l=4 (odd l=5,...,15).     

Emden-chandrasekhar axisymmetric,solid-body rotating polytropes

According to the general results of a previous work (Caimmi, 1980; hereafter referred to as Paper I), solutions to EC equation, which expresses a necessary and sufficient condition for equilibrium of Emden-Chandresekhar axisymmetric, solid-body rotating polytropes (EC polytropes), are taken into consideration, of the type $$\vartheta (\xi ,\mu ) = A_0 \vartheta _0 (\upsilon ,\xi ) + \sum\limits_l^\infty {_l {\rm A}_{2l} (\upsilon )\vartheta _{2l} (\xi )P_{2l} (\mu ),} $$ with ? _2l later defined as the EC associated function of degree 2l. Thus the EC equation, involving (?, μ), is found to be equivalent to the infinite set of EC associated equations, involving ? _2l (μ). We approximate g (?, μ) by neglecting all terms of degree higher than 2 which appear in the above expression, and then search power series solutions to EC associated equations of degree 0 and 2, corresponding to any choice ofn (polytropic index, related to density distribution) andv (related to rotational distorsion). To this aim, we extend the methods used by Seidov and Kuzakhmedov (1977), and Mohan and Al-Bayaty (1980), to construct power series of the type outlined above, related to solid-body rotating configurations and originating both inside and outside the radial boundary (defined as the first zero of ?₀(μ)=0). The corresponding expressions of ?₀ and ?₂ may serve to derive an approximate expression of, and future work becomes possible concerning the determination of some physical parameters (such as volume, mass, potential energy, angular momentum) related to any choice ofn andv. Computations have been performed forn=k/4 (0≤k≤20, i.e. 0≤n≤5) andv=0,v≈v _R/2,v≈v _R, withv _R lowest value ofv leading to balance between gravitation and centrifugal force at the equator of the system. An upper limit to the error, ε^*(μ), done in computing ? _2l , ?? _2l , and ?? _2l at any point ? for a given choice ofn andv, is estimated, ranging from large values (ε^*=1E-2) forn close enough to 0 and ? close enough or outside the radial boundary, to low values (ε^*=1E-10) forn far enough from 0 and no constraint on ?. Comparison between results of this paper and the accurate results by Linnell (1977, 1981) obtained using a different approach and available forn=2,v=0, andn=3,v=0, lead to a fair agreement (up to (1E?5?1E?6). It is apparent that the method followed here continues to hold when the first EC associated functions up to degree 2l are taken into account, leading — at least in way of principle — to a more refined approximation to the EC function; this would only make the related calculations much more complicated.     

A study of the variability of the Delta Scuti-stars

A total of 321 observations of the Delta Scuti star BD –6°4932, obtained in 1968 by Hall and Mallama (1970), are analyzed. We find four frequencies which represent the light curves satisfactorily.The three periods:P ₁=0 _. ^d 240,P ₃=0 _. ^d 182 andP ₄=0 _. ^d 114 seem to correspond to the radial modes of pulsation withK=0, 1, and 3, respectively. The last periodP ₂=0 _. ^d 220 can be related to a non-radial mode.     

Spectral investigation of some Be stars for NRP variability

Spectroscopic observations of the Be stars Eri, Oph, 66 Oph, and Ori for the period 1982–1988 are reported. The NRP hypothesis was verified on the ground of rapid line profile variability, radial velocities, and equivalent widths. The star Eri is pulsating in bothl=2 andl=8 with period 0 _. ^d 7. Pulsation in modesl=2 andl=4 are observed in Hei profiles of Oph for May 1982. For radial velocities has been obtained a period 0 _. ^d 913. The H and H lines of 66 Oph for April–August 1983 are in emission state with two clearly expressed components with intensity variations. All the parameters measured have the same period of variation — 0 _. ^d 025. For Ori variations in line profiles for component Ab have been observed and a period of 0 _. ^d 463 found for the radial velocities.     

THE MOST VIOLENT SUPER-ACTIVE REGIONS IN THE 22nd AND 23rd CYCLES

We survey 14 super-active regions (SARs) in the 22nd cycle and 15 SARs in the 23rd cycle. Each produced major flares and major solar storms. Among them, the 25 most violent super active regions (VSARs) are selected based on five parameters: the largest area of sunspots, X-ray flare index (XRI), 10.7 cm radio flux, proton flux and geomagnetic A _p index. In order to understand the VSARs, we have investigated a few key magnetic properties of those regions, i.e., net magnetic flux, tilt angle and force-free parameter _best. The following results are found: (1) Most VSARs (84%) in our samples have net magnetic flux greater than 10²¹ Mx, implying that those are seriously unbalanced flux regions. Unbalanced flux active regions probably provide a nest to relate the small-scale to the large-scale magnetic field. (2) Most of the VSARs (68%) are of abnormal magnetic structure, violating the Hale–Nicholson Law. For most of the normal VSARs, the tilt angles are larger than 40°. 84% of the VSARs follow the hemispheric helicity rule. Generally, they have large magnetic twist and writhe helicity. (3) We also enlarge our samples to study the locations of VSARs by adding the top 10 of the major flares, proton events and severe magnetic storms from 1976 to 2001. It is found that 77% in our 30 samples of VSARs were preferentially located in 4 longitude bands, i.e., l _c=80°±15° l _c=170°±15° l _c=260°±15° and l _c=350°±15°. The interval of those longitude bands is roughly 90°. From the above results, we suggest that there probably is a special magnetic environment in the sub-photosphere of the four longitude bands where it is preferred to produce abnormal and complex active regions which easily produce major flares and major solar storms. Area, magnetic class, net magnetic flux, Carrington longitude and tilt angle of an active region may serve to predict likelihood of the active region producing hazarded space weather.     

The photometric solution of cc com by use of the

The contact binary system CC Com (=12^h09^m33^s.8, =+22°4339, (1950);V _max=11.31, (B-V)_max=1.24) is a W UMa-type system with the shortest known period. The photometric solution of CC Com is presented using the Wilson and Devinney method. The results show that the CC Com belongs to the late-type eclipsing binary with the spectral type K5V and K6V, low temperatureT ₁=4300 K,T ₂=4265 K, the mass ratioq=0.5873±0.0021, and the inclinationi=87°.719±1°.44. The best regions of the gravity darkening exponents , the bolomotric albedov, and the limb-darkening coefficients are tested. It is found that 0.1250.065, 0.1v0.5, =0.5 are better regions for CC Com. The third body ofl ₃ is not found to be significant. The results are combined with the spectroscopic results of Rucinski to provide an estimate of the absolute parameters.     

Singularitäten in der Allgemeinen Relativitätstheorie

“Regular solutions of EINSTEIN 's equations” mean very different things. In the case of the empty-space equations, R_ik = 0, such solutions must be metrics g_ik(x^l) without additionaly singular “field sources” (EINSTEIN 's “Particle problem”). – However the “phenomenological matter” is defined by the EINSTEIN equations R_ik – 1/2g_ikR =–xT_ik itselves. Therefore if 10 regular functions g_ik(x^l) are given (which the inequalities of LORENTZ -signature fulfil) then these g_ik define 10 functions T_ik(x^l) without singularities. But, the matter-tensor T_ik must fulfil the two inequalities T ≥ 0, T ≥ 1/2 T only and therefore the EINSTEIN -equations with “phenomenological matter” mean the two inequalities R ≥ 0, R ≤ 0 which are incompatible with a permanently regular metric with LORENTZ -signature, generally.     

Effect of fluctuations in Doppler width on the center of strong absorption lines

Stochastic temperatures and turbulence are characterized by average velocities u _th and < u _turb  > ≡ u ₀ and fluctuations u￠_th {u'_{th}} and u′ (<u′ > = 0). Thus, the Doppler width of a line also has a fluctuating component Dl￠_D \Delta {\lambda '_D} . Observed spectra correspond to the radiative flux averaged over time and over a star’s surface, <H_λ>. Usually, only the average velocities u _th and u ₀ are taken into account in photospheric models and these yield the Doppler width Dl_D⁽⁰⁾ \Delta \lambda_D^{(0)} of a line in the customary way. The fluctuations Dl￠_D \Delta {\lambda '_D} mean that near a line center the average absorption coefficient < α_λ > is larger than the usual α_λ, which depends only on the average velocities u _th and u ₀. This enhances the absorption line near the center and is not explained by the photospheric models. This new statistical effect depends on the wavelength of the line. A comparison of observed lines with model profiles yields an estimate for the average level of fluctuations in the Doppler width, h = á | Dl￠_D | ñ

Global modes constituting the solar magnetic cycle

The sunspot occurrence probability defined in Paper I is used to determine the Legendre-Fourier (LF) terms in the rate of emergence of toroidal magnetic flux,Q(, t), above the photosphere per unit latitude interval, per unit time. Assuming that the magnetic flux tubes whose emergence yields solar activity are produced by interference of global MHD waves in the Sun, we determine how the amplitudes and phases of the LF terms in the toroidal magnetic fieldB , representing the waves, will be related to those of the LF terms inQ(, t). The set of LF terms in Q that represents the set of waves whose interference produces most of the observed sunspot activity is {l = 1, 3, , 13;v =nv _*,n = 1, 3, 5}, wherev _* = 1/21.4 yr^–1. However, among the shapes of sunspot cycles modeled using various sets of the computed LF terms the best agreement with the observed shape, for each cycle, is given by the set {l = 3 orl = 3, 5; andn = 1, 3 orn = 1, 3, 5}. The sets of terms: {l = 1, 3, 5, 7;n = 1}, {l = 1, 3, 5, 7;n = 3}, {l = 9, 11, 13, 15;n = 1} and {l = 9, 11, 13, 15;n = 3} seem to represent four modes of global MHD oscillation. Correlations between the amplitudes (and phases) of LF terms in different modes suggest possible existence of cascade of energy from constituent MHD waves of lowerl andn to those of higherl andn. The spectrum of the MHD waves trapped in the Sun may be maintained by the combined effect of this energy cascade and the loss of energy in the form of the emerging flux tubes. The primary energy input into the spectrum may be occurring in the mode {l = 1, 3, 5, 7;n = 1). As expected from the above phenomenological model, the size of a sunspot cycle and its excess over the previous cycle are well correlated (e.g., 90%) to the phase-changes of the two most dominant oscillation modes during the previous one or two cycles. These correlations may provide a physical basis to forecast the cycle sizes.     

Allan Sandage and the cosmic expansion

This is an account of Allan Sandage’s work on (1) The character of the expansion field. For many years he has been the strongest defender of an expanding Universe. He later explained the CMB dipole by a local velocity of 220±50 km s⁻¹ toward the Virgo cluster and by a bulk motion of the Local supercluster (extending out to ∼3500 km s⁻¹) of 450–500 km s⁻¹ toward an apex at l=275, b=12. Allowing for these streaming velocities he found linear expansion to hold down to local scales (∼300 km s⁻¹). (2) The calibration of the Hubble constant. Probing different methods he finally adopted—from Cepheid-calibrated SNe Ia and from independent RR Lyr-calibrated TRGBs—H ₀=62.3±1.3±5.0 km s⁻¹ Mpc⁻¹.     

Numerical determination of families of three-dimensional double-symmetric periodic orbits in the restricted three-body problem. I.

New families of three-dimensional double-symmetric periodic orbits are determined numerically in the Sun-Jupiter case of the restricted three-body problem. These families bifurcate from the vertical-critical orbits ( _v = –1,c _v ),c _v=0) of the basic plane familiesi,g ₁,g ₂,h,a,m andl. Further the numerical procedure employed in the determination of these families has been described and interesting results have been pointed out. Also, computer plots of the orbits of these families have been shown in conical projections.     

Dynamical instability of the envelope of red supergiants and the lower mass limit for carbon detonation supernovae

We investigated the lower mass limitM _l for the carbon detonation supernovae by testing the dynamical instability of the envelopes of red supergiants. It was found that the dependence ofM _l on the mixing lengthl of convection is appreciable. As a smaller value ofl is assumed,M _l becomes larger. It may be as large as 8M ifl is a third of the pressure scale-height. This is one of the ways to remove the difficulty of overproduction of iron-peak elements involved in the model of the carbon detonation supernovae.     

Earth Tides and Lense–Thirring Effect

The general relativistic Lense—Thirring effect can be measured by inspecting a suitable combination of the orbital residuals of the nodes of LAGEOS and LAGEOS II and the perigee of LAGEOS II. The solid and ocean Earth tides affect the recovery of the parameter by means of which the gravitomagnetic signal is accounted for in the combined residuals. Thus an extensive analysis of the perturbations induced on these orbital elements by the solid and ocean Earth tides is carried out. It involves the l=2 terms for the solid tides and the l=2,3,4 terms for the ocean tides. The perigee of LAGEOS II turns out to be very sensitive to the l=3 part of the ocean tidal spectrum, contrary to the nodes of LAGEOS and LAGEOS II. The uncertainty in the solid tidal perturbations, mainly due to the Love number k ₂, ranges from 0.4% to 1.5%, while the ocean tides are uncertain at 5–15% level. The obtained results are used in order to check in a preliminary way which tidal constituents the Lense-Thirring shift is sensitive to. In particular it is tested if the semisecular 18.6-year zonal tide really does not affect the combined residuals. It turns out that, if modeled at the level of accuracy worked out in the paper, the l=2,4 m=0 and also, to a lesser extent, the l=3, m=0 tidal perturbations cancel out.This revised version was published online in October 2005 with corrections to the Cover Date.     

An integrable case of a rotational motion analogous to that of Lagrange and Poisson for a gyrostat in a Newtonian force field

The scope of the present paper is to provide analytic solutions to the problem of the attitude evolution of a symmetric gyrostat about a fixed point in a central Newtonian force field when the potential function isV ⁽²⁾.We assume that the center of mass and the gyrostatic moment are on the axis of symmetry and that the initial conditions are the following: (t ₀)=₀, (t ₀)=₀, (t ₀)=(t ₀)=₀, ₁(t ₀)=0, ₂(t ₀)=0 and ₃(t ₀)= ₃ ⁰ .The problem is integrated when the third component of the total angular momentum is different from zero (B ₁ 0). There now appear equilibrium solutions that did not exist in the caseB ₁=0, which can be determined in function of the value ofl ₃ ^r (the third component of the gyrostatic momentum).The possible types of solutions (elliptic, trigonometric, stationary) depend upon the nature of the roots of the functiong(u). The solutions for Euler angles are given in terms of functions of the timet. If we cancel the third component of the gyrostatic momentum (l ₃ ^r =0), the obtained solutions are valid for rigid bodies.     

NSV 01450: a New Bright Classical Cepheid in the Milky Way Field

Photometric data extracted from more than 5000 CCD observations of theMilky Way field star NSV 01450, with galactic co-ordinates l =145^°.9and b=4^° .9, show that this object is a bright classical Cepheid witha period of 12.639 ± 0.020 days and an average V magnitude of 11.045± 0.016. The computed colour excess is E(B-V)= 0.553± 0.056, yielding a distance modulus m _V -M_V of 15.537 ± 0.212.In addition, it was found that the star GSC 3730_0797 in the field ofNSV 01450 is also variable.     

CO observations,geometry, and galactic structure

2.6 mm carbon monoxide and 21 cm hydrogen line observations of the Galaxy define six directions where gas is seen tangentially along a spiral feature and four directions where extended structures cross theR=R ₀ circle. These directions provide a geometrical framework for any spiral structure pattern. Also simple equations are derived to define the normalized radiusR/R ₀ of gas with a particular radial velocity in terms of the observed tangential velocity of gas with the sameR/R ₀=sinl _T, without specific knowledge of (R), ₀ orR ₀.Paper presented at the IAU Third Asian-Pacific Regional Meeting, held in Kyoto, Japan, between 30 September–6 October, 1984.