The N-gon is not a local minimum of U 2 I for N > 7

A theorem of Palmore's concerning coplanar central configurations of equal mass bodies was shown to be false for all even N 6 by Slaminka and Woerner. Using a variation of that argument I prove that Palmore's Theorem is false for all N 6.Northwestern University     

A note on the correspondence between Mathieu's transformations and redundant variables in lagrangian mechanics

It is shown that, to any change of variables:q _i=q_i(r, t) (i=1,..., n; =1,...,n+m; mn) increasing the number of variables, it is possible to associate a Mathieu's transformation and conversely. The results are applied to the theory of the osculating plane of motion.

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Resumé On montre qu'à toute transformation:q _i=q_i(r, t)(i=1,..., n; =1,...,n+m; mn) augmentant le nombre de variables, on peut associer une transformation de Mathieu et réciproquement. Les résultats sont appliqués à la théorie du plan osculateur du mouvement.

     

Kelvin-Helmholtz instability of composite plasma in an oblique magnetic field with resistivity

Analytic structure of high-density steady isothermal spheres is discussed using the TOV equation of hydrostatic equilibrium which satisfies an equation of state of the kind:P = K _g , = _g c ².Approximate analytical solutions to the Tolman-Oppenheimer-Volkoff (TOV) equations of hydrostatic equilibrium in (, ), (,U) and (u, v) phase planes in concise and simple form useful for short computer programmes or on small calculator, have been given. In Figures 1, 2, and 3, respectively, we display the qualitative behaviours of the ratio of gas density _g to the central density _gc , _g / _gc ; pressureP to the _gc ,P/ _gc ; and the metric componente ^–, for three representative general relativistic (GR) isothermal configurations =0.1, 0.2, and 0.3. Figure 4 shows the solution curve (, ) for =0.1, 0.2, and 0.3 (=0 represents the classical (Newtonian) curve). Numerical values of physical quantitiesv (=4r ² P ^*(r)), in steps ofu (=M(r)/r)=0.03, and the mass functionU, in steps of =0.2 (dimensionless radial distance), are given, respectively, in Tables I and II. Other interesting features of the configurations, such as ratio of gravitational radius 2GM/c ² to the coordinate radiusR, mass distributionM(r)/M, pressure (or density) distributionP/P _c , binding energy (B.E.), etc., have also been incorporated in the text. It has further been shown that velocity of sound inside the configurations is always less than the velocity of light.Part of the work done at Azerbaijan State University, Baku, U.S.S.R., and Mosul University, Mosul, Iraq, 1985-1986     

Expanding and superdense astrophysical systems in general relativistic hierarchical cosmology

A semi-continuous hierarchy, (i.e., one in which there are galaxies outside clusters, clusters outside superclusters etc.), is examined using an expression of the field equations of general relativity in a form due to Podurets, Misner and Sharp. It is shown (a) that for a sufficiently populous hierarchy, the thinning factor( _i+1/ _i [r _i /r _i+1] is approximately equal to the exponentN in a continuous density law (=aR ^–N) provided (r _i /r _i+1)^3-1; (b) that a hierarchical Universe will not look decidedly asymmetric to an observer like a human being because such salient observers live close to the densest elements of the hierarchy (viz stars), the probability of the Universe looking spherically symmetric (dipole anisotropy0.1 to such an observer being of order unity; (c) the existence of a semi-continuous or continuous hierarchy (Peebles) requires that 2 if galaxies, not presently bound to clusters were once members of such systems; (d) there are now in existence no less than ten arguments for believing 2, though recent number counts by Sandageet al. seem to be in contradiction to such a value; (e) Hubble's law, withH independent of distance, can be proved approximately in a relativistic hierarchy provided (i)N=2, (ii)2GM(R)/c ² R1; (iii)Rc (iv)M0 in a system of massM, sizeR (f) Hubble's law holds also in a hierarchy with density jumps; (g)H100 km s^–1 Mpc^–1; (h) objects forming the stellar level of the hierarchy (in a cosmology of the Wilson type) must once have had 2GM/c ² R1; (i) there is a finite pressurep=2Ga in all astrophysical systems (a=R ^N ,N2); (j) for the Galaxy, theory predictsp _G7×10^–12 dyn cm^–2, observation givesp _G5×10^–12 dyn cm^–2; (k) if the mass-defect (or excess binding energy) hypothesis is taken as a postulate, all non-collapsed astrophysical systems must be non-static, and any non-static, p0 systems must in any case be losing mass; (1) the predicted mass-loss rate from the Sun is 10¹² g s^–1, compared to 10¹¹ g s^–1 in the observed solar wind; (m) the mass-loss rates known by observation imply timescales of 5×10⁹ years for the Sun and 10¹⁰ years for other astrophysical systems; (n) degenerate superdense objects composed of fermions must haveN-2 if they were ever at their Schwarzschild radii and comprised a finite numberN _B of baryons; (o)N _B10⁵⁷N for degenerate fermion and boson systems; (p)285-4; (q) the metric coefficients for superdense bodies give equations of motion that imply equal maximum luminosities for all evolving superdense bodies (L _max10⁵⁹ erg s^–1); (r) larger bodies have longer time-scales of energy radiation atL _max (10^–5 s for stars,1 h for QSO's) (s) expansion velocities are c soon after the initial loss of equilibrium in a superdense object; (t) if the density parametera(t) in aR ^–N isa=a (non-atomic constants of physicsc, G, A), andA, thenN=2; (u) N2 is necessary to giveMM at the stellar level of the hierarchy;(v) systems larger than, and including, galaxies must have formed by clumping of smaller systems and not (as advocated by Wertz and others) in a multiple big bang.     

The interpretation of absorption-line shifts in the solar spectrum

The shifts of Fraunhofer lines of different chemical elements in a homogeneous medium with a plane monochromatic progressive adiabatic sound waves are derived. The calculations indicate that lines of neutral elements (6 ₀ 14) with lower excitation potentials ₀ i= 0–2 eV are red shifted, those with excitation potential ₀ i= 4–12 eV are blue shifted, and with ₀ i= 3 eV are both blue and red shifted. The lines of ions are shifted toward the blue. The shifts of Fraunhofer lines are found to decrease from the centre of the solar disk to the limb. These results agree qualitatively and quantitatively with observations.     

Сопоставление модели карлсона с моделью сплошной среды

, (1962, 1963) . , (1) (2) .     

ОJ 287: Поляризационное и фотометрическое поведение В 1971–1976 гг

J 287 1972–1976. , . . J ( ), () . (₀=80°) . J 287 ¶ . p=42.8%, ₀=101°. (50%).     

Радиоизлучение Быстрых космических ионов

, . () . , , , . ( ), , , . . (2.7). ( ₁ k ₁ ,V — , — .) (k ₁) (k) §2 ( (2.14)). , (3.6) (3.4), (3.8) . (3.9)–(3.13) ( (3.9), (3.10) (3.11) , (3.12)–(3.13) ). (3.14), (3.16)–(3.19). - . (3.15). ( (4.14)–(4.15)). (4.23)–(4.25). (4.26)–(4.28). §5. , . ((5.5)–(5.6)). , . (5.10) .     

Возможный механизм радиоизлучения пульсаров

, , , , , . , , . - . — , , .     

The geometry of the Roche coordinates

The exact geometry of the Roche curvilinear coordinates (, , ) in which corresponds to the zero-velocity surfaces is investigated numerically in the plane, as well as in the spatial, case for various values of the mass-ratio between the two point-masses (m ₁,m ₂) constituting a binary system.The geometry of zero-velocity surfaces specified by -values at the Lagrangian points are first discussed by taking their intersections with various planes parallel to thexy-, xz- andyz-planes. The intersection of the zero-velocity surface specified by the -value at the Lagrangian equilateral-triangle pointsL _4,5 with the planex=1/2 discloses two invariable curves passing through the pointsL _4,5 and situated symmetrically with respect to thexy-plane whose form is independent of the mass-ratio.The geometry of the remaining two coordinates (, ) orthogonal to the zero-velocity surfaces is investigated in thexy- andxz-planes from extensive numerical integrations of differential equations generated from the orthogonality relations among the coordinates. The curves (x, y)=constant in thexy-plane are found to be separated into three families by definite envelopes acting as boundaries whose forms depend upon the mass-ratio only: the inner -constant curves associated with the masspointm ₁, the inner -constant curves associated with the mass-pointm ₂ and the outer -constant curves. All the -constant curves in thexy-plane coalesce at either of the Lagrangian equilateraltriangle pointsL _4,5, except for a limiting case coincident with thex-axis. The curves (x, z)=constant in thexz-plane are also separated by definite envelopes depending upon the mass-ratio into different families: the inner -constant curves associated with the mass-pointm ₁, the inner -constant curves associated with the mass-pointm ₂ and the outer -constant curves on both sides out of the envelopes. For larger values ofz, the curves =constant tend asymptotically to the line perpendicular to thex-axis and passing through the centre of mass of the system, except for a limiting case coincident with thex-axis. The geometrical aspects of the envelopes for the curves (x, y)=constant in thexy-plane and the curves (x, z)=constant in thexz-plane are also discussed independently.In the three-dimensional space, the Roche coordinates can be conveniently defined in such a way as to correspond to the polar coordinates in the immediate neighbourhood of the origin, and to the cylindrical coordinates at great distances. From numerical integrations of simultaneous differential equations generating spatial curves orthogonal to the zero-velocity surfaces, the surfaces (x, y, z)=constant and the surfaces (x, y, z)=constant are constructed as groups of such spatial curves with common values of some parameters specifying the respective surfaces.On leave of absence from the University of Tokyo as an Honorary Fellow of the Victoria University of Manchester.     

The center-limb behavior of solar molecular lines

The center-limb behavior of C₂, CH, CN, CO, and MgH lines have been analyzed using five photospheric models. A three-stream model developed by G. Elste gave the most satisfactory results, providing evidence for the existence of inhomogeneities in the photospheric layers - 2.5 log 5000 -0.5.     

Об устойчивости лагранжевых решений пространственной эллиптической задачи трех тел

. . ,e, , . . e, . , .

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Stability of the librational triangular points of the three-dimensional elliptic restricted three-body problem is studied. The problem is solved in the non-linear statement at the small values of eccentricity.For all values ofe, , besides ones which correspond to the resonances of the third and the fourth order the librational points are stable taking into account the terms up to the fourth order in the normal form of the Hamiltonian function of the perturbed motion.At sufficiently smalle and the non-stability in sense of Liapunov has been proved. The approximate equations of the boundary of the stability area in the planee, has been obtained. The cause of the non-stability is an equality of the rotational period of the principal attracting masses in the elliptic orbit and the period of oscillation of indefinitely small mass along the direction perpendicular to the plane of their motion.

     

Three-body problem

In the first part of this paper [Marchal, Yoshida, Sun Yi-Sui 1985] we have analyzed three-body systems satisfying the condition rkR where k is a suitable constant, r the mutual distance of the two masses of the binary and R the distance between the center of mass of the binary and the third mass.That condition rkR puts limits on the acceleration of the third mass and these limits allow us to determine the corresponding escape velocities.In this second part we look for initial conditions under which the inequality rkR will remain forever satisfied and we develop the corresponding tests of escape and their applications.This leads to a major improvement of the knowledge of the nature of three-body motions especially in the vicinity of triple close approaches.The region of bounded motions is much smaller than was generally expected and numerical computations of particular solutions show that we approach very near to the true limit.     

Some applications of Jacobi dynamics to the stellar evolution

The stars in the Main Sequence are seen as a hierarchy of objects with different massesM and effective dynamical radiiR _eff=R/ given by the stellar radii and the coefficients for the inner structure of the stars.As seen in a previous work (Paper I), during the lifetime in the Main SequenceR _eff(t) remains a near invariant when compared to the variation in the time ofR(t) and (t).With such an effectiveR _eff one obtains the amounts of actionA _c(M), the effective densities _eff(M)=(M)³(M), the densities of action and of energy (or mean presures in the stellar interior)a _c(M),e _c(M), and the potential energiesE _p(M).The amounts of action areA _cM ^k withk1.87 for the M stars,k5/3 for the KGF stars, andk1.83 for the A and earlier stars, representing very simples conditions for the other dynamical parameters. For instancek5/3 means a near invariant effective density _eff for the KGF stars, while for such stars the mean densities and coefficients present the strongest variations with masses (M)M ^–1.81, (M)M^0.6.The cases for the M stars (e _c(M)M ^–1) and for the A and earlier stars (betweena _c(M)=constant and _eff(M)M ^–1) and also discussed. These conditions for the earlier stars also represent reasonable mean values for the whole stellar hierarchy in the range of masses 0.2M M25M .With all this, one can build dynamical HR diagrams withA _c(M), E_p(M), _eff M ^–p , etc., whose characteristics are analogous to these in the photometrical HR diagram. A comparison is made betweenA _c(M) from the models here and the HR diagram with the best known stars of luminosity classes IV, V, and white dwarfs.The comparison of the potential energiesE _p(M)M ^–p according to the stellar models used here and the observed frequency function (MM _–q (number of stars in a given interval of masses) from different authors suggests the possibility that the productE _p(M)(M) is a constant, but this must be confirmed with further studies of the function (M) and its fine structure.There are analogies between the formulation used here for the stellar hierarchy and other physical processes, for instance, in modified forms of the Kolmogorov law of turbulence and in the formulation used for the hierarchy of molecular clouds in gravitational equilibrium. Besides, the function of actionA _c(M) for the stars has analogous properties to the relations of angular momenta and massesJ(M) for different types of objects. The cosmological implications of all this are discussed.     

Statistical survey of planetary nebulae: Distances,masses, and distribution in the galaxy

On the basis of empirical (D)-dependency at the frequency of 5 GHz constructed using 15 planetary nebulae with the independently measured distances (10^–171×10^–20 W m^–2 Hz^–1 ster^–1), we evaluated distances of 335 objects. Independent evidence of the correctness of the accepted scale are given. Then(D)-dependency is constructed and it is shown that atD<0.08 pc the mean electron density is higher than the one determined by the Seaton method. We showed that the filling factor diminishes with the increase of the PN diameter (1 atD0.08 pc and 0.2 atD0.4 pc). the ionized mass of 33 PNs is determined. With the diameter increase the ionized mass grows and atD0.4 pc reaches the valueM0.07M . We used the new distance scale when investigating the space distribution of PNs. The mean scale height =130±15 pc and the mean gradient of the change of surface densitym=0.37, which allowed us to estimate the total number of nebulae in the GalaxyN4×10⁴. We divided the PNs according to their velocities (withV _LSR>35 km s^–1 andV _LSR<35 km s^–1) and permitted us to confirm that the PN belong to different sub-systems of the Galaxy. The estimated local formation rate of PNs [=(4.6±2.2)×10^–12 pc^–3 yr^–1] is a little higher than the one of the white dwarfs. That can be explained by a large number of PNs having binary cores, which used in our sample. The statistical estimation of PN expansion velocity showed thatV _ex increases from 5–7 km s^–1 (atD0.03 pc) to 40–50 km s^–1 (atD0.8 pc).     

On a more precise calculation of the electric conductivity in the photosphere and in sunspots

On the basis of a review of data on the effective cross-sections of particle interactions, computations of the effective electron-neutron interaction, of the cross-section values for H, H₂ and He were carried out in the range 0.8 1.7 and 2.5 logP _g 6.5. The mean cross-sections for the solar mixture used by Zwaan were computed (see Table II). Using these results, the values of the electric conductivity and of its anisotropy coefficient were computed (see Tables VII and VIII), as well as other auxiliary quantities. The conductivity values obtained are in the range from 5 × 10⁹ to 5 × 10¹² CGSE units for the assumed physical conditions.On leave from the Astronomical Institute of the Czechoslovak Academy of Sciences, Ondejov, Czechoslovakia.     

A study of massive classical polytropes in general relativity

A detailed study of classical polytropes in general relativity has been presented for _O ((dP/dE)_O) 1.0 and _O((P/E _O)_O. The behaviour of various structural parameters with _O/_O, _O and _O are the values ofP/E and dP/dE at the centre) has been studied. The most important result of this study is the fact the qualitative behaviour of all the structural parameters depends only on the value of µ_O for the various assigned _O values. The maximum value of surface red shift occurs when µ_O=0.6 and for _O=1.0 it equals 0.618. These structures are gravitationally bound for µ_O0.8 and most so for µ_O=0.4. The maximum value of binding coefficient comes out to be 0.181 when _O=1.0. These structures have been used to model neutron stars. The maximum mass of neutron star based upon such a model comes out to be 2.55M (for µ_O=0.4 and _O=1.0) and maximum size comes out to be 15.0 km (for µ_O=0.2 and _O=1.0). It is also seen that the structures are pulsationally stable for µ0.6.     

Hollow beam distribution of energetic electrons and higher harmonics of electron cyclotron maser

The variations of the growth rates of ECM at first four harmonics in X-, Z-, and O-modes excited by a hollow beam distribution of weakly relativistic electrons with a parameter _p / _e are presented in this paper. The results show that the second harmonic of ECM in X-mode dominates the instability if < 1, and if 1.2 , 2 or 2.2 3 the third or fourth harmonic will dominate. The second and third harmonics of Z-mode waves grow faster only if 2 2.2 and 3 3.2, respectively, so it would not be a competition in most cases. It is suggested that the ECM emission at these harmonics in X-mode is a possible mechanism to produce solar spike emissions with high brightness temperature at shorter and longer decimetric wavelengths.Proceedings of the Workshop on Radio Continua during Solar Flares, held at Duino (Trieste), Italy, 27–31 May, 1985.On leave from the Department of Astronomy, Nanjing University, Nanjing, The People's Republic of China.     

Characteristics of electron and high-energy proton flares

High-energy proton (E _p > 55 MeV) and electron (E _e > 50 keV) events were observed by University of Iowa experiments on the satellites Explorer 33 and 35. The solar X-ray (2–12 Å) flares associated with the energetic proton events were found to have in general higher peak fluxes, considerably longer decay times (t) and smaller rise to decay time ratios (r) than the X-ray flares associated with the electron events. The most common decay times and rise to decay time ratios are: 80 t 100 min, 0.1 r 0.2 for the proton X-ray flares and t 20 min, 0.3 r 0.7 for the electron ones.     

A role of cosmic rays in generation of radio and optical radiation by plasma mechanisms

The radiation of ultrarelativistic particles is examined in a quasi-uniform magnetic field superimposed by a wide spectrum of magnetic, electric, and electron density inhomogeneities created in a turbulent plasma. The radiation spectrum from a particle of a given energy is shown to acquire a high-frequency power-law tail with the same spectral index as the index of small-scale turbulence. For a power-law spectrum of ultrarelativistic electrons, dN()/d ~ ^–, with a cut-off at some energy _max, the radiation spectrum consists of a few power-law ranges; the radiation intensity may suffer jumps at frequencies which separate these ranges.In the high-frequency range the spectral index is determined by small-scale magnetic and electric fields. At intermediate frequencies the main contribution comes from the synchrotron radiation in a large-scale field; the radiation spectrum has an index =(–1)/2. The same index may be produced by large-scale Langmuir waves. At lower frequencies the radiation spectrum increases owing to the transition radiation caused by electron density fluctuations; in this case the spectral index is equal to +1–.The possibility of diagnostics of high-frequency cosmic plasma turbulence from radiation of high-energy particles is discussed. It is shown that the proposed theory may explain some features in the spectra of several cosmic objects.