Cosmic-ray antimatter: A primary origin hypothesis

We examine the possibility that the observed cosmic-ray protons are of primary extragalactic origin. The present \(\bar p\) data are consistent with a primary extragalactic component having \(\bar p\) /p?3.2±0.7 x 10^-4 independent of energy. Following the suggestion that most extragalactic cosmic rays are from active galaxies, we propose that most of the observed \(\bar p\) 's are alos from the same sites. This would imply the possibility of destroying the corresponding \(\bar \alpha \) 'sat the source, thus leading to a flux ratio \(\bar \alpha \) /α< \(\bar p\) /p. We further predict an estimate for \(\bar \alpha \) α～10^-5, within the range of future cosmic-ray detectors. the cosmological implications of this proposal are discussed.     

Symmetric and regularized coordinates on the plane triple collision manifold

The planar problem of three bodies is described by means of Murnaghan's symmetric variables (the sidesa _j of the triangle and an ignorable angle), which directly allow for the elimination of the nodes. Then Lemaitre's regularized variables \(\alpha _j = \sqrt {(\alpha ^2 - \alpha _j )}\) , where \(\alpha ^2 = \tfrac{1}{2}(a_1 + a_2 + a_3 )\) , as well as their canonically conjugated momenta are introduced. By finally applying McGehee's scaling transformation \(\alpha _j = r^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}} \tilde \alpha _j\) , wherer ² is the moment of inertia a system of 7 differential equations (with 2 first integrals) for the 5-dimensional triple collision manifold \(T\) is obtained. Moreover, the zero angular momentum solutions form a 4-dimensional invariant submanifold \(N \subset T\) represented by 6 differential equations with polynomial right-hand sides. The manifold \(N\) is of the topological typeS ²×S ² with 12 points removed, and it contains all 5 restpoint (each one in 8 copies). The flow on \(T\) is gradient-like with a Lyapounov function stationary in the 40 restpoints. These variables are well suited for numerical studies of planar triple collision.     

Asymmetries in stokes profiles of magnetic lines: A linear analysis in terms of velocity gradients

A linear analysis of the asymmetries in Stokes profiles of magnetic lines is performed. The asymmetries in the linear and circular polarization profiles are characterized by suitable quantities, \(\delta \tilde Q\) and \(\delta \tilde V\) , strictly related to observed profiles. The response functions of \(\delta \tilde Q\) and \(\delta \tilde V\) to velocity fields are introduced and computed for various configurations of the magnetic field vector in a Milne-Eddington atmosphere. Some conclusions are drawn as to the importance of the asymmetries in Stokes profiles for recovering the velocity gradients from observations.     

Sur un modele explicitant les differents types morphologiques des galaxies

In the now classical Lindblad-Lin density-wave theory, the linearization of the collisionless Boltzmann equation is made by assuming the potential functionU expressed in the formU=U ₀ + \(\tilde U\) +... WhereU ₀ is the background axisymmetric potential and \(\tilde U<< U_0 \) . Then the corresponding density distribution is \(\rho = \rho _0 + \tilde \rho (\tilde \rho<< \rho _0 )\) and the linearized equation connecting \(\tilde U\) and the component \(\tilde f\) of the distribution function is given by $$\frac{{\partial \tilde f}}{{\partial t}} + \upsilon \frac{{\partial \tilde f}}{{\partial x}} - \frac{{\partial U_0 }}{{\partial x}} \cdot \frac{{\partial \tilde f}}{{\partial \upsilon }} = \frac{{\partial \tilde U}}{{\partial x}}\frac{{\partial f_0 }}{{\partial \upsilon }}.$$ One looks for spiral self-consistent solutions which also satisfy Poisson's equation $$\nabla ^2 \tilde U = 4\pi G\tilde \rho = 4\pi G\int {\tilde f d\upsilon .} $$ Lin and Shu (1964) have shown that such solutions exist in special cases. In the present work, we adopt anopposite proceeding. Poisson's equation contains two unknown quantities \(\tilde U\) and \(\tilde \rho \) . It could be completelysolved if a second independent equation connecting \(\tilde U\) and \(\tilde \rho \) was known. Such an equation is hopelesslyobtained by direct observational means; the only way is to postulate it in a mathematical form. In a previouswork, Louise (1981) has shown that Poisson's equation accounted for distances of planets in the solar system(following to the Titius-Bode's law revised by Balsano and Hughes (1979)) if the following relation wasassumed $$\rho ^2 = k\frac{{\tilde U}}{{r^2 }} (k = cte).$$ We now postulate again this relation in order to solve Poisson's equation. Then, $$\nabla ^2 \tilde U - \frac{{\alpha ^2 }}{{r^2 }}\tilde U = 0, (\alpha ^2 = 4\pi Gk).$$ The solution is found in a classical way to be of the form $$\tilde U = cte J_v (pr)e^{ - pz} e^{jn\theta } $$ wheren = integer,p =cte andJ _v (pr) = Bessel function with indexv (v ² =n ² + α²). By use of the Hankel function instead ofJ _v (pr) for large values ofr, the spiral structure is found to be given by $$\tilde U = cte e^{ - pz} e^{j[\Phi _v (r) + n\theta ]} , \Phi _v (r) = pr - \pi /2(v + \tfrac{1}{2}).$$ For small values ofr, \(\tilde U\) = 0: the center of a galaxy is not affected by the density wave which is onlyresponsible of the spiral structure. For various values ofp,n andv, other forms of galaxies can be taken into account: Ring, barred and spiral-barred shapes etc. In order to generalize previous calculations, we further postulateρ ₀ =kU ₀/r ², leading to Poisson'sequation which accounts for the disc population $$\nabla ^2 U_0 - \frac{{\alpha ^2 }}{{r^2 }}U_0 = 0.$$ AsU ₀ is assumed axisymmetrical, the obvious solution is of the form $$U_0 = \frac{{cte}}{{r^v }}e^{ - pz} , \rho _0 = \frac{{cte}}{{r^{2 + v} }}e^{ - pz} .$$ Finally, Poisson's equation is completely solvable under the assumptionρ =k(U/r ². The general solution,valid for both disc and spiral arm populations, becomes $$U = cte e^{ - pz} \left\{ {r^{ - v} + } \right.\left. {cte e^{j[\Phi _v (r) + n\theta ]} } \right\},$$ The density distribution along the O _z axis is supported by Burstein's (1979) observations.     

On Broucke's velocity-related series expansions in the two-body problem

This short article supplements a recent paper by Dr R. Broucke on velocity-related series expansions in the two-body problem. The derivations of the Fourier and Legendre expansions of the functionsF(v), \(\sqrt {F(\upsilon )} \) and \(\sqrt {{1 \mathord{\left/ {\vphantom {1 {F(\upsilon )}}} \right. \kern-0em} {F(\upsilon )}}} \) are given, where $$F(\upsilon ) = (1 - e^2 )/(1 + 2e\cos \upsilon + e^2 ), e< 1$$ In the two-body problem,v is identified with the true anomaly,e the eccentricity andF(v) equals (an/V)². Some interesting relations involving Legendre polynomials are also noted.     

General vacuum-universe solutions in Brans-Dicke cosmology

By a rescalation of the scalar field ? of the Jordan-Brans and Dicke cosmology, the general solutions of the Friedmannian ‘vacuum’ Universe are obtained. Only the flat space solution was previously known. Each solution is caracterized by the sign of the second time derivative of the rescaled field ψ≡?R ³ (R being the scale factor of the Robertson-Walker line-element): \(\ddot \psi\) = 0 (flat space), \(\ddot \psi\) < 0 (closed space), and \(\ddot \psi\) > 0 (open space), so that the solutions are mutually exclusive. Of these, the open space one is damped-oscillatory andR attains its absolute minimum, equal to zero, in only one of the two ‘extreme’ cycles. Otherwise,R min remains positive. If the ?-field is dominant near the singularity, these solutions may have physical significance. Also obtained, by the method mentioned above, is the general flat space solution for a ‘dust’ Universe and from it a closed space ‘dust’ solution. Both were found before by different authors, each one using a different method and, therefore, seemed up to now unrelated.     

Collisionless tearing modes in the presence of shear flow

Tearing modes in a plane collisionless current sheet with shear bulk flow are studied. An analytic expression for the growth rate is obtained for the case \(M^2 = (1 - \varepsilon {\text{ sech}}^m \bar z)\) , whereM is the Mach number,m the shear flow index, ε a positive constant less than unity, and \(\bar z\) the (normalized) co-ordinate normal to the current sheet. The growth rates are large and the unstable wave number domain is increased as compared to the case without flow. The relevance of these results to time-dependent reconnection processes in the Earth's magnetosphere is discussed.     

Apsidal motion in the eclipsing binary AS Cam

Multi-colourWBVR photoelectric observations of the eclipsing binary AS Cam have been carried out and the photometric elements, absolute dimensions, and the angular velocity of a periastron motion ( \(\mathop \omega \limits^ \cdot _{obs}\) ) are determined. The obtained value of \(\mathop \omega \limits^ \cdot _{obs}\) is almost three times smaller than that theoretically predicted.     

Sur un theoreme de stabilite pour un potentiel homogene

The McGehee's study of the triple collision of the 3-body problem is here applied for the stability of an equilibrium. Let us consider the homogeneous Lagrangian: $$L = \frac{{\dot x^2 + \dot y^2 }}{2} + U(x,y)$$ whereU is polynomial, with degreek. We establish a necessary and sufficient condition onU for the stability of \(\omega (x = y = \dot x = \dot y = 0)\) .     

Multiple scattering and the phase variation of equivalent widths for phase functions of typeP(cosθ)=ϖ+ϖ1 P 1(cosθ)+ϖ2 P 2(cosθ)

After computing theH-functions for 21 different phase functions corresponding to various combinations of \(\bar \varpi \) ₁=?₁/?and \(\bar \varpi \) ₂=?₂/?along with 15 values of ?, variations of equivalent widths with phase angle have been obtained for these cases for lines with Lorentz profile with the continuum albedo ? _c =0.99. It is found that: (i) The absolute values of equivalent width at any phase angle are Inversely related to the value of phase function for that angle; (ii) The usual inverse phase effect occurs whenever the phase function has a maximum at α=0 and a dip somewhere between α=0 and α=180; (iii) Whenever the phase function has minima at α=0 and α=180 one obtains an incipient inverse phase effect at large phase angles; and (iv) The total variations are larger for weaker lines.     

Loi de titius-bode et formalisme ondulatoire

Several authors (Basano and Hughes, 1979; ter Haar and Cameron, 1963, Dermott, 1968; Prentice, 1976) give the revised Titius-Bode law in the form $$r_n = r_o C^n ,$$ wherer _n stands for the distance of thenth planet from the Sun;r _o andC are constant. They pointed out, in addition, that regular satellites systems around major planets obey also that law. It is now generally thought that the Kant-laplace primeval nebula accounts for the origin and evolution of the solar system (Reeves, 1976). Furthermore, it is shown (Prentice, 1976) that rings, which obey the Titius-Bode law, are formed through successive contractions of the solar nebula. Among difficulties encountered by Prentice's theory, the formation of regular satellites similar to the planatery system is the most important one. Indeed, the starting point of the planetary system is a rotating flattened circular solar nebula, whereas a gaseous ring must be the starting point of satellites systems. As far as the Titius-Bode law is concerned, we have the feeling that orbits of planets around the Sun and of satellites around their primaries do not depend on starting conditions. That law must be inherent to gravitation, in the same manner that electron orbits depend only on the atomic law instead of the starting conditions under which an electron is captured. If it is correct, then one may expect to formulate similarity between the T-B law and the Bohr law in the early quantum theory. Such a similarity is found (Louise, 1982) by using a postulate similar to the Bohr-Sommerfeld one — i.e., $$\int_{r_o }^{r_n } {U(r) dr = nk,}$$ whereU(r)=GM _⊙ /r is the potential created by the Sun,k is a constant, andn a positive integer. This similarity suggests the existence of an unknown were process in the solar system. The aim of the present paper is to investigate the possibility of such a process. The first approach is to study a steady wave encountered in special membrane, showing node rings similar to the Prentice's rings (1976) which obey the T-B law. In the second part, we try to apply the now classical Lindblad-Lin density wave theory of spiral galaxies to the solar nebula case. This theory was developed since 1940 (Lindblad, 1974) in order to account for the persistence of spiral structure of galaxies (Lin and Shu, 1964; Lin, 1966; Linet al., 1969; Contopoulos, 1973). Its basic assumption concerns the potential functionU expressed in the form $$U = U_0 + \tilde U,$$ whereU _o stands for the background axisymmetric potential due to the disc population, and ?«U _o is responsible of spiral density wave. Then, the corresponding mass-density distribution is \(\rho = \rho _o + \tilde \rho\) , with \(\tilde \rho \ll \rho _o\) . Both quantities ? and \(\tilde \rho\) must satisfy the Poisson's equation $$\nabla ^2 \tilde U + 4\pi G\tilde \rho = 0.$$ It is shown by direct observations that most spiral arms fit well with a logarithmic spiral curve (Danver, 1942; Considère, 1980; Mulliard mand Marcelin, 1981). From the physical point of view, they are represented by maxima of ? (or \(\tilde \rho\) ) which is of the form $$\tilde U = cte cos (q log_e r - m\theta ),$$ wherem is an integer (number of arms),q=cte, andr and θ are polar coordinates. The distancer is expressed in an arbitrary unit (r=d/d_o). In the case of an axisymmetric solar nebula (m=0), successive maxima of \(\tilde U\) are rings showing similar T-B law $$d = d_o C^n ,$$ withC=e ^{2 π/q} constant, andn is a positive integer. It is noted, in addition, that the steady wave equation within the special membrane quoted above and the new expression of the Poisson's equation derived from (5) are quite similar and expressed in the form $$\nabla ^2 \tilde U + cte\tilde U/r^2 = 0.$$ This suggests that both spiral structure of galaxies and Prentice's rings system result from a wave process which is investigated in the last section. From Equation (2) it is possible to derive the wavelength of the assumed wave ‘χ’, by using a procedure similar to the one by L. De Broglie (1923). The velocity of the wave ‘χ’ process is discussed in two cases. Both cases lead to a similar Planck's relation (E=hv).     

Emden-chandrasekhar axisymmetric,rigidly rotating polytropes

We determine equilibrium configuration of Emden-Chandrasekhar axisymmetric, solid-body rotating polytropes, defined as EC polytropes, for polytropic indices ranging from 0 (homogeneous bodies) to 5 (Roche-type bodies). To this aim, we improve Chandrasekhar's method to determine equilibrium configurations on two respects: namely, (a) no distinction exists between undistorted and distorted terms in the expression of the potential, and (b) the comparison between the expressions of gravitational potential and its first derivatives inside and outside the body has to be made on the boundary of a sphere of radius Ξ_E, which does not necessarily coincide with the undistorted Emden's sphere of radius \(\bar \xi _0 \geqslant \Xi _{\text{E}} \) . We also allow different values of \(\bar \xi _0 \) for different physical parameters, and choose a special set which best fits more refined results (involving more complicated and more expensive computer codes) by James (1964). We find an increasing agreement with increasing values of polytropic indexn and vice-versa, while a large discrepancy arises for 0≤n<1, which makes the approximations used here too much rough tobe accepted in this range. A real slight non-monotonic trend is exhibited by axial rations and masses related to rotational equilibrium configurations — i.e., when gravity at the equator is balanced by centrifugal force-with extremum points for 4.8<n<4.85 in both cases. The same holds for masses related to spherical configurations, as already pointed out by Seidov and Kuzakhmedov (1978). Finally, it is shown that isotrophic, one-component models of this paper might provide the required correlation between the ratio of a typical rotation velocity to a typical peculiar velocity and the ellipticity, for about \(\tfrac{3}{4}\) of elliptical systems for which observations are available.     

Gaussian variational equations for osculating elements of an arbitrary separable reference orbit

If a satellite orbit is described by means of osculating Jacobi α's and β's of a separable problem, the paper shows that a perturbing forceF makes them vary according to $$\dot \alpha _\kappa = {\text{F}} \cdot \partial {\text{r/}}\partial \beta _k {\text{ }}\dot \beta _k = {\text{ - F}} \cdot \partial {\text{r/}}\partial \alpha _k ,{\text{ (}}k = 1,2,3).{\text{ (A1)}}$$ Herer is the position vector of the satellite andF is any perturbing force, conservative or non-conservative. There are two special cases of (A1) that have been previously derived rigorously. If the reference orbit is Keplerian, equations equivalent to (A1), withF arbitrary, were derived by Brouwer and Clemence (1961), by Danby (1962), and by Battin (1964). IfF=?gradV ₁(t), whereV ₁ may or may not depend explicitly on the time, Equations (A1) reduce to the well known forms (e.g. Garfinkel, 1966) $$\dot \alpha _\kappa = {\text{ - }}\partial V_1 {\text{/}}\partial \beta _k {\text{ }}\dot \beta _k = \partial V_1 {\text{/}}\partial \alpha _k ,{\text{ (}}k = 1,2,3).{\text{ (A2)}}$$ holding for all separable reference orbits. Equations (A1) can of course be guessed from Equations (A2), if one assumes that \(\dot \alpha _k (t)\) and \(\dot \beta _k (t)\) depend only onF(t) and thatF(t) can always be modeled instantaneously as a potential gradient. The main point of the present paper is the rigorous derivation of (A1), without resort to any such modeling procedure. Applications to the Keplerian and spheroidal reference orbits are indicated.     

Triple approaches in the plane isosceles equal-mass three-body problem

We analyze flyby-type triple approaches in the plane isosceles equal-mass three-body problem and in its vicinity. At the initial time, the central body lies on a straight line between the other two bodies. Triple approaches are described by two parameters: virial coefficient k and parameter $\mu = \dot r/\sqrt {\dot r^2 + \dot R^2 }$ , where $\dot r$ is the relative velocity of the extreme bodies and $\dot R$ is the velocity of the central body relative to the center of mass of the extreme bodies. The evolution of the triple system is traceable until the first turn or escape of the central body. The ejection length increases with closeness of the triple approach (parameter k). The longest ejections and escapes occur when the extreme bodies move apart with a low velocity at the time of triple approach. We determined the domain of escapes; it corresponds to close triple approaches (k>0.8) and to μ in the range ?0.2<μ<0.7. For small deviations from the isosceles problem, the evolution does not differ qualitatively from the isosceles case. The domain of escapes decreases with increasing deviations. In general, the ejection length increases for wide approaches and decreases for close approaches.     

Changes in binding energy of interacting galaxies

It is shown that the fractional increase in binding energy of a galaxy in a fast collision with another galaxy of the same size can be well represented by the formula $$\xi _2 = 3({G \mathord{\left/ {\vphantom {G {M_2 \bar R}}} \right. \kern-\nulldelimiterspace} {M_2 \bar R}}) ({{M_1 } \mathord{\left/ {\vphantom {{M_1 } {V_p }}} \right. \kern-\nulldelimiterspace} {V_p }})^2 e^{ - p/\bar R} = \xi _1 ({{M_1 } \mathord{\left/ {\vphantom {{M_1 } {M_2 }}} \right. \kern-\nulldelimiterspace} {M_2 }})^3 ,$$ whereM ₁,M ₂ are the masses of the perturber and the perturbed galaxy, respectively,V _p is the relative velocity of the perturber at minimum separationp, and \(\bar R\) is the dynamical radius of either galaxy.     

Continuum spectrum of the star Θ1 Ori C

Published photoelectric measurements over a wide wavelength range (0.36–18 µm) are used to study the continuum spectrum of the star Θ¹ Ori C. The model that assumes the following three radiation sources is consistent with observations: (1) a zero-age main-sequence O7 star (object 1) of mass M ₁=20M _⊙, radius R ₁=7.4R _⊙, effective temperature T ₂=37 000 K, and absolute bolometric magnitude $M\mathop {bol}\limits^1 = - 7\mathop .\limits^m 7$ ; (2) object 2 with M ₂=15M _⊙, R ₂=16.2R _⊙, T ₂=4000 K, and $M\mathop {bol}\limits^2 = - 5\mathop .\limits^m 1$ ; and (3) object 3 with R ₃10 700 R _⊙, T ₃=190 K, and $M\mathop {bol}\limits^3 = - 0\mathop .\limits^m 6$ . The visual absorption toward the system is $A_V = 0\mathop .\limits^m 95$ and obeys a normal law. The nature of objects 2 and 3 has not been elucidated. It can only be assumed that object 2 is a companion of the primary star, its spectral type is K7, and it is in the stage of gravitational contraction. Object 3 can be a cocoon star and a member of the system, but can also be a dust envelope surrounding the system as a whole.     

The collision singularity in a perturbed two-body problem

It is shown that, in the neighborhood of a collision singularity, the motion in a perturbed two-body problem \(\ddot r = - \mu r^{ - 3} r + P\) , whereP remains bounded, has the same basic properties as the motion in the neighborhood of a collision in the unperturbed two-body problemP=0.     

Implicit single-sequence methods for integrating orbits

The solutions of \(\ddot x = F(x,t)\) , and also \(\dot x = F(x,t)\) , are developed in truncated series in timet whose coefficients are found empirically. The series ending in thet ⁶ term yields a position at a final prechosen time that is accurate through 9th order in the sequence size. This is achieved by using Gauss-Radau and Gauss-Lobatto spacings for the several substeps within each sequence. This timeseries method is the same in principle as implicit Runge-Kutta forms, including some not described previously. In some orders these methods are unconditionally stable (A-stable). In the time-series formulation the implicit system converges rapidly. For integrating a test orbit the method is found to be about twice as fast as high-order explicit Runge-Kutta-Nyström-Fehlberg methods at the same accuracies. Both the Cowell and the Encke equations are solved for the test orbit, the latter being 35% faster. It is shown that the Encke equations are particularly well-adapted to treating close encounters when used with a single-sequence integrator (such as this one) provided that the reference orbit is re-initialized at the start of each sequence. This use of Encke equations is compared with the use of regularized Cowell equations.     

A relation between Giorgilli-Galgani's and Deprit's algorithms

If \(T = \sum\nolimits_{i = 1}^\infty {\varepsilon ^i } T_i\) and \(W = \sum\nolimits_{n = 1}^\infty {n\varepsilon ^{n - 1} } W^{\left( n \right)}\) are respectively the generators of Giorgilli-Galgani's and Deprit's transformations, we show that the change of variables generated byT is the inverse of the one generated byW, ifT _i =W ⁽ⁱ⁾ for anyi. The method used is to show that the recurrence which defines the first algorithm can also be obtained with the second one.     

The formation of ring galaxies

The conditions under which a head-on collision between a disk galaxy and a spherical galaxy can lead to ring formation are investigated, using the impulsive approximation. The spherical galaxy is modeled as a polytrope of indexn=4 and radiusR _S and the disk galaxy as an exponential disk whose surface density is given by \(\sigma (r) = \sigma _c e^{ - 4r/R_D } \) , where σ _c is the central density andR _D is the radius of the disk. The formation and properties of the rings are closely related to the fractional change in binding energy of the disk galaxy, given by ΔU/?U?=γ _D β _D , where (GM _S ² R _D )/(V ² M _D R _S ² ),M _S andM _D being the masses of the spherical and disk galaxies, respectively, and β _D ≡β _D (n, σ, ?,i) is a function of the models of the two galaxies, the ratio of the radii of the two galaxies ?=R _S /R _D , and the angle of inclinationi, of the disk to the direction of relative motion of the two galaxies. Calculations are made for the caseR _S =R _D . Since practically the entire mass of the spherical galaxy, for the chosen model, lies within 1/3 of its radius, the radius of the spherical galaxy is effectively \(\tfrac{1}{3}\) that of the disk galaxy. It is found that as a result of the collision, the innermost and the outer parts of the disk galaxy are not much affected, but the intermediate region expands and gets evacuated, leading to the crowding of stars in a preferential region forming a ring structure. The rings are best formed for a normal, on-axis collision. For this case, rings form when ΔU/|U| lies between \(\tfrac{1}{2}\) and 2, while they are very sharp and bright when ΔU/|U| lies between \(\tfrac{1}{2}\) and 1. Within this range, as ΔU/|U| increases, the rings become sharper and their positions shift outwards with respect to the centre of the disk galaxy. The relationship $$\gamma _D = 0.0016 + 0.045s_{{\text{max}}}^2 ,$$ wheres _max is the radial distance of the density maximum of the ring from the centre of the disk galaxy (measured in terms of the radius of the disk galaxy as unit) enables us to finds _max from γ _D and vice versa, and interpret some prominent ring galaxies. The effect of introducing a bulge to the disk is to distribute the tidal disruptive effects more evenly and, hence, reduce the sharpness of the ring.