A note on theH-functions of transfer problems in multiplying media

Some useful results and remodelled representations ofH-functions corresponding to the dispersion function $$T\left( z \right) = 1 - 2z^2 \sum\limits_1^n {\int_0^{\lambda r} {Y_r } \left( x \right){\text{d}}x/\left( {z^2 - x^2 } \right)} $$ are derived, suitable to the case of a multiplying medium characterized by $$\gamma _0 = \sum\limits_1^n {\int_0^{\lambda r} {Y_r } \left( x \right){\text{d}}x > \tfrac{1}{2} \Rightarrow \xi = 1 - 2\gamma _0< 0} $$      

Static spherically-symmetric scalar-field theory in general relativity

A spherically-symmetric static scalar field in general relativity is considered. The field equations are defined by $$\begin{gathered} R_{ik} = - \mu \varphi _i \varphi _k ,\varphi _i = \frac{{\partial \varphi }}{{\partial x^i }}, \varphi ^i = g^{ik} \varphi _k , \hfill \\ \hfill \\ \end{gathered} $$ where ?=?(r,t) is a scalar field. In the past, the same problem was considered by Bergmann and Leipnik (1957) and Buchdahl (1959) with the assumption that ?=?(r) be independent oft and recently by Wyman (1981) with the assumption ?=?(r, t). The object of this paper is to give explicit results with a different approach and under a more general condition $$\phi _{;i}^i = ( - g)^{ - 1/2} \frac{\partial }{{\partial x^i }}\left[ {( - g)^{1/2} g^{ik} \frac{\partial }{{\partial x^k }}} \right] = - 4\pi ( -g )^{ - 1/2} \rho $$ where ?=?(r, t) is the mass or the charge density of the sources of the field.     

Nonlinear newtonian theory and the effect of time dilatation

The fact that the energy density ρ_g of a static spherically symmetric gravitational field acts as a source of gravity, gives us a harmonic function \(f\left( \varphi \right) = e^{\varphi /c^2 } \) , which is determined by the nonlinear differential equation $$\nabla ^2 \varphi = 4\pi k\rho _g = - \frac{1}{{c^2 }}\left( {\nabla \varphi } \right)^2 $$ Furthermore, we formulate the infinitesimal time-interval between a couple of events measured by two different inertial observers, one in a position with potential φ-i.e., dt _φ and the other in a position with potential φ=0-i.e., dt ₀, as $${\text{d}}t_\varphi = f{\text{d}}t_0 .$$ When the principle of equivalence is satisfied, we obtain the well-known effect of time dilatation.     

Some identities for spheroidal harmonics

The spheroidal harmonics expressions $$\left[ {P_{2k}^{2s} \left( {i\xi } \right)P_{2k - 2r}^{2s} \left( \eta \right) - P_{2k - 2r}^{2s} \left( {i\xi } \right)P_{2k}^{2s} \left( \eta \right)} \right]e^{i2s\theta } $$ and $$\left[ {\eta ^2 P_{2k}^{2s} \left( {i\xi } \right)P_{2k - 2r}^{2s} \left( \eta \right) + \xi ^2 P_{2k - 2r}^{2s} \left( {i\xi } \right)P_{2k}^{2s} \left( \eta \right)} \right]e^{i2s\theta } $$ , have ξ²+η² as a factor. A method is presented for obtaining for these two expressions the coefficient of ξ²+η² in the form of a linear combination of terms of the formP _2m ^2s (iξ)P _2n ^2s (η)e ^i2sθ. Explicit formulae are exhibited for the casesr=1, 2, 3 and any positive or zero integersk ands. Such identities are useful in gravitational potential theory for ellipsoidal distributions when matching Legendre function expansions are employed.     

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.     

A new integration theory for conservative two degree-of-freedom systems

A two degree-of-freedom, conservative system is reduced to a single degree-of-freedom, kinematic system with Hamiltonian integral under the change of independent variable: $$dt = \zeta dt (\zeta = \upsilon _x - \upsilon _y )$$ where ζ is the curl (or vorticity) of the velocity field with cartesian inertial componentsu(x, y, t) andv(x, y, t). In the autonomous case whenu _t=v _t=0, orbits are globally represented by the level curves of an autonomous Hamiltonian functionH(x,y) satisfying a second-order quasilinear partial differential equation (Szebehely's Equation): $$2(H + U)\left( {H_{xx} H_y^2 - 2H_{xy} H_x H_y + H_{yy} H_x^2 } \right) + (H_x U_x + H_y U_y )\left( {H_x^2 + H_y^2 } \right) = 0$$ whereU(x, y) is the autonomous potential function. An inversion of dependent and independent variables reduces this equation to a second-order, ordinary differential equation for a function specifying the orbital curve. The true time variable is recovered by evaluating a quadrature. Fundamental differences exist between this approach and Hamilton-Jacobi theory.     

Isovortical orbits of autonomous,conservative, two degree-of-freedom dynamical systems

For an autonomous, conservative, two degree-of-freedom dynamical system, vorticity (the curl of velocity) is constant along the orbit if the velocity field is divergence-free such that: $$u\left( {x, v} \right) - \psi _y , v\left( {x, y} \right) = - \psi _x .$$ Isovortical orbits in configuration space are level curves of a scalar autonomous function Ψ (x, v) satisfying a second-order, non-linear partial differential equation of the Monge-Ampere type: $$2\left( {\psi _{xx} \psi _{yy} - \psi _{xy}^2 } \right) + U_{xx} + U_{yy} = 0,$$ where U(x. y) is the autonomous potential function. The solution Soc the time variable is reduced to a quadrature following determinatio of Ψ. Self-similar solutions of the Monge-Ampere equation under Birkhoff's one-parameter transformation group are derived for homogeneous (power-law) potential functions. It is shown that Keplerian orbits belong to the class of planar isovortical flows.     

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.     

Apsidal motion in the close binary IT cassiopeiae

In 1982 and 1993, we carried out highly accurate photoelectric WBVR measurements for the close binary IT Cas. Based on these measurements and on the observations of other authors, we determined the apsidal motion $\left[ {\dot \omega _{obs} = {{(11\mathop .\limits^ \circ 0 \pm 2\mathop .\limits^ \circ 5)} \mathord{\left/ {\vphantom {{(11\mathop .\limits^ \circ 0 \pm 2\mathop .\limits^ \circ 5)} {100 years}}} \right. \kern-0em} {100 years}}} \right]$ . This value is in agreement with the theoretically calculated apsidal motion for these stars $\left[ {\dot \omega _{th} = {{(14^\circ \pm 3^\circ )} \mathord{\left/ {\vphantom {{(14^\circ \pm 3^\circ )} {100 years}}} \right. \kern-0em} {100 years}}} \right]$ .     

Global solution of the ideal resonance problem

If a dynamical problem ofN degress of freedom is reduced to the Ideal Resonance Problem, the Hamiltonian takes the form 1 $$\begin{array}{*{20}c} {F = B(y) + 2\mu ^2 A(y)\sin ^2 x_1 ,} & {\mu \ll 1.} \\ \end{array} $$ Herey is the momentum-vectory _k withk=1,2?N, x ₁ is thecritical argument, andx _k fork>1 are theignorable co-ordinates, which have been eliminated from the Hamiltonian. The purpose of this Note is to summarize the first-order solution of the problem defined by (1) as described in a sequence of five recent papers by the author. A basic is the resonance parameter α, defined by 1 $$\alpha \equiv - B'/\left| {4AB''} \right|^{1/2} \mu .$$ The solution isglobal in the sense that it is valid for all values of α² in the range 1 $$0 \leqslant \alpha ^2 \leqslant \infty ,$$ which embrances thelibration and thecirculation regimes of the co-ordinatex ₁, associated with α² < 1 and α² > 1, respectively. The solution includes asymptotically the limit α² → ∞, which corresponds to theclassical solution of the problem, expanded in powers of ε ≡ μ², and carrying α as a divisor. The classical singularity at α=0, corresponding to an exact commensurability of two frequencies of the motion, has been removed from the global solution by means of the Bohlin expansion in powers of μ = ε^1/2. The singularities that commonly arise within the libration region α² < 1 and on the separatrix α² = 1 of the phase-plane have been suppressed by means of aregularizing function 1 $$\begin{array}{*{20}c} {\phi \equiv \tfrac{1}{2}(1 + \operatorname{sgn} z)\exp ( - z^{ - 3} ),} & {z \equiv \alpha ^2 } \\ \end{array} - 1,$$ introduced into the new Hamiltonian. The global solution is subject to thenormality condition, which boundsAB″ away from zero indeep resonance, α² < 1/μ, where the classical solution fails, and which boundsB′ away from zero inshallow resonance, α² > 1/μ, where the classical solution is valid. Thedemarcation point 1 $$\alpha _ * ^2 \equiv {1 \mathord{\left/ {\vphantom {1 \mu }} \right. \kern-\nulldelimiterspace} \mu }$$ conventionally separates the deep and the shallow resonance regions. The solution appears in parametric form 1 $$\begin{array}{*{20}c} {x_\kappa = x_\kappa (u)} \\ {y_1 = y_1 (u)} \\ {\begin{array}{*{20}c} {y_\kappa = conts,} & {k > 1,} \\ \end{array} } \\ {u = u(t).} \\ \end{array} $$ It involves the standard elliptic integralsu andE((u) of the first and the second kinds, respectively, the Jacobian elliptic functionssn, cn, dn, am, and the Zeta functionZ (u).     

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 the stability of hierarchical four-body systems

Generalized Jacobian coordinates can be used to decompose anN-body dynamical system intoN-1 2-body systems coupled by perturbations. Hierarchical stability is defined as the property of preserving the hierarchical arrangement of these 2-body subsystems in such a way that orbit crossing is avoided. ForN=3 hierarchical stability can be ensured for an arbitrary span of time depending on the integralz=c ² h (angular momentum squared times energy): if it is smaller than a critical value, defined by theL ₂ collinear equilibrium configuration, then the three possible hierarchical arrangements correspond to three disconnected subsets of the invariant manifold in the phase space (and in the configuration space as well; see Milani and Nobili, 1983a). The same definitions can be extended, with the Jacobian formalism, to an arbitrary hierarchical arrangement ofN≥4 bodies, and the main confinement condition, the Easton inequality, can also be extended but it no longer provides separate regions of trapped motion, whatever is the value ofz for the wholeN-body system,N≥4. However, thez criterion of hierarchical stability applies to every 3-body subsystem, whosez ‘integral’ will of course vary in time because of the perturbations from the other bodies. In theN=4 case we decompose the system into two 3-body subsystems whosec ² h ‘integrals’,z ₂₃ andz ₃₄, att=0 are assumed to be smaller than the corresponding critical values \(\tilde z_{23} \) and \(\tilde z_{34} \) , so that both the subsystems are initially hierarchically stable. Then the hierarchical arrangement of the 4 bodies cannot be broken until eitherz ₂₃ orz ₃₄ is changed by an amount \(\tilde z_{ij} - z_{ij} \left( 0 \right)\) ; that is the whole system is hierarchically stable for a time spain not shorter than the minimum between \(\Delta t_{23} = {{\left( {\tilde z_{23} - z_{23} \left( 0 \right)} \right)} \mathord{\left/ {\vphantom {{\left( {\tilde z_{23} - z_{23} \left( 0 \right)} \right)} {\dot z_{23} }}} \right. \kern-0em} {\dot z_{23} }}\) and \(\Delta t_{34} = {{\left( {\tilde z_{34} - z_{34} \left( 0 \right)} \right)} \mathord{\left/ {\vphantom {{\left( {\tilde z_{34} - z_{34} \left( 0 \right)} \right)} {\dot z_{34} }}} \right. \kern-0em} {\dot z_{34} }}\) . To estimate how long is this stability time, two main steps are required. First the perturbing potentials have to be developed in series; the relevant small parameters are some combinations of mass ratios and length ratios, the? _ij of Roy and Walker. When an appropriate perturbation theory is based on the? _ij , the asymptotic expansions are much more rapidly decreasing than the usual expansions in powers of the mass ratios (as in the classical Lagrange perturbation theory) and can be extended also to cases such as lunar theory or double binaries. The second step is the computation of the time derivatives \(\dot z_{ij} \) (we limit ourselves to the planar case). To assess the long term behaviour of the system, we can neglect the short-periodic perturbations and discuss only the long-periodic and the secular perturbations. By using a Poisson bracket formalism, a generalization of Lagrange theorem for semimajor axes and a generalization of the classical first order theories for eccentricities and pericenters, we prove that thez _ij do not undergo any secular perturbation, because of the interaction with the other subsystem, at the first order in the? _ik . After the long-periodic perturbations have been accounted for, and apart from the small divisors problems that could arise both from ordinary and secular resonances, only the second order terms have to be considered in the computation of Δt ₂₃, Δt ₃₄. A full second order perturbative theory is beyond the scope of this paper; however an order-of-magnitude lower estimate of the Δt _ij can be obtained with the very pessimistic assumption that essentially all the second order terms affect in a secular way thez _ij . The same method could be applied also toN≥5 body systems. Since almost everyN-body system existing in nature is strongly hierarchical, the product of two? _ij is very small for almost all the real astronomical problems. As an example, the hierarchical stability of the 4-body system Sun, Mercury, Venus, and Jupiter is investigated; this system turns out to be stable for at least 110 million years. Although this hierarchical stability time is ～10 times less than the real age of the Solar System, taking into account that many pessimistic assumptions have been done we can conclude that the stability of the Solar System is no more a forbidden problem for Celestial Mechanics.     

Sur un formalisme explicitant la loi des distances des planetes

The well-known Titius-Bode law (T-B) giving distances of planets from the Sun was improved by Basano and Hughes (1979) who found: $$a_n = 0.285 \times 1.523^n ;$$ a _n being the semi-major axis expressed in astronomical units, of then-th planet. The integern is equal to 1 for Mercury, 2 for Venus etc. The new law (B-H) is more natural than the (T-B) one, because the valuen=?∞ for Mercury is avoided. Furthermore, it accounts for distances of all planets, including Neptune and Pluto. It is striking to note that this law:

1. does not depend on physical parameters of planets (mass, density, temperature, spin, number of satellites and their nature etc.).
2. shows integers suggesting an unknown, obscure wave process in the formation of the solar system.

In this paper, we try to find a formalism accounting for the B-H law. It is based on the turbulence, assumed to be responsible of accretion of matter within the primeval nebula. We consider the function $$\psi ^2 (r,t) = |u^2 (r,t) - u_0^2 |$$ , whereu ²(r, t) stands for the turbulence, i.e., the mean-square deviation velocities of particles at the pointr and the timet; andu ₀ ² is the value of turbulence for which the accretion process of matter is optimum. It is obvious that Ψ²(r _n,t₀) = 0 forr _n=0.285×1.523 ⁿ at the birth timet ₀ of proto-planets. Under these conditions, it is easily found that $$\psi ^2 (r,t_0 ) = \frac{{A^2 }}{r}\sin ^2 [\alpha log r - \Phi (t_0 )]$$ With α=7.47 and Φ(t ₀)=217.24 in the CGS system, the above function accounts for the B-H law. Another approach of the problem is made by considering fluctuations of the potentialU(r, t) and of the density of matter ρ(r, t). For very small fluctuations, it may be written down the Poisson equation $$\Delta \tilde U(r,t_0 ) + 4\pi G\tilde \rho (r,t_0 ) = 0$$ , withU(r, t)=U ₀(r)+?(r, t ₀ ) and \(\tilde \rho (r,t_0 )\) . It suffices to postulate \(\tilde \rho (r,t_0 ) = k[\tilde U(r,t_0 )/r^2 ](k = cte)\) for finding the solution $$\tilde U(r,t_0 ) = \frac{{cte}}{{r^{1/2} }}\cos [a\log r - \zeta (t_0 )]$$ . Fora=14.94 and ζ(t ₀)=434.48 in CGS system, the successive maxima of ?(r,t ₀) account again for the B-H law. In the last approach we try to write Ψ(r, t) under a wave function form $$\Psi ^2 (r,t) = \frac{{A^2 }}{r}\sin ^2 \left[ {\omega \log \left( {\frac{r}{v} - t} \right)} \right].$$ It is emphasized that all calculations are made under mathematical considerations.     

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.     

The Chandra X-ray galaxy clusters at z<1.4: constraints on the evolution of L X −T−M g relations

We analyzed the luminosity-temperature-mass of gas (L _X ?T?M _g ) relations for a sample of 21 Chandra galaxy clusters. We used the standard approach (β?model) to evaluate these relations for our sample that differs from other catalogues since it considers galaxy clusters at higher redshifts (0.4<z<1.4). We assumed power-law relations in the form $L_{X} \sim(1 +z)^{A_{L_{X}T}} T^{\beta_{L_{X}T}}$ , $M_{g} \sim(1 + z)^{A_{M_{g}T}} T^{\beta_{M_{g}T}}$ , and $M_{g} \sim(1 + z)^{A_{M_{g}L_{X}}} L^{\beta_{M_{g}L_{X}}}$ . We obtained the following fitting parameters with 68 % confidence level: $A_{L_{X}T} = 1.50 \pm0.23$ , $\beta_{L_{X}T} = 2.55 \pm0.07$ ; $A_{M_{g}T} = -0.58 \pm0.13$ and $\beta_{M_{g}T} = 1.77 \pm0.16$ ; $A_{M_{g}L_{X}} \approx-1.86 \pm0.34$ and $\beta_{M_{g}L_{X}} = 0.73 \pm0.15$ , respectively. We found that the evolution of the M _g ?T relation is small, while the M _g ?L _X relation is strong for the cosmological parameters Ω _m =0.27 and Ω _Λ =0.73. In overall, the clusters at high-z have stronger dependencies between L _X ?T?M _g correlations, than those for clusters at low-z. For most of galaxy clusters (first of all, from MACS and RCS surveys) these results are obtained for the first time.     

On the measure-preserving mappings with odd dimension

In this paper, using two methods: LCN'S (Lyapunov characteristic numbers) method and slice cutting method, we study numerically two mappings with odd dimension: $$T_1 :\left\{ {\begin{array}{*{20}c} {x_{n + 1} = x_n + z_n ,} \\ {y_{n + 1} = y_n + x_{n + 1} , (\bmod 2\pi )} \\ {z_{n + 1} = z_n + A\sin y_{n + 1} ,} \\ \end{array} } \right. T_2 :\left\{ {\begin{array}{*{20}c} {x_{n + 1} = x_n + y_n + B \sin z_n ,} \\ {y_{n + 1} = y_n + A \sin x_{n + 1} , (\bmod 2\pi ),} \\ {z_{n + 1} = z_n + B \sin y_{n + 1} ,} \\ \end{array} } \right.$$ whereA, B are parameters. For the mappingT ₁ the whole region is stochastic; however, we find two-dimensional invariant manifolds for the mappingT ₂.     

Sur les solutions Lagrangiennes du problème des trois corps solides avec la loi de Weber

In this work we consider the problem of translational-rotational motion of three solid bodies, for which the elementary particles attract each other according to different Weber's laws for each pair of bodies. This problem represents a special case of the generalized problem of three solids considered in a previous work, (Dubochin, 1974) and it gives an example of the verification of the existence conditions for the Lagrangian solutions. In these solutions, the centers of mass always for m an equilateral triangle. Each body has axial symmetry with the plane of symmetry perpendicular to the axis of symmetry rotates uniformly around this axis, which at any instant stays perpendicular to the plane of the triangle formed by the centers of mass. According to Weber's law (Tisserand, 1896) the elementary particles of two bodiesT _i andT _j (i, j=0, 1, 2) are attracted by forces which are proportional to the function $$F_{ij} (W) = \frac{{f_{ij} }}{{\Delta _{ij^2 } }}\left[ {1 - a_{ij} \dot \Delta _{ij^2 } + 2a_{ij} \Delta _{ij} \ddot \Delta _{ij} } \right]$$ wheref _ij anda _ij (in generalf _ji ≠f _ij anda _ji ≠a _ij ) are functions of the timet, and where the real quantities Δ_ij are the mutual distances between the particles of the bodiesT _i andT _j , and where \(\dot \Delta _{ij} \) and \(\ddot \Delta _{ij} \) are their derivatives with respect to the time. The analysis of the general conditions for the Lagrangian solutions gives the following results for the case of Weber's laws.

1. Only the invariant Lagrangian solutions, (the traingle of the centres of mass does not change in time) are possible in this problem.
2. Besides the conditions (NL) obtained in the case of the Newton-Coulomb law, (all thea _ij are zero), the complementary conditions (WL) must be satisfied.

In particular, if all the bodies are spheres or homogeneous ellipsoids, they must necessarily have the same dimensions, but they can have different masses.     

Ultra-high energy neutrino fluxes from supermassive AGN black holes

We compute the ultra-high energy (UHE) neutrino fluxes from plausible accreting supermassive black holes closely linking to the 377 active galactic nuclei (AGNs). They have well-determined black hole masses collected from the literature. The neutrinos are produced via simple or modified URCA processes, even after the neutrino trapping, in superdense proto-matter medium. The resulting fluxes are ranging from: (1) (quark reactions)— $J^{q}_{\nu\varepsilon}/(\varepsilon_{d}\ \mathrm{erg}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}\,\mathrm{sr}^{-1})\simeq8.29\times 10^{-16}$ to 3.18×10^?4, with the average $\overline{J}^{q}_{\nu\varepsilon}\simeq5.53\times 10^{-10}\varepsilon_{d}\ \mathrm{erg}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}\,\mathrm{sr}^{-1}$ , where ε _d ～10^?12 is the opening parameter; (2) (pionic reactions)— $J^{\pi}_{\nu\varepsilon} \simeq0.112J^{q}_{\nu\varepsilon}$ , with the average $J^{\pi}_{\nu\varepsilon} \simeq3.66\times 10^{-11}\varepsilon_{d}\ \mathrm{erg}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}\,\mathrm{sr}^{-1}$ ; and (3) (modified URCA processes)— $J^{URCA}_{\nu\varepsilon}\simeq7.39\times10^{-11} J^{q}_{\nu\varepsilon}$ , with the average $\overline{J}^{URCA}_{\nu\varepsilon} \simeq2.41\times10^{-20} \varepsilon_{d}\ \mathrm{erg}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}\,\mathrm{sr}^{-1}$ . We conclude that the AGNs are favored as promising pure neutrino sources, because the computed neutrino fluxes are highly beamed along the plane of accretion disk, peaked at high energies and collimated in smaller opening angle θ～ε _d .     

On the connection between the Nekhoroshev theorem and Arnold diffusion

The analytical techniques of the Nekhoroshev theorem are used to provide estimates on the coefficient of Arnold diffusion along a particular resonance in the Hamiltonian model of Froeschlé et al. (Science 289:2108–2110, 2000). A resonant normal form is constructed by a computer program and the size of its remainder ||R _opt || at the optimal order of normalization is calculated as a function of the small parameter ${\epsilon}$ . We find that the diffusion coefficient scales as ${D \propto ||R_{opt}||^3}$ , while the size of the optimal remainder scales as ${||R_{opt}|| \propto {\rm exp}(1/\epsilon^{0.21})}$ in the range ${10^{-4} \leq \epsilon \leq 10^{-2}}$ . A comparison is made with the numerical results of Lega et al. (Physica D 182:179–187, 2003) in the same model.     

Observational constraints on G-corrected holographic dark energy using a Markov chain Monte Carlo method

We constrain holographic dark energy (HDE) with time varying gravitational coupling constant in the framework of the modified Friedmann equations using cosmological data from type Ia supernovae, baryon acoustic oscillations, cosmic microwave background radiation and X-ray gas mass fraction. Applying a Markov Chain Monte Carlo (MCMC) simulation, we obtain the best fit values of the model and cosmological parameters within 1σ confidence level (CL) in a flat universe as: $\varOmega_{b}h^{2}=0.0222^{+0.0018}_{-0.0013}$ , $\varOmega_{c}h^{2}=0.1121^{+0.0110}_{-0.0079}$ , $\alpha_{G}\equiv \dot{G}/(HG) =0.1647^{+0.3547}_{-0.2971}$ and the HDE constant $c=0.9322^{+0.4569}_{-0.5447}$ . Using the best fit values, the equation of state of the dark component at the present time w _d0 at 1σ CL can cross the phantom boundary w=?1.