Normality condition in the ideal resonance problem

The Ideal Resonance Problem, defined by the Hamiltonian $$F = B(y) + 2\mu ^2 A(y)\sin ^2 x,\mu \ll 1,$$ has been solved in Garfinkelet al. (1971). As a perturbed simple pendulum, this solution furnishes a convenient and accurate reference orbit for the study of resonance. In order to preserve the penduloid character of the motion, the solution is subject to thenormality condition, which boundsAB" andB' away from zero indeep and inshallow resonance, respectively. For a first-order solution, the paper derives the normality condition in the form $$pi \leqslant max(|\alpha /\alpha _1 |,|\alpha /\alpha _1 |^{2i} ),i = 1,2.$$ Herep _i are known functions of the constant ‘mean element’y', α is the resonance parameter defined by $$\alpha \equiv - {\rm B}'/|4AB\prime \prime |^{1/2} \mu ,$$ and $$\alpha _1 \equiv \mu ^{ - 1/2}$$ defines the conventionaldemarcation point separating the deep and the shallow resonance regions. The results are applied to the problem of the critical inclination of a satellite of an oblate planet. There the normality condition takes the form $$\Lambda _1 (\lambda ) \leqslant e \leqslant \Lambda _2 (\lambda )if|i - tan^{ - 1} 2| \leqslant \lambda e/2(1 + e)$$ withΛ ₁, andΛ ₂ known functions of λ, defined by $$\begin{gathered} \lambda \equiv |\tfrac{1}{5}(J_2 + J_4 /J_2 )|^{1/4} /q, \hfill \\ q \equiv a(1 - e). \hfill \\ \end{gathered}$$      

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.     

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.     

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.     

An extended Ideal Resonance Problem

An Extended Resonance Problem is defined by the Hamiltonian, $$F = B(y) + 2\mu ^2 A(y)[\sin x + \lambda (y)]^2 \mu<< 1,\lambda = O(\mu ).$$ It is noted here that the phase-plane trajectories exhibit adouble libration, enclosing two centers, for the initial conditions of motion satisfying the inequality $$1 - |\lambda |< |\alpha |< 1 + |\lambda |,$$ where α is the usualresonance parameter. A first order solution for the case of double libration is constructed here by a generalization of the procedure previously used in solving the Ideal Resonance Problem with λ=0. The solution furnishes a reference orbit for a Perturbed Ideal Problem if a double libration occurs as a result of perturbations.     

On The Ideal Resonance Problem

The Ideal Resonance Problem in its normal form is defined by the Hamiltonian (1) $$F = A (y) + 2B (y) sin^2 x$$ with (2) $$A = 0(1),B = 0(\varepsilon )$$ where ? is a small parameter, andx andy a pair of canonically conjugate variables. A solution to 0(?^1/2) has been obtained by Garfinkel (1966) and Jupp (1969). An extension of the solution to 0(?) is now in progress in two papers ([Garfinkel and Williams] and [Hori and Garfinkel]), using the von Zeipel and the Hori-Lie perturbation methods, respectively. In the latter method, the unperturbed motion is that of a simple pendulum. The character of the motion depends on the value of theresonance parameter α, defined by (3) $$\alpha = - A\prime /|4A\prime \prime B\prime |^{1/2} $$ forx=0. We are concerned here withdeep resonance, (4) $$\alpha< \varepsilon ^{ - 1/4} ,$$ where the classical solution with a critical divisor is not admissible. The solution of the perturbed problem would provide a theoretical framework for an attack on a problem of resonance in celestial mechanics, if the latter is reducible to the Ideal form: The process of reduction involves the following steps: (1) the ration ₁/n₂ of the natural frequencies of the motion generates a sequence. (5) $$n_1 /n_2 \sim \left\{ {Pi/qi} \right\},i = 1, 2 ...$$ of theconvergents of the correspondingcontinued fraction, (2) for a giveni, the class ofresonant terms is defined, and all non-resonant periodic terms are eliminated from the Hamiltonian by a canonical transformation, (3) thedominant resonant term and itscritical argument are calculated, (4) the number of degrees of freedom is reduced by unity by means of a canonical transformation that converts the critical argument into an angular variable of the new Hamiltonian, (5) the resonance parameter α (i) corresponding to the dominant term is then calculated, (6) a search for deep resonant terms is carried out by testing the condition (4) for the function α(i), (7) if there is only one deep resonant term, and if it strongly dominates the remaining periodic terms of the Hamiltonian, the problem is reducible to the Ideal form.     

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.     

UBV photometry and light-curve analysis of Algol

Photoelectric observations of the eclipsing variable β Per, were obtained inUBV standard system, and new elements for the primary minimum were determined as $$J.D. = 2445641.5135,O - C = 0_.^d 0.009.$$ The light curves of the system were analysed using Fourier techniques in the frequency-domain. The fractional radii of both components are $$r_1 = 0.217 \pm 0.002,r_2 = 0.233 \pm 0.002andi = 85.5 \pm 0.5.$$ Absolute elements were derived and the effective temperatures are $$T_1 = 11800K,T_2 = 5140K.$$      

Comparison of the classical and the global solutions of the Ideal Resonance Problem

The Ideal Resonance Problem is defined by the Hamiltonian $$F = B(y) + 2\varepsilon A(y) \sin ^2 x,\varepsilon \ll 1.$$ The classical solution of the Problem, expanded in powers of ε, carries the derivativeB′ as a divisor and is, therefore, singular at the zero ofB′, associated with resonance. With α denoting theresonance parameter, defined by $$\alpha \equiv - B'/|4AB''|^{1/2} \mu ,\mu = \varepsilon ^{1/2} ,$$ it is shown here that the classical solution is valid only for $$\alpha ^2 \geqslant 0(1/\mu ).$$ In contrast, the global solution (Garfinkelet al., 1971), expanded in powers ofμ=ε^1/2, removes the classical singularity atB′=0, and is valid for all α. It is also shown here that the classical solution is an asymptotic approximation, for largeα ², of the global solution expanded in powers ofα ^?2. This result leads to simplified expressions for resonancewidth and resonantamplification. The two solutions are compared with regard to their general behavior and their accuracy. It is noted that the global solution represents a perturbed simple pendulum, while the classical solution is the limiting case of a pendulum in a state offast circulation.     

Experimental testing of relativistic effects,variability of the gravitational constant and topography of Mercury surface from radar observations 1964–1989

The new analysis of radar observations of inner planets for the time span 1964–1989 is described. The residuals show that Mercury topography is an important source of systematic errors which have not been taken into account up to now. The longitudinal and latitudinal variations of heights of Mercury surface were found and an approximate map of equatorial zone |?|≤120° was constructed. Including three values characterizing global nonsphericity of Mercury surface into the set of parameters under determination allowed to improve essentially all estimates. In particular, the variability of the gravitational constantG was evaluated: $$\dot G/G = (0.47 \pm 0.47) \times 10^{ - 11} yr^{ - 1} $$ . The correction to Mercury perihelion motion: $$\Delta \dot \pi = - 0''.017 \pm 0''.052 cy^{ - 1} $$ and linear combination of the parameters of PPN formalism: $$\upsilon = (2 + 2\gamma - \beta )/3 = 0.9995 \pm 0.0013$$ were determined; they are in a good agreement with General Relativity predictions. The obtained values Δ.π and ν correspond to the negligible solar oblateness, the estimate of solar quadrupole moment being: $$J_2 = ( - 0.13 \pm 0.41) \times 10^{ - 6} $$ .     

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.     

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).     

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 $$\gamma _0 = \sum\limits_1^n {\int_0^{\lambda r} {Y_r } \left( x \right){\text{d}}x > \tfrac{1}{2} \Rightarrow \xi = 1 - 2\gamma _0< 0} $$      

On the Maximum Separation of Visual Binaries

In this paper, an efficient algorithm is established for computing the maximum (minimum) angular separation ρ _max(ρ _min), the corresponding apparent position angles ( $\theta|_{\rho_{\rm max}}$ , $\theta|_{\rho_{\rm min}}$ ) and the individual masses of visual binary systems. The algorithm uses Reed’s formulae (1984) for the masses, and a technique of one-dimensional unconstrained minimization, together with the solution of Kepler’s equation for $(\rho_{\rm max}, \theta|_{\rho_{\rm max}})$ and $(\rho_{\rm min}, \theta|_{\rho_{\rm min}})$ . Iterative schemes of quadratic coverage up to any positive integer order are developed for the solution of Kepler’s equation. A sample of 110 systems is selected from the Sixth Catalog of Orbits (Hartkopf et al. 2001). Numerical studies are included and some important results are as follows: (1) there is no dependence between ρ _max and the spectral type and (2) a minor modification of Giannuzzi’s (1989) formula for the upper limits of ρ _max functions of spectral type of the primary.     

Orbites periodiques dans un potentiel a trois dimensions II. Bifurcations

In a previous paper, Hayliet al. (1983), two families of periodic orbits in the three-dimensional potential $$U = \frac{1}{2}(Ax^2 + By^2 + Cz^2 ) - \varepsilon xz^2 - nyz^2 $$ with \(\sqrt A :\sqrt B :\sqrt C = 6:4:3\) and ?=0.5 were described. It was found empirically that the characteristic curves of the two families intersect in the space (x₀, y₀, η) for |η|?0.2. This property is demonstrated in the present paper by writing explicitely the Poincaré mapping and by giving an approximation directly comparable with the numerical results obtained in Hayliet al. (1983). It is thus shown that one family bifurcates off the other.     

A Statistical Study on Photospheric Magnetic Nonpotentiality of Active Regions and Its Relationship with Flares During Solar Cycles 22?–?23

A statistical study is carried out on the photospheric magnetic nonpotentiality in solar active regions and its relationship with associated flares. We select 2173 photospheric vector magnetograms from 1106 active regions observed by the Solar Magnetic Field Telescope at Huairou Solar Observing Station, National Astronomical Observatories of China, in the period of 1988??C?2008, which covers most of the 22nd and 23rd solar cycles. We have computed the mean planar magnetic shear angle ( $\overline{\Delta\phi}$ ), mean shear angle of the vector magnetic field ( $\overline{\Delta\psi}$ ), mean absolute vertical current density ( $\overline{|J_{z}|}$ ), mean absolute current helicity density ( $\overline{|h_{\mathrm{c}}|}$ ), absolute twist parameter (|?? _av|), mean free magnetic energy density ( $\overline{\rho_{\mathrm{free}}}$ ), effective distance of the longitudinal magnetic field (d _E), and modified effective distance (d _Em) of each photospheric vector magnetogram. Parameters $\overline{|h_{\mathrm{c}}|}$ , $\overline{\rho_{\mathrm{free}}}$ , and d _Em show higher correlations with the evolution of the solar cycle. The Pearson linear correlation coefficients between these three parameters and the yearly mean sunspot number are all larger than 0.59. Parameters $\overline {\Delta\phi}$ , $\overline{\Delta\psi}$ , $\overline{|J_{z}|}$ , |?? _av|, and d _E show only weak correlations with the solar cycle, though the nonpotentiality and the complexity of active regions are greater in the activity maximum periods than in the minimum periods. All of the eight parameters show positive correlations with the flare productivity of active regions, and the combination of different nonpotentiality parameters may be effective in predicting the flaring probability of active regions.     

Solving Kepler’s equation via Smale’s \alpha -theory

We obtain an approximate solution $\tilde{E}=\tilde{E}(e,M)$ of Kepler’s equation $E-e\sin (E)=M$ for any $e\in [0,1)$ and $M\in [0,\pi ]$ . Our solution is guaranteed, via Smale’s $\alpha $ -theory, to converge to the actual solution $E$ through Newton’s method at quadratic speed, i.e. the $n$ -th iteration produces a value $E_n$ such that $|E_n-E|\le (\frac{1}{2})^{2^n-1}|\tilde{E}-E|$ . The formula provided for $\tilde{E}$ is a piecewise rational function with conditions defined by polynomial inequalities, except for a small region near $e=1$ and $M=0$ , where a single cubic root is used. We also show that the root operation is unavoidable, by proving that no approximate solution can be computed in the entire region $[0,1)\times [0,\pi ]$ if only rational functions are allowed in each branch.     

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.