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1.
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.  相似文献   

2.
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.  相似文献   

3.
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.  相似文献   

4.
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.  相似文献   

5.
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.  相似文献   

6.
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.  相似文献   

7.
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.  相似文献   

8.
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.  相似文献   

9.
The fact that the energy density ρg 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.  相似文献   

10.
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.  相似文献   

11.
For the conservative, two degree-of-freedom system with autonomous potential functionV(x,y) in rotating coordinates; $$\dot u - 2n\upsilon = V_x , \dot \upsilon + 2nu = V_y $$ , vorticity (v x -u y ) is constant along the orbit when the relative velocity field is divergence-free such that: $$u(x,y,t) = \psi _y , \upsilon (x,y,t) = - \psi _x $$ . Unlike isoenergetic reduction using the Jacobi, integral and eliminating the time,non-singular reduction from fourth to second-order occurs when (u,v) are determined explicitly as functions of their arguments by solving for ψ (x, y, t). The orbit function ψ satisfies a second-order, non-linear partial differential equation of the Monge Ampere type: $$2(\psi _{xx} \psi _{yy} - \psi _{xy}^2 ) - 2(\psi _{xx} + \psi _{yy} ) + V_{xx} + V_{yy} = 0$$ . Isovortical orbits in the rotating frame arenot level curves of ψ because it contains time explicitly due to coriolis effects. Rather, (x, y) coordinates along the orbit are obtained, from (u, v) either by numerical integration of the kinematic equations, or by partial differentiation of the Legendre transform ? of ψ. In the latter case, ? is shown to satisfy a non-linear, second-order partial differential equation in three independent variables, derived from the Monge-Ampere Equation. Complete reduction to quadrature is possible when space-time symmetries exist, as in the case of central force motion.  相似文献   

12.
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.  相似文献   

13.
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.  相似文献   

14.
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 $$\begin{gathered} \lambda \equiv |\tfrac{1}{5}(J_2 + J_4 /J_2 )|^{1/4} /q, \hfill \\ q \equiv a(1 - e). \hfill \\ \end{gathered}$$   相似文献   

15.
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 ΞE, 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.  相似文献   

16.
The projection of an axially symmetric satellite's orbit on a plane perpendicular to the rotation axis (z=const.) is given by the second-order differential equation. $$\frac{{y''}}{{1 + y'^2 }} = \bar \Psi _y - y'\bar \Psi _{x,}$$ where the prime denotes the derivative with respect tox and \(\bar \Psi (x,y)\) is a known function. Two integrability cases have been investigated and it has been shown that for these two cases the integration can be carried out either by quadratures or reduced to a first-order differential equation. Analytical and physical properties are expressed, and it is shown that the equation can be derived from the calssical plane eikonal equation of geometric optics.  相似文献   

17.
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/do). 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).  相似文献   

18.
Four colour contrasts have been studied for 104 lunar regions between wavelengths 4000 Å and 8000 Å. Distinct colour differences are found. The greatest contrast between topographs occurs at longer wavelengths. The regional colour differences between the two short wavelengths 4035 Å and 5538 Å show a general trend of increasing reddening with age. For the other three colour differences \(\Delta (\tfrac{{{\text{4 7 6 5}}}}{{{\text{7 9 2 2}}}}), \Delta (\tfrac{{{\text{4 7 6 5}}}}{{{\text{6 6 9 2}}}}) and \Delta (\tfrac{{{\text{6 6 9 2}}}}{{{\text{7 9 2 2}}}})\) , a trend of variation with albedo is detected, and the less scatter on points for the colour differences above the wavelength 6000 Å.  相似文献   

19.
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 \(\omega (x = y = \dot x = \dot y = 0)\) .  相似文献   

20.
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.  相似文献   

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