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

3.
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   

4.
Using γ-ray data detected by Fermi Large Area Telescope (LAT) and multi-wave band data for 35 TeV blazars sample, we have studied the possible correlations between different broad band spectral indices ( $\alpha_{\rm r.ir}$ , $\alpha_{\rm{r.o}}$ , $\alpha_{\rm r.x}$ , $\alpha_{\rm r.\gamma}$ , $\alpha_{\rm{ir.o}}$ , $\alpha_{\rm ir.x}$ , $\alpha_{\rm ir.\gamma}$ , $\alpha_{\rm o.x}$ , $\alpha_{\rm o.\gamma}$ , $\alpha_{\rm r.x}$ , $\alpha_{\rm x.\gamma}$ ) in all states (average/high/low). Our results are as follows: (1) For our TeV blazars sample, the strong positive correlations were found between $\alpha_{\rm r.ir}$ and $\alpha_{\rm{r.o}}$ , between $\alpha_{\rm r.ir}$ and $\alpha_{\rm r.x}$ , between $\alpha_{\rm r.ir}$ and $\alpha_{\rm r.\gamma}$ in all states (average/high/low); (2) For our TeV blazars sample, the strong anti-correlations were found between $\alpha_{\rm r.ir}$ and $\alpha_{\rm x.\gamma}$ , between $\alpha_{\rm{r.o}}$ and $\alpha_{\rm ir.\gamma}$ , between $\alpha_{\rm{r.o}}$ and $\alpha_{\rm o.\gamma}$ , between $\alpha_{\rm{r.o}}$ and $\alpha_{\rm x.\gamma}$ , between $\alpha_{\mathrm{ir.o}}$ and $\alpha_{\rm o.\gamma}$ , between $\alpha_{\rm r.x}$ and $\alpha_{\rm x.\gamma}$ , between $\alpha_{\rm ir.x}$ and $\alpha_{\rm x.\gamma}$ in all states (average/high/low). The results suggest that the synchrotron self-Compton radiation (SSC) is the main mechanism of high energy γ-ray emission and the inverse Compton scattering of circum-nuclear dust is likely to be a important complementary mechanism for TeV blazars. Our results also show that the possible correlations vary from state to state in the same pair of indices, Which suggest that there may exist differences in the emitting process and in the location of the emitting region for different states.  相似文献   

5.
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: $$J_2 = ( - 0.13 \pm 0.41) \times 10^{ - 6} $$ .  相似文献   

6.
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   

7.
A solution of the transfer equation for coherent scattering in stellar atmosphere with Planck's function as a nonlinear function of optical depth, viz., $$B_v (T) = b_0 + b_1 {\text{ }}e^{ - \beta \tau } $$ is obtained by the method of discrete ordinates originally due to Chandrasekhar.  相似文献   

8.
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9.
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.  相似文献   

10.
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 = ε1/2. 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).  相似文献   

11.
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   

12.
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.  相似文献   

13.
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.  相似文献   

14.
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   

15.
A popular model of a cometary plasma is hydrogen (H+) with positively charged oxygen (O+) as a heavier ion component. However, the discovery of negatively charged oxygen (O?) ions enables one to model a cometary plasma as a pair-ion plasma (of O+ and O?) with hydrogen as a third ion constituent. We have, therefore, studied the stability of the ion-acoustic wave in such a pair-ion plasma with hydrogen and electrons streaming with velocities $V_{d\mathrm{H}^{+}}$ and V de , respectively, relative to the oxygen ions. We find the calculated frequency of the ion-acoustic wave with this model to be in good agreement with the observed frequencies. The ion-acoustic wave can also be driven unstable by the streaming velocity of the hydrogen ions. The growth rate increases with increasing hydrogen density $n_{\mathrm{H}^{+}}$ , and streaming velocities $V_{d\mathrm{H}^{+}}$ and V de . It, however, decreases with increasing oxygen ion densities $n_{\mathrm{O}^{+}}$ and $n_{\mathrm{O}^{-}}$ .  相似文献   

16.
Using pseudo-time-dependent models and three different reaction networks, a detailed study of the dominant reaction pathways for the formation of cyanopolyynes and their abundances in TMC-1 is presented. The analysis of the chemical reactions show that for the formation of cyanopolyynes there are two major chemical regimes. First, early times of less than ～104 yrs when ion-molecule reactions are dominant, the main chemical route for the formation of larger cyanopolyynes is $$C_n H^ + \xrightarrow{N}C_n N^ + \xrightarrow{{H_2 }}HC_n N^ + \xrightarrow{{H_2 }}H_2 C_n N^ + \xrightarrow{{e^ - }}HC_n N$$ wheren=5, 7, and 9. Second, at times greater than 104 yrs, when neutral-neutral reactions become dominant, two major reaction routes for the formation of cyanopolyynes are (a), $$HCN\xrightarrow{{C_2 H}}HC_3 N\xrightarrow{{C_2 H}}HC_5 N\xrightarrow{{C_2 H}}HC_7 N\xrightarrow{{C_2 H}}HC_9 N$$ and (b) $$C_n H_2 + CN \to HC_{n + 1} N + H,{\text{ }}n = 4,6, and 8$$ depending on the reaction network used. The results indicate that for route (a) large abundances ofC 2 H (fractional abundances of ～10?7), and for route (b) large abundances ofC 2 H 2 are required in order to reproduce the observed abundances of cyanopolyynes. The calculated abundances of cyanopolyynes show great sensitivity to the value of extinction particularly att?5×105 yrs (i.e. photochemical timescale). The effect of other physical parameters, such as the cosmic-ray ionization abundances are also examined. In general, the model calculations show that the observed abundances of cyanopolyynes can be achieved by pseudo-time-dependent models at late times of several million years.  相似文献   

17.
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=ε1/2, 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.  相似文献   

18.
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.  相似文献   

19.
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   

20.
The development of the post-nova light curve of V1500 Cyg inUBV andHβ, for 15 nights in September and October 1975 are presented. We confirm previous reports that superimposed on the steady decline of the light curve are small amplitude cyclic variations. The times of maxima and minima are determined. These together with other published values yield the following ephemerides from JD 2 442 661 to JD 2 442 674: $$\begin{gathered} {\text{From}} 17 {\text{points:}} {\text{JD}}_{ \odot \min } = 2 442 661.4881 + 0_{^. }^{\text{d}} 140 91{\text{n}} \hfill \\ \pm 0.0027 \pm 0.000 05 \hfill \\ {\text{From}} 15 {\text{points:}} {\text{JD}}_{ \odot \max } = 2 442 661.5480 + 0_{^. }^{\text{d}} 140 89{\text{n}} \hfill \\ \pm 0.0046 \pm 0.0001 \hfill \\ \end{gathered} $$ with standard errors of the fits of ±0 . d 0052 for the minima and ±0 . d 0091 for the maxima. Assuming V1500 Cyg is similar to novae in M31, we foundr=750 pc and a pre-nova absolute photographic magnitude greater than 9.68.  相似文献   

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