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1.
A solution of the transfer equation for coherent scattering in stellar atmosphere with Planck's function as a nonlinear function of optical depth, viz. $$B{\text{ }}_v (T) = b_0 + b_1 {\text{ }}e^{ - \beta \tau } $$ is obtained by the method developed by Busbridge (1953).  相似文献   

2.
The equation of transfer for interlocked multiplets has been solved by Laplace transformation and the Wiener-Hopf technique developed by Dasgupta (1978) considering two nonlinear forms of Planck function: i.e., (a) $$B{\text{ }}_{\text{v}} (T) = B(t) = b_0 + b_1 {\text{ }}e^{ - \alpha t} ,$$ (b) $$B{\text{ }}_{\text{v}} (T) = B(t) = b_0 + b_1 t + b_2 E_2 (t).$$ Solutions obtained by Dasgupta (1978) or by Chandrasekhar (1960) may be obtained from our solutions by dropping the nonlinear terms.  相似文献   

3.
An exact solution of the transfer equation for coherent scattering in stellar atmospheres with Planck's function as a nonlinear function of optical depth, of the form $$B_v (T) = b_0 + b_1 {\text{ }}e^{ - \beta \tau } $$ is obtained by the method of the Laplace transform and Wiener-Hopf technique.  相似文献   

4.
An approximate solution of the transfer equation for coherent scattering in stellar atmospheres with Planck's function as a nonlinear function of optical depth, viz., $$B_v \left( T \right) = b_0 + b_1 e^{ - \beta \tau } $$ is obtained by Eddington's method. is obtained by Eddington's method.  相似文献   

5.
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   

6.
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   

7.
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   

8.
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   

9.
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   

10.
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   

11.
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   

12.
The quintessence dark energy model with a kinetic coupling to gravity within the Palatini formalism is studied in this paper. Two different coupling forms: $\hat{R}\partial^{\mu}\phi\partial_{\mu}\phi$ and $\hat {R}_{\mu\nu}\partial^{\mu}\phi\partial^{\nu}\phi$ are analyzed, respectively. We find that both the model with the $\hat{R}\partial^{\mu}\phi\partial_{\mu}\phi$ coupling and the one with the $\hat{R}_{\mu\nu}\partial^{\mu}\phi\partial^{\nu}\phi$ coupling can realize the phantom divide line crossing from phantom to quintessence at late time for its effective equation-of-state. Furthermore, the former can behave like phantom. These features are different from those found in the $\hat {R}\phi^{2}$ coupling case.  相似文献   

13.
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   

14.
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.  相似文献   

15.
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   

16.
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   

17.
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.  相似文献   

18.
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.  相似文献   

19.
The equilibrium points and the curves of zero-velocity (Roche varieties) are analyzed in the frame of the regularized circular restricted three-body problem. The coordinate transformation is done with Levi-Civita generalized method, using polynomial functions of n degree. In the parametric plane, five families of equilibrium points are identified: \(L_{i}^{1}, L_{i}^{2}, \ldots, L_{i}^{n}\) , \(i\in\{ 1,2,\ldots,5 \}, n \in\mathbb{N}^{*}\) . These families of points correspond to the five equilibrium points in the physical plane L 1,L 2,…,L 5. The zero-velocity curves from the physical plane are transformed in Roche varieties in the parametric plane. The properties of these varieties are analyzed and the Roche varieties for n∈{1,2,…,6} are plotted. The equation of the asymptotic variety is obtained and its shape is analyzed. The slope of the Roche variety in \(L_{1}^{1}\) point is obtained. For n=1 the slope obtained by Plavec and Kratochvil (1964) in the physical plane was found.  相似文献   

20.
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   

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