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1.
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   

2.
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   

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.
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   

5.
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   

6.
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   

7.
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   

8.
We investigate the ‘equilibrium’ and stability of spherically-symmetric self-similar isothermal blast waves with a continuous post-shock flow velocity expanding into medium whose density varies asr ?ω ahead of the blast wave, and which are powered by a central source (a pulsar) whose power output varies with time ast ω?3. We show that:
  1. for ω<0, no physically acceptable self-similar solution exists;
  2. for ω>3, no solution exists since the mass swept up by the blast wave is infinite;
  3. ? must exceed zero in order that the blast wave expand with time, but ?<2 in order that the central source injects a finite total energy into the blast wave;
  4. for 3>ωmin(?)>ω>ωmax(?)>0, where $$\begin{gathered} \omega _{\min } (\varphi ){\text{ }} = {\text{ }}2[5{\text{ }} - {\text{ }}\varphi {\text{ }} + {\text{ }}(10{\text{ }} + {\text{ 4}}\varphi {\text{ }} - {\text{ 2}}\varphi ^2 )^{1/2} ]^2 [2{\text{ }} + {\text{ (10 }} + {\text{ 4}}\varphi {\text{ }} - {\text{ 2}}\varphi ^2 {\text{)}}^{{\text{1/2}}} ]^{ - 2} , \hfill \\ \omega _{\max } (\varphi ){\text{ }} = {\text{ }}2[5{\text{ }} - {\text{ }}\varphi {\text{ }} - {\text{ }}(10{\text{ }} + {\text{ 4}}\varphi {\text{ }} - {\text{ 2}}\varphi ^2 )^{1/2} ]^2 [2{\text{ }} - {\text{ (10 }} + {\text{ 4}}\varphi {\text{ }} - {\text{ 2}}\varphi ^2 {\text{)}}^{{\text{1/2}}} ]^{ - 2} , \hfill \\ \end{gathered} $$ two critical points exist in the flow velocity versus position plane. The physically acceptable solution must pass through the origin with zero flow speed and through the blast wave. It must also pass throughboth critical points if \(\varphi > \tfrac{5}{3}\) , while if \(\varphi< \tfrac{5}{3}\) it must by-pass both critical points. It is shown that such a solution exists but a proper connection at the lower critical point (for ?>5/3) (through whichall solutions pass with thesame slope) has not been established;
  5. for 3>ω>ωmin(?) it is shown that the two critical points of (iv) disappear. However a new pair of critical points form. The physically acceptable solution passing with zero flow velocity through the origin and also passing through the blast wave mustby-pass both of the new critical points. It is shown that the solution does indeed do so;
  6. for 3>ωmin(?)>ωmax(?)>ω it is shown that the dependence of the self-similar solution on either ω or ? is non-analytic and therefore, inferences drawn from any solutions obtained in ω>ωmax(?) (where the dependence of the solutionis analytic on ω and ?) are not valid when carried over into the domain 3>ωmin(?)>ωmax(?)>ω;
  7. all of the physically acceptable self-similar solutions obtained in 3>ω>0 are unstable to short wavelength, small amplitude but nonself-similar radial velocity perturbations near the origin, with a growth which is a power law in time;
  8. the physical self-similar solutions are globally unstable in a fully nonlinear sense to radial time-dependent flow patterns. In the limit of long times, the nonlinear growth is a power law in time for 5<ω+2?, logarithmic in time for 5>ω+2?, and the square of the logarithm in time for 5=ω+2?.
The results of (vii) and (viii) imply that the memory of the system to initial and boundary values does not decay as time progresses and so the system does not tend to a self-similar form. These results strongly suggest that the evolution of supernova remnants is not according to the self-similar form.  相似文献   

9.
Dynamical systems with three degrees of freedom can be reduced to the study of a fourdimensional mapping. We consider here, as a model problem, the mapping given by the following equations: $$\left\{ \begin{gathered} x_1 = x_0 + a_1 {\text{ sin (}}x_0 {\text{ + }}y_0 {\text{)}} + b{\text{ sin (}}x_0 {\text{ + }}y_0 {\text{ + }}z_{\text{0}} {\text{ + }}t_{\text{0}} {\text{)}} \hfill \\ y_1 = x_0 {\text{ + }}y_0 \hfill \\ z_1 = z_0 + a_2 {\text{ sin (}}z_0 {\text{ + }}t_0 {\text{)}} + b{\text{ sin (}}x_0 {\text{ + }}y_0 {\text{ + }}z_{\text{0}} {\text{ + }}t_{\text{0}} {\text{) (mod 2}}\pi {\text{)}} \hfill \\ t_1 = z_0 {\text{ + }}t_0 \hfill \\ \end{gathered} \right.$$ We have found that as soon asb≠0, i.e. even for a very weak coupling, a dynamical system with three degrees of freedom has in general either two or zero isolating integrals (besides the usual energy integral).  相似文献   

10.
An attempt has been made to obtain an expression for the rate of stellar mass loss using dimensional analysis. The best expression for O and B stars is of the form: $$\dot M = A'{\text{ }}\left( {\frac{1}{{G^{1/2} c^4 }}} \right){\text{ }}L^{\text{2}} {\text{ (}}R/M)^{{\text{3/2}}} .$$ It is also found thatA′ increases as one goes from B→O stars and from O→O(f)→O(f)), but is not sensitive to luminosity.  相似文献   

11.
The ratio between the Earth's perihelion advance (Δθ) E and the solar gravitational red shift (GRS) (Δø s e)a 0/c 2 has been rewritten using the assumption that the Newtonian constant of gravitationG varies seasonally and is given by the relationship, first found by Gasanalizade (1992b) for an aphelion-perihelion difference of (ΔG)a?p . It is concluded that $$\begin{gathered} (\Delta \theta )_E = \frac{{3\pi }}{e}\frac{{(\Delta \phi _{sE} )_{A_0 } }}{{c^2 }}\frac{{(\Delta G)_{a - p} }}{{G_0 }} = 0.038388 \sec {\text{onds}} {\text{of}} {\text{arc}} {\text{per}} {\text{revolution,}} \hfill \\ \frac{{(\Delta G)_{a - p} }}{{G_0 }} = \frac{e}{{3\pi }}\frac{{(\Delta \theta )_E }}{{(\Delta \phi _{sE} )_{A_0 } /c^2 }} = 1.56116 \times 10^{ - 4} . \hfill \\ \end{gathered} $$ The results obtained here can be readily understood by using the Parametrized Post-Newtonian (PPN) formalism, which predicts an anisotropy in the “locally measured” value ofG, and without conflicting with the general relativity.  相似文献   

12.
Available satellite, rocket and balloon observations on cosmic X- and gamma ray background are critically examined to understand the spectral characteristics of the radiation. Appropriate corrections have been applied to the balloon observations to account for the multiple Compton scattering of X-rays in the atmosphere. It is shown that within the experimental uncertainties, all the available observations of cosmic X- and gamma ray background in the energy range 1 keV-1 MeV are consistent with a single spectrum of type $${\text{d}}N/{\text{d}}E = 30 E^{ - 2.0 \pm 0.2} {\text{photons cm}}^{{\text{ - 2}}} {\text{s}}^{{\text{ - 1}}} {\text{sr}}^{{\text{ - 1}}} {\text{keV}}^{{\text{ - 1}}} $$ .  相似文献   

13.
The possibility of chemical ‘trapping’ of the Ar+ ion in the reaction $$v{\text{ }} + {\text{ }}^{{\text{37}}} {\text{Cl}} \to {\text{ }}^{{\text{37}}} {\text{Ar}}^{\text{ + }} + {\text{ e}}^ - ,$$ when it takes place in tetrachloroethylene (C2Cl4) liquid, is examined in detail. It is concluded that if trapping does take place, the rate is much smaller than the charge neutralization rate. Therefore, this niechanism cannot explain the observed small rate of Ar production in the Brookhaven solar neutrino experiment. A detailed examination of a number of experiments which are sensitive to possible trapping lends strong support to this conclusion.  相似文献   

14.
Non-linear stability of the libration point L 4 of the restricted three-body problem is studied when the more massive primary is an oblate spheroid with its equatorial plane coincident with the plane of motion, Moser's conditions are utilised in this study by employing the iterative scheme of Henrard for transforming the Hamiltonian to the Birkhoff's normal form with the help of double D'Alembert's series. It is found that L 4 is stable for all mass ratios in the range of linear stability except for the three mass ratios: $$\begin{gathered} \mu _{c1} = 0.0242{\text{ }}...{\text{ }}{}^{{\text{\_\_}}}0.1790{\text{ }}...{\text{ }}A_1 , \hfill \\ \mu _{c2} = 0.0135{\text{ }}...{\text{ }}{}^{{\text{\_\_}}}0.0993{\text{ }}...{\text{ }}A_1 , \hfill \\ \mu _{c3} = 0.0109{\text{ }}...{\text{ }}{}^{{\text{\_\_}}}0.0294{\text{ }}...{\text{ }}A_1 . \hfill \\ \end{gathered} $$   相似文献   

15.
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   

16.
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   

17.
The dark energy model with the equation of state \(p_{\mathit{DE}} = {-} \rho_{\mathit{DE}} - A\rho_{\mathit{DE}}^{\alpha} \) is studied in Kaluza-Klein space time. The model comprises and provides realization of several types of singularities in different parameter regimes. We discuss the finite-time singularities into four classes and explicitly present the models which give rise to these singularities by assuming the form of the equation of state of dark energy. Also, we discussed the models in terms of the cosmological redshift and some observational parameters.  相似文献   

18.
  相似文献   

19.
This paper summarises an investigation of chaos in a toy potential which mimics much of the behaviour observed for the more realistic triaxial generalisations of the Dehnen potentials, which have been used to model cuspy triaxial galaxies both with and without a supermassive black hole. The potential is the sum of an anisotropic harmonic oscillator potential, ${\text{V}}_{\text{0}} = \frac{1}{2}\left( {a^2 x^2 + b^2 y^2 + c^2 z^2 } \right)$ , and aspherical Plummer potential, ${\text{V}}_{\text{P}} = M_{BH} /\sqrt {r^2 + \varepsilon ^2 } $ , with $r^2 = x^2 + y^2 + z^2$ . Attention focuses on three issues related tothe properties of ensembles of chaotic orbits which impact on chaotic mixing and the possibility of constructing self-consistent equilibria:(1) What fraction of the orbits are chaotic? (2) How sensitive are the chaotic orbits, that is, how large are their largest (short time) Lyapunov exponents? (3) To what extent is the motion of chaotic orbits impeded by Arnold webs, that is, how 'sticky' are the chaotic orbits? These questions are explored as functions of the axis ratio a: b: c, black hole mass M BH, softening length ε, and energy E with the aims of understanding how the manifestations of chaos depend onthe shape of the system and why the black hole generates chaos. The simplicity of the model makes it amenable to a perturbative analysis. That it mimics the behaviour of more complicated potentials suggests that much of this behaviour should be generic.  相似文献   

20.
Nonlinear magnetosonic waves propagating in a magnetic neutral sheet are investigated within the framework of a fluid model. It is shown that the behavior of the magnetosonic waves is governed by a ‘modified Burgers equation’ with an additional termc(η)? due to the relevant slowly varying background plasma parameter (density or magnetic field), $$\frac{{\partial \phi }}{{\partial \eta }}$$ where ?(ξ, η) is the amplitude of the wave, \(\xi = \int {k_x } {\text{d}}x + k_y y - \omega t\) , and η=εx is the coordinate stretched by a smallness parameter ε. When we consider fast magnetosonic waves propagating toward the neutral region across the magnetic field, they grow and undergo rapid steepening after passing through the neutral region; i.e., shock formation is promoted by the background inhomogeneity. By the numerical computation of the above equation, the time evolution is examined for two initial disturbances, the pulse type (gaussian) and the wave train type (sinusoidal wave). The relevance of the interactions between the magnetosonic shock waves and the neutral sheet plasma to a triggering mechanism of sympathetic flares is also suggested.  相似文献   

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