Some identities for spheroidal harmonics     

R. G. Langebartel 《Astrophysics and Space Science》1991,175(1):51-56

The spheroidal harmonics expressions $$\left[ {P_{2k}^{2s} \left( {i\xi } \right)P_{2k - 2r}^{2s} \left( \eta \right) - P_{2k - 2r}^{2s} \left( {i\xi } \right)P_{2k}^{2s} \left( \eta \right)} \right]e^{i2s\theta } $$ and $$\left[ {\eta ^2 P_{2k}^{2s} \left( {i\xi } \right)P_{2k - 2r}^{2s} \left( \eta \right) + \xi ^2 P_{2k - 2r}^{2s} \left( {i\xi } \right)P_{2k}^{2s} \left( \eta \right)} \right]e^{i2s\theta } $$ , have ξ²+η² as a factor. A method is presented for obtaining for these two expressions the coefficient of ξ²+η² in the form of a linear combination of terms of the formP _2m ^2s (iξ)P _2n ^2s (η)e ^i2sθ. Explicit formulae are exhibited for the casesr=1, 2, 3 and any positive or zero integersk ands. Such identities are useful in gravitational potential theory for ellipsoidal distributions when matching Legendre function expansions are employed.  相似文献   

Optical spectral decomposition of NGC 4051     

Zhen Yang  Wei-Hao Bian  YueFeng Wang 《Astrophysics and Space Science》2013,348(2):517-526

Considering the host galaxy contribution, a spectral decomposition method is used to reanalyzed the archive data of optical spectra for a narrow line Seyfert 1 galaxy, NGC 4051. The light curves of the continuum f _λ (5100 Å), and Hβ, He ii, Fe ii emission lines are given. We find strong flux correlations between line emissions of Hβ, He ii, Fe ii and the continuum f _λ (5100 Å). These low-ionization lines (Hβ, Fe ii, He ii) have “inverse” intrinsic Baldwin effects. Using the methods of the cross-correlation function and the Monte Carlo simulation, we find the time delays, with respect to the continuum, are $3.45^{+12.0}_{-0.5}~\mbox{days}$ with the probability of 34 % for the intermediate component of Hβ, $6.45^{+13.0}_{-1.0}~\mbox{days}$ with the probability of 65 % for the intermediate component of He ii. From these intermediate components of Hβ and He ii, the calculated central black hole masses are $0.86^{+4.35}_{-0.33}\times 10^{6}$ and $0.82^{+3.12}_{-0.45}\times 10^{6}~M_{\odot }$ . We also find that the time delays for Fe ii are $9.7^{+3.0}_{-5.0}~\mbox{days}$ with the probability of 36 %, $8.45^{+1.0}_{-2.0}~\mbox{days}$ with the probability of 18 % for the total epochs and “subset 1” data, respectively. It seems that the Fe ii emission region is outside of the Hβ emission region.  相似文献   

Photoionization Cross-Section of Chlorine-like Iron     

Sunny Aggarwal  Jagjit Singh  A. K. S. Jha  Man Mohan 《Journal of Astrophysics and Astronomy》2012,33(3):291-301

Semi-relativistic calculations are performed for the photoionization of Fe X (an important coronal ion) from its ground state 3s²3p $^{5}(^{2}P^{0}_{3/2})$ and the first two excited states 3s²3p $^{5}(^{2}P^{0}_{1/2})$ and 3s3p $^{6}(^{2}S_{1/2})$ using the Breit?CPauli R-matrix method. A lowest 41 state eigenfunction expansion for Fe XI is employed to ensure an extensive treatment of auto ionizing resonances that affect the effective cross-sections. In the present calculations, we have considered all the important physical effects like channel coupling, exchange and short range correlation. The present calculations using the lowest 41 target levels of Fe XI in the LSJ coupling scheme are reported and we expect that the present results should enable more accurate modelling of the emission spectrum of highly excited plasma from the optical to the far UV region.  相似文献   

Ultra-high energy neutrino fluxes from supermassive AGN black holes     

G. Ter-Kazarian 《Astrophysics and Space Science》2014,349(2):919-938

We compute the ultra-high energy (UHE) neutrino fluxes from plausible accreting supermassive black holes closely linking to the 377 active galactic nuclei (AGNs). They have well-determined black hole masses collected from the literature. The neutrinos are produced via simple or modified URCA processes, even after the neutrino trapping, in superdense proto-matter medium. The resulting fluxes are ranging from: (1) (quark reactions)— $J^{q}_{\nu\varepsilon}/(\varepsilon_{d}\ \mathrm{erg}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}\,\mathrm{sr}^{-1})\simeq8.29\times 10^{-16}$ to 3.18×10^?4, with the average $\overline{J}^{q}_{\nu\varepsilon}\simeq5.53\times 10^{-10}\varepsilon_{d}\ \mathrm{erg}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}\,\mathrm{sr}^{-1}$ , where ε _d ～10^?12 is the opening parameter; (2) (pionic reactions)— $J^{\pi}_{\nu\varepsilon} \simeq0.112J^{q}_{\nu\varepsilon}$ , with the average $J^{\pi}_{\nu\varepsilon} \simeq3.66\times 10^{-11}\varepsilon_{d}\ \mathrm{erg}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}\,\mathrm{sr}^{-1}$ ; and (3) (modified URCA processes)— $J^{URCA}_{\nu\varepsilon}\simeq7.39\times10^{-11} J^{q}_{\nu\varepsilon}$ , with the average $\overline{J}^{URCA}_{\nu\varepsilon} \simeq2.41\times10^{-20} \varepsilon_{d}\ \mathrm{erg}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}\,\mathrm{sr}^{-1}$ . We conclude that the AGNs are favored as promising pure neutrino sources, because the computed neutrino fluxes are highly beamed along the plane of accretion disk, peaked at high energies and collimated in smaller opening angle θ～ε _d .  相似文献   

Observational constraints on G-corrected holographic dark energy using a Markov chain Monte Carlo method     

Hamzeh Alavirad  Mohammad Malekjani 《Astrophysics and Space Science》2014,349(2):967-974

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   

Normality condition in the ideal resonance problem     

Boris Garfinkel 《Celestial Mechanics and Dynamical Astronomy》1972,6(2):151-166

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   

A family of zero-velocity curves in the restricted three-body problem     

R. Roman  I. Szücs-Csillik 《Astrophysics and Space Science》2014,352(2):481-491

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   

Stability of ion-acoustic waves in a pair-ion plasma with a third species of ions: application to cometary plasmas     

Noble P. Abraham  Sijo Sebastian  G. Sreekala  Savithri E. Devi  G. Renuka  Venugopal Chandu 《Astrophysics and Space Science》2014,349(1):49-55

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   

The Chandra X-ray galaxy clusters at z<1.4: constraints on the evolution of L X −T−M g relations     

I. Babyk  I. Vavilova 《Astrophysics and Space Science》2014,349(1):415-421

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   

Changes in binding energy of interacting galaxies     

M. Narasimha Rao  K. S. V. S. Narasimhan 《Astrophysics and Space Science》1992,194(2):245-250

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   

The stability of differential rotation with non-axisymmetric perturbations     

Chao-Ho Sung 《Astrophysics and Space Science》1975,33(1):127-140

A plane-wave analysis on a simplified scheme based on the Boussinesq approximation and shallow convection is used to establish the necessary conditions for stability of a differentiallyrotating, compressible flow between two coaxial cylinders subject to non-axisymmetric perturbations. To test the adequateness of this simplification, the sufficient conditions for stability are again established which agree with those obtained by a normal-mode analysis on an exact scheme in an earlier paper by the author. This model is applicable to stellar models with rotation Ω=Ω(ω), where ω is the radial distance from the axis of rotation (thez-axis). A necessary condition for stability, in the non-dissipative case, is found to be that $$\frac{1}{\varrho }G_\varpi S_\varpi + \frac{{k_z^2 }}{M}\Phi - \frac{1}{4}\frac{{m^2 }}{M}\left( {D\Omega } \right)^2 \geqslant 0$$ everywhere. Here,m andk _z are the wave numbers in the ø- andz-direction, \(M \equiv k_z^2 + m^2 /\varpi ^2 ,D \equiv d/d\varpi ,G_\varpi \equiv - \varrho ^{ - 1} Dp,\varrho \) the density,p the pressure,S _ω and Φ the Schwarzschild and the Rayleigh discriminants defined as \(S_\varpi \equiv \left( {\gamma p/\varrho } \right)^{ - 2} Dp - D\varrho and \Phi \equiv ^{ - 3} d\left( {\varpi ^4 \Omega ^2 } \right)/d\varpi \) respectively, γ the ratio of specific heats. This condition is also a sufficient one. Some conjectures regarding the stabilizing influence of uniform rotation and the destabilizing influence of differential rotation are also verified. The most striking instability mechanism introduced by shear forces and by radiative dissipation is the excitation of the stable motion of small oscillations into that of oscillations with growing amplitude, i. e., overstability. In the case of radiative dissipation and axisymmetric perturbations, the Goldreich-Schubert criterion is only necessary but not sufficient for stability. Instability sets in as soon as the Schwarzschild criterion is violated. When the perturbations are non-axisymmetric, instability always sets in as overstability as long as rotation is differential. This may explain the convective turbulence in the upper atmosphere where the radiation is active.  相似文献   

On the connection between the Nekhoroshev theorem and Arnold diffusion     

Christos Efthymiopoulos 《Celestial Mechanics and Dynamical Astronomy》2008,102(1-3):49-68

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   

On pulsar-driven isothermal blast-wave models of supernova remanants     

I. Lerche 《Astrophysics and Space Science》1977,48(2):335-356

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>ω_min(?)>ω>ω_max(?)>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>ω_min(?) 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>ω_min(?)>ω_max(?)>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>ω_max(?) (where the dependence of the solutionis analytic on ω and ?) are not valid when carried over into the domain 3>ω_min(?)>ω_max(?)>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   

A relation between Giorgilli-Galgani's and Deprit's algorithms     

Michel Rapaport 《Celestial Mechanics and Dynamical Astronomy》1982,28(3):291-293

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   

Investigation of X-ray cavities in the cooling flow system Abell 1991     

M. B. Pandge  N. D. Vagshette  S. S. Sonkamble  M. K. Patil 《Astrophysics and Space Science》2013,345(1):183-193

We present results based on the systematic analysis of Chandra archive data on the X-ray bright Abell Richness class-I type cluster Abell 1991 with an objective to investigate properties of the X-ray cavities hosted by this system. The unsharp masked image as well as 2-d β model subtracted residual image of Abell 1991 reveals a pair of X-ray cavities and a region of excess emission in the central ～12 kpc region. Both the cavities are of ellipsoidal shape and exhibit an order of magnitude deficiency in the X-ray surface brightness compared to that in the undisturbed regions. Spectral analysis of X-ray photons extracted from the cavities lead to the temperature values equal to $1.77_{-0.12}^{+0.19}~\mathrm{keV}$ for N-cavity and $1.53_{-0.06}^{+0.05}~\mathrm{keV}$ for S-cavity, while that for the excess X-ray emission region is found to be equal to $2.06_{-0.07}^{+0.12}~\mathrm{keV}$ . Radial temperature profile derived for Abell 1991 reveals a positive temperature gradient, reaching to a maximum of 2.63 keV at ～76 kpc and then declines in outward direction. 0.5–2.0 keV soft band image of the central 15′′ region of Abell 1991 reveals relatively cooler three different knot like features that are about 10′′ off the X-ray peak of the cluster. Total power of the cavities is found to be equal to ${\sim}8.64\times10^{43}~\mathrm{erg\,s}^{-1}$ , while the X-ray luminosity within the cooling radius is found to be $6.04 \times10^{43}~\mathrm{erg\,s}^{-1}$ , comparison of which imply that the mechanical energy released by the central AGN outburst is sufficient to balance the radative loss.  相似文献   

Global solution of the ideal resonance problem     

Boris Garfinkel 《Celestial Mechanics and Dynamical Astronomy》1973,8(2):207-212

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).  相似文献   

The Na <Emphasis Type="SmallCaps">i</Emphasis> and Sr <Emphasis Type="SmallCaps">ii</Emphasis> Resonance Lines in Solar Prominences     

G. Stellmacher  E. Wiehr 《Solar physics》2017,292(7):83

We estimate the electron density, \(n_{\mathrm{e}}\), and its spatial variation in quiescent prominences from the observed emission ratio of the resonance lines Na?i?5890 Å (D₂) and Sr?ii?4078 Å. For a bright prominence (\(\tau_{\alpha}\approx25\)) we obtain a mean \(n_{\mathrm{e}}\approx2\times10^{10}~\mbox{cm}^{-3}\); for a faint one (\(\tau _{\alpha }\approx4\)) \(n_{\mathrm{e}}\approx4\times10^{10}~\mbox{cm}^{-3}\) on two consecutive days with moderate internal fluctuation and no systematic variation with height above the solar limb. The thermal and non-thermal contributions to the line broadening, \(T_{\mathrm{kin}}\) and \(V_{\mathrm{nth}}\), required to deduce \(n_{\mathrm{e}}\) from the emission ratio Na?i/Sr?ii cannot be unambiguously determined from observed widths of lines from atoms of different mass. The reduced widths, \(\Delta\lambda_{\mathrm{D}}/\lambda_{0}\), of Sr?ii?4078 Å show an excess over those from Na?D₂ and \(\mbox{H}\delta\,4101\) Å, assuming the same \(T_{\mathrm{kin}}\) and \(V_{\mathrm{nth}}\). We attribute this excess broadening to higher non-thermal broadening induced by interaction of ions with the prominence magnetic field. This is suggested by the finding of higher macro-shifts of Sr?ii?4078 Å as compared to those from Na?D₂.  相似文献   

Multi-wavelength correlation studies of diffuse emission from the Aquila Rift     

S. N. Jyothy  N. V. Sujatha  Narayanankutty Karuppath 《Astrophysics and Space Science》2018,363(10):218

We present a multi-wavelength correlation study of diffuse ultraviolet radiation using GALEX observations towards the Aquila Rift. Apart from airglow and zodiacal emissions, we find a diffuse background of \(1300\mbox{--} 3700~\mbox{ph}\,\mbox{cm}^{-2}\,\mbox{s}^{-1}\,\mbox{sr}^{-1}\,\mathring{\mathrm{A}}^{-1}\) in the far-ultraviolet (FUV, 1350–1750 Å) band and \(1300\mbox{--}2800~\mbox{ph}\,\mbox{cm}^{-2}\,\mbox{s}^{-1}\,\mbox{sr}^{-1}\,\mathring{\mathrm{A}}^{-1}\) in the near-ultraviolet (NUV, 1750–2850 Å) band. The observed diffuse UV emissions are saturated with total as well as neutral hydrogen column density in the region due to high optical depth in UV (\(\tau \), 0.91–23.38). Higher values of FUV/NUV ratio in the region, greater than the threshold value of 0.6, along with the positive correlation between the ratio and FUV intensity are due to excess emission in the FUV band which is absent in the NUV band. We estimated the excess emission to be in the range \(\sim 400\mbox{--} 2700~\mbox{ph}\,\mbox{cm}^{-2}\,\mbox{s}^{-1}\,\mbox{sr}^{-1}\,\mathring{\mathrm{A}}^{-1}\), plausibly due to H₂ fluorescence, ion line emissions and two-photon continuum emissions from the region in the FUV band, which also shows saturation in optically thick regions with N(H₂) as well as \(\mbox{H}\alpha \) emissions. Since N(H₂) and \(\mbox{H}\alpha \) emissions spread all over the region, the excess emission from the field is composite in nature and a detailed spectroscopic analysis is needed to disentangle the contribution from individual components.  相似文献