共查询到20条相似文献，搜索用时 31 毫秒
1.
We derive accurate analytic approximations to the solution of the isothermal Lane–Emden equation, a basic equation in Astrophysics that describes the Newtonian equilibrium structure of self-gravitating, isothermal fluid spheres. The solutions we obtain, using analytic arguments and rational approximations, have simple forms, and are accurate over a radial extent that is much larger than that covered by conventional series expansions around the origin. Our best approximation has a maximum error on density of 0.04 % at 10 core radii, and is still within 1 % from an accurate numerical solution at a radius three times larger. 相似文献
2.
In this work, we first establish a simple procedure to obtain with 11-figure accuracy the values of Chandrasekhar’s H-function for isotropic scattering using a closed-form integral representation and the Gauss-Legendre quadrature. Based on the numerical values of the function produced by this method for various combinations of ? 0, the single scattering albedo, and μ, the cosine of the zenith angle θ of the direction of radiation emergent from or incident upon a semi-infinite scattering-absorbing medium, we propose a rational approximation formula with μ 1/4 and \(\sqrt{1-\varpi_{0}}\) as the independent variables. This allows us to reproduce the correct values of H(? 0,μ) within a relative error of 2.1×10?5 without recourse to any iterative procedure or root-finding process. 相似文献
3.
K. Stamnes 《Planetary and Space Science》1980,28(4):427-441
The interaction of a beam of auroral electrons with the atmosphere is described by the linear transport equation, encompassing discrete energy loss, multiple scattering and secondary electrons. A solution to the transport equation provides the electron intensity as a function of altitude, pitch angle (with respect to the geomagnetic field) and energy. A multi-stream (discrete ordinate) approximation to the transport equation is developed. An analytic solution is obtained in this approximation. The computational scheme obtained by combining the present transport code with the energy degradation method of Swartz (1979) conserves energy identically. The theory provides a framework within which angular distributions can be easily calculated and interpreted. Thus, a detailed study of the angular distributions of “non-absorbed” electrons (i.e., electrons that have lost just a small fraction of their incident energy) reveals a systematic variation with incident angle and energy, and with penetration depth. The present approach also gives simple yet accurate solutions in low order multi-stream approximations. The accuracy of the four-stream approximation is generally within a few per cent, whereas two-stream results for backscattered mean intensities and fluxes are accurate to within 10–15%. 相似文献
4.
Bruce Hapke 《Icarus》1984,59(1):41-59
A mathematically rigorous formalism is derived by which an arbitrary photometric function for the bidirectional reflectance of a smooth surface may be corrected to include effects of general macroscopic roughness. The correction involves only one arbitrary parameter, the mean slope angle , and is applicable to surfaces of any albedo. Using physically reasonable assumptions and mathematical approximations the correction expressions are evaluated analytically to second order in . The correction is applied to the bidirectional reflectance function of B. Hapke (1981, J. Geophys. Res.86, 3039–3054). Expressions for both the differential and integral brightnesses are obtained. Photometric profiles on hypothetical smooth and rough planets of low and high albedo are shown to illustrate the effects of macroscopic roughness. The theory is applied to observations of Mercury and predicts the integral phase function, the apparent polar darkening, and the lack of limb brightness surge on the planet. The roughness-corrected bidirectional reflectance function is sufficiently simple that it can be conveniently evaluated on a programmable hand-held calculator. 相似文献
5.
Tõnu Viik 《Astrophysics and Space Science》1993,204(2):213-231
The first two derivatives and the mixed derivative of theH-function with respect to the angular variable and the albedo of single scattering are studied in the case of isotropic scattering using the kernel approximation method. This allows to obtain simple formulae the accuracy of which is estimated to be very good in rather low orders of approximation. Some samples of numerical results are given with eight decimal accuracy. 相似文献
6.
In this work, we studied the Logarithmic Entropy-Corrected Holographic Dark Energy (LECHDE) model in a spatially non-flat universe and in the framework of Ho?ava-Lifshitz cosmology. As infrared cutoff of the system we considered the cut-off recently proposed by Granda and Oliveros which contains two terms, one proportional to H 2 and one to $\dot{H}$ . For the two cases containing non-interacting and interacting Dark Energy (DE) and Dark Matter (DM), we obtained the exact differential equation that determines the evolution of the density parameter. Moreover, we derived the expressions of the deceleration parameter q and, using a parametrization of the equation of state (EoS) parameter ω D of our model as ω D (z)=ω 0+ω 1 z, we derived both the expressions of ω 0 and ω 1 for both non-interacting and interacting cases. All derivations made in this work are done in small redshift approximation and for low redshift expansion of the equation of state (EoS) parameter. 相似文献
7.
V. V. Ivanov 《Astrophysics》2009,52(2):276-287
Multiple Rayleigh scattering is examined in a semi-infinite atmosphere with uniformly distributed primary sources of partially
polarized radiation. The resulting linear polarization is described by a 2×2 matrix transfer equation. A matrix generalization
of Rybicki's two point Q-integral is obtained for this case. It is shown that the Volterra equation for the matrix source function for this problem
is a particular case of our Q integral. Applying the Laplace transform to it yields the matrix form of the Ambartsumyan-Chandrasekhar H-equation. The Volterra equation for Sobolev's matrix resolvent function is another simple consequence of this equation.
Translated from Astrofizika, Vol. 52, No. 2, pp. 301–310 (May 2009). 相似文献
8.
D. Hubert 《Planetary and Space Science》1983,31(1):119-127
The ion distribution function is calculated for the E and the F regions of the auroral latitudes. In these regions the plasma is weakly ionized and there exist convective electric fields which may attain very significant intensities. Boltzmann's equation is solved in the limit where the ion-neutral collision frequency is much lower than the ion gyrofrequency. This solution is obtained in the form of a generalized polynomial series expansion starting from a good zeroth-order approximation. With this weight function, and considering a development to the order of the 32 moments, good approximations are obtained for high electric fields. The resonant charge exchange model and the polarization model are successively considered. The Post-Rosenbluth instability threshold is discussed for the above two models. 相似文献
9.
An exact solution of the transport equation in radiative transfer for an axially symmetric Rayleigh scattering problem in semi-infinite planetary atmosphere both for emergent intensity and intensity at any optical depth has been derived with the help of the Laplace transform and the Wiener-Hopf technique, and by use of the constancy of net flux. Chandrasekhar's results for emergent intensity have been verified. New expressions for theH
l
andH
r
functions have been obtained. 相似文献
10.
The Eddington approximation is often assumed to be useful only for optically thick media having a single-scattering albedo near unity. We present detailed evidence in this paper that, for homogeneous layers illuminated by a beam of radiation, the Eddington approximation predicts albedo and absorptivity reasonably well for all values of optical depth and single-scattering albedo, for several scattering phase functions (Rayleigh, Henyey-Greenstein, and Mie) having asymmetry factors less than or equal to . The worst errors are in the neighborhood of optical depth unity and single-scattering albedo 0.5. The Eddington approximation is further found to maintain good accuracy over almost the full range of incident beam directions and surface albedos. It is least accurate for the Mie phase function example, where one can obtain a dramatic improvement in accuracy by going over to the δ-Eddington approximation; this shows that the forward peak of the Mie phase function, and not its detailed shape, is the primary cause of diminished accuracy in the Eddington approximation. 相似文献
11.
Force-free magnetic fields can be computed by making use of a new numerical technique, in which the fields are represented
by a boundary integral equation based on a specific Green's function. Vector magnetic fields observed on the photospheric
surface can be taken as the boundary conditions of this equation. In this numerical computation, the following two points
are emphasized: (1) A new method for data reduction is proposed, for removing uncertainties in boundary data and determining
the parameter in this Green's function, which is important for solving the boundary integral equation. In this method, the
transverse components of the observed boundary field are calibrated with a linear force-free field model without changing
their azimuth. (2) The computed 3-D fields satisfy the divergence-free and force-free conditions with high precision. The
alignment of these field lines is mostly in agreement with structures in Hα and Yohkoh soft X-ray images. Since the boundary data are calibrated with a linear force-free field model, the computed 3-D magnetic
field can be regarded as a quasi-linear force-free field approximation. The reconstruction of 3-D magnetic field in active
region NOAA 7321 was taken as an example to quantitatively exhibit the capability of our new numerical technique. 相似文献
12.
The measured brightness temperatures of the low-frequency synchrotron radiation from intense extragalactic sources reach 1011–1012 K. If there is some amount of nonrelativistic ionized gas within such sources, it must be heated through induced Compton scattering of the radiation. If cooling via inverse Compton scattering of the same radio radiation counteracts this heating, then the plasma can be heated up to mildly relativistic temperatures kT～10–100 keV. In this case, the stationary electron velocity distribution can be either relativistic Maxwellian or quasi-Maxwellian (with the high-velocity tail suppressed), depending on the efficiency of Coulomb collisions and other relaxation processes. We derive several simple approximate expressions for the induced Compton heating rate of mildly relativistic electrons in an isotropic radiation field, as well as for the stationary electron distribution function and temperature. We give analytic expressions for the kernel of the integral kinetic equation (one as a function of the scattering angle, and the other for an isotropic radiation field), which describes the photon redistribution in frequency through induced Compton scattering in thermal plasma. These expressions can be used in the parameter range [in contrast to the formulas written out previously in Sazonov and Sunyaev (2000), which are less accurate]. 相似文献
13.
We investigate the multiple scattering of radiation in semi-infinite homogeneous atmosphere when the sources of the radiation are distributed inhomogeneous, for example, are created by restricted beams penetrating into the medium. The case of isotropic scattering is considered. It is shown that the density of radiation and the intensity of outgoing radiation for any forms of the sources can be represented as some integrals with the real and imaginary parts of the universal H-function, which satisfies the nonlinear integral equation. We calculated the intensity of radiation emerging from the surface after multiple scattering for the case when a beam with a finite radius incident perpendicular on the medium surface. The results allowed us to estimate quantitatively when the intensity of outgoing radiation in the center of a beam coincides with that for the classical case of unbounded flux (the case considered by Chandrasekhar et al.). We compared our exact solutions with those in the diffusion approximation. For conservative medium the difference is ?20–30%, depending on the particular forms of the radiation sources. For absorbing medium the difference is much larger. Our exact semi-analytical solution can be generalized for the cases of multiple anisotropic scattering of the polarized beams. The presented simple theory can be used at the consideration of close binary systems, flare stars etc. 相似文献
14.
N. Hiotelis 《Astrophysics and Space Science》2013,343(2):755-761
We studied the formation of dark matter haloes assuming a Fractional Brownian Motion (FBM) of trajectories in the plane (S,δ). S is the variance of the smoothed overdensity δ. The values of δ are calculated using a filter of radius R that defines a mass scale M, and thus S is considered as a function of mass M. Focusing on a specific point of the initial Universe and calculating δ as a function of the radius R of the filter a random walk on the plane (S,δ) is established. If the above evolution is a FBM motion then there exist correlations between various mass scales. These correlations depend on a Hurst exponent H. Various mass scales are not correlated for H=1 and the evolution is reduced to the classical Brownian motion. Following Zhang and Hui (Astrophys. J. 641:641, 2006), we constructed a Volterra integral equation for the distribution of trajectories after a first crossing of an ellipsoidal barrier. The integral equation is solved numerically and multiplicity functions of dark matter haloes were derived and compared to the results of N-body simulations. Our results show that for H=1.05 the resulting multiplicity functions are in excellent agreement, much better than the agreement predicted by any other model, with the predictions of N-body simulations. This shows that FBM model is a very promising tool for the study of the formation of structures in the Universe. 相似文献
15.
Albert P. Linnell 《Astrophysics and Space Science》1977,48(1):165-183
Chandrasekhar's (1933) paper on rotational distortion of polytropes contained a perturbation term in the potential which was
linear inv, the rotation parameter. The same paper, and subsequent papers by various authors, developed an analytic expression for the
boundary also linear inv. The latter expression is equivalent to a two term Taylor series about the unperturbed boundary, and is in error by 12% near
critical rotation, for a polytropic index 3.0. The boundary can be located directly from the functions representing density,
potential, and the potential gradient. The boundary error by this procedure is 0.2% near critical rotation. 相似文献
16.
A. R. Bestman 《Astrophysics and Space Science》1991,179(2):217-222
The flow in the boundary layer of a very hot two-component plasma is analysed when the radiative heat flux is given by the exact integral equation expression. The basic nonlinear integro-differential equation is solved by perturbing it about the differential approximation for radiation. In this way some light is shed on the order of accuracy of the differential approximation of radiation. In fact an error of about 4% may be incurred by invoking the differential approximation. 相似文献
17.
Paolo Lanzano 《Earth, Moon, and Planets》1986,34(3):283-304
We examine the problem of the thermoelastic deformation of a spherical Earth with constant elastic parameters heated from within by the spontaneous decay of radiogenic elements.The problem consists of the simultaneous solution of the Navier-Stokes equation and the heat conduction equation. We reach an integrodifferential equation which we solve by means of the Laplace transform and the Green's function approach.We obtain analytic solutions for the temperature distribution and radial deformation as infinite series of functions of the radial distance and time, depending also on a sequence of eigenvalues. We provide particular solutions for the case when the two specific heats C
p and C
v are approximately equal. p ]We believe that our analytic results are applicable to the study of the oceanic lithosphere deformations. Our approach could be successfully applied to ascertain the deformation according to other regimes of internal heating. 相似文献
18.
The behavior of quantum dust ion-acoustic (QDIA) shocks in a plasma including inertialess quantum electrons and positrons, classical cold ions and stationary negative dust grains are studied, using a quantum hydrodynamic model (QHD). The effect of dissipation due to the viscosity of ions is taken into account. The propagation of small but finite amplitude QDIA shocks is governed by the Kortoweg-de Vries-Burgers (KdVB) equation. The existence regions of oscillatory and monotonic shocks will depend on the quantum diffraction parameter (H) and dust density (d) as well as dissipation parameter (η 0). The effect of plasma parameters (d,H,η 0), on these structures is investigated. Results indicate that the thickness and height of monotonic shocks; oscillation amplitude of the oscillatory shock wave and it’s wavelength effectively are affected by these parameters. Additionally, the possibility of propagation of both compressive and rarefactive shocks is investigated. It is found that depending on some critical value of dust density (d c ), which is a function of H, compressive and rarefactive shock waves can’t propagate in model plasma. The present theory is applicable to analyze the formation of nonlinear structures at quantum scales in dense astrophysical objects. 相似文献
19.
The structure of a hydrogen atom situated in an intense magnetic field is investigated. Three approaches are employed. An elementary Bohr picture establishes a crucial magnetic field strength,H a ?5×109G. Fields in excess ofH a are intense in that they are able to modify the characteristic atomic scales of length and binding energy. A second approach solves the Schrödinger equation by a combination of variational methods and perturbation theory. It yields analytic expressions for the wave functions and energy eigenvalues. A third approach determines the energy eigenvalues by reducing the Schrödinger equation to a one-dimensional wave equation, which is then solved numerically. Energy eigenvalues are tabulated for field strengths of 2×1010G and 2×1012 G. It is found that at 2×1012 G the lowest energy eigenvalue is changed from ?13.6 eV to about ?180 eV in agreement with previous variational computations. 相似文献
20.
Jabbar R. Jabbar 《Astrophysics and Space Science》1984,100(1-2):447-449
Emden's differential equation for polytropic gas spheres, as well as Chandrasekhar's equation for associated Emden function of zero order and indexn=0(0.1)4.9 have been integrated numerically with a CDC 7600 automatic computer to 16 decimal places; and the surface values of the respective functions compared with previous work. 相似文献