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1.
Robert A. Van Gorder 《Celestial Mechanics and Dynamical Astronomy》2011,109(2):137-145
We apply the δ-expansion method to a transformed Lane–Emden equation. The results are then transformed back, and we recover analytical solutions
to the Lane–Emden equation of the second kind (which describes Bonnor–Ebert gas spheres) in a special case. The rapid convergence
of the method results in qualitatively accurate solutions in relatively few iterations, as we see when we compare the obtained
analytical solutions to numerical results. 相似文献
2.
We apply the Boubaker Polynomials Expansion Scheme (BPES) in order to obtain analytical–numerical solutions to two separate Lane–Emden problems: the Lane–Emden initial value problem of the first kind (describing the gravitational potential of a self-gravitating spherically symmetric polytropic gas), the Lane–Emden initial value problem of the second kind (describing isothermal gas spheres embedded in a pressurized medium at the maximum possible mass allowing for hydrostatic equilibrium). Both types of problems are simultaneously singular and nonlinear, and hence can be challenging to solve either numerically or analytically. We find that the BPES allows us to compute numerical solutions to both types of problems, and an error analysis demonstrates the accuracy of the method. In all cases, we demonstrate that relative error can be controlled to less than 1%. Furthermore, we compare our results to those of Hunter (2001). [Hunter, C., 2001. Series solutions for polytropes and the isothermal sphere. Monthly Notices of the Royal Astronomical Society, 328 839–847] and Mirza (2009). Approximate analytical solutions of the Lane–Emden equation for a self-gravitating isothermal gas sphere. Monthly Notices of the Royal Astronomical Society, 395 2288–2291. in order to demonstrate the accuracy of our method. 相似文献
3.
Robert A. Van Gorder 《New Astronomy》2011,16(2):65-67
Perturbation solutions are obtained for the Lane–Emden equation of the second kind which describe Bonnor–Ebert gas spheres. In particular, we employ the field-theoretic perturbative procedure due to Bender et al. to obtain analytical solutions to the nonlinear initial value problem. We find that the method allows one to construct perturbation solutions which converge rapidly to the true solutions in many cases, as it allows one to more accurately represent the influence of nonlinear terms in the linearized equations. The rapid convergence of the method results in qualitatively accurate solutions in relatively few iterations. 相似文献
4.
《New Astronomy》2013
In this paper, we present a new second kind Chebyshev (S2KC) operational matrix of derivatives. With the aid of S2KC, an algorithm is described to obtain numerical solutions of a class of linear and nonlinear Lane–Emden type singular initial value problems (IVPs). The idea of obtaining such solutions is essentially based on reducing the differential equation with its initial conditions to a system of algebraic equations. Two illustrative examples concern relevant physical problems (the Lane–Emden equations of the first and second kind) are discussed to demonstrate the validity and applicability of the suggested algorithm. Numerical results obtained are comparing favorably with the analytical known solutions. 相似文献
5.
The quasisimilar theory is used to investigate the solution of the blast wave problem with generalized geometries in a non-ideal
gas satisfying the equation of state of the Van der Waals type. Here it is assumed that the distribution of normalized velocity,
pressure and density are nearly similar in the narrow range of the shock strength. A comparison between approximate analytical
solution and numerical solution of the problem is presented for the cylindrical geometry. The numerical solutions are presented
for the generalized geometry in a non-ideal gas. It is also assessed as to how the non-idealness of the gas affects the behavior
of the flow parameters. 相似文献
6.
Babur M. Mirza 《Monthly notices of the Royal Astronomical Society》2009,395(4):2288-2291
The isothermal Lane–Emden equation arises in many astrophysical problems, in particular in modelling of a self-gravitating, polytropic gas in a spherically symmetric configuration. In this work, the isothermal Lane–Emden equation is investigated using the fractional approximation technique. The method provides an efficient and accurate way of obtaining approximate analytic solution to the Lane–Emden equation thus is useful in the modelling of self-gravitating gaseous spheres in astrophysics. 相似文献
7.
Equilibrium masses of slowly rotating interstellar gas clouds have been calculated using an equation of state for the interstellar gas developed by Penston and Brown. The rotation is found to increase the equilibrium masses, but still the cloud masses are not as large as indicated by other considerations. 相似文献
8.
Gas phases of the interstellar medium (ISM) coexist locally, penetrate each other and mix by means of dynamical and plasmaphysical
processes. E.g. heat conduction from the hot to the cooler gas leads to energy and mass exchange between the gas phases. Analytical
solutions exist under which evaporation of cloudy material or condensation of hot gas onto the clouds' surface dominate. Since
these results are derived for stationary and static conditions and under ideal assumptions, they do not necessarily hold for
a dynamical ISM. On the other hand, the mass and energy exchange between the gas phases is of great importance for the energy
budget of the ISM and by this influences the evolution of galaxies. This led us to investigate the evolution of interstellar
clouds in a hot gas by means of numerical simulations. At first, we compare static models with the analytical results and
found that interstellar clouds with parameters requiring analytically evaporation are, in contrast, accreting surrounding
material if self-gravitation and cooling are implied. For the more realistic case, where clouds are embedded in a streaming
hot gas, the models show that Kelvin-Helmholtz instability which leads to the disruption of the clouds is suppressed by heat
conduction so that the clouds are stabilized to survive.
This revised version was published online in September 2006 with corrections to the Cover Date. 相似文献
9.
Self-similar solutions for one-dimensional unsteady flow of a non-ideal gas behind an exponential shock propagating into a uniform atmosphere are investigated. The equation of state in the suitable form for such gases is found to be fairly accurate at low-density region. 相似文献
10.
Similarity solutions for propagation of plane relativistic shock waves through a medium of decreasing nucleon density and approaching the edge of the gas as well as for the subsequent motion of the gas after the shock front arrives at the vacuous boundary are studied in this paper. The medium in the pre-disturbed stage is assumed cold and in the disturbed stage its equation of state is taken as that of a photonic gas. 相似文献
11.
We propose a novel mathematical method to construct an exact polytropic sphere in self-gravitating hydrostatic equilibrium, improving the non-linear Poisson equation. The central boundary condition for the present equation requires a ratio of gas pressure to total one at the centre, which is uniquely identified by the whole mass and molecular weight of the system. The special solution derived from the Lane–Emden equation can be reproduced. This scheme is now available for modelling the molecular cloud cores in interstellar media. The mass–radius relation of the first core is found to be consistent with the recent results of radiation hydrodynamic simulations. 相似文献
12.
The paper presents an analytical theory for propagation of induced star formation. The model takes into consideration a nonlocal influence in the interactions of gas clouds and stellar systems and time-delay processes in formation of new stars in clouds which have experienced the influence of previous generations of stars. The analytical solutions for propagating waves of star formation are obtained. Numerical simulations of basic equations demonstrate the formation of stationary waves and confirm the analytical results.In two-dimensional case the formation of circular propagating waves of star formation is demonstrated. These structures can be associated with observed rings of enhanced star formation (Appleton and Struck-Marcel, 1987). 相似文献
13.
Numerical calculations have been made of the gravitational axisymmetric collapse of isothermal gas clouds endowed with angular momentum. The evolutionary study is based on the so-called Fluid-in-Cell method coupled to an efficient algebraic algorithm which allows the Poisson equation to be integrated by means of block tri-diagonal matrices. The results, at ages slight larger than the initial free-fall time, indicate that flattened disk-shaped structures are formed in the central region of the clouds-in good agreement with the previous analytical results predicted by the authors. 相似文献
14.
Similarity solutions are obtained for spherical radiation-driven shock waves propagating in a non-uniform atmosphere at rest obeying a density power law. Approximate analytical solutions are also obtained and found to be in good agreement with the numerical solutions. The effect of the parameter characterizing the initial density distribution of the gas on solutions of the flow field is studied in detail. It is also shown analytically that the shock wave propagates as an overdriven detonation. 相似文献
15.
《New Astronomy》2022
Theoretical work addressing the role of external pressure with magnetic fields in collapsing molecular clouds is important in building a comprehensive theory of star formation(SF). In many SF studies turbulence, magnetic fields, and self-gravity are described as the key dynamical processes involved in SF. However, the importance of external pressure in collapsing strongly magnetized clouds has not yet been particularly explained. Magnetic fields transport excess angular momentum from the central core while external pressure compresses the cloud. Thus the outflow of angular momentum, and on the other hand, the compression made by external pressure are the cause for matter falling onto the central core from the envelope. Therefore, external pressure facilitates the collapse of gas and the inflow of matter. In this work, we show theoretically how the strong magnetic field is dragged inward by the external pressure during the core collapse and formulate the critical mass of the core in the presence of external pressure. 相似文献
16.
T. Penz N.V. Erkaev H. Lammer H. Gunell S. Barabash A. Milillo 《Planetary and Space Science》2004,52(13):1157-1167
Mars Global Surveyor detected cold electrons above the Martian ionopause, which can be interpreted as detached ionospheric plasma clouds. Similar observations by the Pioneer Venus Orbiter electron temperature probe showed also extreme spatial irregularities of electrons in the form of plasma clouds on Venus, which were explained by the occurrence of the Kelvin-Helmholtz instability. Therefore, we suggest that the Kelvin-Helmholtz instability may also detach ionospheric plasma clouds on Mars. We investigate the instability growth rate at the Martian ionopause resulting from the flow of the solar wind for the case where the interplanetary magnetic field is oriented normal to the flow direction. Since the velocity shear near the subsolar point is very small, this area is stable with respect to the Kelvin-Helmholtz instability. We found that the highest flow velocities are reached at the equatorial flanks near the terminator plane, while the maximum plasma density in the terminator plane appears at the polar areas. By comparing the instability growth rate with the magnetic barrier formation time, we found that the instability can evolve into a non-linear stage at the whole terminator plane but preferably at the equatorial flanks. Escape rates of O+ ions due to detached plasma clouds in the order of about 2×1023-3×1024 s-1 are found. Thus, atmospheric loss caused by the Kelvin-Helmholtz instability should be comparable with other non-thermal loss processes. Further, we discuss our results in view of the expected observations of heavy ion loss rates by ASPERA-3 on board of Mars Express. 相似文献
17.
Joel E. Tohline 《Icarus》1985,61(1):10-21
The scalar virial equation can be used to elucidate many interesting properties of equilibrium gas clouds when the effects of surface pressure, rotation, self-gravity, and internal isothermal gas pressure are considered simultaneously. Details regarding the internal structure of rotating isothermal gas clouds are ignored in order to obtain an analytical expression describing global cloud properties. Excellent agreement is obtained between the simple analytical model and other previously published, more detailed models in physical regimes where other models have been constructed. For the first time, a physical connection is drawn between the surface-pressure-dominated equilibrium models of S. W. Stahler (1983, Astrophys. J.268, 155–184) and the rotation-dominated models of C. Hayashi, S. Narita, and S. M. Miyama (1982, Prog. Theor. Phys.68, 1949–1966). Stable axisymmetric models of any mass and angular momentum can be constructed. Using the analytic expression for virial equilibrium as a foundation, all rotating, isothermal collapse calculations can now be well understood. Limiting properties of isothermal clouds are outlined, and realistic “starting” models for cloud collapse are proposed. 相似文献
18.
Kazunari Iwasaki Toru Tsuribe 《Monthly notices of the Royal Astronomical Society》2008,387(4):1554-1562
A new self-similar solution describing the dynamical condensation of a radiative gas is investigated under a plane-parallel geometry. The dynamical condensation is caused by thermal instability. The solution is applicable to generic flow with a net cooling rate per unit volume and time ∝ρ2 T α , where ρ, T and α are the density, temperature and a free parameter, respectively. Given α, a family of self-similar solutions with one parameter η is found in which the central density and pressure evolve as follows: ρ( x = 0, t ) ∝ ( t c − t )−η/(2−α) and P ( x = 0, t ) ∝ ( t c − t )(1−η)/(1−α) , where t c is the epoch at which the central density becomes infinite. For η∼ 0 the solution describes the isochoric mode, whereas for η∼ 1 the solution describes the isobaric mode. The self-similar solutions exist in the range between the two limits; that is, for 0 < η < 1 . No self-similar solution is found for α > 1 . We compare the obtained self-similar solutions with the results of one-dimensional hydrodynamical simulations. In a converging flow, the results of the numerical simulations agree well with the self-similar solutions in the high-density limit. Our self-similar solutions are applicable to the formation of interstellar clouds (H i clouds and molecular clouds) by thermal instability. 相似文献
19.
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
20.
In this paper we propose Sinc-Collocation method for solving Lane–Emden equation which is a nonlinear ordinary differential equation on a semi-infinite interval. It is found that Sinc procedure converges with the solution at an exponential rate. This method is utilized to reduce the computation of this problem to some algebraic equations. We also compare this solution with some well-known results and show that it is accurate. 相似文献