Accelerated power series solution of polytropic and isothermal gas spheres

Nouh, M. I.

doi:10.1016/j.newast.2004.02.003

Cited by 67 publications

(54 citation statements)

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“…It is this singularity that prevents real power series solutions about the center to converge to the outer surface once the condition n > 1.9121 is satisfied. However, as shown in Nouh (2004), an Euler transformation gives power series that do converge up to the outer radius. Moreover, the Euler-transformed series converge significantly faster than the series obtained in Roxburgh & Stockman (1999), which are limited to finite radii whenever n > 5 by a complex conjugate pair of singularities.…”

Section: The Case Of the Arbitrary Polytropic Index Nmentioning

confidence: 98%

“…(3) have been intensively studied in the astrophysical and mathematical literature (Mohan & Al-Bayati 1980;Roxburgh & Stockman 1999;Hunter 2001;Nouh 2004). The series solutions are represented as θ = ∞ k=0 a k ξ 2k and θ n = ∞ k=0 b k ξ 2k , respectively, with a 0 = b 0 = 1 (Nouh 2004). One can define the radius of convergence of these series as the distance from ξ = 0 to the closest singularity of θ(ξ) in the complex ξ-plane.…”

Section: The Case Of the Arbitrary Polytropic Index Nmentioning

confidence: 99%

“…One can define the radius of convergence of these series as the distance from ξ = 0 to the closest singularity of θ(ξ) in the complex ξ-plane. Non-linear ordinary differential equations, such as the LaneEmden equation for n > 1, can have two kinds of singularities, fixed and movable (Nouh 2004). The LaneEmden equation for polytropic index n > 1 and its n → ∞ limit, corresponding to the limit of the isothermal sphere equation, are singular at some negative value of the radius squared (Nouh 2004).…”

Section: The Case Of the Arbitrary Polytropic Index Nmentioning

confidence: 99%

“…Non-linear ordinary differential equations, such as the LaneEmden equation for n > 1, can have two kinds of singularities, fixed and movable (Nouh 2004). The LaneEmden equation for polytropic index n > 1 and its n → ∞ limit, corresponding to the limit of the isothermal sphere equation, are singular at some negative value of the radius squared (Nouh 2004). It is this singularity that prevents real power series solutions about the center to converge to the outer surface once the condition n > 1.9121 is satisfied.…”

Section: The Case Of the Arbitrary Polytropic Index Nmentioning

confidence: 99%

“…Thus the Lane-Emden Eq. (3) was solved by using a power series method in (Mohan & Al-Bayati 1980;Roxburgh & Stockman 1999;Hunter 2001;Nouh 2004), where the convergence of the solutions was also studied.…”

Section: Introductionmentioning

confidence: 99%

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Exact power series solutions of the structure equations of the general relativistic isotropic fluid stars with linear barotropic and polytropic equations of state

Harkó

Mak

2016

Astrophys Space Sci

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Exact power series solutions of the structure equations of the general relativistic isotropic fluid stars with linear barotropic and polytropic equations of state T. Harko 1,2 • M. K. Mak 3Abstract Obtaining exact solutions of the spherically symmetric general relativistic gravitational field equations describing the interior structure of an isotropic fluid sphere is a long standing problem in theoretical and mathematical physics. The usual approach to this problem consists mainly in the numerical investigation of the Tolman-Oppenheimer-Volkoff and of the mass continuity equations, which describes the hydrostatic stability of the dense stars. In the present paper we introduce an alternative approach for the study of the relativistic fluid sphere, based on the relativistic mass equation, obtained by eliminating the energy density in the Tolman-Oppenheimer-Volkoff equation. Despite its apparent complexity, the relativistic mass equation can be solved exactly by using a power series representation for the mass, and the Cauchy convolution for infinite power series. We obtain exact series solutions for general relativistic dense astrophysical objects described by the linear barotropic and the polytropic equations of state, respectively. For the polytropic case we obtain the exact power series solution corresponding to arbitrary values of the polytropic index n. The explicit form of the solution is presented for the polytropic index n = 1, and for the indexes n = 1/2 and n = 1/5, respectively. The case of n = 3 is also considered. In each case the exact power series solution is compared with the exact numerical solutions, which are reproduced by the power series solutions truncated to seven

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Section: The Case Of the Arbitrary Polytropic Index Nmentioning

confidence: 98%