Hans A. Buchdahl (7.7.1919–7.1.2010)

Goenner, Hubert

doi:10.1007/s10714-010-0965-9

Cited by 4 publications

(4 citation statements)

References 21 publications

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“…Goenner & Havas (2000). No solutions other than for n = 0 and 1 have been found for our particular parameters in closed form and a discussion using symmetry transformations suggests that they do not exist, (Goenner 2001). In the limiting case n = 0, this equation provides an alternative, but less transparent, method to the one presented above for treating homogeneous matter to lowest order.…”

Section: Polytropic Ringsmentioning

confidence: 82%

Uniformly rotating homogeneous and polytropic rings in Newtonian gravity

Petroff¹,

Horatschek²

2008

Monthly Notices of the Royal Astronomical Society

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An analytical method is presented for treating the problem of a uniformly rotating, self‐gravitating ring without a central body in Newtonian gravity. The method is based on an expansion about the thin ring limit, where the cross‐section of the ring tends to a circle. The iterative scheme developed here is applied to homogeneous rings up to the 20th order and to polytropes with the index n= 1 up to the third order. For other polytropic indices no analytic solutions are obtainable, but one can apply the method numerically. However, it is possible to derive a simple formula relating mass to the integrated pressure to leading order without specifying the equation of state. Our results are compared with those generated by highly accurate numerical methods to test their accuracy.

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Section: Polytropic Ringsmentioning

confidence: 82%

Uniformly rotating homogeneous and polytropic rings in Newtonian gravity

Petroff¹,

Horatschek²

2008

Monthly Notices of the Royal Astronomical Society

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“…For the applications of symmetries (Lie group analysis) to differential equations, the interested reader is referred to [39][40][41]. Goenner and Havas [42] and Goenner [43] studied the so-called generalized LaneEmden equation of the first kind,…”

Section: Introductionmentioning

confidence: 99%

“…In Goenner [43], symmetries of eq. (3) were uncovered to explain integrability of (3) for certain values of the parameters considered in Goenner and Havas [42].…”

Section: Introductionmentioning

confidence: 99%

Exact solutions of the generalized Lane–Emden equations of the first and second kind

Muatjetjeja

Khalique

2011

Pramana - J Phys

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In this paper we discuss the integrability of the generalized Lane-Emden equations of the first and second kinds. We carry out their Noether symmetry classification. Various cases for the arbitrary functions in the equations are obtained for which the equations have Noether point symmetries. First integrals of such cases are obtained and also reduction to quadrature of the corresponding Lane-Emden equations are presented. New cases are found.Keywords. Lane-Emden equations of the first and second kinds; Noether point symmetries; first integrals.

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“…The Lane–Emden equation in this case is only approximately solvable, and various numerical or perturbative techniques are used to attack the problem (Natarajan & Lynden‐Bell 1997; McKee & Holliman 1999; Roxburgh & Stockman 1999; Hunter 2001; Honda & Honda 2003; Nouh 2004). In particular, dimensional analysis and symmetry consideration have been partially successful in giving physically interesting results (Goenner 2001). Dimensional analysis has shown that the central pressure in a isothermal gas sphere is independent of the temperature given as P c = GM 2 / R 4 .…”

Section: Introductionmentioning

confidence: 99%

Approximate analytical solutions of the Lane-Emden equation for a self-gravitating isothermal gas sphere

Mirza¹

2009

Monthly Notices of the Royal Astronomical Society

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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.

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Hans A. Buchdahl (7.7.1919–7.1.2010)

Cited by 4 publications

References 21 publications

Uniformly rotating homogeneous and polytropic rings in Newtonian gravity

Uniformly rotating homogeneous and polytropic rings in Newtonian gravity

Exact solutions of the generalized Lane–Emden equations of the first and second kind

Approximate analytical solutions of the Lane-Emden equation for a self-gravitating isothermal gas sphere

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