The Lane-Emden equation and its generalizations

Abstract

where r is a constant, models the thermal behaviour of a spherical cloud of gas acting under the mutual attraction of its molecules and subject to the classical laws of thermodynamics. This equation was proposed by Lane (1) (see also (3)) and studied in detail by Emden (2). Fowler (4; 5) considered a generalization of Eq. (1), called Emden-Fowler equation (6), where the last term is replaced by x ν−1 y r .The Lane-Emden equation (1) also models the equilibria of nonrotating fluids in which internal pressure balances self-gravity. When spherically symmetric solutions of Eq. (1) appeared in (7), they got the attention of astrophysicists. In the latter half of the twentieth century, some interesting applications of the isothermal solution (singular isothermal sphere) and its nonsingular modifications were used in the structures of collisionless systems such as globular clusters and early-type galaxies (8; 9).The work of Emden (2) also got the attention of physicists outside the field of astrophysics who investigated the generalized polytropic forms of the Lane-Emden equation (1) for specific polytropic indices r. Some singular solutions for r = 3 were produced by Fowler (4; 5) and the Emden-Fowler equation in the literature was established, while the works of Thomas (10) and Fermi (11) (29)). Usually, for r = 5, only a one-parameter family of solutions is presented. A more general form of (1), in which the