Self-gravitating instability of a gas-core ¯uid cylinder pervaded by magnetic ®elds Ahmed E. Radwan * Department of Mathematics, Faculty of Science, Ain-Shams University, Abbassia, Cairo, Egypt Received 29 January 1997 Abstract The self-gravitating instability of gas-core ¯uid cylinder pervaded by magnetic ®elds has been investigated. The eigenvalue relation is derived, discussed analytically, and the results are con®rmed numerically. The gravitational instability is decreasing with increasing S (0 < S < 1), neutral stability occurs when S = 1, and a surprising result is that the model is unstable not only for perturbation with long wavelengths, but also for those of short wavelengths as 1 < S < 1, where S = s 2 /s 1 (s 1 is the gas density and s 2 is the ¯uid density). This is physically interpreted with correlation of spiral arms of galaxies instabilities and also of destruction of interstellar clouds. The inclusion of the electromagnetic force in¯uence leads to the modi®cation of the eigenvalue relation, and improves the gravitational instability, in several cases it suppressed it completely. The magnetic ®eld always has a stabilizing in¯uence for all wavelengths in all symmetric and asymmetric modes of disturbances. As the model is acting upon the combined eect of the self-gravitating and the electromagnetic forces, the densities gas- ¯uid ratio S is strongly stabilizing as 0 < S < 1, but it is still destabilizing as 1 < S < 1, as for the case in which the model is acting upon the self-gravitating only, where the ®eld intensity H o =0. Several results for the cases (S = 0, H o $ 0), (S = 0, H o =0) and (S $ 0, H = 0) are well documented in the literature and are reviewed here. # 1998 Elsevier Science Ltd. All rights reserved. 1. Introduction The response of a self-gravitating ¯uid cylinder to small axisymmetric disturbances has been carried out by Chandrasekhar and Fermi [1] using the principle of energy. Meanwhile, Ogansian [2] investigated the problem using the normal mode analysis, see also Chandrasekhar [3]. Chandrasekhar [4] (p. 516) has also studied the stability of a self- gravitating full ¯uid cylinder submerged in a self-gravitating vacuum, not only to axisymmetric International Journal of Engineering Science 37 (1999) 123±141 0020-7225/98/$19.00 # 1998 Elsevier Science Ltd. All rights reserved. PII: S0020-7225(98)00055-X PERGAMON * Tel.: 00 202 415 3788.