# P r p δpp r 23495 impl ıcito na derivac ao desta

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∂P 0 ∂r 0 - P 0 ∂ δP/P 0 ∂r 0 (23.495) Impl´ ıcito na deriva¸c˜ ao desta equa¸c˜ ao est˜ao as condi¸c˜ oes ¨ r 0 = 0 e ˙ r 0 = 0, j´a que o estado de equil´ ıbrio ´ e completamente est´atico. Neste ponto da an´alise tomamos o caminho tradicional em teoria de perturba¸c˜ ao e assumimos que todos as perturba¸c˜ oes prefixadas por δ pode ser decompostas nas componentes de Fourier com o elemento de tempo re- presentado por exponenciais. Desta maneira, introduzimos a componente espacial do deslocamento relativo do fluido, ζ ( r 0 ), como δr r 0 ( t, r 0 ) = ζ ( r 0 ) e iσt (23.496) onde a exponencial representa a descri¸c˜ ao da evolu¸c˜ ao temporal do desloca- mento e ζ ( r 0 ), que depende somente de r 0 (isto ´ e, do elemento de massa), pode ser considerado como a forma do deslocamente no instante zero de 531
d ( δP/P ) dr = - d ln P dr 4 ζ + σ 2 r 3 GM r ζ + δP P (23.500) onde o fator r 3 /GM r aparece como resultado de usarmos a equa¸c˜ ao do equil´ ıbrio hidrost´atico para eliminar os termos contendo dP/dr .

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