# sol5 - Astro 405/505, fall semester 2004 Homework, 5th set,...

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Astro 405/505, fall semester 2004 Homework, 5th set, solutions. Problem 1: Electrostatic waves The momentum conservation equation (14.5 in the notes) now reads m e n e ~ V e ∂t + ( ~ V e · ∇ ) ~ V e + ~ P = - e n e ~ E Be careful, because c s will also be perturbed! Using ∂P ∂r = ∂P ∂ρ ∂ρ ∂r ∂P ∂ρ = c 2 s ∂P ∂r = c 2 s A Z m p ∂n e ∂r we derive the perturbation equation in Fourier representation. n e, 0 ı ω v 1 = e n e, 0 m e E 1 + c 2 s A Z m p m e ı k n e, 1 which must be solved together with the three unchanged equations. ı k E 1 = - 4 π e n e, 1 ı ω E 1 = - 4 π e v 1 n e, 0 ω n e, 1 = n e, 0 k v 1 The solution is ω 2 = ω 2 p + c 2 s A Z m p m e k 2 = ω 2 p + γ κT m e k 2 so the mean thermal velocity of the electrons enters the dispersion relation as a characteristic scale. The phase velocity is v φ = ω k = v u u t ω 2 p k 2 + γ κT m e ' ω p k small k q γ κT m e large k The group velocity is v gr

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## This note was uploaded on 02/29/2012 for the course PHYS 227 taught by Professor Rabe during the Fall '08 term at Rutgers.

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sol5 - Astro 405/505, fall semester 2004 Homework, 5th set,...

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