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Consider a non-rotating circular thin disc of gas of radius R. The only forces present in the system are pressure forces within the disc and its self-gravity. The disc is surrounded by empty space. In the disc is present a surface density perturbation of the type 01 = 010e (wt-kr) where σ10 is the amplitude of the perturbation, t represents time, r the radial coordi- nate from the centre of the disc, w is the angular frequency of the perturbation and k its wavenumber. Under the influence of the above perturbation, the linear stability of the disc is determined by the following dispersion relation w² = u²k² - 2πGook, where u is the sound speed in the disc, σ the surface density of the disc, and G is the gravitational constant. 1. Using the dispersion relation and appropriate definitions derive an expression of the group velocity of the small perturbations as a function of u, σo and their wavelength. 2. State the criterion for the disc to be stable and then show that the disc is stable if R< u² 4Gσo

Consider a non-rotating circular thin disc of gas of radius R. The only forces present in the system are pressure forces within the disc and its self-gravity. The disc is surrounded by empty space. In the disc is present a surface density perturbation of the type 01 = 010e (wt-kr) where σ10 is the amplitude of the perturbation, t represents time, r the radial coordi- nate from the centre of the disc, w is the angular frequency of the perturbation and k its wavenumber. Under the influence of the above perturbation, the linear stability of the disc is determined by the following dispersion relation w² = u²k² - 2πGook, where u is the sound speed in the disc, σ the surface density of the disc, and G is the gravitational constant. 1. Using the dispersion relation and appropriate definitions derive an expression of the group velocity of the small perturbations as a function of u, σo and their wavelength. 2. State the criterion for the disc to be stable and then show that the disc is stable if R< u² 4Gσo

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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Consider a non-rotating circular thin disc of gas of radius R. The only forces present in the system are pressure forces within the disc and its self-gravity. The disc is surrounded by empty space. In the disc is present a surface density perturbation of the type 01 = 010e (wt-kr) where σ10 is the amplitude of the perturbation, t represents time, r the radial coordi- nate from the centre of the disc, w is the angular frequency of the perturbation and k its wavenumber. Under the influence of the above perturbation, the linear stability of the disc is determined by the following dispersion relation w² = u²k² - 2πGook, where u is the sound speed in the disc, σ the surface density of the disc, and G is the gravitational constant. 1. Using the dispersion relation and appropriate definitions derive an expression of the group velocity of the small perturbations as a function of u, σo and their wavelength. 2. State the criterion for the disc to be stable and then show that the disc is stable if R< u² 4Gσo
Transcribed Image Text:Consider a non-rotating circular thin disc of gas of radius R. The only forces present in the system are pressure forces within the disc and its self-gravity. The disc is surrounded by empty space. In the disc is present a surface density perturbation of the type 01 = 010e (wt-kr) where σ10 is the amplitude of the perturbation, t represents time, r the radial coordi- nate from the centre of the disc, w is the angular frequency of the perturbation and k its wavenumber. Under the influence of the above perturbation, the linear stability of the disc is determined by the following dispersion relation w² = u²k² - 2πGook, where u is the sound speed in the disc, σ the surface density of the disc, and G is the gravitational constant. 1. Using the dispersion relation and appropriate definitions derive an expression of the group velocity of the small perturbations as a function of u, σo and their wavelength. 2. State the criterion for the disc to be stable and then show that the disc is stable if R< u² 4Gσo
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