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
Classical Dynamics of Particles and Systems
5th Edition
ISBN:9780534408961
Author:Stephen T. Thornton, Jerry B. Marion
Publisher:Stephen T. Thornton, Jerry B. Marion
Chapter9: Dynamics Of A System Of Particles
Section: Chapter Questions
Problem 9.56P
Related questions
Question
![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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb360a251-a88e-4e24-b6e1-ceeebc35a45e%2F9957d11f-66f7-4faa-8633-3ba5fc0626b5%2F15whekm_processed.jpeg&w=3840&q=75)
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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