Consider the dispersion relation of a linear spiral density wave perturbation (equation 4.45 in Chapter 3 of the lecture notes). The Toomre's parameter is (V2)²K defined as Q = = 1/2 лGσ。 , where (V2) 1/2 is the root-mean-square turbulent plus thermal velocity, the epicyclic frequency, σ the unperturbed surface density of the disc and G the gravitational constant. If Q = 0.7, for each of the the following values of the radial wavenumber k choose whether the perturbation makes the spiral pattern stable or unstable. k k = II = 2 4 K K 1/2 stable (1/2 stable (V2)1/2 Κ k = (V unstable 2 1/2 ÷ → k = 2. K (V2)1/2 unstable → k = K (V2)1/2 unstable →
Consider the dispersion relation of a linear spiral density wave perturbation (equation 4.45 in Chapter 3 of the lecture notes). The Toomre's parameter is (V2)²K defined as Q = = 1/2 лGσ。 , where (V2) 1/2 is the root-mean-square turbulent plus thermal velocity, the epicyclic frequency, σ the unperturbed surface density of the disc and G the gravitational constant. If Q = 0.7, for each of the the following values of the radial wavenumber k choose whether the perturbation makes the spiral pattern stable or unstable. k k = II = 2 4 K K 1/2 stable (1/2 stable (V2)1/2 Κ k = (V unstable 2 1/2 ÷ → k = 2. K (V2)1/2 unstable → k = K (V2)1/2 unstable →
College Physics
11th Edition
ISBN:9781305952300
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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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Transcribed Image Text:Consider the dispersion relation of a linear spiral density wave perturbation (equation 4.45 in Chapter 3 of the lecture notes). The Toomre's parameter is
(V2)²K
defined as Q = =
1/2
лGσ。
,
where (V2) 1/2 is the root-mean-square turbulent plus thermal velocity, the epicyclic frequency, σ the unperturbed surface
density of the disc and G the gravitational constant. If Q = 0.7, for each of the the following values of the radial wavenumber k choose whether the
perturbation makes the spiral pattern stable or unstable.
k
k
=
II
=
2
4
K
K
1/2 stable
(1/2 stable
(V2)1/2
Κ
k = (V unstable
2 1/2
÷
→
k = 2.
K
(V2)1/2
unstable →
k =
K
(V2)1/2
unstable →
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