Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 321 / 1 / d

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- Solution d

Solution

Setting

ν = ν_{b} = 0

, the dispersion relation reduces to

λ (λ^{2} + Ω^{2} - 2 π G Σ_{0} k + v_{s}^{2} k^{2}) = 0.

An axisymmetric density mode grows when

2 π G Σ_{0} k - v_{s}^{2} k^{2} > Ω^{2} .

The left-hand side is a concave quadratic function of

k

, with maximum

(π G Σ_{0})^{2} / v_{s}^{2}

at

k = π G Σ_{0} / v_{s}^{2}

. Growth is therefore possible precisely when the Toomre stability criterion is violated:

Q = \frac{v _{s} Ω}{π G Σ _{0}} < 1 .

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