P = n k T

κ ∝ \frac{ρ}{T^{3.5}}

λ = 4L

f = \frac{v_{s}}{λ} = \frac{v_{s}}{4L}

period = \frac{1}{f} = 4\frac{L}{v_{s}}

period ∼ 4 (\frac{R}{v_{s}} )

v_{s} = \sqrt{\frac{T}{μ}}

v_{s} ∼ \sqrt{\frac{P}{ρ}}

P ∼ \frac{GMρ}{R}

v_{s} ∼ \sqrt{\frac{GM}{R}}

period ∼ \frac{4R}{\sqrt{\frac{GM}{R}}} = 4 (\frac{GM}{R^{3}} )^{−1/2}

L = (4πR^{2}) σT^{4}

L ∝ R^{2}  →  R ∝ L^{1/2}

period ∝ M^{−1/2} R^{3/2}

period ∝ M^{−1/2} L^{3/4}

L ∼ M^{4}  →  M ∼ L^{1/4}

period ∝ (L^{1/4})^{−1/2} L^{3/4} ∼ L^{5/8}