Let

c_{s}^{2} \sim P_{g} / ρ

. The assumed magnetic pressure gives

\frac{P _{m}}{P _{g}} \sim \frac{ρ r Ω c _{s}}{ρ c _{s}^{2}} = \frac{r Ω}{c _{s}} = Ma,

so the plasma beta obeys

β = P_{g} / P_{m} \sim Ma^{- 1}

. In the magnetically dominated regime, vertical hydrostatic equilibrium gives

P_{m} / ρ \sim Ω^{2} H^{2}

. Combining this with

P_{m} / ρ \sim r Ω c_{s}

yields

\frac{H}{r} \sim (\frac{c _{s}}{r Ω})^{1/2} \sim Ma^{- 1/2} .

ii

Solution

P \sim ρ Ω^{2} H^{2}, F \sim α P Ω H, F \sim \frac{σ T ^{4}}{κ ρ H}, ρ \sim \frac{Σ}{H} .

Magnetic support also gives

Ω^{2} H^{2} \sim r Ω T^{1/2}

, so

T \sim Ω^{2} H^{4} / r^{2}

. Eliminating

P, F, ρ, T

gives

H^{14} \propto Σ^{2} r^{8} Ω^{- 5} .

For Keplerian

Ω \propto r^{- 3/2}

H \propto r^{31/28} Σ^{1/7} .

\overset{ν}{ˉ} \sim α P / (ρ Ω) \sim α Ω H^{2}

, and therefore

\overset{ν}{ˉ} \propto r^{5/7} Σ^{2/7}, a = \frac{5}{7}, b = \frac{2}{7} .

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