Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 321 / 1 / c

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Solution

The corotation radius is where the stellar spin matches Keplerian rotation:

r_{co} = (\frac{GM}{Ω _{m}^{2}})^{1/3} .

(1)

A parcel forced into corotation at

r_{m}

has outward centrifugal acceleration

Ω_{m}^{2} r_{m}

. This exceeds the inward gravitational acceleration when

r_{m} > r_{co} ⟺ Ω_{m} > Ω_{K} (r_{m}) .

(2)

This is the onset condition for a centrifugal propeller effect. It does not by itself guarantee escape to infinity. For example, a parcel released with only azimuthal speed

Ω_{m} r_{m}

has nonnegative specific orbital energy only if

\frac{1}{2} Ω_{m}^{2} r_{m}^{2} - \frac{GM}{r _{m}} \geq 0 ⟺ r_{m} \geq 2^{1/3} r_{co} .

(3)

Additional magnetic work, radial motion or pressure can alter that escape condition.

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Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 321 / 1 / c

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