Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 314 / 2 / a / i / Solution

The collapse is perpendicular to the initial field, so magnetic flux freezing preserves the mass-to-flux ratio of each material flux tube:

\frac{B _{z}}{ρ} = \frac{B _{0}}{ρ _{0}}, B_{z} = \frac{B _{0}}{ρ _{0}} ρ .

With no toroidal field, radial magnetostatic equilibrium is

\frac{d}{d R} (P + \frac{B _{z}^{2}}{8 π}) = - ρ \frac{d Φ}{d R} .

Define the effective polytropic constant

K_{m e ff} = K + \frac{B _{0}^{2}}{8 π ρ _{0}^{2}} .

Then gas and magnetic pressure combine as

P + B_{z}^{2} / (8 π) = K_{eff} ρ^{2}

. Dividing equilibrium by

ρ

, differentiating, and using the cylindrical Poisson equation

\frac{1}{R} \frac{d}{d R} (R \frac{d Φ}{d R}) = 4 π Gρ

gives

\frac{1}{R} \frac{d}{d R} (R \frac{d ρ}{d R}) + \frac{2 π G}{K _{eff}} ρ = 0 .

Put

a^{2} = K_{eff} / (2 π G)

. This is the order-zero Bessel differential equation, and regularity on the axis together with

ρ (0) = ρ_{1}

yields

ρ (R) = ρ_{1} J_{0} (R / a) .

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