The physical filament ends when the <mass density> first reaches zero. If $x_1\simeq2.4$ is the first positive zero of the <Bessel function> $J_0$, then
$$
\boxed{
R_f=x_1a
=x_1\sqrt{\frac{K+B_0^2/(8\pi\rho_0^2)}{2\pi G}}}.
$$
The first zero must be used because continuing to the next lobe would make the density negative.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-314/2/a/ii/solution