In a steady axisymmetric outflow, <mass conservation> makes $2\pi r\Sigma(r)v_r(r)$ independent of $r$. The <steady radial dust outflow> therefore has
$$
\boxed{\Sigma(r)\propto\frac1{rv_r}
\propto\frac1{\sqrt{(x-1)[(2\beta-1)x+1]}}},
\qquad x=\frac r{a_p}.
$$
It has an integrable $(r-a_p)^{-1/2}$ pile-up just outside the source ring, where the radial speed starts from zero, and approaches $Sigma\propto r^{-1}$ at large radius.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 316 / 2 / vi / Solution

For an axisymmetric steady dust outflow with surface density $\Sigma$ and radial speed $v_r$, number or mass conservation gives $2\pi r\Sigma v_r=\text{constant}$.


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 Incoming links: Steady radial dust outflow