## Appendix A Derivation of Arrival Time to Circular Plumes
Multiplying Equation (100a) by \(\cos(m\theta)\) and integrating over \((r,\theta)\in(R^{-1}-\varepsilon,R^{-1}+\varepsilon)\times[0,2\pi)\), we find in the limit as \(\varepsilon\to 0\) that  
\[\lim_{\varepsilon\to 0}\int_{r=R^{-1}-\varepsilon}^{R^{-1}+\varepsilon}\int_{ \theta=0}^{2\pi}\cos(m\theta)\frac{\partial}{\partial r}\left(r\frac{\partial \hat{p}}{\partial r}\right)drd\theta=-1. \tag{102}\]  
The orthogonality of \(\{\cos m\theta\}_{m=0}^{\infty}\) over \([0,2\pi)\) yields the jump condition  
\[\frac{A_{n}c}{R}\left[\int_{\theta=0}^{2\pi}\cos^{2}(n\theta)\, d\theta\right]\left[\left(\frac{I_{n}(cR^{-1})}{K_{n}(cR^{-1})}-\frac{I_{n}(c)}{K _{n}(c)}\right)K_{n}^{\prime}(cR^{-1})\right.\\ \left.-I_{n}^{\prime}(cR^{-1})+\frac{I_{n}(c)}{K_{n}(c)}K_{n}^{ \prime}(cR^{-1})\right]=-1. \tag{103}\]  
The jump condition (103) can be simplified with the Wronskian identity  
\[K_{n}(z)I_{n}^{\prime}(z)-I_{n}(z)K_{n}^{\prime}(z)=\frac{1}{z}, \tag{104}\]  
to find that  
\[A_{n}=K_{n}(cR^{-1})\left[\int_{\theta=0}^{2\pi}\cos^{2}(n\theta)\,d\theta \right]^{-1}= \begin{cases}\frac{K_{0}(cR^{-1})}{2\pi},&n=0,\\ \frac{K_{n}(cR^{-1})}{\pi},&n\geq 1.\end{cases}  \tag{105}\]  
The Laplace transform of the flux over the disk is then given by  
\[\frac{\partial\hat{p}}{\partial r}\Big{|}_{r=1}=\sum\limits_{n=0}^{\infty} \frac{A_{n}}{K_{n}(c)}\cos(n\theta)=\frac{1}{2\pi}\frac{K_{0}(cR^{-1})}{K_{0}( c)}+\frac{1}{\pi}\sum\limits_{n=1}^{\infty}\frac{K_{n}(cR^{-1})}{K_{n}(c)} \cos n\theta. \tag{106}\]  
Returning to the variable to \(\sqrt{s}=c\), we now apply the inverse Laplace transform of Equation (106) and evaluate the contour integrals:  
\[\frac{1}{2\pi i}\int_{\Gamma}\frac{K_{n}(\sqrt{s}R^{-1})}{K_{n}(\sqrt{s})}e^{ \pi x}\,ds,\qquad n=0,1,2,\dots, \tag{107}\]  
where the integral is evaluated over the complex curve \(\Gamma=\{\gamma+iy\mid y\in(-\infty,\infty)\}\). The parameter \(\gamma=\text{Re}(\Gamma)\) is chosen such that singularities of the integrand lie to the left of \(\Gamma\). Since the integrand hasno poles and its only singularity is a branch cut on the negative real axis, we deform the contour to a hairpin along \(\mathrm{Re}(s)<0\) and introduce the substitution \(s=-w^{2}\). The integral becomes  
\[\frac{1}{\pi i}\int_{\omega=0}^{\infty}\Bigg{[}\frac{K_{n}(i\omega R ^{-1})}{K_{n}(i\omega)}-\frac{K_{n}(-i\omega R^{-1})}{K_{n}(-i\omega)}\Bigg{]} \omega e^{-\omega^{2}\tau}\,d\omega\\ =\frac{2}{\pi}\int_{\omega=0}^{\infty}\Bigg{[}\frac{J_{n}(\omega )Y_{n}(\omega R^{-1})-J_{n}(\omega R^{-1})Y_{n}(\omega)}{Y_{n}^{2}(\omega)+J_{ n}^{2}(\omega)}\Bigg{]}\omega e^{-\omega^{2}\tau}\,d\omega, \tag{100}\]  
where we have used the identity (NIST DLMF ([88] SS 10.27))  
\[K_{n}(-iz)=\frac{\pi}{2}\Big{[}-Y_{n}(z)+ij_{n}(z)\Big{]}, \tag{101}\]  
to write the integrand in terms of the Bessel functions of the first and second kinds, \(J_{n}(z)\) and \(Y_{n}(z)\), respectively. The expression for the flux \(\mathcal{J}(\tau,\theta)=p_{r}|_{r=1}\) is now  
\[\mathcal{J}(\tau,\theta)=\frac{1}{\pi^{2}}\int_{\omega=0}^{\infty }\Bigg{[}\frac{J_{0}(\omega)Y_{0}(\omega R^{-1})-J_{0}(\omega R^{-1})Y_{0}( \omega)}{Y_{0}^{2}(\omega)+J_{0}^{2}(\omega)}\Bigg{]}\,\omega e^{-\omega^{2} \tau}\,d\omega\\ +\frac{2}{\pi^{2}}\sum_{n=1}^{\infty}\cos(n\theta)\int_{\omega=0} ^{\infty}\Bigg{[}\frac{J_{n}(\omega)Y_{n}(\omega R^{-1})-J_{n}(\omega R^{-1}) Y_{n}(\omega)}{Y_{n}^{2}(\omega)+J_{n}^{2}(\omega)}\Bigg{]}\omega e^{-\omega^{2} \tau}\,d\omega. \tag{102}\]  
The total flux to the trapping region is given by  
\[S(\tau)=\int_{\theta=0}^{2\pi}\mathcal{J}(\tau,\theta)\,d\theta=\frac{2}{\pi} \int_{\omega=0}^{\infty}\Bigg{[}\frac{J_{0}(\omega)Y_{0}(\omega R^{-1})-J_{0} (\omega R^{-1})Y_{0}(\omega)}{Y_{0}^{2}(\omega)+J_{0}^{2}(\omega)}\Bigg{]} \omega e^{-\omega^{2}\tau}\,d\omega. \tag{103}\]  
This gives the distribution of arrival times to the inner disk. For a fixed time \(\tau=\tau^{*}\), the distribution of arrival angles \(\theta\) is then given by \(\mathcal{J}(\tau^{*},\theta)\).