Collisions in a system of conical jet/counterjet outflows

TL;DR

This work develops a parsimonious geometric-statistical framework to estimate jet-jet collisions among protostellar outflows in star-forming clusters. By combining Monte Carlo simulations for two-outflow encounters with analytic extensions to $N_j$ outflows, the authors derive a practical expression for the expected number of collisions as a function of the jet opening angle $\alpha$ and the jet-to-separation ratio $L_j/R_c$, and they validate the approach against 3D gasdynamics simulations. Application to real systems (NGC 1333 and the Orion Nebula Cluster) demonstrates how observed outflow morphologies can be used to infer collision probabilities and interaction rates, with angular width emerging as a dominant parameter. The study also introduces the volume filling factor $f_V(t)$ to connect cumulative jet activity to the evolving dynamical state of the cluster, highlighting how increasing jet activity promotes a highly interactive regime.

Abstract

Stars predominantly form in compact, non-hierarchical clusters. The gas outflows ejected by protostars can intersect and interact with each other, resulting in complex interactions that affect the dynamics, morphology, and evolution of these outflows. Determining the probability of an encounter between them requires a Bayesian approach that considers the collimation, length (or age), and separation between young stellar objects in the clusters. In this study, we employ a Monte Carlo approach to estimate this probability as a function of the jet opening angle and the ratio between the jet length and the separation between stars. We propose a function that predicts the number of interactions within a cluster based on the opening angle of the gas outflows ejected by protostars.

Figures (5)

• Figure 1: The probability of collision between two "double cone" outflows as a function of the $L_j/L_{12}$ jet length to source separation ratio. The results of sets of $10^4$ "Bernoulli experiments" for double cones with half-opening angles $\alpha=2.5^\circ$ (crosses), $5^\circ$ (squares), $10^\circ$ (full circles) and $15^\circ$ (open circles) are shown. The solid curves are obtained from the analytic fit of equation (\ref{['p']}) for the same $\alpha$ values.
• Figure 2: Configuration of a system of $N_j=5$ biconical bipolar outflows with a half-opening angle $\alpha=10^\circ$. The outflow directions are randomly chosen, and the source positions are obtained by sampling a uniform source distribution within a sphere of radius $R_c$. The projection of the cones on the $xy$-plane is shown in green, and the regions of physical (3D) superpositions between cones are shown in white. The outflow cones have $L_j=3R_c$ lengths.
• Figure 3: Top: Number of jet superpositions as a function of the half-opening angle $\alpha$ for a system of $N_j=10$ bi-conical outflows. The results for (single) Bernoulli experiments of randomly directed jets from sources with a uniform spatial distribution within a sphere of radius $R_c$ have been computed for cones of length $R_j=R_c$ (blue crosses) and $R_j=3R_c$ (red circles). The solid curves correspond to the results obtained with the analytic fit of equation (\ref{['nc']}), and the dashed curves are the corresponding $\pm \sigma$ envelopes. Bottom: number of jet superpositions for a system of cones with $\alpha=5^\circ$ as a function of the number $N_j$ of jets with sources within a sphere of radius $R_c$. The results for jets of lengths $R_j=R_c$ (blue crosses and curves) and $R_j=3R_c$ (red circles and curves) are shown.
• Figure 4: Column density time frames for $t=0$ (the initial condition, top frame), 200 (center) and 400 yr time-integration (bottom) obtained from the 3D gasdynamic simulation described in the text. The column densities are displayed with the logarithmic scale given (in cm$^{-2}$ by the top bar. The column densities correspond to integrations along the $z$-axis, and the $(x,y)$ coordinates are given in units of the $R_c=5\times 10^{16}$ cm radius of the outflow source position distribution. The source position and direction values for the 5 bipolar outflows are the same as the ones of the geometrical model of Figure 2.
• Figure 5: Total expected number of jet-jet collisions, $N_{col}$, as function of the volume filling factor, $f_V(t)$, for the NGC 1333 region. The curve corresponds to a system of $N_j =7$ bipolar outflows, each with an initial jet length of $0.033\,\mathrm{pc}$, confined within a spherical cluster of radius $R_c = 0.038\,\mathrm{pc}$, and adopting a half-opening angle of $\alpha = 3^\circ$. The initial and final points are highlighted in red. The dashed vertical line marks the point where the volume filling factor reaches unity ($f_V = 1$), indicating that jets fully occupy the cluster volume. Text boxes indicate key physical parameters, including the jet velocity, characteristic scales, and corresponding jet lengths at initial and final times. This graph illustrates how collision rates increase as the jets dynamically expand and progressively fill the available volume.