Sink Particles in Astrophysical Simulations

• Sink particles are discrete Lagrangian objects that model unresolved gravitational collapse and serve as proxies for protostellar or cluster formation.
• They form when local gas densities exceed the Jeans length threshold, ensuring physical collapse criteria are met while preventing artificial fragmentation.
• They accrete mass, momentum, and angular momentum from surrounding gas, enabling efficient simulation of feedback processes and stellar population mapping.

A sink particle is a discrete, Lagrangian object introduced into hydrodynamical simulations once local gas collapse exceeds the numerical resolution limit. In both grid-based and particle-based codes, sink particles allow simulations to bypass artificially unresolved gravitational collapse, acting as subgrid models representing protostars, stellar clusters, or, more generally, unresolved point-mass accretors. Sinks dynamically interact with gas and with each other via gravity, accrete gas from their surroundings, and serve as proxies for star-formation and feedback physics in computational astrophysics.

1. Motivations for Introducing Sink Particles

The formation of sink particles is necessitated by the inability of computational grids or particle sets to resolve arbitrarily small Jeans lengths ($\lambda_J = (\pi c_s^2/G\rho)^{1/2}$). Without intervention, simulations suffer from artificial fragmentation or excessive computational cost as density peaks become singular. Sink particles replace regions exceeding a threshold density—subject to additional physical criteria on convergence, boundedness, and potential minima—by a Lagrangian particle that accretes further gas and tracks mass, momentum, and angular momentum of the unresolved object ($M_{\rm sink}$, $\mathbf{v}_{\rm sink}$, $\mathbf{J}_{\rm sink}$) (Gong et al., 2012, Federrath et al., 2010, Federrath et al., 2010). The initial mass of the sink is typically set by the gas mass within an accretion/control volume or sphere ($r_{\rm acc}$) and continues to grow as more bound gas is accreted.

For AMR and mesh codes, sinks prevent violations of the Truelove condition ($\lambda_J / \Delta x > N_J$); in SPH, they prevent over-condensation within a fixed set of smoothing kernels. This ensures dynamical fidelity without requiring prohibitively fine resolution (Federrath et al., 2010, Gong et al., 2012).

2. Physical and Numerical Criteria for Sink Particle Formation

Sink creation requires rigorous filtering to ensure only self-gravitating, physically collapsing regions generate sinks, thus avoiding spurious sink particles from transient or shock-driven overdensities (Federrath et al., 2010, Gong et al., 2012, Hubber et al., 2013):

• Density threshold: $\rho_{\rm cell} > \rho_{\rm th}$, with $\rho_{\rm th}$ set by the local Jeans length marginally resolved by the maximum allowed resolution.
• Potential minimum: New sinks are only created at local minima of the gravitational potential within a control volume.
• Converging flows: The collapse region must satisfy $\nabla\cdot\mathbf{v}<0$ (global convergence) and, in some algorithms, $\partial_i v_i < 0$ in all principal directions.
• Boundness: Gravitational, kinetic, thermal, and (if present) magnetic energies are summed to demand $$E_{\rm tot} = E_{\rm kin}+E_{\rm th}+E_{\rm mag}+E_{\rm grav}<0$$.
• Jeans instability: The region must be thermally/magnetically Jeans-unstable, e.g., $|E_{\rm grav}|>2\,E_{\rm th}$.
• Refinement/proximity checks: Sink creation is only permitted at the highest refinement, and not within $r_{\rm acc}$ of an existing sink (Federrath et al., 2010, Federrath et al., 2010, Hubber et al., 2013, Bleuler et al., 2014).

In SPH with the "NewSink" algorithm (Hubber et al., 2013), there are further requirements: the candidate must be at a potential minimum (Eq. 4), outside the Hill radius of other sinks, and inside a density peak that dominates its local gravity.

3. Accretion, Dynamics, and Merging of Sink Particles

Accretion

Sinks accrete from nearby gas if the following hold: (i) $\rho > \rho_{\rm th}$ in local cells or particles, (ii) the gas is gravitionally bound to the sink plus local gas, (iii) the flow is radially inward relative to the sink (Gong et al., 2012, Federrath et al., 2010, Hubber et al., 2013).

Accretion occurs by either:

• Control volume summation (AMR/grid): Gas mass and momentum within $r_{\rm acc}$ is periodically transferred to the sink (Gong et al., 2012, Federrath et al., 2010).
• Lagrangian particle consumption (SPH): SPH particles entering $r_{\rm acc}$ and passing energy/boundness criteria are swallowed.
• Regulated accretion (NewSink): Mass is removed gradually from interaction-zone SPH particles according to calculated radial (free-fall) and/or viscous (disc) timescales, preserving smooth gradients and preventing spurious accretion (Hubber et al., 2013).

Angular momentum may be either absorbed by the sink (standard), or (in improved schemes) continually transferred back to the surrounding medium to prevent unrealistic spin-up (Hubber et al., 2013, Bleuler et al., 2014).

Gravitational Interactions

Sink–sink and sink–gas gravity is typically handled via:

• Particle-mesh deposition and force interpolation (grid codes): Sinks are cloud-in-cell or TSC deposited onto the mesh for Poisson gravity solver (Gong et al., 2012, Bleuler et al., 2014).
• Softened/Plummer-law pairwise forces for close sink–sink encounters; PM methods for larger distances.
• Subcycling: Close approaches can require substepping the N-body integration at shorter timesteps than the hydro update (Federrath et al., 2010, Gong et al., 2012).

Merging

If two sinks approach within a specified merging radius (often $2\,r_{\rm acc}$) and are converging and bound, they may be merged into a new sink with mass and momentum by conservation laws (Gong et al., 2012, Bleuler et al., 2014).

4. Algorithms for Mapping Sink Mass to Stellar Content

At resolutions where each sink represents a cluster or stellar population, it is necessary to statistically assign individual stellar masses ("convert sinks to stars"). The methodology in "A simple method to convert sink particles into stars" (Sormani et al., 2016) and closely related grouped star formation prescriptions (2112.05158) is as follows:

• Discretize the IMF into $N$ mass bins labeled by $i=1..N$ with representative mass $m_i$ and fractional contribution $f_i$.
• Poisson sampling: For a sink of effective mass $M$, draw $n_i\sim{\rm Poisson}(f_i M/m_i)$ independently in each bin.
• Assignment: The total stellar mass created is $M_* = \sum_i n_i m_i$. Each $n_i$ is interpreted as the number of stars of mass $m_i$ in the sink.
• Continuous IMF limit: Alternatively, draw $\mathcal{N}\sim{\rm Poisson}(M/\langle m\rangle)$ and sample $\mathcal{N}$ individual stellar masses from the cumulative IMF (e.g., Kroupa).

The variance in $M_*$ about $M$ scales as $\sqrt{\bar m / M}$ with $\bar m = \sum_i f_i m_i$; thus, statistical fluctuations become negligible for massive sinks but remain physically meaningful for small clusters. The method is additive under accretion and allows modular updates whenever a sink accretes additional mass (Sormani et al., 2016).

Grouped prescriptions combine several neighboring sinks' masses into a "group" and sample stars from the combined mass, then probabilistically assign the resulting stars to sinks proportionally to their individual masses, enforcing exact mass conservation (2112.05158).

5. Feedback Calculations from Sink-Assigned Stellar Populations

Once the stellar content of each sink is specified, radiative, mechanical, and chemical feedback rates are derived by summing over the assigned population:

• Radiative luminosity, $$L_{\rm bol}(t) = \sum_i n_i L_{\rm bol}(m_i, t - t_{\rm birth, i})$$
• Ionizing photon rate, $$\dot N_\gamma(t) = \sum_i n_i Q_{\rm H}(m_i, t - t_{\rm birth, i})$$
• Supernova rate: For stars with mass $m_i$ and lifetime $\tau_{\rm SN}(m_i)$, $$dN_{\rm SN}/dt = \sum_i n_i \delta(t - t_{\rm birth, i} - \tau_{\rm SN}(m_i))$$
• Yield and momentum injection at SN, $y(m_i)$ summed over all stars at time of death.

These feedback parameters are then provided to the simulation's radiation, chemistry, and momentum-injection modules, ensuring physically consistent, IMF-sampled input at all scales (Sormani et al., 2016).

6. Limitations and Best Practices

The statistical mapping approach retains stochastic fluctuations in the stellar content of small sinks (clusters), with relative scatter $\propto \sqrt{m_i / (f_i M)}$ especially pronounced for rare bins (massive stars in small clusters). The only constraint is that $M_* > M$ can occur stochastically, but in practice $M$ is interpreted as a dynamical not exact baryonic mass, and star-formation efficiency $\epsilon<1$ can be enforced by scaling $M\to\epsilon M$ (Sormani et al., 2016).

To resolve high-mass IMF sampling accurately, grouping prescriptions increase the effective mass before sampling, while maintaining approximate spatial and kinematic realism (2112.05158). For small clusters, the method simply produces zero or one massive star stochastically, representing genuine sampling fluctuations.

For robust, converged results, it is essential to ensure:

• The full set of physical collapse checks is applied for sink formation (Federrath et al., 2010, Hubber et al., 2013).
• Accretion radii and gravity softening are physically matched to code resolution and local Jeans length.
• All mass, momentum, and (where feasible) angular momentum are exactly conserved across accretion, merging, and feedback routines.
• The sink-to-star mapping and subsequent feedback are modular and adapted to the dynamical mass in each sink (IMF truncation, grouping scale, etc.) (Sormani et al., 2016, 2112.05158).

7. Alternative Contexts and Generalizations

Beyond star-formation modeling, "sink particle" models exist across many simulation disciplines, including planetary accretion, feedback from accreting black holes, and (in a distinct context) as point-sinks in fluidic aether models for gravity (Wang, 2018). In such models, the sink term typically injects a source or sink of mass, momentum, or other conserved quantities into the continuum field, serving as a subgrid closure for unresolved physical processes.

In summary, sink particles provide a rigorously founded, physically motivated, and numerically efficient tool for tracking unresolved compact objects, their subsequent growth, and their impact on the computational domain. Modern developments in formation criteria, regulated accretion, and stellar population sampling have resulted in reliable, reproducible predictions for star formation, feedback, and mass functions in computational astrophysics (Gong et al., 2012, Federrath et al., 2010, Sormani et al., 2016, 2112.05158, Hubber et al., 2013).