Star Formation

Overview

The star formation module adds star particles through a lightweight stochastic prescription that plugs into the Stochastic Stellar Population specialisation.

  • Runs once per hydro timestep and evaluates each cell independently.
  • Always spawns a low-mass star particle when a trial succeeds.
  • Adds high-mass particles probabilistically so the Chabrier (2005) initial mass function (IMF) is satisfied in expectation.

Jeans instability filter

Eligible cells are first identified through a Jeans-length check before any stochastic sampling occurs.

  • Compute the Jeans length \(\lambda_J = c_s / \sqrt{G \rho}\) in every cell.
  • Mark the cell as eligible when \(\lambda_J < J \cdot \Delta x\) with \(J = 0.5\).
  • Only eligible cells continue to the sampling steps below.

Controlling the formation rate

Two efficiency parameters tune how aggressively eligible gas is converted into star particles during each hydro step.

  • \(\epsilon_{ff}\): efficiency per free-fall time, defined through \(\epsilon_{ff} = (\dot M_{\star} \, t_{ff}) / (M_{cell} \, \Delta t)\).
  • \(\epsilon_{\star}\): fraction of the cell mass used when a particle is spawned.
  • The target stellar mass for the step is \(\epsilon_{ff} M_{cell} (\Delta t / t_{ff})\).
  • Bernoulli probability for spawning: \( P = \frac{\epsilon_{ff}}{\epsilon_{\star}} \frac{\Delta t}{t_{ff}} \).
  • The expectation value \(\langle M_{\star} \rangle = P \epsilon_{\star} M_{cell}\) matches the target mass provided \(\Delta t < t_{ff}\); the CFL condition typically enforces that inequality.

Sampling the stellar population

Once a cell passes the filter and the Bernoulli draw succeeds, we construct the composite stellar population represented by the spawned particles.

  • Every accepted draw creates one low-mass particle that represents all stars with \(M < 8 M_{\odot}\).
  • High-mass stars follow the Chabrier (2005) IMF: log-normal below \(1 M_{\odot}\) and a slope of \(2.35\) above it.
  • Pre-computed IMF integrals provide the mass fraction \(f_{\star,high}\) and mean mass \(\langle m \rangle_{\star,high}\) of the high-mass component.
  • The expected number of massive stars is \(f_{\star,high} \epsilon_{\star} M_{cell} / \langle m \rangle_{\star,high}\); a Poisson variate with that mean sets the actual count.
  • Each massive star draws its mass from the high-mass end of the IMF, while the low-mass particle retains the remaining fraction \(1 - f_{\star,high}\) of the spawned mass.
  • If the Poisson draw returns zero, only the low-mass particle is inserted and it inherits the local gas velocity.

Assigning particle velocities

Sampled star particles receive velocities that combine the local bulk flow with an isotropic runaway kick.

  • Each massive star draws a speed from a power-law distribution \(p(v) \propto v^{-1.8}\) truncated between 3 and 385 km s\(^{-1}\), then converts it to cgs units.
  • A random direction is chosen by sampling \(\cos \theta\) uniformly in \([-1, 1]\) and \(\phi\) in \([0, 2\pi)\); the resulting kick is added to the gas velocity of the parent cell.
  • The total momentum of all massive stars is accumulated, and the low-mass composite particle receives the opposite momentum so that the cell-level particle system conserves momentum.

Practical considerations

A few implementation notes help interpret corner cases and limitations of the current recipe.

  • Star formation is operator-split from the hydrodynamics. When \(t_{ff}\) is unresolved (\(\Delta t \gtrsim t_{ff}\)), the true star formation rate is not captured, and this scheme provides one possible approximation; no explicit limiter is enforced beyond the CFL-controlled hydro step.
  • All spawned particles are inserted at the cell centre. Other physics modules are responsible for any subsequent repositioning or feedback coupling.