# Star Formation and Feedback

This page covers the gas-to-stars conversion step: which star-formation
prescription is used, how supernova feedback returns gas to the hot/CGM
reservoir and ejects some beyond the halo, and how feedback-free burst (FFB)
galaxies bypass the SN loop entirely.

Source: [`src/model_starformation_and_feedback.c`](https://github.com/MBradley1985/SAGE26/blob/main/src/model_starformation_and_feedback.c)

Called from: [Per-halo physics loop](../core_build_model.md) -- step 9 of the
substep ordering.

## Per-substep orchestration -- `starformation_and_feedback()`

The function executes the following block once per galaxy per substep:

1. **FFB early exit.** If `FeedbackFreeModeOn >= 1` and the galaxy is in FFB
   regime (`FFBRegime == 1`, set in `core_build_model.c`), control jumps to
   `starformation_ffb()` and the standard SF/feedback path is skipped.
2. **Compute the SFR** via the prescription selected by `SFprescription`
   (see below). Returns a star formation rate `strdot` in code units.
3. **Compute H1/H2 split** for H2-tracking prescriptions (1, 3, 4, 5, 6, 7),
   storing `H2gas` and `H1gas` on the galaxy.
4. **Compute reheated and ejected masses.** Standard Croton+06 budget, or
   the FIRE-scaling alternative (Muratov+15) when `FIREmodeOn = 1`.
5. **Apply mass and metal updates** via `update_from_star_formation()` and
   `update_from_feedback()`.
6. **Record the SFR** in the fixed-size `SfrDisk[step]` history bin.
7. **Check disk instability** (`check_disk_instability()` if
   `DiskInstabilityOn`).
8. **Produce metals** via the instantaneous-recycling approximation and
   route the fraction that leaves the disk to either `CGMgas` (Regime 0)
   or `HotGas` (Regime 1) using the Krumholz & Dekel (2011) Eq. 22
   suppression factor `FracZleaveDisk * exp(-Mvir / 3e11)`.

## Star formation prescriptions

The `SFprescription` parameter selects one of eight recipes. All share the
same effective star-forming radius `r_eff = 3.0 * DiskScaleRadius` (Milky
Way calibration) and dynamical time `t_dyn = r_eff / V_vir`.

| Value | Recipe | What sets the SFR |
|-------|--------|-------------------|
| 0 | Croton+2006 | Kauffmann+96 cold-gas threshold; `SFR = epsilon * (ColdGas - cold_crit) / t_dyn` |
| 1 | Blitz & Rosolowsky 2006 (BR06) | H2 from pressure-based fitting; `SFR = epsilon * H2gas / t_dyn` |
| 2 | Somerville+25 (no H2) | Density-modulated efficiency; `SFR = epsilon_cl * f_dense * ColdGas / t_dyn` |
| 3 | Somerville+25 + H2 | Same as 2, but with `ColdGas` replaced by `H2gas` from BR06 |
| 4 | Krumholz & Dekel 2012 (KD12) | Metallicity-dependent H2 fraction; `SFR = epsilon * H2gas / t_dyn` |
| 5 | Krumholz, McKee & Tumlinson 2009 (KMT09) | Photodissociation-balance H2; same form as KD12 |
| 6 | Krumholz 2013 (K13) | Two-phase H2 model |
| 7 | Gnedin & Draine 2014 (GD14) | UV-modulated H2; same form as KMT09 |

The H2-tracking prescriptions compute `H2gas` and `H1gas` either via a
single-slab calculation (cheaper) or a radial ring integration
(`H2RadialIntegrationOn = 1`, more accurate). The single-slab disk area is
selected by `H2DiskAreaOption`:

| Value | Area |
|-------|------|
| 0 | pi * r_disk^2 |
| 1 | pi * (3 r_disk)^2 (default; matches Milky Way calibration) |
| 2 | 2 pi * r_disk^2 |

The radial mode uses `H2RadialNBins` rings out to `H2RadialRMaxFactor *
r_disk`.

## Supernova feedback

Two reheating/ejection budgets are available, selected by `FIREmodeOn`.

### Standard Croton+2006 budget (`FIREmodeOn = 0`)

- **Reheated mass:** `reheated_mass = FeedbackReheatingEpsilon * stars`.
  Transferred from `ColdGas` to `HotGas` (or `CGMgas` in Regime 0).
- **Ejected mass:** energy balance with the unused SN energy:
  `ejected_mass = (FeedbackEjectionEfficiency * eta_SN * E_SN / V_vir^2
                    - FeedbackReheatingEpsilon) * stars`.
  Transferred from `HotGas` (or `CGMgas`) to `EjectedMass`. Clipped to zero
  if negative (deeper potentials retain all the reheated gas).

### FIRE scaling (`FIREmodeOn = 1`)

Replaces the fixed coefficients with the Muratov et al. (2015) velocity-
and redshift-dependent scaling:

```
fire_scaling = (1 + z)^RedshiftPowerLawExponent * (V_vir / V_crit)^beta
```

with `V_crit = 60 km/s` and the broken-power-law exponent
`beta = -3.2` below `V_crit`, `-1.0` above. The reheating mass becomes
`eta_reheat = FeedbackReheatingEpsilon * fire_scaling`, and the ejection
mass is computed from the surplus SN energy after lifting the reheated
gas (Hirschmann+2016 energy budget):

```
E_FB = FeedbackEjectionEfficiency * fire_scaling * 0.5 * stars * eta_SN * E_SN
E_lift = 0.5 * reheated_mass * V_vir^2
ejected_mass = max(E_FB - E_lift, 0) / (0.5 * V_vir^2)
```

The FIRE coefficient is computed once per call and reused for both reheating
and ejection.

## `update_from_star_formation()` and `update_from_feedback()`

These two helpers do the actual reservoir bookkeeping.

**`update_from_star_formation()`** removes `(1 - RecycleFraction) * stars`
from `ColdGas` (the rest is instantaneously recycled), increments
`StellarMass`, and tracks metallicity. H1 and H2 are clamped so they remain
consistent with the post-SF cold gas.

**`update_from_feedback()`** transfers `reheated_mass` from `ColdGas` to
the central's hot reservoir (regime-aware: `CGMgas` if Regime 0,
`HotGas` if Regime 1), then transfers `ejected_mass` from that same
reservoir to `EjectedMass`. Metals follow the gas in both transfers.

## Metal production and routing

After SF and feedback the function produces new metals via the
instantaneous-recycling approximation: `metals_new = Yield * stars`. A
fraction stays in the disk and the rest leaves to the hot reservoir,
controlled by:

```
FracZleaveDiskVal = FracZleaveDisk * exp(-Mvir / 30)
```

(Krumholz & Dekel 2011 Eq. 22; mass in `10^10 Msun/h`, so the
characteristic scale is `3 x 10^11 Msun/h`). The leaving fraction routes
to `MetalsCGMgas` in Regime 0 and `MetalsHotGas` in Regime 1
(or always `MetalsHotGas` when `CGMrecipeOn = 0`). If `ColdGas` is
exhausted, all metals leave the disk.

## Feedback-free burst mode -- `starformation_ffb()`

When the FFB regime classification flags a galaxy as bursty, the SFR is
set directly by the FFB efficiency rather than the usual Kauffmann
threshold:

```
SFR = FFBMaxEfficiency * gas_for_sf / t_dyn
```

The "feedback-free" label refers to the **physical regime**, not to the
code path. The mechanism (Li+2024, Boylan-Kolchin+2025): in dense compact
gas reservoirs the SN energy escapes the cloud before it can disrupt the
burst, so the *star formation efficiency* reaches
`epsilon_FFB ~ 0.2-1.0` instead of the usual few percent. The code still
bookkeeps the resulting SN feedback; it just does not let feedback
throttle the SFR.

Specifically, the only differences between `starformation_ffb()` and the
standard `starformation_and_feedback()` path are:

1. **SFR formula:** `epsilon_FFB * gas_for_sf / t_dyn` -- no cold-gas
   threshold, no mediation by molecular fraction in `ColdGas` modes.
2. **No disk-instability check** after SF -- rapid burst SF is assumed
   to stabilise the disk.

Everything else still runs:

- SN reheating: `reheated_mass = FeedbackReheatingEpsilon * stars` (or
  the FIRE-scaled `eta_reheat * stars` when `FIREmodeOn = 1`).
- SN ejection: the same energy-balance budget (Croton or FIRE form) as
  the main path.
- `update_from_feedback()` transfers reheated and ejected mass with
  full regime-aware routing.
- Metal production and routing via the same Krumholz & Dekel (2011)
  Eq. 22 factor, regime-aware to `MetalsCGMgas` or `MetalsHotGas`.

`gas_for_sf` is either the full `ColdGas` (modes 1-5) or the molecular
fraction `H2gas` (modes 6-7). For H2 modes the H2 calculation is run
inline using whichever underlying SFprescription (BR06, KD12, KMT09,
K13, GD14) is set.

The seven sub-modes of `FeedbackFreeModeOn` (1-7) control which threshold
classifies a galaxy as FFB-eligible (Li+2024 vs Boylan-Kolchin+2025, sigmoid
vs sharp, concentration source). The classification itself lives in
`determine_and_store_ffb_regime()`; this function only consumes the
`FFBRegime` flag.

## What is NOT in this module

- **Quasar-mode AGN.** BH accretion driven by mergers is handled in
  `model_mergers.c`. See [Mergers and disruption](mergers_and_disruptions.md).
- **Radio-mode AGN.** Handled in `model_cooling_heating.c`. See
  [Cooling and AGN heating](cooling_and_heating.md).
- **Disk instability.** Called from this module (step 7 above) but
  implemented in `model_disk_instability.c`. See the dedicated
  [Disk instability](disk_instability.md) page.

## Switches and parameters

| Parameter | Effect |
|-----------|--------|
| `SFprescription` | Select SF recipe (0-7). |
| `SfrEfficiency` | Efficiency factor in `epsilon * (Cold or H2) / t_dyn`. Used by SFprescription 0, 1, 4, 5, 7 always, and by 6 in slab mode only. Unused by 2 and 3 (Somerville+25) and by 6 in radial mode. |
| `SupernovaRecipeOn` | 0 disables SN feedback entirely. |
| `FeedbackReheatingEpsilon` | Mass-loading scaling for the reheating term. |
| `FeedbackEjectionEfficiency` | Fraction of SN energy that drives ejection. |
| `EnergySN`, `EtaSN` | Energy per SN and number of SN per solar mass. |
| `Yield`, `RecycleFraction` | Metal yield and instantaneous recycling fraction. |
| `FracZleaveDisk` | Fraction of new metals that leave the disk (modulated by halo mass). |
| `FIREmodeOn` | Switch to FIRE/Muratov+15 reheating and ejection. |
| `RedshiftPowerLawExponent` | Redshift exponent in the FIRE scaling. |
| `FeedbackFreeModeOn` | FFB regime classification mode (0-7); 0 disables the FFB path. |
| `FFBMaxEfficiency` | SF efficiency in FFB mode. |
| `H2DiskAreaOption`, `H2RadialIntegrationOn`, `H2RadialNBins`, `H2RadialRMaxFactor` | H2 surface-density geometry. |
| `DiskInstabilityOn` | Run the Toomre check after SF. |

See [`parameters.md`](../parameters.md) for full descriptions and defaults.

## References

- Croton et al. (2006), MNRAS 365, 11 -- original SAGE star formation and
  feedback recipes.
- Kauffmann (1996), MNRAS 281, 487 -- cold-gas threshold.
- Blitz & Rosolowsky (2006), ApJ 650, 933 -- pressure-based H2 fraction.
- Krumholz, McKee & Tumlinson (2009), ApJ 693, 216 -- photodissociation
  H2 model.
- Krumholz, Dekel & McKee (2011), ApJ 745, 69 -- metal-leaving-disk
  scaling (Eq. 22).
- Krumholz & Dekel (2012), ApJ 753, 16 -- KD12 H2 prescription.
- Krumholz (2013), MNRAS 436, 2747 -- two-phase H2.
- Gnedin & Draine (2014), ApJ 795, 37 -- UV-modulated H2.
- Somerville et al. (2025) -- density-modulated SF efficiency.
- Muratov et al. (2015), MNRAS 454, 2691 -- FIRE wind mass loading.
- Hirschmann et al. (2016), MNRAS 461, 1760 -- energy-balance ejection.
- Li et al. (2024) -- feedback-free burst threshold.
- Boylan-Kolchin (2025) -- FFB at high redshift.