Disk Instability

This page describes the Toomre disk-instability channel that grows the instability-driven bulge component and feeds the central black hole.

Source: src/model_disk_instability.c (with the Tonini+2016 radius update in src/model_misc.c)

Called from: starformation_and_feedback() after star formation, when DiskInstabilityOn = 1. Not run in the FFB path (Star formation and feedback).

What disk instability does

After each star-formation event, the disk’s combined mass of cold gas and disk stars is compared against the Mo, Mao & White (1998) Toomre critical mass:

M_crit = V_max^2 * 3 * DiskScaleRadius / G

(the factor of 3 comes from the disk half-mass radius being ~1.68 r_s and the Mo-Mao-White stability analysis for an exponential disk; see TOOMRE_DISK_FACTOR in the source). If the disk mass exceeds M_crit, the excess is “unstable” and must be redistributed.

Splitting the unstable mass

The unstable excess is split between gas and stars in proportion to the current cold-gas fraction:

gas_fraction   = ColdGas / disk_mass
unstable_gas   = gas_fraction       * (disk_mass - M_crit)
unstable_stars = (1 - gas_fraction) * (disk_mass - M_crit)

where disk_mass = ColdGas + (StellarMass - BulgeMass).

Where the unstable stars go

Unstable stars are transferred straight into the instability-driven bulge channel:

• BulgeMass += unstable_stars

• InstabilityBulgeMass += unstable_stars (separate channel from merger-driven bulge growth – see Mergers and disruption)

• MetalsBulgeMass is updated with the disk-star metallicity

InstabilityBulgeRadius is then updated via the Tonini+2016 incremental radius evolution (Eq. 15), using a snapshot of DiskScaleRadius taken before any mass transfer in this event:

R_new = (R_old * M_old + delta_M * 0.2 * R_disc) / (M_old + delta_M)

The 0.2 factor (TONINI16_DISK_FRAC in the source) is the fraction of the disk scale radius at which instability-driven bulge stars are deposited. On the first instability event when M_old == 0, the radius is initialised directly as 0.2 * R_disc. This update only fires when BulgeSizeOn = 3 (Tonini mode); other bulge models leave InstabilityBulgeRadius unchanged.

The disk scale radius itself is not changed by the instability: the remaining disk retains the same specific angular momentum per unit mass, so DiskScaleRadius stays constant.

Where the unstable gas goes

If there is unstable gas in the disk:

1. Black hole accretion (if AGNrecipeOn > 0):

   grow_black_hole(p, unstable_gas_fraction, ...)
   

   uses the same quasar-mode channel as mergers, with the unstable gas fraction playing the role of the merger mass ratio. This both grows the BH and triggers quasar_mode_wind() (see Mergers and disruption).

2. Collisional starburst (mode 1):

   collisional_starburst_recipe(unstable_gas_fraction, p, centralgal,
                                time, dt, halonr,
                                /*mode=*/1, step,
                                /*burst_to_merger_bulge=*/0,
                                DiskScaleRadius, galaxies, run_params)
   

   Mode 1 makes the burst efficiency eburst = mass_ratio (i.e. the entire unstable gas fraction is consumed) rather than the Cox-thesis power law used by mergers. The burst_to_merger_bulge = 0 flag routes burst stars into the instability bulge, consistent with the secular origin of the event.

The starburst itself applies SN feedback through the standard update_from_feedback() path, with FIRE scaling when FIREmodeOn = 1.

Order of operations and bookkeeping

Inside the function the order matters because the bulge-radius update needs the pre-event disk radius:

1. Compute M_crit and clip if it exceeds the actual disk mass.

2. Compute unstable_gas and unstable_stars from the gas fraction.

3. Save old_disk_radius = DiskScaleRadius before any mass transfers.

4. Transfer unstable stars to the bulge; call update_instability_bulge_radius() with old_disk_radius.

5. Burst the unstable gas (BH growth + starburst).

The “save disk radius first” step is the reason this function passes old_disk_radius explicitly into both helper paths rather than letting them read the live value.

What is NOT in this module

• Toomre Q itself. The criterion is implemented as a mass comparison against M_crit, not a stability parameter Q.

• Pseudobulge classification. The instability bulge is treated as a classical bulge component for radius and ICS purposes.

• BulgeRadius derivation. Done in model_misc.c via get_bulge_radius() as a mass-weighted combination of InstabilityBulgeRadius and MergerBulgeRadius (when BulgeSizeOn = 3).

Switches and parameters

Parameter	Effect
DiskInstabilityOn	0 disables the entire channel.
AGNrecipeOn	0 skips the BH accretion step but still runs the gas burst.
BulgeSizeOn	Must be 3 (Tonini+2016 multi-channel) for the incremental InstabilityBulgeRadius update to fire.
FIREmodeOn	Applies FIRE scaling to the burst SN feedback.
StarburstColdGasOn	Controls whether the burst uses stored H2gas or recomputes from ColdGas.
BlackHoleGrowthRate	Scaling for grow_black_hole() accretion in the instability path.
QuasarModeEfficiency	Coupling between BH accretion and wind energy in quasar_mode_wind().

See parameters.md for full descriptions and defaults.

References

• Mo, Mao & White (1998), MNRAS 295, 319 – Toomre critical mass for an exponential disk; the form M_crit = V_max^2 * 3 R_d / G used here.

• Toomre (1964), ApJ 139, 1217 – the underlying axisymmetric stability criterion for thin disks.

• Tonini et al. (2016), MNRAS 459, 4109 – two-channel bulge formation with separate radius evolution for instability- and merger-driven components (Eq. 15 used here).

• Croton et al. (2006), MNRAS 365, 11 – original SAGE disk-instability treatment.