Toomre Q parameter disk stability calculator

The Toomre Q parameter calculator determines gravitational stability of differentially rotating disk (galactic, protoplanetary, accretion) against fragmentation and substructure formation, per the formulation by Estonian-American physicist Alar Toomre 1964 in Astrophysical Journal "On the gravitational stability of a disk of stars" (vol. 139, p. 1217) at MIT. The classic formula for gas disk: Q = c_s · κ / (π · G · Σ), while for stellar disk: Q = σ_R · κ / (3.36 · G · Σ), with c_s sound speed m/s, σ_R stellar radial velocity dispersion m/s, κ epicyclic frequency s⁻¹ (κ² = R · dΩ²/dR + 4Ω²), G gravitational constant (6.674×10⁻¹¹), Σ disk surface density kg/m². Toomre stability criterion: Q > 1 locally stable (no fragmentation); Q = 1 marginally stable (star formation threshold); Q < 1 unstable with gravitational fragmentation → protostars, spiral arms, bars, protoplanetary disks planetesimals. For two-component gas+stars disk (Wang-Silk 1994 + Romeo 2017): 1/Q_eff = 2 (1/Q_g + 1/Q_*) / [1 + (Q_g/Q_*)²].

Characteristic epicyclic frequency: Keplerian disk (Ω ∝ R⁻³/²) κ = Ω; flat rotation (V_circ const, Ω ∝ R⁻¹) κ = √2 · Ω; solid body (Ω constant) κ = 2Ω. Italian astrophysical applications:

• Milky Way spiral arms (Q ≈ 1.5 galactic stellar disk, Q < 1 in HII active star formation regions like W3, Sagittarius B2, Orion)
• M51 NGC 5194-5195 spiral galaxy INAF Bologna Loiano 1936 (Q variable 0.5-3 along arms)
• barred galaxies NGC 1300, M83 INAF (Q < 1 in bar, bar instability)
• Andromeda M31 INAF Trieste 1898 (Q ≈ 2 stable disk)
• T-Tauri protoplanetary disks ALMA INAF Padua 1761 (Q ≈ 1 planetesimal fragmentation 0.1 AU, giant planet formation)
• AGN/black hole accretion disks Cygnus X-1 INAF Brera Milan 1762 (variable Q self-regulating star formation in AGN disks). Italian MHD disk simulations: CINECA Casalecchio Bologna 1969 Marconi100 30 PFlops supercomputer + Leonardo HPC pre-exascale 250 PFlops 2023 (5th TOP500). Italian galactic institutions: INAF Italian National Astrophysics Institute Rome 2002 (16 observatories); SISSA Trieste 1978 astrophysics; University of Padua 1222 Galileo Galilei; Sapienza Rome 1303; Roberto Maiolino Cambridge ex-INAF Arcetri Florence first galaxies JWST; Andrea Ferrara Scuola Normale Pisa 1810 first stars cosmic reionization. Italian galactic telescopes: TNG National Galileo Telescope La Palma 1998 3.58 m (HARPS-N + GIANO-B); LBT Mount Graham Arizona 2005 2×8.4 m INAF 25% (LUCI MOS spectrograph); VST Cerro Paranal Chile 2011 2.6 m INAF Capodimonte (KIDS Kilo-Degree Survey 1,500 deg² weak lensing dark matter); SRT Sardinia Radio Telescope Cagliari 2013 64 m (FRB Fast Radio Burst distant galaxies)

Toomre Q disk parameter tables

(Source: Toomre 1964 ApJ 139, 1217 + Romeo & Wiegert 2011 + Binney & Tremaine "Galactic Dynamics" Princeton 2008 + INAF Rome 2002).

Italian Toomre Q calculation examples

• Milky Way Sun 8.3 kpc INAF Bologna Loiano 1936: c_s = 10 km/s cold HI gas; κ = 30 km/s/kpc rotation curve (V_circ = 220 km/s); Σ = 50 M_sun/pc² Sun surface. Q_gas = (10×10³ × 30×10³/3.086×10¹⁹) / (π × 6.674×10⁻¹¹ × 50 × 1.989×10³⁰ / (3.086×10¹⁶)²) = Q ≈ 1.5 (marginally stable, observation consistent)
• Inner spiral arms Milky Way W3 INAF: c_s = 5 km/s molecular clouds, Σ = 100 M_sun/pc². Q = 0.8 unstable, active star formation confirmed UV/Hα
• Andromeda M31 galactic disk 778 kpc INAF Trieste 1898: c_s = 8 km/s, κ = 25 km/s/kpc, Σ = 30 M_sun/pc². Q = 2.0 stable
• NGC 1300 barred galaxy INAF Capodimonte Naples 1812: c_s = 15 km/s gas, Σ = 500 M_sun/pc² central bar. Q = 0.4 bar instability bar driven gas inflow + AGN feeding
• T-Tauri ρ Ophiuchi protoplanetary disk ALMA INAF Padua 1761 + Arcetri 1872: c_s = 0.5 km/s 30 K cold gas, Keplerian κ = 200 km/s/kpc 1 M_sun, Σ = 1,000 g/cm² massive disk. Q = 0.9 planetesimal fragmentation Jupiter-like at 50 AU formation (ALMA 0.1" resolution = 6 AU)
• M87 SMBH AGN accretion disk EHT 2019: c_s = 20 km/s, κ = 500 km/s/kpc, Σ = 10,000 M_sun/pc². Q = 0.3 star formation in disk (Sirko-Goodman 2003)
• Cygnus X-1 BH disk INAF Brera Milan 1762: c_s = 10⁶ m/s, Ω = 10³ s⁻¹, Σ = 10⁹ kg/m². Q = 0.1 unstable
• M83 NGC 5236 barred spiral INAF Padua Asiago Copernico 182 cm: c_s = 12 km/s, Σ = 200 M_sun/pc² center. Q = 0.6 arms/bar instability

Italian galactic astrophysics

Italian galactic astrophysics:

• INAF Italian National Institute of Astrophysics Rome 2002 Monte Porzio Catone (16 observatories, 1,700 researchers, €250M annual budget): Padua 1761 Galileo (Padua University 1222 mathematics chair 1592-1610); Brera Milan 1762 Schiaparelli; Capodimonte Naples 1812; Arcetri Florence 1872 Galileo; Rome 1786; Trieste 1898; Catania-Etna 1880; Bologna Loiano 1936; Palermo 1790; Cagliari 1899 SRT 64 m radio
• Galactic telescopes INAF: TNG National Galileo Telescope La Palma 1998 3.58 m; LBT Mount Graham Arizona 2005 2×8.4 m INAF 25%; VST Cerro Paranal Chile 2011 2.6 m + ESO; SRT 64 m Sardinia 2013; BIRTH Brera Tenerife 0.8 m
• ASI Galactic missions Rome 1988: BeppoSAX 1996-2002, Cassini-Huygens 1997-2017, Mars Express 2003, Rosetta 2004-2016, BepiColombo 2018, Cheops 2019, Euclid 2023, JWST 2021 NIRSpec
• Italian Physics Nobel Prizes: Marconi 1909, Fermi 1938, Segrè 1959, Carlo Rubbia 1984, Riccardo Giacconi 2002. Andrea Ghez UCLA + Reinhard Genzel MPE 2020 Nobel Sgr A*
• SISSA Trieste 1978 astrophysics MHD; ICRA Rome Sapienza 1985 + ICRANet Pescara 2003 Remo Ruffini; Scuola Normale Pisa 1810 Andrea Ferrara first stars; CINECA Bologna 1969 Marconi100 30 PFlops + Leonardo pre-exascale 250 PFlops 2023 (5th TOP500 worldwide). Italian R&D tax bonuses 2025: R&D Tax Credit L. 160/2019 up to 25%; Patent Box L. 178/2020 50%; PNRR Mission 4 Component 2 €6.9B + National HPC Center CN1 Bologna

Updated July 2026