Free-Fall Time Gravitational Collapse Calculator

Free-fall time t_ff = sqrt(3pi/32Grho) gravitational collapse molecular cloud calc. GMC Orion. INAF Arcetri Palla, ALMA, Herschel SPIRE.

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Free-fall time gravitational collapse calculator

The free-fall time calculator determines the gravitational collapse time of a uniform spherical cloud in free fall from rest (no pressure, no rotation, no magnetic field) toward the center of mass, per classical Newtonian mechanics derivation 1687 + astrophysical application Larson 1969 MNRAS protostellar collapse + Shu 1977 ApJ "Self-similar collapse of isothermal spheres" Caltech. The classic formula: t_ff = sqrt(3 · pi / (32 · G · rho)), with G = 6.674×10⁻¹¹ m³ kg⁻¹ s⁻² Henry Cavendish 1798 torsion balance experiment, rho mean cloud density kg/m³, t_ff free-fall time seconds (Penston 1969 MNRAS mass-independence: depends only on rho). Physics: radial equation of motion d²r/dt² = -G·M(r)/r² with M(r)=(4/3)pi·r³·rho constant (uniform), integration → central singularity time t_ff. Equivalent formula H₂ clouds: rho = n_H · m_H · mu, with m_H = 1.673×10⁻²⁷ kg hydrogen atomic mass, mu ≈ 2.33 mean molecular weight (incl. helium He cosmic abundance 24%); n_H number density cm⁻³. 5 Galactic ISM molecular cloud categories:

  • GMC Giant Molecular Cloud n_H = 10²-10³ cm⁻³, M = 10⁴-10⁶ M_sun, T = 10-30 K, t_ff = 1-3 Myr (Orion B, Taurus, Mon R2)
  • Dense prestellar core n_H = 10⁴-10⁵ cm⁻³, M = 1-10 M_sun, T = 10 K, t_ff = 0.1-0.3 Myr (L1544, Barnard 68)
  • Hot protostellar core n_H = 10⁶-10⁸ cm⁻³, T = 100-300 K, hot corino chemistry COMs ALMA (Sgr B2, IRAS 16293)
  • Bok globule n_H = 10³-10⁴ cm⁻³ isolated Bart Bok 1947, M = 1-50 M_sun (Barnard 68 photometric)
  • Diffuse cloud n_H = 10-10² cm⁻³ HI atomic (no collapse). Astrophysical applications:
  • Star formation Schmidt-Kennicutt SFR ∝ rho_gas^1.4
  • Jeans mass instability M_J = (5kT/G·mu·m_H)^(3/2) · (3/(4·pi·rho))^(1/2) Sir James Jeans 1902 Phil. Trans. Roy. Soc.
  • Pre-main-sequence Hayashi track Chushiro Hayashi 1961 PASJ quasi-static contraction post-Larson 1969
  • Spherical Bondi accretion Hermann Bondi 1952 MNRAS rate dM/dt = pi·rho·c_s²·G·M²/c_s³
  • Cosmological structure formation Press-Schechter 1974 ΛCDM dark matter free-fall onto density peaks
  • Black hole accretion Eddington L_Edd
  • Gould Belt 500 pc local star-forming complex. Italian applications:
  • GMC Orion B 500 M_sun INAF Arcetri Florence 1872 Francesco Palla star formation legacy
  • Taurus dark cloud 140 pc INAF Padua 1761
  • ALMA Atacama 2013 Italian contribution band 9 INAF Arcetri high-frequency 600-720 GHz hot corino IRAS 16293 COMs
  • Herschel Space Observatory ESA 2009-2013 SPIRE INAF Gould Belt Survey 500 deg² 50,000 prestellar cores
  • SRT Sardinia 64 m INAF 2013 NH₃ maser hot cores massive star formation
  • Noto 32 m INAF CO line Galactic plane surveys
  • JWST 2021 NIRCam INAF first stars Class 0 protostars
  • LBT Mount Graham Arizona 25% INAF 2005 LUCI MOS T Tauri spectroscopy
  • Planck Surveyor 2009-2018 INAF Trieste 1898 dust thermal emission cold Galactic cores

Italian molecular cloud free-fall table

Sourcen_H (cm⁻³)rho (kg/m³)T (K)t_ff (Myr)Italian application
Diffuse HI cloud502×10⁻¹⁹504.7SRT INAF Sardinia 2013
GMC Orion B2008×10⁻¹⁹202.3INAF Arcetri 1872 Palla
GMC Taurus5002×10⁻¹⁸101.5INAF Padua 1761
Pipe Nebula2501×10⁻¹⁸152.1Herschel SPIRE INAF
Hot Bok globule1,0004×10⁻¹⁸101.1INAF Capodimonte 1812
Barnard 68 globule5,0002×10⁻¹⁷100.5Herschel Gould Belt INAF
Dense core L154410,0004×10⁻¹⁷70.33ALMA INAF band 9
Class 0 protostar10⁶4×10⁻¹⁵300.033JWST NIRCam INAF
Hot corino IRAS 1629310⁸4×10⁻¹³2000.0034ALMA INAF Arcetri 1872
Inner protostellar disk10¹²4×10⁻⁹1,500~3,400 yrSRT NH₃ maser INAF

(Source: Larson 1969 MNRAS protostellar collapse + Shu 1977 ApJ self-similar isothermal + Penston 1969 MNRAS + Bart Bok 1947 globules + INAF Arcetri Florence 1872 Francesco Palla + Herschel Gould Belt Survey 2009-2013).

Italian free-fall calculation examples

  • GMC Orion B 500 M_sun INAF Arcetri Florence 1872 Francesco Palla star formation legacy: rho = 8×10⁻¹⁹ kg/m³ (n_H = 200 cm⁻³ molecular H₂ mu=2.33 m_H=1.673×10⁻²⁷). t_ff = sqrt(3·pi/(32·6.674×10⁻¹¹·8×10⁻¹⁹)) = 7.5×10¹³ s = 2.4 Myr. Star formation rate Galaxy SFR=1-2 M_sun/yr (Schmidt-Kennicutt MW). Orion Nebula M42 OB proto-cluster
  • Taurus dark cloud 140 pc INAF Padua 1761: rho = 2×10⁻¹⁸ kg/m³ (n_H = 500 cm⁻³). t_ff = 1.5 Myr. ~300 T Tauri pre-main-sequence stars Hayashi track. INAF Padua Galileo 1761 Asiago 1979 Copernico 182 cm spectroscopy
  • Pipe Nebula Herschel SPIRE INAF Gould Belt Survey 2009-2013: rho = 1×10⁻¹⁸ kg/m³ (n_H = 250 cm⁻³). t_ff = 2.1 Myr. 500 deg² mapped 50,000 prestellar cores 70-500 micron
  • Bok globule Barnard 68 INAF Capodimonte VLT: rho = 2×10⁻¹⁷ kg/m³ (n_H = 5,000 cm⁻³ photometric Alves+ 2001 Nature). t_ff = 0.5 Myr. M = 2 M_sun isolated Bart Bok 1947 silhouette
  • Class 0 protostellar disk ALMA INAF band 9 IRAS 16293-2422 hot corino: rho = 4×10⁻¹⁵ kg/m³ (n_H = 10⁶ cm⁻³). t_ff = sqrt(3·pi/(32·6.674×10⁻¹¹·4×10⁻¹⁵)) = 1.06×10¹² s = 33 kyr. Class 0 protostar embedded envelope 10⁵ AU. ALMA 600-720 GHz INAF Arcetri 1872 Roberto Maiolino Cambridge ex-INAF
  • Solar System formation rho = 3×10⁻¹⁸ kg/m³ (n_H = 750 cm⁻³ presolar nebula). t_ff ≈ 1.2 Myr. Sun 4.567 Gyr formed (Connelly 2012 CAI Allende meteorite Pb-Pb)
  • Dense core L1544 Taurus INAF + ALMA: rho = 4×10⁻¹⁷ kg/m³ (n_H = 10⁴ cm⁻³). t_ff = 0.34 Myr. Pre-stellar core deuterium fractionation NH₂D/NH₃=0.01 (low T 7 K CO freeze-out)
  • Hot core Sgr B2 N1 ALMA INAF band 9 600 GHz Galactic Center: rho = 4×10⁻¹³ kg/m³ (n_H = 10⁸ cm⁻³). t_ff = 3,400 yr = 0.0034 Myr. COMs glycine glycolaldehyde detected 2014

Italian star formation astrophysics

Italian star formation molecular cloud astrophysics:

  • INAF Italian National Institute of Astrophysics Rome 2002 Monte Porzio Catone (16 observatories, 1,700 researchers): Padua 1761 Galileo + Asiago 1979 Copernico 182 cm T Tauri YSO spectroscopy; Brera Milan 1762 Schiaparelli; Capodimonte Naples 1812; Arcetri Florence 1872 Giancarlo Tozzi + Francesco Palla star formation legacy (Palla & Stahler 1990 ApJ pre-main-sequence isochrone, Palla 2008 IMF molecular cloud); Rome 1786 Monte Porzio; Trieste 1898 Planck CMB; Catania-Etna 1880 T Tauri spectroscopy; Bologna Loiano 1936; Palermo 1790; Cagliari 1899 SRT 64 m
  • Italian INAF star formation telescopes: TNG Telescopio Nazionale Galileo La Palma 1998 3.58 m INAF (HARPS-N exoplanets young T Tauri); LBT Mount Graham Arizona 2005 2×8.4 m INAF 25% (LUCI MOS spectroscopy massive YSO); VST Cerro Paranal Chile 2011 INAF Capodimonte; SRT Sardinia 2013 64 m radio NH₃ maser hot cores massive star formation 22 GHz H₂O maser; Noto 32 m INAF CO line Galactic plane surveys; ALMA Atacama Chile 2013 Italian contribution band 9 600-720 GHz INAF Arcetri hot corino COMs IRAS 16293-2422
  • Italian star formation missions ASI Rome 1988: Herschel Space Observatory ESA 2009-2013 SPIRE instrument Italian contribution INAF Gould Belt Survey 500 deg² 50,000 prestellar cores 70-500 micron mapping; Planck Surveyor 2009-2018 INAF Trieste 1898 dust thermal emission cold Galactic cores; JWST 2021 NIRCam Italian INAF contribution first stars Class 0 protostars (Roberto Maiolino Cambridge ex-INAF Arcetri ERC Advanced Grant); Spitzer Space Telescope NASA 2003-2020 IRAC photometry YSO; BeppoSAX 1996-2002 ASI X-ray T Tauri flares
  • Italian Physics Nobel Prizes: Marconi 1909, Fermi 1938, Segrè 1959, Carlo Rubbia 1984, Riccardo Giacconi 2002 X-ray astronomy
  • Italian research centers: SISSA Trieste 1978 astrophysics MHD turbulence Larson 1981 scaling laws sigma_v ∝ L^0.38; Scuola Normale Pisa 1810 Andrea Ferrara cosmic dawn first stars Pop III proto-clusters z=20-30 reionization; ICRA Rome Sapienza 1985 + ICRANet Pescara 2003 Remo Ruffini gravitational collapse; CINECA Bologna 1969 Marconi100 30 PFlops + Leonardo HPC pre-exascale 250 PFlops 2023 (4th TOP500 Europe, 5th world) MHD turbulence Larson 1981 scaling laws + AREPO RAMSES protostellar collapse Truelove resolution criterion. 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

Updated July 2026

Frequently Asked Questions

What is free-fall time?

Gravitational collapse time of a uniform spherical cloud from rest, no pressure, no rotation. Formula t_ff = sqrt(3·pi/(32·G·rho)) Larson 1969 MNRAS + Penston 1969 (mass-independent, depends only on rho). G = 6.674×10⁻¹¹ m³/kg/s² Cavendish 1798. Applied Shu 1977 ApJ self-similar isothermal sphere collapse Caltech. Used star formation, Bondi accretion, cosmology.

Free-fall time formula?

t_ff = sqrt(3·pi/(32·G·rho)) where rho mean cloud density kg/m³, G = 6.674×10⁻¹¹ m³ kg⁻¹ s⁻². For molecular H₂ cloud: rho = n_H · m_H · mu with m_H = 1.673×10⁻²⁷ kg, mu = 2.33 mean molecular weight (with He). GMC n_H=10² cm⁻³ rho=4×10⁻¹⁹ → t_ff=3.3 Myr. Dense core n_H=10⁴ → 0.33 Myr. Protostellar n_H=10⁶ → 33 kyr. Mass-independent.

GMC Orion B Italian example?

GMC Orion B 500 M_sun INAF Arcetri Florence 1872 Francesco Palla star formation legacy: rho = 8×10⁻¹⁹ kg/m³ (n_H=200 cm⁻³ H₂ mu=2.33 m_H=1.673×10⁻²⁷). t_ff = sqrt(3·pi/(32·6.674×10⁻¹¹·8×10⁻¹⁹)) = 7.5×10¹³ s = 2.4 Myr. Star formation rate Milky Way SFR=1-2 M_sun/yr (Schmidt-Kennicutt). M42 Orion Nebula OB proto-cluster Hayashi track.

Italian astrophysics applications?

Italian star formation applications: 1) GMC Orion B INAF Arcetri 1872 Francesco Palla legacy; 2) Taurus dark cloud INAF Padua 1761; 3) ALMA Atacama 2013 band 9 INAF Arcetri hot corino IRAS 16293; 4) Herschel ESA 2009-2013 SPIRE INAF Gould Belt 50,000 prestellar cores; 5) SRT Sardinia 64 m INAF NH₃ maser hot cores; 6) JWST 2021 NIRCam INAF first stars Class 0 protostars.

Jeans mass and free-fall?

Jeans mass M_J = (5kT/G·mu·m_H)^(3/2) · (3/(4·pi·rho))^(1/2) Sir James Jeans 1902 Phil. Trans. Roy. Soc.: critical mass beyond which cloud collapses gravitationally. For GMC T=10 K mu=2.33: M_J ≈ 1-10 M_sun. When M_cloud > M_J → collapse in t_ff ~ Myr. Schmidt-Kennicutt SFR ∝ rho_gas^1.4 from Larson 1981 MNRAS scaling sigma_v ∝ L^0.38 turbulence.

Protostellar star formation phases?

5 star formation phases: 1) Diffuse cloud HI 50 cm⁻³ no collapse; 2) GMC 100-1,000 cm⁻³ M=10⁴-10⁶ M_sun T=10-30 K t_ff=1-3 Myr; 3) Dense prestellar core 10⁴-10⁵ cm⁻³ Class -1 t_ff=0.3 Myr; 4) Class 0/I protostar 10⁶-10⁸ cm⁻³ disk + outflow t_ff=0.03 Myr; 5) Class II/III T Tauri Hayashi track Chushiro Hayashi 1961 PASJ pre-main-sequence.

Italian star formation telescopes?

Italian INAF star formation telescopes: 1) TNG La Palma 1998 3.58 m HARPS-N; 2) LBT Mount Graham Arizona 25% INAF 2005 LUCI MOS YSO spectroscopy; 3) SRT Sardinia 64 m 2013 NH₃ maser hot cores 22 GHz H₂O maser; 4) Noto 32 m CO line Galactic plane surveys; 5) ALMA Atacama 2013 band 9 INAF Arcetri 1872 hot corino COMs IRAS 16293; 6) Herschel SPIRE Gould Belt INAF.

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