How to calculate Jeans mass star formation astrophysics

Calculate Jeans mass M_J + Jeans length lambda_J + free-fall time gravitational instability star formation molecular clouds ISM. James Jeans UK 1877-1946 1902. NASA + ALMA + JWST + Hubble + Mt Wilson.

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How to calculate the Jeans mass (M_J) fundamental scale astrophysics star formation determining minimum mass of self-gravitating gas cloud for gravitational collapse and star formation: M_J = (4/3) * pi * (lambda_J/2)^3 * rho where lambda_J = c_s * sqrt(pi/(G*rho)) Jeans length. c_s = sqrt(5*k_B*T/(3*mu*m_H)) isothermal sound speed gas cloud + T temperature K (10 K dense molecular cloud; 100 K HI cloud; 8000 K warm ionized ISM; 1e7 K solar corona); rho density kg/m^3 (dense molecular cloud 1e-15; GMC giant molecular cloud 1e-21; HI cloud 1.7e-23; warm ISM 5e-25; cosmological 1e-26); mu mean molecular weight (2.3 H2 molecular; 1.3 HI atomic; 0.6 fully ionized HII); m_H = 1.673e-27 kg hydrogen mass; G = 6.674e-11 N*m^2/kg^2 gravitational constant; k_B = 1.381e-23 J/K Boltzmann.

Free-fall time: t_ff = sqrt(3*pi / (32*G*rho)) + gravitational collapse time + depends only on rho + typical dense molecular cloud 1e5 yrs + GMC 1e7 yrs + classical reference.

Jeans 1902 instability criterion: M > M_J + lambda > lambda_J + cloud gravitationally collapses forming stars + classical reference James Hopwood Jeans 1902 Phil Trans Roy Soc A 199:1 'The stability of a spherical nebula'.

Star types from mass: M < 0.08 M_sun brown dwarf no hydrogen fusion; 0.08 < M < 0.5 M_sun M-type red dwarf; 0.5 < M < 1.5 M_sun sun-like G-K-F; 1.5 < M < 8 M_sun A-B-O type massive; 8 < M < 25 M_sun massive blue stars supernova type II; > 25 M_sun Wolf-Rayet hypergiants supernova GRB.

IMF Initial Mass Function: dN/dM ~ M^(-2.35) Salpeter 1955 + Kroupa 2001 + Chabrier 2003 + classical reference modern + low-mass stars dominant + high-mass rare.

Related numbers: Jeans length lambda_J; Jeans wavenumber k_J = 2*pi/lambda_J; free-fall time t_ff; thermal pressure P_th = n*k_B*T; gravitational pressure P_g = G*M^2/R^4; virial parameter alpha_vir = 5*sigma_v^2*R/(G*M); turbulent pressure; magnetic pressure B^2/(2*mu_0).

Pressure support models: thermal isothermal Jeans classical 1902; turbulent Larson 1981 + classical reference; magnetic Mestel-Spitzer 1956 + Mouschovias 1976 + classical reference; rotation bar instability + classical reference.

Hierarchical structure: diffuse clouds 100 pc 1000 K HI; GMC giant molecular cloud 50 pc 10^5 M_sun; clump 1 pc 1000 M_sun; core 0.1 pc 10 M_sun protostar; star cluster 10^3-10^6 stars + classical reference. History: James Hopwood Jeans (Ormskirk UK 1877 - Dorking 1946) Cambridge Trinity College + Princeton + classical reference astrophysicist popularizer; 1902 paper Phil Trans Roy Soc A 199:1 'The stability of a spherical nebula' classic Jeans instability + Jeans criterion + 1929 'The Universe Around Us' + 1930 'The Mysterious Universe' classical reference popular astronomy + 1928 'Astronomy and Cosmogony' Cambridge + classical reference + Jeans law Rayleigh-Jeans + Knighted 1928 + Royal Society + Member US National Academy + Pope Pius XI Pontifical Academy + Time Magazine cover.

Lord Rayleigh John William Strutt (UK 1842-1919) Cambridge Cavendish 1879-1884; 1900 paper Phil Mag 49:539 Rayleigh-Jeans law classical reference thermal radiation + Nobel Prize Physics 1904 argon discovery.

Edwin Hubble (1889-1953) Mount Wilson Observatory 1919-1953; 1929 PNAS 15:168 Hubble law universe expansion + 1936 'The Realm of the Nebulae' + Hubble Space Telescope eponym.

Subrahmanyan Chandrasekhar (India-USA 1910-1995) Univ Chicago + Yerkes 1937-1995; 1931 ApJ 74:81 Chandrasekhar limit 1.4 M_sun + 1939 'An Introduction to the Study of Stellar Structure' classical + Nobel Prize Physics 1983 stellar structure.

Edwin Salpeter (Vienna-USA 1924-2008) Cornell; 1955 ApJ 121:161 Salpeter IMF classic initial mass function + Salpeter eponym. Legacy: Stahler-Palla 'The Formation of Stars' 2004 Wiley 865 pages classic textbook + Kippenhahn-Weigert-Weiss 'Stellar Structure and Evolution' 2nd 2012 Springer 604 pages classical + Klessen-Glover 'Physical Processes in the Interstellar Medium' 2014 Wiley + Krumholz 'Star Formation' 2017 World Scientific 489 pages classical reference modern + Shu 'The Physical Universe' 1982 University Science 614 pages + Binney-Tremaine 'Galactic Dynamics' 2nd 2008 Princeton 920 pages classical galaxy + Spitzer 'Physical Processes in the Interstellar Medium' 1978 Wiley 318 pages classic.

USA research Jeans + star formation + ISM: NASA Goddard Marshall JPL Ames 17000 + Hubble Space Telescope HST + James Webb Space Telescope JWST world-leading + Spitzer Space Telescope + Kepler + classical reference; NSF Physics Astronomy 12000; NSF Geosciences 4500; DOE labs Argonne ORNL LANL 35000; Space Telescope Science Institute STScI Baltimore 800 + HST JWST world-leading; Mount Wilson Observatory 1500 + Edwin Hubble legacy; Carnegie Observatories Pasadena 100 + Las Campanas Chile; Caltech Owens Valley + Submillimeter 200 + classical reference; NRAO National Radio Astronomy Observatory Charlottesville 600 + VLA + ALMA + GBT world-leading; NOAO National Optical Astronomy Observatory Tucson 600; SAO Smithsonian Astrophysical Observatory Cambridge 600.

USA universities astrophysics + star formation: Princeton 5400 + Tomislav Zwitter + Romeel Dave + Anatoly Spitkovsky cosmology; MIT 11500 + classical reference; Caltech 8800 + Phil Hopkins star formation + Sterl Phinney; Stanford 11000; UC Berkeley 8500 + Chris McKee turbulence world-leading + Eliot Quataert; Harvard 22000 + Avi Loeb world-leading + Lars Hernquist cosmology; Yale 13000; Univ Chicago 9000 + S. Chandrasekhar legacy + Don York astronomy; Cornell 5200; UCLA 31000; Carnegie Mellon CMU 14000; Penn State 8500; Univ Texas Austin 50000.

USA industry telescope astronomy: Northrop Grumman Falls Church 95000 + James Webb mirrors world-leading + classical reference; Ball Aerospace Boulder CO 5500 + Spitzer Kepler instruments; Lockheed Martin Space 122000 + HST instruments; L3Harris Melbourne FL 47000 + telescope mirrors; SpaceX Hawthorne 13000 + Falcon Heavy launch; ULA Centennial 4500 + Atlas V Delta IV.

USA industry radio telescopes: NRAO 600 + ALMA Chile world-leading + VLA Socorro NM + GBT Green Bank world-leading; SKA US partnership 200 + Square Kilometre Array; Caltech Owens Valley 200; SETI Institute 100.

USA industry cosmological simulation: Argonne ALCF Aurora 3500 + cosmological simulation HACC GADGET; ORNL OLCF Frontier exascale + classical reference world-leading; NCAR 1500; NSF Frontera TACC Texas 50; Princeton Vesuvius cosmology; NASA Pleiades Ames.

Jeans mass formula

Definition: M_J = (4/3) * pi * (lambda_J/2)^3 * rho. lambda_J Jeans length + rho density.

Jeans length: lambda_J = c_s * sqrt(pi/(G*rho)) + c_s = sqrt(5*k_B*T/(3*mu*m_H)) isothermal sound speed.

Free-fall time: t_ff = sqrt(3*pi/(32*G*rho)).

Jeans 1902 criterion: M > M_J + lambda > lambda_J cloud gravitationally collapses forming star.

Typical values: dense molecular cloud T=10 K rho=1e-15 + M_J ~ 1 M_sun protostar; GMC T=10 K rho=1e-21 + M_J ~ 1000 M_sun cluster; warm ISM T=8000 K rho=5e-25 + M_J ~ 1e6 M_sun cosmological.

History and context

James Hopwood Jeans (Ormskirk UK 1877 - Dorking 1946) Cambridge Trinity + Princeton + classical reference astrophysicist popularizer; 1902 paper Phil Trans Roy Soc A 199:1 'The stability of a spherical nebula' classic Jeans instability + Jeans criterion + 1929 'The Universe Around Us' + 1930 'The Mysterious Universe' classical popular astronomy + 1928 'Astronomy and Cosmogony' Cambridge + classical + Jeans law Rayleigh-Jeans + Knighted 1928 + Royal Society + Member US National Academy + Pope Pius XI Pontifical Academy + Time Magazine cover.

Lord Rayleigh (1842-1919) Cambridge Cavendish; 1900 Phil Mag 49:539 Rayleigh-Jeans law + Nobel Prize Physics 1904 argon.

Edwin Hubble (1889-1953) Mount Wilson; 1929 PNAS 15:168 Hubble law + Hubble Space Telescope eponym.

Chandrasekhar (1910-1995) Chicago + Yerkes; 1931 Chandrasekhar limit + 1939 Stellar Structure + Nobel Prize Physics 1983.

Edwin Salpeter (1924-2008) Cornell; 1955 Salpeter IMF classic. Legacy: Stahler-Palla 2004 Wiley 865 + Kippenhahn-Weigert-Weiss 2012 Springer 604 + Klessen-Glover 2014 Wiley + Krumholz 2017 World Sci 489 + Shu 1982 + Binney-Tremaine 2008 Princeton 920 + Spitzer 1978 Wiley 318 classic.

USA research: NASA 17000 + HST + JWST world-leading; NSF 12000; STScI 800 + HST JWST; Mount Wilson 1500 + Hubble legacy; Carnegie Observatories 100 + Las Campanas; Caltech Owens Valley 200; NRAO 600 + ALMA VLA GBT world-leading; NOAO 600; SAO 600.

Universities: Princeton 5400 + Spitkovsky cosmology; Caltech 8800 + Hopkins SF + Phinney; UC Berkeley 8500 + McKee turbulence world-leading + Quataert; Harvard 22000 + Loeb world-leading + Hernquist cosmology; Chicago 9000 + Chandrasekhar legacy + York; UCLA 31000; CMU 14000; Penn State 8500; Texas Austin 50000.

Industry telescope: Northrop Grumman 95000 + JWST mirrors world-leading; Ball Aerospace 5500; Lockheed Martin 122000; L3Harris 47000; SpaceX 13000; ULA 4500.

Radio: NRAO 600 + ALMA + VLA + GBT world-leading; SKA 200; Caltech 200; SETI 100.

Cosmology simulation: Argonne ALCF 3500 + Aurora HACC; ORNL OLCF 6000 + Frontier exascale world-leading; NCAR 1500; TACC 50; Princeton; NASA Pleiades.

Jeans values table

  • M_J = (4/3)*pi*(lambda_J/2)^3*rho: definition
  • lambda_J = c_s*sqrt(pi/(G*rho)): Jeans length
  • c_s = sqrt(5*k_B*T/(3*mu*m_H)): sound speed
  • t_ff = sqrt(3*pi/(32*G*rho)): free-fall
  • Criterion: M>M_J collapse
  • Dense cloud: M_J~1 M_sun protostar
  • GMC: M_J~1000 M_sun cluster
  • HI cloud: M_J~1e6 M_sun
  • Warm ISM: cosmological
  • Salpeter IMF: dN/dM ~ M^(-2.35)

5 numerical examples

  1. Dense molecular cloud core: T=10 K + rho=1e-15 kg/m^3 + mu=2.3 + c_s=0.18 km/s + lambda_J=0.5 pc + M_J ~ 1 M_sun + t_ff ~ 1e5 yrs protostar.
  2. GMC giant molecular cloud: T=10 K + rho=1e-21 kg/m^3 + lambda_J=15 pc + M_J ~ 1000 M_sun + t_ff ~ 1e7 yrs cluster + ALMA NRAO world-leading.
  3. HI cloud: T=100 K + rho=1.7e-23 + mu=1.3 + c_s=0.97 km/s + lambda_J=550 pc + M_J ~ 1e6 M_sun.
  4. Protostar dense core: T=30 K + rho=1e-13 + lambda_J=0.05 pc + M_J ~ 0.1 M_sun + t_ff ~ 1e4 yrs + ALMA world-leading.
  5. Warm ionized ISM: T=8000 K + rho=5e-25 + mu=0.6 + c_s=12 km/s + lambda_J=15000 pc + M_J ~ 1e8 M_sun cosmological.

Limitations

Isothermal sound speed: c_s = sqrt(5*k_B*T/(3*mu*m_H)) + non adiabatic + Stahler-Palla 2004.

Spherical symmetry: real cloud filamentary + sheet + Larson 1981 turbulent.

No turbulence: Larson 1981 turbulent support sigma_v >> c_s + virial parameter alpha_vir.

No magnetic field: Mestel-Spitzer 1956 + Mouschovias 1976 magnetic critical mass M_phi.

No rotation: bar instability + accretion disk.

Common mistakes

  • lambda_J vs M_J: lambda_J Jeans length linear scale + M_J Jeans mass volume rho*lambda_J^3 distinct + Stahler 2004.
  • Mean molecular weight: H2 molecular mu=2.3 + HI atomic 1.3 + HII fully ionized 0.6 + Spitzer 1978.
  • Sound speed isothermal vs adiabatic: c_s_iso = sqrt(P/rho) vs c_s_ad = sqrt(gamma*P/rho).
  • Turbulent support: sigma_v >> c_s + virial parameter + Larson 1981 + Krumholz 2017.
  • Magnetic critical: Mouschovias 1976 mass-to-flux ratio + Klessen 2014.

Applications

  1. Star formation molecular clouds: NRAO ALMA world-leading + JCMT + HST JWST + Spitzer Kepler + dense cores 1 M_sun protostar.
  2. Giant molecular clouds GMC: NRAO + IRAM 30m + 1000 M_sun star cluster formation.
  3. Galactic structure ISM: classical reference + warm ionized HII.
  4. Cosmological large scale structure: Argonne ALCF Aurora + ORNL Frontier exascale world-leading + GADGET HACC.
  5. Protoplanetary disks: ALMA + JWST + planet formation gravitational instability.
  6. Brown dwarf formation: M < 0.08 M_sun + Stahler 2004.
  7. Globular cluster formation: 10^5 M_sun + classical reference.
  8. Galaxy formation: Princeton + Caltech + UC Berkeley + Harvard cosmology + N-body.
  9. Population III first stars: classical reference + early universe + JWST world-leading.
  10. Pulsar formation neutron stars: 25-40 M_sun progenitor + LIGO Virgo gravitational waves Nobel 2017.

Updated July 2026

Frequently Asked Questions

What does the Jeans mass M_J measure?

Jeans mass M_J = (4/3)*pi*(lambda_J/2)^3*rho measures critical minimum mass of self-gravitating gas cloud for gravitational collapse and star formation. Typically dense molecular cloud M_J ~ 1 M_sun protostar; GMC ~ 1000 M_sun cluster; warm ISM ~ 1e6 M_sun. James Hopwood Jeans UK 1877-1946 1902 Phil Trans Roy Soc A 199:1 'The stability of a spherical nebula' classic + Knighted 1928 + Royal Society + Pope Pius XI.

What is the Jeans length lambda_J?

Jeans length lambda_J = c_s*sqrt(pi/(G*rho)) critical spatial scale beyond which density perturbation in self-gravitating gas cloud diverges and eventually collapses. c_s isothermal sound speed sqrt(5*k_B*T/(3*mu*m_H)). Typical dense molecular cloud lambda_J = 0.5 pc + GMC 15 pc + HI cloud 550 pc + classical reference Stahler-Palla 2004 Wiley 865 pages textbook + Krumholz 2017 World Sci 489.

What is free-fall time t_ff?

Free-fall time t_ff = sqrt(3*pi/(32*G*rho)) characteristic gravitational collapse time of self-gravitating cloud without thermal pressure + depends only on rho density. Typical dense molecular cloud t_ff ~ 1e5 yrs + GMC ~ 1e7 yrs + HI cloud 1e8 yrs + warm ISM 1e9 yrs cosmological. Independent of geometry. Classical reference Stahler-Palla 2004 + Binney-Tremaine 'Galactic Dynamics' 2008 Princeton 920 pages classic.

What is Salpeter 1955 IMF Initial Mass Function?

IMF Initial Mass Function dN/dM ~ M^(-2.35) Edwin Salpeter 1955 paper ApJ 121:161 'The luminosity function and stellar evolution' describes frequency distribution of stellar masses in newly formed cluster. Power-law: low-mass stars dominant + high-mass rare. Extensions Kroupa 2001 broken + Chabrier 2003 lognormal + classical reference modern + classical reference Stahler-Palla 2004 + Krumholz 2017.

Difference Jeans thermal vs turbulent criterion?

Jeans thermal classic 1902 c_s isothermal sound speed vs turbulent Larson 1981 sigma_v turbulent velocity dispersion >> c_s + virial parameter alpha_vir = 5*sigma_v^2*R/(G*M) > 1 unbound. Real molecular clouds sigma_v ~ 1 km/s vs c_s ~ 0.2 km/s + turbulent support dominant + classical reference Krumholz 2017 'Star Formation' World Scientific 489 pages.

What is magnetic critical mass-to-flux ratio?

Magnetic critical mass-to-flux ratio Mestel-Spitzer 1956 + Mouschovias 1976: cloud is magnetically supported if M*Phi/B < critical = 0.5*sqrt(G*Phi^2/M^2) + otherwise collapses. Phi magnetic flux + B magnetic field + classical reference + Klessen-Glover 2014 'Physical Processes in the Interstellar Medium' Wiley + classical reference modern + Mouschovias eponym.

Jeans star formation applications?

Applications: star formation molecular clouds ALMA 66 antennas world-leading + JCMT + Mt Wilson + Mt Palomar + dense cores 1 M_sun protostar; GMC + IRAM 30m + 1000 M_sun cluster; cosmological large scale structure Argonne Aurora + ORNL Frontier exascale world-leading + GADGET HACC simulation; protoplanetary disks ALMA + JWST + planet formation; brown dwarfs M < 0.08 M_sun; globular cluster 1e5 M_sun; galaxy formation Princeton + Caltech + Berkeley cosmology; Population III JWST world-leading; pulsar neutron star LIGO Virgo Nobel 2017.

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