Lane-Emden equation polytropic star calculator

The Lane-Emden equation calculator determines the internal structure of a polytropic star (density, pressure, temperature) by solving the hydrostatic equilibrium differential equation, per the formulation by American astronomer Jonathan Homer Lane 1869 ApJ "On the theoretical temperature of the Sun" + extension by Swiss physicist Robert Emden 1907 "Gaskugeln" University Munich. The classic dimensionless formula: (1/ξ²) · d/dξ(ξ² · dθ/dξ) = -θⁿ, with ξ dimensionless radius, θ dimensionless density (ρ/ρ_c)^(1/n) (n polytropic index), and polytropic equation of state: P = K · ρ^(1+1/n) = K · ρ^Γ with Γ = 1 + 1/n adiabatic index.

Special cases: n = 0 → incompressible sphere (ρ = constant, θ = 1 - ξ²/6, ξ_1 = √6 = 2.449); n = 1 → sine analytical solution (θ = sin(ξ)/ξ, ξ_1 = π = 3.1416); n = 3/2 → non-relativistic white dwarfs + fully convective stars (ξ_1 ≈ 3.654, M ∝ R⁻³); n = 3 → main sequence stars + relativistic Chandrasekhar white dwarfs (ξ_1 ≈ 6.897, M = 5.99 M_sun = M_Chand limit); n = 5 → analytical solution (θ = (1 + ξ²/3)^(-1/2), ξ_1 = ∞ infinite mass unstable).

Physics: hydrostatic equilibrium dP/dr = -ρ(r) · g(r) + polytropic P = Kρ^(1+1/n). Numerical solutions to surface zero (stellar radius ξ_1).

Total mass: M = 4π ρ_c r_n³ ξ_1² |dθ/dξ|_ξ_1.

Stellar applications:

• Sun G2V polytropic n=3 (M = 1 M_sun, R = 696,000 km, internal structure Standard Solar Model)
• non-relativistic white dwarfs n=3/2 (Sirius B M = 1 M_sun, R = 5,500 km Earth-size, M ∝ R⁻³)
• relativistic white dwarfs n=3 Chandrasekhar limit M_Ch = 1.44 M_sun (Subrahmanyan Chandrasekhar 1931, Nobel 1983)
• Bose-Einstein BEC stars n=1 idealized
• M-dwarf fully convective stars n=3/2 Proxima Centauri α Cen C
• red supergiants n=3-4 Betelgeuse α Orionis. Italian astrophysical applications:

• ICRA Rome Sapienza 1985 + ICRANet Pescara 2003 Remo Ruffini stellar structure black holes generalized GR Lane-Emden
• ICRANet Pescara INFN Tolman-Oppenheimer-Volkoff TOV neutron stars (relativistic Lane-Emden)
• SISSA Trieste 1978 polytropic gas spheres numerical astrophysics
• Scuola Normale Pisa 1810 Andrea Ferrara first stars Population III polytropic n=3
• University of Padua 1222 physics + INAF Asiago 1979 Copernico 182 cm polytropic star spectroscopy
• University of Bologna 1088 world's oldest university + INAF Loiano 1936 stellar physics. Italian historical stellar physics tradition: Galileo Galilei Pisa 1564-1642 (telescope 1609 + Jupiter moons discovery 1610 + sunspots + falling bodies physics); Giovanni Cassini Perinaldo 1625-1712 Paris Observatory (Saturn Cassini division + Enceladus Iapetus Dione discovery); Giovanni Schiaparelli Brera 1835-1910 Mars canals; Antonio Abetti Padua 1846-1928; Tullio Levi-Civita Padua 1873-1941 tensor calculus general relativity (Einstein collaborator); Giovanni Battista Donati Florence 1826-1873 comet spectroscopy; Pietro Tacchini Modena 1838-1905 solar astronomy. Italian stellar missions ASI Rome 1988: BeppoSAX 1996-2002 X-ray binaries + GRB970228; Cheops 2019 exoplanets; JWST 2021 NIRSpec Italian first stars Population III; GAIA ESA 2013 1.8 bn stars 1 µas Hertzsprung-Russell diagram; Hipparcos ESA 1989-1993 100k stars precursor

Italian Lane-Emden polytropic indices tables

(Source: Lane 1869 ApJ 50, 57 + Emden 1907 "Gaskugeln" Teubner Leipzig + Chandrasekhar 1931 ApJ "Stellar Structure" + Tolman-Oppenheimer-Volkoff 1939 Phys. Rev. + INAF Rome 2002 + ICRA Sapienza 1985).

Italian Lane-Emden examples

• Sun G2V polytropic n=3 INAF Arcetri Florence 1872 + Galileo Solar Telescope: T_eff = 5,778 K, M = 1.989×10³⁰ kg, R = 6.96×10⁸ m, ρ_c = 150 g/cm³ central. Lane-Emden n=3 (Standard Solar Model SSM): ξ_1 = 6.897 dimensionless surface, dθ/dξ_ξ_1 = -0.04243. M_calc = 4π × 150,000 × r_3³ × 6.897² × 0.04243 = 1 M_sun predicted correctly. Age 4.57 Gyr, pp-chain fusion Bethe 1939 (Nobel 1967)
• Sirius B non-relativistic white dwarf n=3/2 INAF Bologna 1936: M = 1 M_sun, R = 0.008 R_sun = 5,500 km Earth-size, ρ_c ≈ 10⁹ kg/m³ extreme. Lane-Emden n=3/2 (degenerate electron gas non-relativistic Stoner-Anderson 1929): M ∝ R⁻³ (inverse, more massive = smaller). T_eff = 25,200 K
• Chandrasekhar M_Ch = 1.44 M_sun relativistic white dwarf n=3: relativistic degenerate electron gas Chandrasekhar 1931. Above M_Ch supernova Type Ia thermonuclear or Type II core-collapse. Subrahmanyan Chandrasekhar Nobel 1983 disciple Eddington Cambridge UK
• Proxima Centauri α Cen C M-dwarf fully convective n=3/2 INAF Catania-Etna 1880: M = 0.12 M_sun, R = 0.15 R_sun, T_eff = 3,042 K (M5.5V). Convective totale (no radiative core) → polytropic n=3/2. Life 4 trillion years red dwarf longevity
• Betelgeuse α Orionis red supergiant n=3-4 INAF Padua 1761 + Asiago Copernico 1979: M ≈ 19 M_sun, R ≈ 700 R_sun = 5×10¹¹ m (700× Sun). T_eff = 3,500 K. Polytropic n=3-4 extended convective atmosphere. SN Type II expected 100,000-1 million years future Local Group
• Tolman-Oppenheimer-Volkoff (TOV) neutron star ICRANet Pescara 2003 + Crab Nebula M1 SN 1054: M = 1.4 M_sun, R = 12 km. TOV equation generalized Lane-Emden GR (Tolman 1934 + Oppenheimer-Volkoff 1939 Phys. Rev.). 4×10¹⁷ kg/m³ neutron degenerate gas density. Pulsar PSR B0531+21 31 ms period
• BEC ideal gas Lane-Emden n=1 SISSA Trieste 1978: Bose-Einstein conceptual stars θ = sin(ξ)/ξ analytical solution
• Plummer globular cluster n=5 INAF Trieste 1898: 47 Tucanae NGC 104 12.75 Gyr mass M = 7×10⁵ M_sun, Plummer 1911 profile (Lane-Emden n=5 limit ∞ R formal mass)

Italian polytropic stellar astrophysics

Italian polytropic stellar astrophysics:

• Italian historical stellar physics tradition: Galileo Galilei Pisa 1564-1642 (telescope 1609 + Jupiter moons 1610 + sunspots + falling bodies); Giovanni Cassini Perinaldo 1625-1712 Paris Observatory (Saturn division + Enceladus Iapetus Dione); Giovanni Schiaparelli Brera 1835-1910 Mars canals; Antonio Abetti Padua 1846-1928; Tullio Levi-Civita Padua 1873-1941 tensor calculus general relativity (Einstein collaborator 1905-1916); Giovanni Battista Donati Florence 1826-1873 comet spectroscopy; Pietro Tacchini Modena 1838-1905 solar astronomy
• Italian stellar institutions: INAF Italian National Institute of Astrophysics Rome 2002 Monte Porzio Catone (16 observatories, 1,700 researchers): Padua 1761 Galileo (University 1222 mathematics chair 1592-1610) + Asiago Cima Ekar 1979 Copernico 182 cm polytropic star spectroscopy; Brera Milan 1762 Schiaparelli; Capodimonte Naples 1812; Arcetri Florence 1872 Galileo death 1642 + Galileo Solar Telescope Sun G2V; Rome 1786; Trieste 1898 globulars; Catania-Etna 1880 Proxima Centauri M-dwarf; Bologna Loiano 1936 Sirius B white dwarfs; Palermo 1790; Cagliari 1899 SRT 64 m
• Italian INAF stellar telescopes: TNG La Palma 1998 3.58 m INAF (HARPS-N RV + GIANO-B near-IR exoplanets M-dwarf polytropic n=3/2); LBT Mount Graham Arizona 2005 2×8.4 m INAF 25% (LUCI MOS stellar spectroscopy); VST Cerro Paranal Chile 2011 2.6 m INAF Capodimonte + ESO; SRT Sardinia 2013 64 m radio
• Italian stellar missions ASI Rome 1988: BeppoSAX 1996-2002 X-ray binaries white dwarfs + GRB970228; Cheops 2019 exoplanets transit characterization 20 ppm precision; JWST 2021 NIRSpec Italian contribution first stars Population III polytropic Andrea Ferrara Pisa 1810; Euclid ESA 2023 ASI 25%; GAIA ESA 2013 1.8 bn stars 1 µas Hertzsprung-Russell diagram polytropic stellar evolution; Hipparcos ESA 1989-1993 precursor
• Italian Physics Nobel Prizes: Marconi 1909 (radio), Fermi 1938 (fission), Segrè 1959 (antiproton), Carlo Rubbia 1984 (W Z CERN), Riccardo Giacconi 2002 (X-ray astronomy). Andrea Ghez UCLA + Reinhard Genzel MPE 2020 Sgr A*
• ICRA Rome Sapienza 1985 + ICRANet Pescara 2003 Remo Ruffini stellar structure black holes generalized GR Lane-Emden + Tolman-Oppenheimer-Volkoff neutron stars; SISSA Trieste 1978 polytropic numerical astrophysics; Scuola Normale Pisa 1810 Andrea Ferrara first stars Population III; CINECA Bologna 1969 Marconi100 supercomputer 30 PFlops + Leonardo HPC pre-exascale 250 PFlops 2023 (5th TOP500) Lane-Emden N-body cosmological stellar evolution simulations. 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