Lane-Emden self-gravitating spheres

Lane-Emden self-gravitating spheres¶

Isothermal Lane-Emden equation for Bonnor-Ebert spheres¶

Status: verified

The isothermal Lane-Emden equation psi’’ + (2/xi) psi’ = e^{-psi} has no closed form and no finite edge; jaxstro integrates it with an adaptive diffrax solve seeded by its origin series, giving the density and enclosed-mass profile of a pressure-bounded isothermal (Bonnor-Ebert) sphere.

Scope¶

solve_isothermal on its documented domain, including the origin series start and the enclosed-mass and monotonicity contracts consumed downstream.

Conventions¶

• psi(0) = 0, psi’(0) = 0; density rho = rho_c e^{-psi}; mass m = xi^2 psi’

• the isothermal sphere has no zero; its edge is set by the confining external pressure

• integration starts at XI_0 = 1e-6 seeded with psi = xi^2/6 - xi^4/120

• xi_max and n_points are static (they size the output grid)

• verification is numerical against the derived origin series (psi = xi^2/6 - xi^4/120, m = xi^3/3 - xi^5/30) with the truncation error’s xi^4 convergence rate checked explicitly, not by page-checking the cited editions

Sources¶

• source - the Bonnor-Ebert sphere as a pressure-truncated isothermal Lane-Emden solution (the isothermal ODE and profile this module integrates; the critical mass is not computed here). Locator: the pressure-bounded isothermal gas sphere

• Ebert, R. (1955), Zeitschrift fuer Astrophysik 37, 217 - the isothermal self-gravitating sphere confined by external pressure. Locator: the isothermal self-gravitating sphere embedded in a pressured medium

Code & validation¶

• code: src/jaxstro/numerics/lane_emden.py::solve_isothermal

• code: src/jaxstro/numerics/lane_emden.py::LaneEmdenSolution

• validation: tests/unit/test_lane_emden.py::TestIsothermalOriginSeries::test_psi_matches_series_near_origin

• validation: tests/unit/test_lane_emden.py::TestIsothermalOriginSeries::test_psi_is_monotonically_increasing

• validation: tests/unit/test_lane_emden.py::TestEnclosedMass::test_isothermal_mass_small_xi

• validation: tests/unit/test_lane_emden.py::TestEnclosedMass::test_isothermal_mass_series_converges_at_expected_order

• validation: tests/unit/test_lane_emden.py::TestEnclosedMass::test_isothermal_mass_strictly_increasing

Polytropic Lane-Emden equation and its exact solutions¶

Status: verified

The polytropic Lane-Emden equation theta’’ + (2/xi) theta’ = -theta^n admits three closed-form solutions (n = 0, 1, 5) and a first zero xi_1 that marks the sphere’s surface; jaxstro integrates it with an adaptive diffrax solve and locates xi_1 as a differentiable event root.

Scope¶

solve_polytrope and polytrope_xi1 on n < 5 (finite radius), including the origin series start and the differentiable first-zero event.

Conventions¶

• theta(0) = 1, theta’(0) = 0; density rho = rho_c theta^n; mass m = -xi^2 theta’

• n relates to the adiabatic index by n = 1/(gamma - 1)

• integration starts at XI_0 = 1e-6 seeded with theta = 1 - xi^2/6 + n xi^4/120

• xi_max and n_points are static; n is traced and differentiable

• verification is numerical against the exact n = 0, 1, 5 solutions (theta = 1 - xi^2/6 with xi_1 = sqrt(6); sin(xi)/xi with xi_1 = pi; (1 + xi^2/3)^(-1/2) with no zero), not by page-checking the cited editions

Sources¶

• Chandrasekhar, S. (1939), An Introduction to the Study of Stellar Structure, Univ. of Chicago Press - the polytropic Lane-Emden equation and its three exact closed-form solutions. Locator: Chapter IV, the Lane-Emden equation and its solutions for n = 0, 1, 5

Code & validation¶

• code: src/jaxstro/numerics/lane_emden.py::solve_polytrope

• code: src/jaxstro/numerics/lane_emden.py::polytrope_xi1

• code: src/jaxstro/numerics/lane_emden.py::LaneEmdenSolution

• validation: tests/unit/test_lane_emden.py::TestPolytropeExactSolutions::test_n0_matches_exact_theta

• validation: tests/unit/test_lane_emden.py::TestPolytropeExactSolutions::test_n1_matches_exact_theta

• validation: tests/unit/test_lane_emden.py::TestPolytropeExactSolutions::test_n5_matches_exact_theta

• validation: tests/unit/test_lane_emden.py::TestPolytropeFirstZero::test_xi1_n0_is_sqrt6

• validation: tests/unit/test_lane_emden.py::TestPolytropeFirstZero::test_xi1_n1_is_pi

• validation: tests/validation/test_lane_emden_gradients.py::test_polytrope_xi1_gradient_matches_finite_difference