---
title: "Lane-Emden self-gravitating spheres"
description: >-
  Generated provenance cards for lane-emden self-gravitating spheres.
---

<!-- GENERATED by scripts/build_provenance_registry.py; DO NOT EDIT. -->

# Lane-Emden self-gravitating spheres

(card-bonnor-ebert-isothermal-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

- <a href="Bonnor, W. B. (1956), Monthly Notices of the Royal Astronomical Society 116, 351">source</a> - 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`
- <a href="Ebert, R. (1955), Zeitschrift fuer Astrophysik 37, 217">source</a> - 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`

(card-lane-emden-polytropic-spheres)=
## 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

- <a href="Chandrasekhar, S. (1939), An Introduction to the Study of Stellar Structure, Univ. of Chicago Press">source</a> - 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`