HPC Tier Module
Lane-Emden Stellar Structure Solver
Integrate the Lane-Emden polytrope with RK4, validated against the exact n=0, 1, 5 solutions.
This tool runs inside GDBS HPC. What it computes and the authorities it applies are below.
The problem
Studying polytropic stellar structure usually means setting up an ODE integrator from scratch and hand-checking the singular origin and the surface zero - easy places to introduce a quiet error.
The local-first solution
This solver integrates the dimensionless Lane-Emden equation with RK4 in your browser, starts off the singular origin with the exact even Taylor series, locates the first zero and mass weight, and validates against the closed-form n=0, 1, 5 solutions. Deterministic f64 math, nothing uploaded.
What it does
RK4 integration of the Lane-Emden first-order system for a given polytropic index n
Exact origin Taylor series start to remove the singular 0/0 at xi=0
First-zero surface detection xi_1 and the dimensionless mass weight -xi_1^2 theta'
Validation against the exact n=0 (sqrt 6), n=1 (pi), and n=5 (no finite zero) solutions
Cited cross-check tables (Chandrasekhar 1939, Horedt 2004) for non-integrable indices
GeoNum drift probe flagging conditioning loss near the origin and the surface
Honest scope
EXACT: the Lane-Emden ODE, the origin Taylor series, RK4 (O(h^4) truncation), and the closed-form n=0/1/5 references; the reported xi_1 and mass weight are the f64 RK4 values with their error versus the analytic answer shown. Tabulated (xi_1, mass) pairs for n=1.5, 3, etc. are cited confirm-by-source values, never invented. NOT modeled: rotation, magnetic fields, GR (TOV) corrections, or multi-zone polytropes. A teaching/verification tool, not stellar-evolution advice.
Authorities cited
- Chandrasekhar, S. (1939). An Introduction to the Study of Stellar Structure, Ch. IV. Univ. of Chicago Press (Dover reprint 1958). - The Lane-Emden equation (IV.1), origin series (IV.10), closed forms for n = 0, 1, 5 (IV.2), and the (xi_1, -xi_1^2 theta') table (Table 4).
- Lane, J. H. (1870). On the Theoretical Temperature of the Sun. American Journal of Science 50, 57-74. - Original formulation of the polytropic gas-sphere equation.
- Emden, R. (1907). Gaskugeln. B. G. Teubner, Leipzig. - Systematic study of polytropic gas spheres (the "Emden functions").
- Horedt, G. P. (2004). Polytropes: Applications in Astrophysics and Related Fields. Astrophysics and Space Science Library 306, Kluwer. - eqn 2.1.2 (Lane-Emden), Table 2.5.1 tabulated (xi_1, -xi_1^2 theta') to high precision.
- Kippenhahn, R., Weigert, A., Weiss, A. (2012). Stellar Structure and Evolution, 2nd ed., Ch. 19 (Polytropic Stellar Models). Springer. DOI 10.1007/978-3-642-30304-3.
- Press, W. H. et al. (2007). Numerical Recipes, 3rd ed., Sec. 17.1 - classical fourth-order Runge-Kutta (RK4), local truncation error O(h^5) / global O(h^4).
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Integrate a polytrope with proof
Run the solve in your browser with nothing uploaded, then save the profile and validation errors to a Sandbox workspace or attach them to a Worklog case.