back to table · edit · history · where entries came from · files · download
5073 bytes, as of the version from 2026-10-30 06:50 (current). Recorded here, not run.
# Radius of Lane–Emden polytropes: argument selection
Table: T613. Family: numberdb-data issue 246. Build: 20261030T005732Z-build.
This is the builder's justification, not the independent critique acceptance.
## Instruction provenance
The archived manifest is `../20261030T005732Z-build-skills/manifest.json`
(relative to the build work directory). Both archived local and live skill
copies were read and are byte-identical, SHA-256
`28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199`.
The manifest and skill evidence are retained with this run.
## Definition and only free parameter
The index n is a rational number in 0 <= n < 5, for the regular,
centrally normalized, three-dimensional Lane–Emden equation
(xi² theta')' = -xi² theta^n, theta(0)=1, theta'(0)=0.
The value is its first positive zero, on the positive real branch before the
surface. There are no freely sampled radial positions, density scales,
perturbation indices or normalizations. The selected exponents in order are
0, 1/2, 1, 3/2, 2, 5/2, 3, 7/2, 4, 9/2.
## Disjoint exhaustive groups (mixed)
* Intrinsic: 0 and 1. The constant-density and linear equations have the
elementary solutions 1-xi²/6 and sin(xi)/xi. Their radii are sqrt(6) and pi.
Source: Yip, Chan and Leung, section 2, equations (7)–(10).
* Documented applications: 3/2 and 3. These model stars with adiabatic
nonrelativistic and ultrarelativistic pressure, respectively. A convective
envelope also supplies the index 3/2. These are idealized stellar models,
not claims that observed stars obey a polytropic law exactly.
Sources: Princeton *Polytropes*, page 3; Wenbin Lu, *Polytropic Star*, page 2.
* Exploratory reference sample: 1/2, 2, 5/2, 7/2, 4, 9/2. The aim is to
recognize normalized radii arising from simple algebraic density–enthalpy
laws, rho/rho_c=theta^n, outside the distinguished cases. No specific
astrophysical application is claimed for each of these exponents.
Source for the family and normalization: Yip, Chan and Leung, section 2;
source for finite/infinite-radius boundary: their introduction and section 2.
## Reference policy
* Purpose: exploratory identification of global surface radii of the simplest
nonlinear algebraic Lane–Emden models, including centrally concentrated
finite profiles beyond the principal stellar indices.
* Bounds: the integer and half-integer exponents from 0 through 9/2, after
assigning 0, 1, 3/2 and 3 to the other groups. There are no additional inputs.
* Coverage rationale: integer monomials and powers that become integer
monomials after theta=t² cover two elementary algebraic classes, from stiff
to soft finite profiles. This breadth precedes adding higher denominators
or approaching the divergent endpoint at five. It is a sparse reference
sample rather than an interpolation mesh or entry-count target.
* Noncanonical choices acknowledged: denominator two and the finite upper
bound 9/2 are practical reference choices, not an intrinsic classification
of all useful models. The upper bound leaves a positive distance from five.
* Notation independent: this uses algebraic exponent complexity, not a decimal
spelling or a choice of numeral base.
* Supporting sources checked: yes, for the equation, normalization,
applications and endpoint behavior, not for an application at every input.
## Breadth and exclusions
The intrinsic finite cases and two stellar application indices are accompanied
by references on both sides of three; retaining only the famous four cases
would discard the deliberately exploratory stiff and soft model coverage.
There is no Cartesian product. Sharing these n values with planned structural
companions means using the same background models, not applying a universal
mesh to unrelated functions. The radius table excludes n=5 because that
solution has no finite surface. No dynamical stability is asserted for n>3.
There is one quantity, xi_1(n), and no parameter choosing other structural
constants. No quantile or statistical critical-value table is involved.
## Sources read
* K. L. S. Yip, T. K. Chan and P. T. Leung, *Perturbative solution to the
Lane–Emden equation: an eigenvalue approach*, MNRAS 465 (2017), 4265–4280,
https://arxiv.org/html/1611.07202, section 2 and introduction.
* Princeton, *Polytropes*, https://www.astro.princeton.edu/~gk/A403/polytrop.pdf,
pages 1–3, equations (poly.4)–(poly.10a) and the application paragraph.
* Wenbin Lu, *Polytropic Star*, https://wenbinlu.github.io/files/LaneEmden.pdf,
pages 1–2, equations (1)–(4) and the convective-envelope paragraph.
The family issue was available. Its linked batch archive in numberdb-runs
returned HTTP 404, so no inaccessible batch text is claimed as checked.
The live corpus searches for Lane, Emden, polytrope and stellar found no
companion. T7, T29 and T187 were read. Only T7 directly supports the radius
identity at n=1; the pressure and tidal links in the family map belong to the
future companion tables. No unbuilt companion is assigned a T-number.