—   axisymmetric branch index ($k\geq1$)

$p_\ell(\xi)=(-i)^\ell P_\ell(i\xi),\qquad q^{\mathrm{obl}}_\ell(\xi)=p_\ell(\xi)\int_\xi^\infty\frac{dt}{(1+t^2)p_\ell(t)^2},\quad \xi>0$. These are the real oblate radial functions of [1], Section 2.

$2p_{2k+2}(\xi_k)q^{\mathrm{obl}}_{2k+2}(\xi_k)=\frac{\xi_k}{p_2(\xi_k)}\left(1-2\xi_kq^{\mathrm{obl}}_2(\xi_k)\right),\quad e_k=(1+\xi_k^2)^{-1/2}$. This is the axisymmetric bifurcation condition in [1], equation (3).

$p_{2k+2}(\xi_k)q^{\mathrm{obl}}_{2k+2}(\xi_k)=\xi_k\left(1-\xi_k\arctan(1/\xi_k)\right)$. This equivalent form follows from $p_2(\xi)=(1+3\xi^2)/2$ and $q^{\mathrm{obl}}_2(\xi)=p_2(\xi)\arctan(1/\xi)-3\xi/2$, with $\arctan(1/\xi)\in(0,\pi/2)$.

The spheroid is a uniformly rotating, homogeneous, incompressible Newtonian fluid equilibrium. Its equatorial and polar semiaxes are $a$ and $c$, with $0<c<a$; its eccentricity is $e=\sqrt{1-c^2/a^2}$. The parameter $\xi=\sqrt{1-e^2}/e$ is positive. These dimensionless shapes are independent of the overall size and mass density.

For each $k\geq1$, $\xi_k$ is the positive root of (2). This is the axisymmetric branch indexing of [1], Section 2, equations (3)–(4). It does not count rings or include the non-axisymmetric Jacobi bifurcation. The first branch continues to the Dyson rings; a bifurcation point is not by itself a claim of stability. [2]

$P_\ell$ is the Legendre polynomial normalized by $P_\ell(1)=1$. The functions $p_\ell$ and $q^{\mathrm{obl}}_\ell$ in (1) use a real integral to fix the oblate convention. They differ from the secondary Legendre polynomials and from the real-argument branch tabulated in Legendre functions of the second kind.

The selected hierarchy includes later branches past the axisymmetric dynamical-instability threshold discussed in [1], Section 1. Those entries describe mathematical equilibria without asserting stability. The nonrotating homogeneous spherical limit is the $n=0$ model in the polytropic moment-of-inertia table.

Sage

# Download generate.py from this table, then run with SageMath.
from generate import enclose
e11 = enclose(11, digits=100)[0]
print(e11)

M. Ansorg, A. Kleinwächter and R. Meinel, Uniformly rotating axisymmetric fluid configurations bifurcating from highly flattened Maclaurin spheroids, Monthly Notices of the Royal Astronomical Society 339 (2003), 515–523. (arXiv) (doi)

Ansorg, Kleinwächter and Meinel, full text, Section 2 and Table 1

Legendre polynomials —   the degree $2k+2$ polynomial determines the neutral axisymmetric mode

Secondary Legendre polynomials —   these polynomials give the polynomial part of the second Legendre solution; they are distinct from the oblate radial functions used here

Legendre functions of the second kind —   values on the real-argument branch, in contrast to the oblate convention in the bifurcation equation

Moments of inertia of polytropic stars —   the homogeneous spherical model is the nonrotating limit of the Maclaurin sequence

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds all first ten axisymmetric branches, $k=1,\ldots,10$, whose neutral perturbations have Legendre degree $2k+2$. This practical, noncanonical prefix covers the branches investigated in [1], including both parity classes and later reference equilibria, and supplies a published coarse comparison for every entry. The infinite hierarchy is not exhausted; argument-selection.md records the coverage rationale)

How they were obtained:

Exact rational polynomial identities establish the real integral convention and certify a unique positive bifurcation root separately for every selected degree. Arb sign bisection on $0<\xi<1$ uses 500-bit arithmetic, then propagates the root enclosure through $e=(1+\xi^2)^{-1/2}$. The largest eccentricity radius is less than $7.68\times10^{-129}$; 100 significant digits are stored. A second run at 700 bits gives identical rational root brackets.

more

Independent validated quadrature evaluates the defining oblate integrals, using explicit factorial coefficients for the Legendre polynomials. The original bifurcation equation has opposite strict signs at the endpoints represented by each stored decimal. Nonzero closed-form controls precede these integral checks. The eccentricities and axis ratios also agree with the rounded values in [1], Table 1. The attached certification.md, checks.py and numerical-evidence.json contain the proof, controls and per-entry evidence.