Dimensionless squared angular momentum at axisymmetric bifurcations of Maclaurin spheroids

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Numbers

$k$

$j_k^2$

0.02174090500810580942637435173470451544670734035282978522337854121237866127261421240922724140369966305

comment: This branch continues to the Dyson-ring sequence [1].

0.02915206629391926502281241701859397277715161737659517112695082330155950007957875070087385502091238105

0.03463777984576747171580905906139692829514856160051217262695900220467039347384472848709709021506371445

0.03903724196983531278164266735932562892675852778274199997280982126331714653369242589533908765540989266

0.04273987354996289318503817766257573647389269204164277062817936139797923649714855642886256369926134530

0.04595660210452526949651760458545217600079315029007096349545551732349434800117965437339447055694419345

0.04881426319572194521700351678079134552169728670903077552354824134515424857072512473621644284003169583

0.05139505806784018940912893756012165044276140670875183484437928283619155596288762642243204230741008547

0.05375533886583299938395964687819251150032850937014257958337807907666060136247842924841552417706699464

10:

0.05593545955260816511585374952575323841953559033600763623855223619072965097873642969350721651143238231

Definition

Each value $j_k^2=J^2\rho^{1/3}/(4\pi GM^{10/3})$ is the dimensionless squared angular momentum of a homogeneous Newtonian Maclaurin spheroid at its $k$th axisymmetric bifurcation, indexed by the neutral perturbation of Legendre degree $2k+2$ [1].

Parameters

$k$

— axisymmetric branch index ($k\geq1$)

Formulas

(1)

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

(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).

(3)

$\displaystyle j_k^2=\frac4{25}\left(\frac3{4\pi}\right)^{4/3}\frac{\omega_k^2}{(1-e_k^2)^{2/3}},\qquad \omega_k^2=\frac{\sqrt{1-e_k^2}(3-2e_k^2)}{2e_k^3}\arcsin(e_k)-\frac{3(1-e_k^2)}{2e_k^2}$. Here $\omega_k^2=\Omega^2/(4\pi G\rho)$ is the dimensionless squared angular velocity at the same bifurcation.

(4)

$\displaystyle M=\frac{4\pi}{3}\rho a^3s,\qquad I=\frac25Ma^2,\qquad J=I\Omega$. These expressions for a homogeneous spheroid give (3) when substituted into the normalization $j^2=J^2\rho^{1/3}/(4\pi GM^{10/3})$ of [1].

(5)

$\displaystyle j_k^2=\frac4{25}\left(\frac3{4\pi}\right)^{4/3}\frac{\beta_k\arcsin(e_k)}{e_k(1-e_k^2)^{1/6}}$, where $\beta_k=T/|W|$ is the kinetic-to-gravitational energy ratio. This follows from $T=I\Omega^2/2$ and $W=-(3GM^2/(5a))\arcsin(e_k)/e_k$.

Comments

(6)

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 mass density is $\rho$, mass is $M$, axial angular momentum is $J$, angular velocity is $\Omega\geq0$, and axial moment of inertia is $I$. Newton's gravitational constant is $G$.

(7)

The eccentricity $e_k$ at the same branch determines $j_k^2$ by (3). Here $e=\sqrt{1-c^2/a^2}$, $s=\sqrt{1-e^2}=c/a$, $\theta=\arcsin(e)\in(0,\pi/2)$, and $\xi=s/e>0$.

(8)

For $k\geq1$, $\xi_k$ is the positive root of (2). This is the indexing of [1], Section 2, equations (3)–(4), where $2k+2$ is the neutral perturbation degree. It does not count rings or include the non-axisymmetric Jacobi bifurcation. The first branch continues to Dyson rings. The later selected branches include mathematical equilibria beyond the axisymmetric dynamical-instability threshold discussed in that source; this table makes no stability claim. [2]

(9)

$P_\ell$ 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.

(10)

The squared convention is that of [1], equation (7). The normalization cancels size, density and $G$. The inertia in (4) uses the equatorial radius $a$; in the nonrotating spherical limit it agrees with the homogeneous $n=0$ entry in the polytropic moment-of-inertia table.

Programs

(P1)

Sage

# Download the attached generate.py to the current directory.
from generate import enclose
for k in range(1, 11):
    print(k, enclose(k, digits=100)[0])

References

[1]

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)

Links

[2]

Ansorg, Kleinwächter and Meinel, full text, Sections 2 and 4 and Table 1

Similar tables

Eccentricities at axisymmetric Maclaurin bifurcations — the same configurations indexed by $k$; their exact eccentricities determine the squared angular momenta by (3)

Dimensionless squared angular velocity at axisymmetric Maclaurin bifurcations — the squared rotation rates of the same configurations, with the conversion in (3)

Kinetic-to-gravitational energy ratios at axisymmetric Maclaurin bifurcations — the energy ratios of the same configurations, related by (5)

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

Secondary Legendre polynomials — the polynomial part of the second Legendre solution is distinct from the oblate radial function used in the bifurcation equation

Legendre functions of the second kind — real-argument values use a different branch from the real oblate integral convention

Moments of inertia of polytropic stars — the homogeneous spherical model supplies the nonrotating limit of the inertia normalization

Data properties

Entries are of type: real number

digits: 100 (Unknown key)

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 angular-momentum 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 certify the real oblate convention and a unique positive bifurcation root for each selected degree. Strict Arb signs bisect $0<\xi<1$ at 500 working bits. The full root enclosure is propagated into the angular-momentum formula, yielding bounds containing the exact value with radius less than $2.23\times10^{-127}$; 100 significant digits are stored. A second computation at 700 bits gives identical exact rational root brackets.

Independent validated quadrature evaluates the defining oblate integrals using explicit factorial coefficients for the Legendre polynomials and certifies opposite signs in the original bifurcation equation at each root-bracket endpoint. A separate physical calculation uses the surface equipotential condition to obtain $\Omega^2$, evaluates volume integrals for $M$ and $I$, and assembles $J^2\rho^{1/3}/(4\pi GM^{10/3})$. Its largest radius is less than $1.43\times10^{-126}$. Nonzero closed-form controls precede these checks. Rescaling length, density and $G$ leaves the result unchanged. Both computations lie inside every written decimal interval. The displayed conversions and all ten coarse angular-momentum values in [1], Table 1, also agree. The attached generator, checks.py, certification.md and numerical evidence describe the proofs, controls and comparisons.