Kinetic-to-gravitational energy ratios at axisymmetric bifurcations of Maclaurin spheroids

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Numbers

$k$

$\beta_k=T/|W|$

0.3589038426086023489212077095394976938733198790188782811837733386976592448431413347536499791124832650

0.4034546560514278875104263937991677914926072257320901229910846733194936819231593658758351340069518789

0.4266350788289110340115277434309860626782686583947364785409904099343559390387146239911605383518065690

0.4408438858453380818788005671156565580641702779828970724093421009921843092520957824079281359528278095

0.4504437511469904155870450255636579921249617349316140669755082541085733629982880546262065081368000199

0.4573637367354507890597402325162575738162407756461796798543286680230306504592551110014315400078797924

0.4625883433024108650841521266487169934337558054636693002729858392788013772044512546831952890458037012

0.4666725198614399817480380279421531070060618084985290769908372082567362898420959009952668600578514947

0.4699528712415190837142322925587485213335326473795846505542923667411204968634900109609866127428395359

10:

0.4726454147635197162171961338556955822870470885280978441091384585974743660295423655066730947687918184

Definition

Each value $\beta_k=T/|W|$ is the ratio of rotational kinetic energy $T$ to the magnitude of negative gravitational potential energy $W$ for a 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)

$\displaystyle \beta_k=\frac{3}{2e_k^2}-1-\frac{3\sqrt{1-e_k^2}}{2e_k\arcsin(e_k)}=\frac{1+3\xi_k^2}{2}-\frac{3\xi_k}{2\arctan(1/\xi_k)}$. The positive real inverse trigonometric branches give $\arcsin(e_k)=\arctan(1/\xi_k)$.

(4)

$\displaystyle M=\frac{4\pi}{3}\rho a^3s,\quad I=\frac25Ma^2,\quad T=\frac12 I\Omega^2,\quad W=-\frac{3GM^2}{5a}\frac{\theta}{e}$, and $\displaystyle \frac{\Omega^2}{4\pi G\rho}=\frac{s(3-2e^2)\theta}{2e^3}-\frac{3s^2}{2e^2}$. These homogeneous-spheroid expressions give (3); the equilibrium and energy conventions are those of [1], Sections 2 and 4.

Comments

(5)

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 $M$, angular velocity is $\Omega\geq0$, and axial moment of inertia is $I$. Newton's gravitational constant is $G$. The ratio uses $|W|=-W$, so it is positive and independent of the overall size and density.

(6)

The eccentricity $e_k$ at the same branch determines the energy ratio 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 $\xi=s/e>0$.

(7)

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]

(8)

$P_\ell$ 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.

(9)

The dimensionless inertia is $I/(Ma^2)=2/5$. In the nonrotating spherical limit, $a$ equals the spherical radius $R$, so this ratio equals the homogeneous $n=0$ entry of the polytropic moment-of-inertia table.

Programs

(P1)

Sage

# Download generate.py from this table, then run with SageMath.
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 $k$, with energy ratios obtained from their exact eccentricities by (3)

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

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 energy-ratio 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. Propagating the complete root enclosure through (3) gives bounds containing the exact ratio; the largest radius is less than $7.96\times10^{-128}$, justifying the stored 100 significant digits. Repetition at 700 bits gives the same rational root brackets.

Independent validated quadrature computes the defining oblate integrals from explicit factorial coefficients of the Legendre polynomials and certifies opposite signs in the original bifurcation equation at every root-bracket endpoint. A separate calculation assembles the surface potential, equilibrium angular velocity, inertia, kinetic energy and gravitational energy using the virial relation in [1]. Its largest energy-ratio radius is less than $1.75\times10^{-125}$. Both enclosures lie inside every written decimal interval. Nonzero closed-form controls precede these checks. Every displayed conversion and all ten coarse energy ratios and eccentricities in Table 1 of [1] are also checked. The attached generator, checks.py, certification.md and numerical evidence document the computation and its verification.