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{"Title": "Kinetic-to-gravitational energy ratios at axisymmetric bifurcations of Maclaurin spheroids", "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 $2k+2$ CITE{AKM}.", "Parameters": {"k": {"type": "Z", "title": "axisymmetric branch index", "display": "$k$", "constraints": "$k\\geq1$"}}, "Tags": ["physics", "fluid dynamics", "bifurcation theory", "dimensionless"], "References": {"AKM": {"bib": "M. Ansorg, A. Kleinw\u00e4chter and R. Meinel, Uniformly rotating axisymmetric fluid configurations bifurcating from highly flattened Maclaurin spheroids, Monthly Notices of the Royal Astronomical Society 339 (2003), 515\u2013523.", "doi": "10.1046/j.1365-8711.2003.06190.x", "arxiv": "astro-ph/0208267"}}, "Links": {"AKM-text": {"title": "Ansorg, Kleinw\u00e4chter and Meinel, full text, Sections 2 and 4 and Table 1", "url": "https://arxiv.org/html/astro-ph/0208267v1"}}, "Data properties": {"type": "R", "digits": "100", "rigour": "proven", "complete": "no", "complete-note": "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 CITE{AKM}, 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", "rigour details": "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 CITE{formula-energy} 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.\n\nIndependent validated quadrature computes the defining oblate integrals from explicit factorial coefficients of the HREF{Legendre_polynomials}[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 CITE{AKM}. 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 CITE{AKM} are also checked. The attached generator, checks.py, certification.md and numerical evidence document the computation and its verification."}, "Display properties": {"number-header": "$\\beta_k=T/|W|$"}, "Numbers": {"1": "0.3589038426086023489212077095394976938733198790188782811837733386976592448431413347536499791124832650", "2": "0.4034546560514278875104263937991677914926072257320901229910846733194936819231593658758351340069518789", "3": "0.4266350788289110340115277434309860626782686583947364785409904099343559390387146239911605383518065690", "4": "0.4408438858453380818788005671156565580641702779828970724093421009921843092520957824079281359528278095", "5": "0.4504437511469904155870450255636579921249617349316140669755082541085733629982880546262065081368000199", "6": "0.4573637367354507890597402325162575738162407756461796798543286680230306504592551110014315400078797924", "7": "0.4625883433024108650841521266487169934337558054636693002729858392788013772044512546831952890458037012", "8": "0.4666725198614399817480380279421531070060618084985290769908372082567362898420959009952668600578514947", "9": "0.4699528712415190837142322925587485213335326473795846505542923667411204968634900109609866127428395359", "10": "0.4726454147635197162171961338556955822870470885280978441091384585974743660295423655066730947687918184"}, "Keywords": ["Maclaurin spheroid", "axisymmetric bifurcation", "rotating homogeneous fluid", "Dyson ring bifurcation", "energy ratio", "T/|W|", "neutral mode"], "Comments": {"comment-model": "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.", "comment-shape": "The HREF{Eccentricities_at_axisymmetric_bifurcations_of_Maclaurin_spheroids}[eccentricity $e_k$ at the same branch] determines the energy ratio by CITE{formula-energy}. Here $e=\\sqrt{1-c^2/a^2}$, $s=\\sqrt{1-e^2}=c/a$, $\\theta=\\arcsin(e)\\in(0,\\pi/2)$, and $\\xi=s/e>0$.", "comment-branch": "For $k\\geq1$, $\\xi_k$ is the positive root of CITE{formula-bifurcation}. This is the indexing of CITE{AKM}, Section 2, equations (3)\u2013(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. CITE{AKM-text}", "comment-functions": "$P_\\ell$ is the HREF{Legendre_polynomials}[Legendre polynomial] normalized by $P_\\ell(1)=1$. The functions $p_\\ell$ and $q^{\\mathrm{obl}}_\\ell$ in CITE{formula-oblate} use a real integral to fix the oblate convention. They differ from the HREF{Secondary_polynomials_of_the_Legendre_polynomials}[secondary Legendre polynomials] and from the real-argument branch tabulated in HREF{Values_of_the_Legendre_functions_of_the_second_kind}[Legendre functions of the second kind].", "comment-inertia": "The inertia $I=2Ma^2/5$ is normalized with the equatorial semiaxis. In the nonrotating spherical limit it agrees with the homogeneous $n=0$ entry of HREF{Moment_of_inertia_of_polytropic_stars#0}[the polytropic moment-of-inertia table], whose radius is spherical."}, "Formulas": {"formula-oblate": "$p_\\ell(\\xi)=(-i)^\\ell P_\\ell(i\\xi),\\qquad q^{\\mathrm{obl}}_\\ell(\\xi)=p_\\ell(\\xi)\\int_\\xi^\\infty\\frac{dt}{(1+t^2)p_\\ell(t)^2},\\quad \\xi>0$. These are the real oblate radial functions of CITE{AKM}, Section 2.", "formula-bifurcation": "$2p_{2k+2}(\\xi_k)q^{\\mathrm{obl}}_{2k+2}(\\xi_k)=\\frac{\\xi_k}{p_2(\\xi_k)}\\left(1-2\\xi_kq^{\\mathrm{obl}}_2(\\xi_k)\\right),\\quad e_k=(1+\\xi_k^2)^{-1/2}$. This is the axisymmetric bifurcation condition in CITE{AKM}, equation (3).", "formula-energy": "$\\displaystyle \\beta_k=\\frac{3}{2e_k^2}-1-\\frac{3\\sqrt{1-e_k^2}}{2e_k\\arcsin(e_k)}=\\frac{1+3\\xi_k^2}{2}-\\frac{3\\xi_k}{2\\arctan(1/\\xi_k)}$. The positive real inverse trigonometric branches give $\\arcsin(e_k)=\\arctan(1/\\xi_k)$.", "formula-physical": "$\\displaystyle M=\\frac{4\\pi}{3}\\rho a^3s,\\quad I=\\frac25Ma^2,\\quad T=\\frac12 I\\Omega^2,\\quad W=-\\frac{3GM^2}{5a}\\frac{\\theta}{e},\\quad \\frac{\\Omega^2}{4\\pi G\\rho}=\\frac{s(3-2e^2)\\theta}{2e^3}-\\frac{3s^2}{2e^2}$. These homogeneous-spheroid expressions give CITE{formula-energy}; the equilibrium and energy conventions are those of CITE{AKM}, Sections 2 and 4."}, "Similar tables": [{"table": "HREF{Eccentricities_at_axisymmetric_bifurcations_of_Maclaurin_spheroids}[Eccentricities at axisymmetric Maclaurin bifurcations]", "relation": "the same configurations indexed by $k$, with energy ratios obtained from their exact eccentricities by CITE{formula-energy}"}, {"table": "HREF{Legendre_polynomials}[Legendre polynomials]", "relation": "the degree $2k+2$ polynomial determines the neutral axisymmetric mode"}, {"table": "HREF{Secondary_polynomials_of_the_Legendre_polynomials}[Secondary Legendre polynomials]", "relation": "the polynomial part of the second Legendre solution is distinct from the oblate radial function used in the bifurcation equation"}, {"table": "HREF{Values_of_the_Legendre_functions_of_the_second_kind}[Legendre functions of the second kind]", "relation": "real-argument values use a different branch from the real oblate integral convention"}, {"table": "HREF{Moment_of_inertia_of_polytropic_stars}[Moments of inertia of polytropic stars]", "relation": "the homogeneous spherical model supplies the nonrotating limit of the inertia normalization"}], "Programs": {"program-sage": {"language": "Sage", "code": "# Download generate.py from this table, then run with SageMath.\nfrom generate import enclose\nfor k in range(1, 11):\n print(k, enclose(k, digits=100)[0])"}}}