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{"Title": "Moment of inertia $I/(MR^2)$ of polytropic stars", "Definition": "The dimensionless axial moment of inertia is $I/(MR^2)$, where $I=(8\\pi/3)\\int_0^R\\rho(r)r^4\\,dr$ is the moment about a diameter of a spherical Newtonian Lane\u2013Emden polytrope with total mass $M$ and finite radius $R$ CITE{Lu}.", "Parameters": {"n": {"type": "Q", "title": "polytropic index", "display": "$n$", "constraints": "$0\\leq n<5$"}}, "Tags": ["physics", "dimensionless"], "Data properties": {"type": "R", "complete": "no", "rigour": "heuristic (agreement-checked)", "complete-note": "it holds $n=0,1/2,1,3/2,2,5/2,3,7/2,4,9/2$. The analytic-profile cases $0,1$ are intrinsic, and $3/2,3$ are stellar application indices. The remaining six indices are exploratory references for axial inertia in simple algebraic density\u2013enthalpy laws $\\rho/\\rho_c=\\theta_n^n$. Integer and half-integer exponents span stiff through centrally concentrated finite models before adding higher denominators or approaching five more closely. Denominator two and the cutoff $9/2$ are noncanonical, notation-independent reference choices. The attached argument-selection.md explains the groups and coverage tradeoffs", "rigour details": "The incompressible value is the exact rational $2/5$. The index-one closed form is evaluated with Arb ball arithmetic. Other entries use central-series initialization and Taylor continuation in $t=\\sqrt{\\theta_n}$, accumulating the axial density moment. Two runs use 150 and 210 working decimal digits, Taylor orders 160 and 210, step caps $1/12$ and $1/18$, and central matching radii $1/2$ and $2/3$. Their worst relative difference over the ten models is less than $3.25\\times10^{-154}$; only 100 significant digits are stored.\n\nThe exact-profile cases are controls for the numerical integration. A separate implementation in the original radial coordinate checks every selected model, and the two independent moment integrals obey the integration-by-parts identity. A published Pad\u00e9 approximation in CITE{YL}, Appendix C.4, provides an additional normalization check on $0\\leq n\\leq3$; its largest observed error on the selected indices is about $0.2142$ percent. The measured error rounds to the paper's $0.21$ percent, which is not used as a strict bound. These checks do not rigorously bound Taylor truncation, propagated ODE error or the radial endpoint approximation. The attached code and numerical evidence record the comparisons and their limitations."}, "Numbers": {"0": {"number": "2/5", "comment": "The incompressible model has constant interior density, so $I/(MR^2)=2/5$."}, "1/2": "0.3259310823229534719464853993336045443105167882219770751853836902950038575618442312675120714335220737", "1": {"number": "0.2613819320973155808911488138277561110492315679776804712842305390899741828177249019018295709641793162", "comment": "The linear profile $\\theta_1(\\xi)=\\sin(\\xi)/\\xi$ gives $I/(MR^2)=2/3-4/\\pi^2$."}, "3/2": "0.2045998945746605070652043768951524258597849801408828983706649136983270028195821891945386765202628756", "2": "0.1548491833299797436513273920407495539775133666845901937560057552323741781915787541384728977134433185", "5/2": "0.1118049209855242369273655988199145244990578627191312536620619176915140576131061617385893780072484024", "3": "0.07535763996014873254377092405034916453997994539003603290673525942206637785303632985364500685648686873", "7/2": "0.04554837776599655713544578366050906131816355218124753039932280832947497556916702630310876768802866242", "4": "0.02257372948677378219084285189762278790604071681480629361867874865366185398525732917669147403534097898", "9/2": "0.006895160488497352250471174725496718319925166762305220139435506503538345980048222767880121665337970500"}, "Keywords": ["Lane-Emden equation", "stellar structure", "polytropic stars", "axial moment of inertia", "radius of gyration"], "Comments": {"comment-model": "The model is three-dimensional, spherical, nonrotating and Newtonian. Its density is $\\rho=\\rho_c\\theta_n^n$, where $(\\xi^2\\theta_n^\\prime)^\\prime=-\\xi^2\\theta_n^n$, $\\theta_n(0)=1$ and $\\theta_n^\\prime(0)=0$. The normalized solution follows its positive real branch to the HREF{Radius_of_Lane_Emden_polytropes}[first zero $\\xi_1(n)$], without continuing fractional powers through negative values CITE{YCL}.", "comment-eos": "For $n>0$, pressure $P$ and density $\\rho$ obey $P=K\\rho^{1+1/n}$ with $K>0$. The index $n=0$ denotes the incompressible model, with constant interior density and a surface density jump CITE{Princeton}.", "comment-normalization": "The normalization is by $MR^2$. Thus the table also gives the squared radius of gyration divided by $R^2$, where the radius of gyration is $\\sqrt{I/M}$. The spherical profile is nonrotating; its moment about a diameter is defined without specifying an angular velocity.", "comment-applications": "The indices $n=3/2$ and $n=3$ describe idealized adiabatic stars supported by nonrelativistic and ultrarelativistic pressure, respectively CITE{Princeton}. A convective envelope also motivates $n=3/2$ CITE{Lu}. Models with $n>3$ are included as mathematical equilibria without a claim of radial dynamical stability.", "comment-endpoint": "The model at $n=5$ has infinite radius and a divergent axial inertia integral. It supplies no finite-radius entry here, although its convergent total mass appears in HREF{Mass_constants_of_Lane_Emden_polytropes#5}[the mass table] CITE{YCL}."}, "Formulas": {"formula-integral": "$\\dfrac{I}{MR^2}=\\dfrac{2J_n}{3\\mu_n\\xi_1(n)^2}$, where $J_n=\\int_0^{\\xi_1(n)}\\xi^4\\theta_n(\\xi)^n\\,d\\xi$, $\\xi_1(n)$ is the HREF{Radius_of_Lane_Emden_polytropes}[dimensionless radius], and $\\mu_n=\\int_0^{\\xi_1(n)}\\xi^2\\theta_n(\\xi)^n\\,d\\xi=-\\xi_1(n)^2\\theta_n^\\prime(\\xi_1(n))$ is the HREF{Mass_constants_of_Lane_Emden_polytropes}[dimensionless mass constant] CITE{Lu}.", "formula-scale": "$R=a\\xi_1(n)$, $M=4\\pi a^3\\rho_c\\mu_n$ and $I=(8\\pi/3)a^5\\rho_cJ_n$, where $r=a\\xi$ and $a>0$ is a length scale. Consequently $I/(MR^2)$ is independent of $a$ and $\\rho_c$ CITE{YL}.", "formula-parts": "$J_n=\\mu_n\\xi_1(n)^2-6\\int_0^{\\xi_1(n)}\\xi^2\\theta_n(\\xi)\\,d\\xi$. This follows by integrating the Lane\u2013Emden equation twice by parts.", "formula-zero": "$\\theta_0(\\xi)=1-\\xi^2/6$ and $\\xi_1(0)=\\sqrt6$, so $I/(MR^2)=2/5$ at $n=0$.", "formula-one": "$\\theta_1(\\xi)=\\sin(\\xi)/\\xi$, with its central value defined by continuity, and $\\xi_1(1)=\\mu_1=$ HREF{Pi}[$\\pi$], so $I/(MR^2)=2/3-4/\\pi^2$ at $n=1$."}, "Similar tables": [{"table": "HREF{Radius_of_Lane_Emden_polytropes}[Radius of Lane\u2013Emden polytropes]", "relation": "gives the surface radius used to normalize the inertia of the same finite models"}, {"table": "HREF{Mass_constants_of_Lane_Emden_polytropes}[Mass constants of Lane\u2013Emden polytropes]", "relation": "gives the mass normalization for the same density profiles"}, {"table": "HREF{Central-to-mean_density_ratio_of_Lane_Emden_polytropes}[Central-to-mean density ratio of Lane\u2013Emden polytropes]", "relation": "describes central concentration for the same models, while axial inertia weights the density by squared distance from the axis"}], "Links": {"Princeton": {"title": "Polytropes, Princeton stellar-structure notes, pp. 1\u20133", "url": "https://www.astro.princeton.edu/~gk/A403/polytrop.pdf"}, "Lu": {"title": "Wenbin Lu, Polytropic Star, equations (1)\u2013(10)", "url": "https://wenbinlu.github.io/files/LaneEmden.pdf"}}, "References": {"YCL": {"bib": "K. L. S. Yip, T. K. Chan and P. T. Leung, Perturbative solution to the Lane\u2013Emden equation: an eigenvalue approach, Monthly Notices of the Royal Astronomical Society 465 (2017), 4265\u20134280.", "arxiv": "1611.07202", "doi": "10.1093/mnras/stw3041"}, "YL": {"bib": "K. L. S. Yip and P. T. Leung, Tidal Love numbers and moment-Love relations of polytropic stars, Monthly Notices of the Royal Astronomical Society 472 (2017), 4965\u20134981.", "arxiv": "1709.02469", "doi": "10.1093/mnras/stx2363"}}, "Programs": {"program-sage": {"language": "Python", "code": "# SageMath, with the attached generate.py and solver.py beside this file.\nfrom generate import PolytropicInertia\ng = PolytropicInertia()\ng.value({'n': '3/2'}, digits=100)"}}, "Display properties": {"number-header": "$I/(MR^2)$"}}