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{"Title": "Central pressure constants of Lane\u2013Emden polytropes", "Definition": "The dimensionless central pressure constant is $W_n=P_cR^4/(GM^2)$ for a spherical, nonrotating Newtonian Lane\u2013Emden polytrope with central pressure $P_c$, total mass $M$, finite zero-pressure radius $R$ and gravitational constant $G$ CITE{Pols}.", "Parameters": {"n": {"type": "Q", "title": "polytropic index", "display": "$n$", "constraints": "$0\\leq n<5$"}}, "Tags": ["physics", "dimensionless"], "Links": {"Pols": {"title": "O. R. Pols, Stellar Structure and Evolution (2011), Table 4.1 and equation (4.17)", "url": "https://www.astro.ru.nl/~onnop/education/stev_utrecht_notes/chapter1-4.pdf"}, "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(6) and (14)\u2013(18)", "url": "https://wenbinlu.github.io/files/LaneEmden.pdf"}}, "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 central-pressure coefficients of 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 two analytic values are evaluated with Arb ball arithmetic. Other entries use fresh central-series initialization and Taylor continuation in $t=\\sqrt{\\theta_n}$. Runs at 150 and 210 working decimal digits use Taylor orders 160 and 210, step caps $1/12$ and $1/18$, and matching radii $1/2$ and $2/3$. Across all ten models, including analytic controls, the worst relative difference is less than $1.71\\times10^{-153}$. Only 100 significant digits are stored.\n\nA separate radial-coordinate implementation accumulates the pressure integral and uses the virial identity to recover $W_n$. It uses 70 working decimal digits, order 80, exact rational central coefficients, and a leading tail correction after reaching positive $\\theta_n<10^{-30}$. The analytic models are nonzero controls and deliberately displaced controls are rejected. The seven overlapping entries also agree with the printed precision of CITE{Pols}, Table 4.1. Readback checks test stored intervals, analytic anchors, radius/mass and concentration identities, and the independent pressure-integral identity.\n\nThese comparisons do not rigorously bound ODE truncation, propagation or endpoint errors. The attached generator, checks, numerical evidence, source manifest and argument selection record the method and its limitations."}, "Display properties": {"number-header": "$W_n$"}, "Numbers": {"0": {"number": "0.1193662073189215018266628225293857715258447343053423365607505080441725982256699013175853520746898145", "comment": "The incompressible profile $\\theta_0(\\xi)=1-\\xi^2/6$ gives $W_0=3/(8\\pi)$."}, "1/2": "0.2122090668847058146514151712294673607965824670515319520774842154499937681418714051577494461214301425", "1": {"number": "0.3926990816987241548078304229099378605246461749218882276218680740384770507857761248285043531677646335", "comment": "The linear profile $\\theta_1(\\xi)=\\sin(\\xi)/\\xi$ gives $W_1=$ HREF{Rational_multiples_of_pi#1/8}[$\\pi/8$]."}, "3/2": "0.7701403713509219860693239538862683630544209233233207499020699321649034335275621508650320009704158844", "2": "1.638183345774008023821355261677060437565586841900123307066213837059046342901006046332720775030527010", "5/2": "3.909061722491728320095517068744889431666761501032459324012515917380241760744477014551488616687436247", "3": "11.05068102346314799304315971504126365736010544040883781458040379045898704778751120597494750858828034", "7/2": "40.90981695757662252639119748667322587515690192200886067796327564754499533836115754134730786475829749", "4": "247.5594395183506235294731524402453676389264301987962555617980889349328389420367998828989004999094764", "9/2": "4921.842229833709556638384550698135185951827314279612453512847716674082968145767852052647305893687408"}, "Comments": {"comment-model": "The normalized density is $\\rho=\\rho_c\\theta_n^n$, where $\\rho_c$ is the central density and $(\\xi^2\\theta_n^\\prime)^\\prime=-\\xi^2\\theta_n^n$, $\\theta_n(0)=1$ and $\\theta_n^\\prime(0)=0$. Follow the positive real solution to its HREF{Radius_of_Lane_Emden_polytropes}[first zero $\\xi_1(n)$], without continuing fractional powers through negative values CITE{Princeton}.", "comment-eos": "For $n>0$, the pressure obeys $P=K\\rho^{1+1/n}$ with $K>0$, and $P=P_c\\theta_n^{n+1}$. The index $n=0$ denotes the incompressible model, with constant interior density and a surface density jump CITE{Princeton}.", "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 $n=5$ model has infinite radius, so the finite-radius normalization supplies no entry here. Its convergent total mass is included in HREF{Mass_constants_of_Lane_Emden_polytropes#5}[the mass table] CITE{Princeton}.", "comment-notation": "The coefficient $W_n$ agrees with the notation in CITE{Pols}, equation (4.17). That source writes $z_n$ for $\\xi_1(n)$ and $\\Theta_n$ for the dimensionless mass $\\mu_n$."}, "Formulas": {"formula-surface": "$W_n=\\dfrac{\\xi_1(n)^4}{4\\pi(n+1)\\mu_n^2}=\\dfrac{1}{4\\pi(n+1)\\theta_n^\\prime(\\xi_1(n))^2}$, where $\\xi_1(n)$ is the HREF{Radius_of_Lane_Emden_polytropes}[dimensionless radius] and $\\mu_n=-\\xi_1(n)^2\\theta_n^\\prime(\\xi_1(n))=\\int_0^{\\xi_1(n)}\\xi^2\\theta_n(\\xi)^n\\,d\\xi$ is the HREF{Mass_constants_of_Lane_Emden_polytropes}[dimensionless mass constant] CITE{Pols}.", "formula-scale": "$R=a\\xi_1(n)$, $M=4\\pi a^3\\rho_c\\mu_n$ and $a^2=(n+1)P_c/(4\\pi G\\rho_c^2)$, where $r=a\\xi$ and $a>0$ is the length scale. Thus $W_n$ is independent of $a$ and $\\rho_c$ CITE{Lu}.", "formula-concentration": "$W_n=\\dfrac{9C_n^2}{4\\pi(n+1)\\xi_1(n)^2}$, where $C_n=\\rho_c/\\bar\\rho=\\xi_1(n)^3/(3\\mu_n)$ is the HREF{Central-to-mean_density_ratio_of_Lane_Emden_polytropes}[central-to-mean density ratio] and $\\bar\\rho=3M/(4\\pi R^3)$.", "formula-virial": "$W_n\\int_0^{\\xi_1(n)}\\xi^2\\theta_n(\\xi)^{n+1}\\,d\\xi=\\dfrac{\\xi_1(n)^3}{4\\pi(5-n)}$. This is the pressure-integral form of the polytropic virial identity CITE{Lu}.", "formula-zero": "$\\theta_0(\\xi)=1-\\xi^2/6$, $\\xi_1(0)=\\sqrt6$ and $\\mu_0=2\\sqrt6$, so $W_0=3/(8\\pi)$.", "formula-one": "$\\theta_1(\\xi)=\\sin(\\xi)/\\xi$, with its central value defined by continuity, and $\\xi_1(1)=\\mu_1=\\pi$, so $W_1=$ HREF{Rational_multiples_of_pi#1/8}[$\\pi/8$]."}, "Similar tables": [{"table": "HREF{Radius_of_Lane_Emden_polytropes}[Radius of Lane\u2013Emden polytropes]", "relation": "gives the surface radius whose fourth power enters the pressure normalization"}, {"table": "HREF{Mass_constants_of_Lane_Emden_polytropes}[Mass constants of Lane\u2013Emden polytropes]", "relation": "gives the dimensionless mass whose square enters the pressure normalization"}, {"table": "HREF{Central-to-mean_density_ratio_of_Lane_Emden_polytropes}[Central-to-mean density ratio of Lane\u2013Emden polytropes]", "relation": "determines the pressure coefficient together with the surface radius and polytropic index"}, {"table": "HREF{Moment_of_inertia_of_polytropic_stars}[Moment of inertia of polytropic stars]", "relation": "gives the normalized axial inertia of the same finite models, complementing their central pressure"}, {"table": "HREF{Rational_multiples_of_pi#1/8}[$\\pi/8$]", "relation": "equals the pressure constant of the index-one model"}], "Keywords": ["Lane-Emden equation", "Lane Emden central pressure", "polytropic stars", "stellar structure", "polytropic pressure coefficient"], "Programs": {"program-sage": {"language": "Python", "code": "# SageMath, with the attached generate.py and solver.py beside this file.\nfrom generate import LaneEmdenPressure\ng = LaneEmdenPressure()\ng.value({'n': '3/2'}, digits=100)"}}}