{
  "before": {
    "Title": "Mass-radius constants of Lane–Emden polytropes",
    "Definition": "The dimensionless mass-radius constant $N_n$ is defined by $K=N_nGM^{(n-1)/n}R^{(3-n)/n}$ for a spherical, nonrotating Newtonian polytrope obeying $P=K\\rho^{1+1/n}$, with pressure $P$, density $\\rho$, total mass $M$, finite zero-pressure radius $R$ and gravitational constant $G$ CITE{Princeton}.",
    "Parameters": {
      "n": {
        "type": "Q",
        "title": "polytropic index",
        "display": "$n$",
        "constraints": "$0<n<5$"
      }
    },
    "Tags": [
      "physics",
      "dimensionless"
    ],
    "Links": {
      "Princeton": {
        "title": "Polytropes, Princeton stellar-structure notes, equations (poly.10)–(poly.11) and (poly.17)",
        "url": "https://www.astro.princeton.edu/~gk/A403/polytrop.pdf"
      },
      "Pols": {
        "title": "O. R. Pols, Stellar Structure and Evolution (2011), Table 4.1 and equation (4.15)",
        "url": "https://www.astro.ru.nl/~onnop/education/stev_utrecht_notes/chapter1-4.pdf"
      },
      "Lu": {
        "title": "Wenbin Lu, Polytropic Star, equations (1)–(7)",
        "url": "https://wenbinlu.github.io/files/LaneEmden.pdf"
      }
    },
    "Data properties": {
      "type": "R",
      "complete": "no",
      "rigour": "heuristic (agreement-checked)",
      "complete-note": "it holds $n=1/2,1,3/2,2,5/2,3,7/2,4,9/2$. The linear model $1$ is intrinsic, and $3/2,3$ are stellar application indices. The remaining six indices are exploratory references for mass-radius coefficients of simple algebraic density–enthalpy 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 incompressible index zero is excluded because this coefficient contains $1/n$. The attached argument-selection.md explains the disjoint groups and coverage tradeoffs",
      "rigour details": "The index-one value $2/\\pi$ is 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 $2/3$. The largest relative difference over all nine coefficients, including the analytic control, is less than $1.82\\times10^{-154}$. Only 100 significant digits are stored.\n\nA separate radial-coordinate implementation accumulates the pressure integral and uses the polytropic virial identity to recover the mass normalization and then $N_n$. The analytic index-zero and index-one backgrounds are nonzero controls; displaced controls are rejected. Every selected coefficient agrees with this calculation to a relative tolerance of $10^{-40}$, and the six overlapping entries agree with the printed precision of CITE{Pols}, Table 4.1. Readback checks test stored intervals, the index-one anchor, the physical scale equations, and the radius, mass, concentration and pressure identities.\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": "$N_n$"
    },
    "Numbers": {
      "1/2": "2.523598217854275902292409216845526433589728626251636294335259942272127183364035855587278723960197685",
      "1": {
        "number": "0.6366197723675813430755350534900574481378385829618257949906693762355871905369061403604552110650123438",
        "comment": "The linear profile $\\theta_1(\\xi)=\\sin(\\xi)/\\xi$ gives $N_1=2/\\pi$."
      },
      "3/2": "0.4242166795999143054300800493765433608497259148955332929829457813948871126514900180650397787129495070",
      "2": "0.3647479420921269201344337345944851502857083143700611518486431326202962906625898698660025640552715746",
      "5/2": "0.3515009408214026965177580190482487628244585207789958095103054761944702191995540492366057830279611529",
      "3": "0.3639390619287273761082466923988154714800583253638038147513513898696629626407076169217773200331871787",
      "7/2": "0.4010435197550305191911372334788411164390397888033057580214159957450177000779555612881345047495246186",
      "4": "0.4771955635610856203008816339649316484467462942052017051026712129331325412861135115392436471447729326",
      "9/2": "0.6579832258419505680044133116057910404190658207306360690587391038783342542472109953935427213151036816"
    },
    "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-convention": "The coefficient $N_n$ has the convention of CITE{Pols}, equation (4.15). That source writes $z_n$ for $\\xi_1(n)$ and $\\Theta_n$ for the dimensionless mass $\\mu_n$. The equation-of-state coefficient $K>0$ is dimensional; $N_n$ is dimensionless.",
      "comment-applications": "The indices $n=3/2$ and $n=3$ describe idealized adiabatic stellar models supported by nonrelativistic and ultrarelativistic pressure, respectively CITE{Princeton}. A convective envelope also motivates $n=3/2$ CITE{Lu}. Models with $n>3$ are mathematical equilibria without a claim of radial dynamical stability.",
      "comment-special": "At fixed $K$, the index-one radius is independent of mass, and the index-three mass is independent of radius CITE{Pols}. The coefficient $N_3$ remains well defined.",
      "comment-endpoints": "The incompressible model at $n=0$ supplies no entry because the defining equation-of-state exponent contains $1/n$. The model at $n=5$ has infinite radius and supplies no finite-radius entry; its convergent dimensionless mass appears in HREF{Mass_constants_of_Lane_Emden_polytropes#5}[the mass table] CITE{Princeton}."
    },
    "Formulas": {
      "formula-surface": "$N_n=\\dfrac{(4\\pi)^{1/n}}{n+1}\\mu_n^{(1-n)/n}\\xi_1(n)^{(n-3)/n}$, 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{Princeton}.",
      "formula-scale": "$R=a\\xi_1(n)$, $M=4\\pi a^3\\rho_c\\mu_n$ and $K=\\dfrac{4\\pi G}{n+1}a^2\\rho_c^{1-1/n}$, where $r=a\\xi$ and $a>0$ is the length scale. Eliminating $a$ and $\\rho_c$ gives the defining mass-radius relation CITE{Lu}.",
      "formula-pressure": "$N_n=W_n\\left(\\dfrac{4\\pi}{3C_n}\\right)^{1+1/n}$, where $W_n=P_cR^4/(GM^2)$ is the HREF{Central_pressure_constants_of_Lane_Emden_polytropes}[central pressure constant], $P_c$ is the central pressure, and $C_n=\\rho_c/\\bar\\rho$ is the HREF{Central-to-mean_density_ratio_of_Lane_Emden_polytropes}[central-to-mean density ratio], with $\\bar\\rho=3M/(4\\pi R^3)$ CITE{Pols}.",
      "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 $N_1=2/\\pi$ and $R=\\sqrt{\\pi K/(2G)}$.",
      "formula-three": "$N_3=\\dfrac{(4\\pi)^{1/3}}{4\\mu_3^{2/3}}$ and $M=\\left(\\dfrac{K}{GN_3}\\right)^{3/2}$ at $n=3$ CITE{Princeton}."
    },
    "Similar tables": [
      {
        "table": "HREF{Radius_of_Lane_Emden_polytropes}[Radius of Lane–Emden polytropes]",
        "relation": "gives the surface radius entering the mass-radius coefficient for the same finite models"
      },
      {
        "table": "HREF{Mass_constants_of_Lane_Emden_polytropes}[Mass constants of Lane–Emden polytropes]",
        "relation": "gives the mass normalization entering the mass-radius coefficient"
      },
      {
        "table": "HREF{Central_pressure_constants_of_Lane_Emden_polytropes}[Central pressure constants of Lane–Emden polytropes]",
        "relation": "combines with central concentration to determine the equation-of-state coefficient"
      },
      {
        "table": "HREF{Central-to-mean_density_ratio_of_Lane_Emden_polytropes}[Central-to-mean density ratio of Lane–Emden polytropes]",
        "relation": "combines with the central pressure constant to determine the mass-radius coefficient"
      },
      {
        "table": "HREF{Moment_of_inertia_of_polytropic_stars}[Moment of inertia of polytropic stars]",
        "relation": "gives the axial inertia of the same finite density profiles"
      }
    ],
    "Keywords": [
      "Lane-Emden equation",
      "Lane Emden mass-radius relation",
      "polytropic stars",
      "stellar structure",
      "polytropic equation of state",
      "polytropic mass-radius coefficient"
    ]
  },
  "after": {
    "Title": "Mass-radius constants of Lane–Emden polytropes",
    "Definition": "The dimensionless mass-radius constant $N_n$ is defined by $K=N_nGM^{(n-1)/n}R^{(3-n)/n}$ for a spherical, nonrotating Newtonian polytrope obeying $P=K\\rho^{1+1/n}$, with pressure $P$, density $\\rho$, total mass $M$, finite zero-pressure radius $R$ and gravitational constant $G$ CITE{Princeton}.",
    "Parameters": {
      "n": {
        "type": "Q",
        "title": "polytropic index",
        "display": "$n$",
        "constraints": "$0<n<5$"
      }
    },
    "Tags": [
      "physics",
      "dimensionless"
    ],
    "Links": {
      "Princeton": {
        "title": "Polytropes, Princeton stellar-structure notes, equations (poly.10)–(poly.11) and (poly.17)",
        "url": "https://www.astro.princeton.edu/~gk/A403/polytrop.pdf"
      },
      "Pols": {
        "title": "O. R. Pols, Stellar Structure and Evolution (2011), Table 4.1 and equation (4.15)",
        "url": "https://www.astro.ru.nl/~onnop/education/stev_utrecht_notes/chapter1-4.pdf"
      },
      "Lu": {
        "title": "Wenbin Lu, Polytropic Star, equations (1)–(7)",
        "url": "https://wenbinlu.github.io/files/LaneEmden.pdf"
      }
    },
    "Data properties": {
      "type": "R",
      "complete": "no",
      "rigour": "heuristic (agreement-checked)",
      "complete-note": "it holds $n=1/2,1,3/2,2,5/2,3,7/2,4,9/2$. The linear model $1$ is intrinsic, and $3/2,3$ are stellar application indices. The remaining six indices are exploratory references for mass-radius coefficients of simple algebraic density–enthalpy 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 incompressible index zero is excluded because this coefficient contains $1/n$. The attached argument-selection.md explains the disjoint groups and coverage tradeoffs",
      "rigour details": "The index-one value $2/\\pi$ is 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 $2/3$. The largest relative difference over all nine coefficients, including the analytic control, is less than $1.82\\times10^{-154}$. Only 100 significant digits are stored.\n\nA separate radial-coordinate implementation accumulates the pressure integral and uses the polytropic virial identity to recover the mass normalization and then $N_n$. The analytic index-zero and index-one backgrounds are nonzero controls; displaced controls are rejected. Every selected coefficient agrees with this calculation to a relative tolerance of $10^{-40}$, and the six overlapping entries agree with the printed precision of CITE{Pols}, Table 4.1. Readback checks test stored intervals, the index-one anchor, the physical scale equations, and the radius, mass, concentration and pressure identities.\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": "$N_n$"
    },
    "Numbers": {
      "1/2": "2.523598217854275902292409216845526433589728626251636294335259942272127183364035855587278723960197685",
      "1": {
        "number": "0.6366197723675813430755350534900574481378385829618257949906693762355871905369061403604552110650123438",
        "comment": "The index-one profile $\\theta_1(\\xi)=\\sin(\\xi)/\\xi$ gives $N_1=2/\\pi$."
      },
      "3/2": "0.4242166795999143054300800493765433608497259148955332929829457813948871126514900180650397787129495070",
      "2": "0.3647479420921269201344337345944851502857083143700611518486431326202962906625898698660025640552715746",
      "5/2": "0.3515009408214026965177580190482487628244585207789958095103054761944702191995540492366057830279611529",
      "3": "0.3639390619287273761082466923988154714800583253638038147513513898696629626407076169217773200331871787",
      "7/2": "0.4010435197550305191911372334788411164390397888033057580214159957450177000779555612881345047495246186",
      "4": "0.4771955635610856203008816339649316484467462942052017051026712129331325412861135115392436471447729326",
      "9/2": "0.6579832258419505680044133116057910404190658207306360690587391038783342542472109953935427213151036816"
    },
    "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-convention": "The coefficient $N_n$ has the convention of CITE{Pols}, equation (4.15). That source writes $z_n$ for $\\xi_1(n)$ and $\\Theta_n$ for the dimensionless mass $\\mu_n$. The equation-of-state coefficient $K>0$ is dimensional; $N_n$ is dimensionless.",
      "comment-applications": "The indices $n=3/2$ and $n=3$ describe idealized adiabatic stellar models supported by nonrelativistic and ultrarelativistic pressure, respectively CITE{Princeton}. A convective envelope also motivates $n=3/2$ CITE{Lu}. Models with $n>3$ are mathematical equilibria without a claim of radial dynamical stability.",
      "comment-special": "At fixed $K$, the index-one radius is independent of mass, and the index-three mass is independent of radius CITE{Pols}. The coefficient $N_3$ remains well defined.",
      "comment-endpoints": "The incompressible model at $n=0$ supplies no entry because the defining equation-of-state exponent contains $1/n$. The model at $n=5$ has infinite radius and supplies no finite-radius entry; its convergent dimensionless mass appears in HREF{Mass_constants_of_Lane_Emden_polytropes#5}[the mass table] CITE{Princeton}."
    },
    "Formulas": {
      "formula-surface": "$N_n=\\dfrac{(4\\pi)^{1/n}}{n+1}\\mu_n^{(1-n)/n}\\xi_1(n)^{(n-3)/n}$, 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{Princeton}.",
      "formula-scale": "$R=a\\xi_1(n)$, $M=4\\pi a^3\\rho_c\\mu_n$ and $K=\\dfrac{4\\pi G}{n+1}a^2\\rho_c^{1-1/n}$, where $r=a\\xi$ and $a>0$ is the length scale. Eliminating $a$ and $\\rho_c$ gives the defining mass-radius relation CITE{Lu}.",
      "formula-pressure": "$N_n=W_n\\left(\\dfrac{4\\pi}{3C_n}\\right)^{1+1/n}$, where $W_n=P_cR^4/(GM^2)$ is the HREF{Central_pressure_constants_of_Lane_Emden_polytropes}[central pressure constant], $P_c$ is the central pressure, and $C_n=\\rho_c/\\bar\\rho$ is the HREF{Central-to-mean_density_ratio_of_Lane_Emden_polytropes}[central-to-mean density ratio], with $\\bar\\rho=3M/(4\\pi R^3)$ CITE{Pols}.",
      "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 $N_1=2/\\pi$ and $R=\\sqrt{\\pi K/(2G)}$.",
      "formula-three": "$N_3=\\dfrac{(4\\pi)^{1/3}}{4\\mu_3^{2/3}}$ and $M=\\left(\\dfrac{K}{GN_3}\\right)^{3/2}$ at $n=3$ CITE{Princeton}."
    },
    "Similar tables": [
      {
        "table": "HREF{Radius_of_Lane_Emden_polytropes}[Radius of Lane–Emden polytropes]",
        "relation": "gives the surface radius entering the mass-radius coefficient for the same finite models"
      },
      {
        "table": "HREF{Mass_constants_of_Lane_Emden_polytropes}[Mass constants of Lane–Emden polytropes]",
        "relation": "gives the mass normalization entering the mass-radius coefficient"
      },
      {
        "table": "HREF{Central_pressure_constants_of_Lane_Emden_polytropes}[Central pressure constants of Lane–Emden polytropes]",
        "relation": "combines with central concentration to determine the equation-of-state coefficient"
      },
      {
        "table": "HREF{Central-to-mean_density_ratio_of_Lane_Emden_polytropes}[Central-to-mean density ratio of Lane–Emden polytropes]",
        "relation": "combines with the central pressure constant to determine the mass-radius coefficient"
      },
      {
        "table": "HREF{Moment_of_inertia_of_polytropic_stars}[Moment of inertia of polytropic stars]",
        "relation": "gives the axial inertia of the same finite density profiles"
      }
    ],
    "Keywords": [
      "Lane-Emden equation",
      "Lane Emden mass-radius relation",
      "polytropic stars",
      "stellar structure",
      "polytropic equation of state",
      "polytropic mass-radius coefficient"
    ]
  },
  "old_generator": "\"\"\"Mass-radius constants of Lane–Emden polytropes -- numberdb.org/T618.\n\nRun with SageMath (install numberdb once with sage -pip install numberdb):\n    sage -python generate.py           # recompute and verify every stored row\n    sage -python generate.py --fill    # fill this existing draft\nDownload solver.py and the other attached files listed in `files` beside this\nscript. Draft reads and writes need NUMBERDB_API_KEY. The campaign wrapper\naccepts NUMBERDB_KEY_FROM_STDIN=1 and NUMBERDB_PUBLISH=1 for filling entries.\nFilling does not offer, review or publish the draft.\n\nN_n = (4*pi)^(1/n)/(n+1) * mu_n^((1-n)/n) * xi_1^((n-3)/n),\nso K = N_n G M^((n-1)/n) R^((3-n)/n) for P=K*rho^(1+1/n).\nThe index-one value is 2/pi, evaluated in Arb ball arithmetic.\nEvery other value integrates the normalized positive Lane–Emden solution\nfreshly. solver.py is the unmodified attachment from T617, itself shared by\nthis family; its download URL and SHA-256 are in source-manifest.json.\nRuns at 150/210 decimal digits use orders 160/210, step caps 1/12 and 1/18,\nand central matching radii 1/2 and 2/3. Their agreement supports only a\nheuristic claim: ODE truncation, propagation and endpoint errors are not\nrigorously bounded. 100 significant digits are retained.\nThe solver supports half-integer indices, not all rational n in the domain.\nNo stored numerical input is used by value(). See argument-selection.md for\nthe mixed argument families and the deliberately noncanonical finite bounds.\n\"\"\"\nimport os\nimport sys\nfrom functools import lru_cache\nimport numberdb.sage as numberdb\nfrom sage.rings.real_arb import RealBallField\nfrom sage.rings.rational_field import QQ\nfrom solver import radius\n\nINDICES=('1/2','1','3/2','2','5/2','3','7/2','4','9/2')\nCONFIG={150:(160,12,'1/2'),210:(210,18,'2/3')}\n\n@lru_cache(maxsize=24)\ndef background(n,working):\n    order,divisor,start=CONFIG[working]\n    return radius(n,working,order,divisor,start,trace=True)\n\ndef coefficient(n,working):\n    x,q,meta=background(n,working)\n    F=x.parent(); nn=F(QQ(n)); one=F(1)\n    return (4*F.pi())**(one/nn)/(nn+1)*q**((1-nn)/nn)*x**((nn-3)/nn)\n\nclass LaneEmdenMassRadius(numberdb.Generator):\n    table='T618'\n    parameters=('n',)\n    type='R'\n    digits=100\n    rigour='heuristic (agreement-checked)'\n    files=('generate.py','solver.py','compute.py','independent_checks.py',\n           'radial_check.py','readback_checks.py','dry_run.py','check.py','argument-selection.md',\n           'numerical-evidence.json','independent-evidence.json',\n           'skill-manifest.json','skill-evidence.json','source-manifest.json')\n\n    def enumerate(self):\n        for n in INDICES:yield {'n':n}\n\n    def value(self,params,digits):\n        n=str(QQ(params['n']))\n        if n not in INDICES:\n            raise ValueError('Use a selected positive half-integer below five.')\n        if digits>100:\n            raise ValueError('This precision policy was validated for at most 100 digits.')\n        if n=='1':\n            B=RealBallField(numberdb.bits(digits,losing=80))\n            return {'number':B(2)/B.pi(),'comment':r'The linear profile $\\theta_1(\\xi)=\\sin(\\xi)/\\xi$ gives $N_1=2/\\pi$.'}\n        return numberdb.agreeing(lambda working:str(coefficient(n,working)),at=(150,210))\n\nif __name__=='__main__':\n    if os.environ.get('NUMBERDB_KEY_FROM_STDIN')=='1':\n        token=sys.stdin.read().strip()\n        if '=' in token and token.split('=',1)[0].replace('_','').isalnum():\n            token=token.split('=',1)[1].strip().strip('\\\"\\'')\n        os.environ['NUMBERDB_API_KEY']=token\n    g=LaneEmdenMassRadius()\n    if '--fill' in sys.argv or os.environ.get('NUMBERDB_PUBLISH')=='1':\n        print(g.publish(message='Add nine agreement-checked polytropic mass-radius coefficients.'))\n    else:\n        report=g.verify(sample=None);print(report);sys.exit(0 if report.ok else 1)\n"
}