Central-to-mean density ratio of Lane–Emden polytropes

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Numbers

$n$

$C_n=\rho_c/\bar\rho$

comment: The incompressible model has constant interior density, so $\rho_c=\bar\rho$.

1/2:

1.835142742470789404397958139747332812550251023548086514109983238953816851225391284973473031432204202

3.289868133696452872944830333292050378437899802413596875471116458740014940806401747667257801239517411

comment: The index-one profile $\theta_1(\xi)=\sin(\xi)/\xi$ gives $C_1=\pi^2/3$.

3/2:

5.990704516303533569432122437352046143543709953439713519283484180934551058311017080294624120362848068

11.40254286195126166312668457624234481985100690198526797428283178503085860670612824078973659908583200

5/2:

23.40646226706270850255127920224583248151521809908861920850824617012777438501855885729369015134657876

54.18248110734076301518016972241244733847845680017920564327776190125397687378097819865577384229324613

7/2:

152.8836629709408051112153108344213443224687316036533327101564226813566769134978244793752597943995386

622.4078822507046597285667570409154261830488462500344603438503906047524589734425078112836397533638132

9/2:

6189.473124953750481499831224751744668134288847868172125266471122643033818206440911891146569271957931

Definition

The central-to-mean density ratio $C_n=\rho_c/\bar\rho$ compares the central density $\rho_c$ with the volume-averaged density $\bar\rho=3M/(4\pi R^3)$ of a finite Lane–Emden polytrope of total mass $M$ and radius $R$ [2].

Parameters

$n$

— polytropic index ($0\leq n<5$)

Formulas

(1)

$C_n=\dfrac{\xi_1(n)^3}{3\mu_n}=-\dfrac{\xi_1(n)}{3\theta_n^\prime(\xi_1(n))}$, where $\xi_1(n)$ is the 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 dimensionless mass constant [2].

(2)

$R=a\xi_1(n)$ and $M=4\pi a^3\rho_c\mu_n$, where $r=a\xi$ is the physical radial coordinate and $a>0$ is its length scale. Consequently $C_n$ is independent of $a$ and $\rho_c$ [3].

(3)

$\theta_0(\xi)=1-\xi^2/6$, $\xi_1(0)=\sqrt6$ and $\mu_0=2\sqrt6$, so $C_0=1$.

(4)

$\theta_1(\xi)=\sin(\xi)/\xi$, with its central value defined by continuity, and $\xi_1(1)=\mu_1=$ $\pi$, so $C_1=\pi^2/3$.

Comments

(5)

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 first zero $\xi_1(n)$, without continuing fractional powers through negative values [1].

(6)

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 [2].

(7)

The indices $n=3/2$ and $n=3$ describe idealized adiabatic stars supported by nonrelativistic and ultrarelativistic pressure, respectively [2]. A convective envelope also motivates $n=3/2$ [3]. Models with $n>3$ are included as mathematical equilibria without a claim of radial dynamical stability.

(8)

The $n=5$ model has infinite radius and no finite central-to-mean density ratio [1]. Its convergent mass is included in the mass table, but it supplies no entry here.

Programs

(P1)

Python

# SageMath, with the attached generate.py and solver.py beside this file.
from generate import LaneEmdenDensityRatio
g = LaneEmdenDensityRatio()
g.value({'n': '3/2'}, digits=100)

References

[1]

K. L. S. Yip, T. K. Chan and P. T. Leung, Perturbative solution to the Lane–Emden equation: an eigenvalue approach, Monthly Notices of the Royal Astronomical Society 465 (2017), 4265–4280. (arXiv) (doi)

Links

[2]

Polytropes, Princeton stellar-structure notes, equations (poly.12)–(poly.13)

[3]

Wenbin Lu, Polytropic Star, equations (1)–(5) and the convective-envelope discussion

Similar tables

Radius of Lane–Emden polytropes — gives the dimensionless radius in the numerator of the concentration formula for the same finite models

Mass constants of Lane–Emden polytropes — gives the dimensionless mass that determines the volume-averaged density for the same models

Data properties

Entries are of type: real number

How well the digits are known: heuristic (agreement-checked)

Table is complete: no (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 concentration in 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, not an interpolation mesh. The attached argument-selection.md explains the groups and coverage tradeoffs)

How they were obtained:

The incompressible ratio is the exact integer 1, and the index-one value is evaluated as $\pi^2/3$ with Arb ball arithmetic. The other eight ratios are calculated from freshly integrated radii and masses, using central-series initialization and Taylor continuation in $t=\sqrt{\theta_n}$. Runs at 150 and 210 working decimal digits use orders 160 and 210, step caps $1/12$ and $1/18$, and matching radii $1/2$ and $2/3$. The largest relative difference of the ten computed ratios, including the analytic controls, is less than $1.38\times10^{-153}$. Only 100 significant digits are stored for the nine approximated entries.

A separate radial-coordinate implementation checks all ten ratios at a relative tolerance of $10^{-40}$ and checks the pressure-integral virial identity. It uses 70 working decimal digits, order 80, exact rational central coefficients, and a leading mass-tail correction after reaching positive $\theta_n<10^{-30}$. The exact models are controls and deliberately displaced values are rejected. The stellar cases also agree with the two-decimal reference values in [2]. Readback checks test the stored decimal intervals, analytic anchors, companion radius/mass identity and independent integral identity.

These comparisons do not rigorously bound ODE truncation, propagated error or endpoint approximation. The attached generator, validation scripts, numerical evidence, argument selection and source manifests document the method and its limitations.