Mass-radius constants of Lane–Emden polytropes

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Numbers

$n$

$N_n$

1/2:

2.523598217854275902292409216845526433589728626251636294335259942272127183364035855587278723960197685

0.6366197723675813430755350534900574481378385829618257949906693762355871905369061403604552110650123438

comment: The index-one profile $\theta_1(\xi)=\sin(\xi)/\xi$ gives $N_1=2/\pi$.

3/2:

0.4242166795999143054300800493765433608497259148955332929829457813948871126514900180650397787129495070

0.3647479420921269201344337345944851502857083143700611518486431326202962906625898698660025640552715746

5/2:

0.3515009408214026965177580190482487628244585207789958095103054761944702191995540492366057830279611529

0.3639390619287273761082466923988154714800583253638038147513513898696629626407076169217773200331871787

7/2:

0.4010435197550305191911372334788411164390397888033057580214159957450177000779555612881345047495246186

0.4771955635610856203008816339649316484467462942052017051026712129331325412861135115392436471447729326

9/2:

0.6579832258419505680044133116057910404190658207306360690587391038783342542472109953935427213151036816

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 $G$ [1].

Parameters

$n$

— polytropic index ($0<n<5$)

Formulas

(1)

$N_n=\dfrac{(4\pi)^{1/n}}{n+1}\mu_n^{(1-n)/n}\xi_1(n)^{(n-3)/n}$, where $\xi_1(n)$ is the 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 dimensionless mass constant [1].

(2)

$R=a\xi_1(n)$, $M=4\pi a^3\rho_c\mu_n$ 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 [3].

(3)

$N_n=W_n\left(\dfrac{4\pi}{3C_n}\right)^{1+1/n}$, where $W_n=P_cR^4/(GM^2)$ is the central pressure constant, $P_c$ is the central pressure, and $C_n=\rho_c/\bar\rho$ is the central-to-mean density ratio, with $\bar\rho=3M/(4\pi R^3)$ [2].

(4)

$\theta_1(\xi)=\sin(\xi)/\xi$, with its central value defined by continuity, and $\xi_1(1)=\mu_1=$ $\pi$, so $N_1=2/\pi$ and $R=\sqrt{\pi K/(2G)}$.

(5)

$N_3=\dfrac{(4\pi)^{1/3}}{4\mu_3^{2/3}}$ and $M=\left(\dfrac{K}{GN_3}\right)^{3/2}$ at $n=3$ [1].

Comments

(6)

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

(7)

The coefficient $N_n$ has the convention of [2], 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.

(8)

The indices $n=3/2$ and $n=3$ describe idealized adiabatic stellar models supported by nonrelativistic and ultrarelativistic pressure, respectively [1]. A convective envelope also motivates $n=3/2$ [3]. Models with $n>3$ are mathematical equilibria without a claim of radial dynamical stability.

(9)

At fixed $K$, the index-one radius is independent of mass, and the index-three mass is independent of radius [2]. The coefficient $N_3$ remains well defined.

(10)

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 the mass table [1].

Links

[1]

Polytropes, Princeton stellar-structure notes, equations (poly.10)–(poly.11) and (poly.17)

[2]

O. R. Pols, Stellar Structure and Evolution (2011), Table 4.1 and equation (4.15)

[3]

Wenbin Lu, Polytropic Star, equations (1)–(7)

Similar tables

Radius of Lane–Emden polytropes — gives the surface radius entering the mass-radius coefficient for the same finite models

Mass constants of Lane–Emden polytropes — gives the mass normalization entering the mass-radius coefficient

Central pressure constants of Lane–Emden polytropes — combines with central concentration to determine the equation-of-state coefficient

Central-to-mean density ratio of Lane–Emden polytropes — combines with the central pressure constant to determine the mass-radius coefficient

Moment of inertia of polytropic stars — gives the axial inertia of the same finite density profiles

Data properties

Entries are of type: real number

Table is complete: no (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)

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

How they were obtained:

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.

A 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 [2], Table 4.1. Readback checks test stored intervals, the index-one anchor, the physical scale equations, and the radius, mass, concentration and pressure identities.

These 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.