—   polytropic index ($0\leq n<5$)

$W_n=\dfrac{\xi_1(n)^4}{4\pi(n+1)\mu_n^2}=\dfrac{1}{4\pi(n+1)\theta_n^\prime(\xi_1(n))^2}$, where $\xi_1(n)$ is the 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].

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

$W_n=\dfrac{9C_n^2}{4\pi(n+1)\xi_1(n)^2}$, where $C_n=\rho_c/\bar\rho=\xi_1(n)^3/(3\mu_n)$ is the central-to-mean density ratio and $\bar\rho=3M/(4\pi R^3)$.

$W_n\int_0^{\xi_1(n)}\xi^2\theta_n(\xi)^{n+1}\,d\xi=\dfrac{\xi_1(n)^3}{4\pi(5-n)}$. This is the pressure-integral form of the polytropic virial identity [3].

$\theta_0(\xi)=1-\xi^2/6$, $\xi_1(0)=\sqrt6$ and $\mu_0=2\sqrt6$, so $W_0=3/(8\pi)$.

$\theta_1(\xi)=\sin(\xi)/\xi$, with its central value defined by continuity, and $\xi_1(1)=\mu_1=\pi$, so $W_1=$ $\pi/8$.

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 [2].

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 [2].

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.

The $n=5$ model has infinite radius, so the finite-radius normalization supplies no entry here. Its convergent total mass is included in the mass table [2].

The coefficient $W_n$ agrees with the notation in [1], equation (4.17). That source writes $z_n$ for $\xi_1(n)$ and $\Theta_n$ for the dimensionless mass $\mu_n$.

Python

# Python 3 with SciPy: a double-precision example for rational 0 <= n < 5.
# Eight significant digits are displayed; no rigorous error bound is claimed.
# Tighten tolerances and check convergence, especially near n = 5.
from fractions import Fraction
from math import isfinite, pi
from scipy.integrate import solve_ivp


def pressure(index, rtol=2e-13, atol=2e-15, start=1e-4, limit=1e5):
    n_exact = Fraction(index)
    if not 0 <= n_exact < 5:
        raise ValueError('Require 0 <= n < 5.')
    n = float(n_exact)
    if not 0 <= n < 5:
        raise ValueError('Index too close to an endpoint for double precision.')
    if not 0 < start < limit:
        raise ValueError('Require 0 < start < limit.')
    # Regular central series through xi^4 and its derivative.
    theta = 1 - start**2 / 6 + n * start**4 / 120
    slope = -start / 3 + n * start**3 / 30

    def rhs(xi, state):
        y, v = state
        # A zero-density extension only locates the surface event;
        # fractional powers are evaluated only on nonnegative arguments.
        density = 1.0 if n == 0 else max(y, 0.0)**n
        return [v, -2 * v / xi - density]

    def surface(xi, state):
        return state[0]

    surface.terminal = True
    surface.direction = -1
    sol = solve_ivp(rhs, (start, limit), [theta, slope], method='DOP853',
                    events=surface, rtol=rtol, atol=atol)
    if sol.status != 1:
        raise RuntimeError('Surface not reached: ' + sol.message)
    v_surface = sol.y_events[0][0, 1]
    result = 1 / (4 * pi * (n + 1) * v_surface**2)
    if not isfinite(result) or result <= 0:
        raise ArithmeticError('Invalid pressure constant.')
    return result


if __name__ == '__main__':
    print(f"{pressure('13/4'):.8g}")

O. R. Pols, Stellar Structure and Evolution (2011), Table 4.1 and equation (4.17)

Polytropes, Princeton stellar-structure notes, pp. 1–3

Wenbin Lu, Polytropic Star, equations (1)–(7) and (17)–(22)

Radius of Lane–Emden polytropes —   gives the surface radius whose fourth power enters the pressure normalization

Mass constants of Lane–Emden polytropes —   gives the dimensionless mass whose square enters the pressure normalization

Central-to-mean density ratio of Lane–Emden polytropes —   determines the pressure coefficient together with the surface radius and polytropic index

Moment of inertia of polytropic stars —   gives the normalized axial inertia of the same finite models, complementing their central pressure

$\pi/8$ —   equals the pressure constant of the index-one model

Entries are of type: real number

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-pressure 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 attached argument-selection.md explains the groups and coverage tradeoffs)

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

How they were obtained:

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.

more

A 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 [1], Table 4.1. Readback checks test stored intervals, analytic anchors, radius/mass and concentration identities, and the independent pressure-integral identity.

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.