—   polytropic index ($0\leq n<5$)

—   multipole degree ($\ell\geq2$)

Write the exterior radial tidal potential as $H_\ell(\xi)=A_\ell\xi^\ell+B_\ell\xi^{-\ell-1}$, with $A_\ell\ne0$. Then $k_\ell(n)=B_\ell/(2A_\ell\xi_1(n)^{2\ell+1})$ [1], equation (15).

For $0<n<5$, the interior field satisfies $H_\ell^{\prime\prime}+2H_\ell^\prime/\xi+[n\theta_n^{n-1}-\ell(\ell+1)/\xi^2]H_\ell=0$ and $H_\ell(\xi)\sim\xi^\ell$ as $\xi\to0$. If $Y_\ell=\xi H_\ell^\prime/H_\ell$, then $k_\ell(n)=[\ell-Y_\ell(\xi_1)]/[2(\ell+1+Y_\ell(\xi_1))]$ [1], equations (14)–(20).

Let $q_n(\xi)=\int_0^\xi s^2\theta_n(s)^n\,ds=-\xi^2\theta_n^\prime(\xi)$ and $y_\ell=Y_\ell-\xi^3\theta_n^n/q_n$. Then $\xi y_\ell^\prime+y_\ell(y_\ell+1)+2\xi^3\theta_n^n(y_\ell+2)/q_n=\ell(\ell+1)$, with $y_\ell(0)=\ell-3$, and $k_\ell(n)=[\ell-y_\ell(\xi_1)]/[2(\ell+1+y_\ell(\xi_1))]$. This continuous modified field also handles $n=0$; $q_n(\xi_1)$ is the dimensionless mass constant [1], equations (23)–(25).

At $n=0$, $Y_\ell(\xi_1^-)=\ell$ and the density jump gives $Y_\ell(\xi_1^+)=\ell-3$, hence $k_\ell(0)=3/[4(\ell-1)]$ [1], equations (18) and (28).

At $n=1$, $H_\ell(\xi)\propto j_\ell(\xi)$ and $\xi_1=$ $\pi$, so $k_\ell(1)=j_{\ell+1}(\pi)/(2j_{\ell-1}(\pi))$. Here $j_\ell(x)=\sqrt{\pi/(2x)}J_{\ell+1/2}(x)$ is the spherical Bessel function, expressed through the Bessel function $J_\nu$ [1], equations (29)–(33). In particular, $k_2(1)=(15-\pi^2)/(2\pi^2)$ and $k_3(1)=(105-10\pi^2)/(6\pi^2)$.

The background is three-dimensional, spherical, nonrotating and Newtonian. Its density is $\rho=\rho_c\theta_n^n$, where $\rho_c$ is central density and $(\xi^2\theta_n^\prime)^\prime=-\xi^2\theta_n^n$, $\theta_n(0)=1$, $\theta_n^\prime(0)=0$. Follow the positive real branch to the first zero $\xi_1(n)$, without continuing fractional powers through negative values [1].

For $n>0$, pressure obeys $P=K\rho^{1+1/n}$ with $K>0$. The index $n=0$ denotes the incompressible model with constant interior density and a surface density jump [2].

The apsidal-motion constant is half the potential Love number in the convention that uses the full induced-to-applied potential ratio. The degree $\ell=2$ is quadrupolar and $\ell=3$ is octupolar. These dimensionless constants are independent of the radius and central-density scales [1], section 5.1.

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

The index-five Lane–Emden model has infinite radius, so it supplies no finite-surface entry here. Its convergent mass appears in the mass table.

The normalized axial inertia describes the same backgrounds. The I–Love relation compares this structural quantity with the quadrupolar response; its approximate insensitivity to the equation of state is not an exact equality [1], section 9.

Python

# SageMath with numberdb: analytic n=0 or 1, integer ell>=2.
# Returns an exact rational or an Arb enclosure with the requested significant digits.
import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.rings.integer_ring import ZZ
from sage.rings.complex_arb import ComplexBallField

def love_analytic(n, ell, digits=100):
    n, ell = QQ(n), QQ(ell)
    if n not in (0, 1) or ell.denominator() != 1 or ell < 2:
        raise ValueError('Use n=0 or 1 and an integer ell>=2.')
    ell = ZZ(ell)
    if n == 0:
        return QQ(3) / (4 * (ell - 1))
    target = numberdb.bits(digits, losing=8)
    precision = target + 80
    while True:
        p = ComplexBallField(precision).pi()
        k = (p.bessel_J(ell + QQ(3)/2)
             / (2 * p.bessel_J(ell - QQ(1)/2))).real()
        if k.is_finite() and k.accuracy() >= target:
            return k
        precision *= 2

print(love_analytic(0, 4))
print(love_analytic(1, 4))

K. L. S. Yip and P. T. Leung, Tidal Love numbers and moment-Love relations of polytropic stars, Monthly Notices of the Royal Astronomical Society 472 (2017), 4965–4981. (arXiv) (doi)

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

Wenbin Lu, Polytropic Star, pp. 1–2

Radius of Lane–Emden polytropes —   locates the free surface at which the tidal response is matched to the exterior potential

Mass constants of Lane–Emden polytropes —   gives the enclosed-mass normalization used in the continuous modified tidal field

Moment of inertia of polytropic stars —   gives the axial inertia compared with the quadrupolar response in the approximate I–Love relation

Central-to-mean density ratio of Lane–Emden polytropes —   measures central concentration of the same density profiles whose tidal deformation is tabulated here

Bessel J values —   holds the function family appearing in the analytic index-one solution

Entries are of type: real number

Table is complete: no (it holds $n=0,1/2,1,3/2,2,5/2,3,7/2,4,9/2$, each at the leading tidal degrees $\ell=2,3$. The analytic cases $0,1$ are intrinsic, and $3/2,3$ are stellar application indices. The remaining six are exploratory references for simple algebraic density–enthalpy laws $\rho/\rho_c=\theta_n^n$. Integer and half-integer powers span stiff through centrally concentrated finite models before increasing denominators or approaching five more closely. Denominator two, the cutoff $9/2$ and the two-degree prefix are practical, noncanonical choices independent of decimal notation. The attached argument-selection.md explains the groups, bounds and coverage tradeoffs)

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

How they were obtained:

The two incompressible values are exact rationals, and the index-one closed forms are evaluated with Arb ball arithmetic. All other entries use central-series initialization and Taylor continuation in $t=\sqrt{\theta_n}$ of the Lane–Emden background and the original-potential Riccati equation. Runs at 150 and 210 working decimal digits use Taylor orders 160 and 210, step caps $1/12$ and $1/18$, and central matching radii $1/2$ and $2/3$. Across all twenty responses, including the analytic controls, the largest relative discrepancy is less than $5.76\times10^{-152}$. Only 100 significant digits are stored.

more

A separate DOP853 integration in the original radial coordinate uses the continuous modified-potential equation, without importing the production solver. It checks every response; its largest relative discrepancy is less than $2.77\times10^{-10}$. Both analytic models are nonzero controls, and deliberately displaced controls are rejected. An independent 140-digit Bessel evaluation checks the index-one formulas. The modified-field transformation is checked as an exact polynomial identity. Readback checks compare the written intervals with the high-precision results, repeat exterior matching and the radial comparisons on the stored values, and compare every background radius and mass with the companion tables.

These checks do not rigorously bound Taylor truncation, propagated ODE error or the numerical surface location, so the ODE values are heuristic (agreement-checked). The attachments retain the reproducible generator, independent checks, numerical evidence, argument-selection justification, source hashes and the coordinated skill manifest.