Radius of Lane–Emden polytropes

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Numbers

$n$

$\xi_1(n)$

2.449489742783178098197284074705891391965947480656670128432692567250960377457315026539859433104640235

comment: The incompressible profile is $\theta_0(\xi)=1-\xi^2/6$, so $\xi_1(0)=\sqrt6$.

1/2:

2.752698054064987853153265640401943056124867186763253690388959098437368899789337892734061373570562118

3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068

comment: The linear profile is $\theta_1(\xi)=\sin(\xi)/\xi$, so $\xi_1(1)=$ $\pi$.

3/2:

3.653753736219122424609094280459265817024670389249320696565999634402741397141273940694348477626987161

4.352874595946124676973570061526142628112365363213008835302151694123582204698220445791064674344506140

5/2:

5.355275459010779459909360002973632473230913893154022969235420929234514705181751415720738708551783532

6.896848619376960375454528187123121276972189114184943529563510118684484825312554196016663314890901153

7/2:

9.535805344244850444104742625789381201938299334375507881439175962001748965444840018273455974203489162

14.97154634883809509761106621836539574691859481089739149722087527931063319472756524579210010458397593

9/2:

31.83646324469428526438280928066527469188343951039946735959427977287447144393742413890540998153078740

Definition

The dimensionless radius $\xi_1(n)$ is the first positive zero of the regular Lane–Emden solution $\theta_n$, defined by $(\xi^2\theta_n^\prime)^\prime=-\xi^2\theta_n^n$, $\theta_n(0)=1$ and $\theta_n^\prime(0)=0$ [1]. The solution follows the positive real branch until it reaches this zero.

Parameters

$n$

— polytropic index ($0\leq n<5$)

Formulas

(1)

$\theta_0(\xi)=1-\xi^2/6$, hence $\xi_1(0)=\sqrt6$.

(2)

$\theta_1(\xi)=\sin(\xi)/\xi$, with the value at $\xi=0$ defined by continuity, hence $\xi_1(1)=$ $\pi$.

Comments

(3)

The model is three-dimensional, spherical, nonrotating and Newtonian. For $n>0$, its 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]. The dimensionless radius does not specify a physical stellar radius without a length scale.

(4)

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

(5)

The normalized model at $n=5$ has an infinite radius [1], so it supplies no finite entry here.

Programs

(P1)

Python

# SageMath, with the attached generate.py and solver.py beside this file.
from generate import LaneEmdenRadius
g = LaneEmdenRadius()
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, pp. 1–3

[3]

Wenbin Lu, Polytropic Star, pp. 1–2

Similar tables

$\pi$ — equals the first positive zero of the index-one Lane–Emden solution

Mass constants of Lane–Emden polytropes — gives $\mu_n=-\xi_1(n)^2\theta_n^\prime(\xi_1(n))$, the dimensionless mass evaluated at this first zero for $0\leq n<5$. Its additional $n=5$ mass is defined by an integral to infinity and has no finite-radius counterpart

Data properties

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 exact-profile cases $0,1$ are intrinsic, and $3/2,3$ are stellar application indices. The remaining six indices are exploratory references for simple algebraic density–enthalpy laws $\rho/\rho_c=\theta_n^n$, where $\rho_c$ is the central density. Integer and half-integer exponents cover 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 tradeoffs)

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

How they were obtained:

The two analytic radii are evaluated with Arb ball arithmetic. All other entries use central-series initialization followed by Taylor continuation in $t=\sqrt{\theta_n}$. The two runs use 150 and 210 working decimal digits, Taylor orders 160 and 210, step caps $1/12$ and $1/18$, and matching radii $1/2$ and $2/3$. The worst relative difference of the internal results is less than $4.31\times10^{-154}$; only 100 significant digits are stored.

A separately implemented integration in the original radial coordinate checks every selected model to at least 35 digits. It starts with exact rational central coefficients, stops at positive $\theta_n<10^{-20}$ and extrapolates the homogeneous exterior solution. The $n=0,1$ exact radii are positive controls, and deliberately displaced values are rejected. Stored decimal intervals are checked against the high-precision results and the analytic anchors after readback. These computations do not rigorously bound Taylor truncation, propagated ODE error or the exterior extrapolation. The attached sources and numerical evidence record the checks and their limitations.