Mass constants of Lane–Emden polytropes

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Numbers

$n$

$\mu_n$

4.898979485566356196394568149411782783931894961313340256865385134501920754914630053079718866209280470

comment: The incompressible profile $\theta_0(\xi)=1-\xi^2/6$ gives $\mu_0=2\sqrt6$.

1/2:

3.788651184884005725858991073566130020744363998379329944557197732582689807929270932828329973951954331

3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068

comment: The linear profile $\theta_1(\xi)=\sin(\xi)/\xi$ gives $\mu_1=$ $\pi$.

3/2:

2.714055120108645719024533278803231148479459311782922475514579402693523737124833330828170776250164148

2.411046012096893783648442744671402891869445416349857344362294844576927205112899840572090568314543516

5/2:

2.187199565517078953219520989736020888783031739987630276411128754986014658996631670101010148848846475

2.018235950966228402812813170057969585302183202862202577717886005970707684353269150167163413499249995

7/2:

1.890557093443116393908585305853529349420010746280017377769029608808829461844377579917045884022964872

1.797229914439249961494315490419345024774164486195427513723180341891886413612268336853789885759673062

9/2:

1.737798867666032348873310638765409225494717541030522402615940804084994539027274227100259087881527067

1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576

comment: The profile $\theta_5(\xi)=(1+\xi^2/3)^{-1/2}$ has infinite radius and finite mass $\mu_5=\sqrt3$.

Definition

The dimensionless mass constant is $\mu_n=\int_0^{\xi_1(n)}\xi^2\theta_n(\xi)^n\,d\xi$, where $\theta_n$ is the normalized regular Lane–Emden solution and $\xi_1(n)$ is its first positive zero [2]. At $n=5$, the upper limit is infinity.

Parameters

$n$

— polytropic index ($0\leq n\leq5$)

Formulas

(1)

$\mu_n=-\xi_1(n)^2\theta_n^\prime(\xi_1(n))$ for $0\leq n<5$, where $\xi_1(n)$ is the dimensionless radius [2].

(2)

$M=4\pi a^3\rho_c\mu_n$, where $M$ is the total mass, $r=a\xi$ is the physical radial coordinate and $a^2=(n+1)P_c/(4\pi G\rho_c^2)$. Here $P_c$ and $\rho_c$ are the central pressure and density, and $G$ is Newton's gravitational constant [2].

(3)

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

(4)

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

(5)

$\theta_5(\xi)=(1+\xi^2/3)^{-1/2}$ and $\int_0^\infty \xi^2(1+\xi^2/3)^{-5/2}\,d\xi=\sqrt3$, so $\mu_5=\sqrt3$ [1].

(6)

$\int_0^{\xi_1(n)}\xi^2\theta_n(\xi)^{n+1}\,d\xi=\frac{n+1}{5-n}\frac{\mu_n^2}{\xi_1(n)}$ for $0\leq n<5$. This is the dimensionless pressure-integral form of the polytropic virial identity [3].

Comments

(7)

The normalization is $(\xi^2\theta_n^\prime)^\prime=-\xi^2\theta_n^n$, $\theta_n(0)=1$ and $\theta_n^\prime(0)=0$. The solution follows its positive real branch to the first zero for $n<5$, without continuation of fractional powers through negative values [1].

(8)

The model is three-dimensional, spherical, nonrotating and Newtonian. For $n>0$, pressure $P$ and density $\rho$ obey $P=K\rho^{1+1/n}$ with $K>0$. The index $n=0$ denotes an incompressible model with constant interior density and a surface density jump [2].

(9)

The density is $\rho=\rho_c\theta_n^n$, where $\rho_c$ is the central density. The mass normalization omits the solid-angle factor $4\pi$: the dimensionless function $m(n)$ in [1], equation (64), equals $4\pi\mu_n$. The scaled-coordinate second moment also denoted $\mu(n)$ in that paper is a different quantity.

(10)

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

(11)

For $n<5$, the dimensionless radius and the surface derivative determine $\mu_n$. The $n=5$ profile has no finite surface but its density decreases sufficiently rapidly for the mass integral to converge [1].

Programs

(P1)

Python

# SageMath, with the attached generate.py and solver.py beside this file.
from generate import LaneEmdenMass
g = LaneEmdenMass()
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, equations (1)–(4) and (16)–(18)

Similar tables

Radius of Lane–Emden polytropes — gives the first zero at which the surface mass is evaluated for the same finite models

$\pi$ — equals the mass constant of the index-one model

Data properties

Entries are of type: real number

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

Table is complete: no (it holds $n=0,1/2,1,3/2,2,5/2,3,7/2,4,9/2,5$. The analytic cases $0,1,5$ are intrinsic, and $3/2,3$ are stellar application indices. The remaining six indices are exploratory references for simple algebraic density–enthalpy laws $\rho/\rho_c=\theta_n^n$. Integer and half-integer exponents span stiff through centrally concentrated finite models before adding higher denominators or approaching five more closely; the finite-mass endpoint $5$ is included separately. Denominator two and the cutoff $9/2$ are noncanonical, notation-independent reference choices, not an interpolation mesh. The attached argument-selection.md explains the disjoint groups and tradeoffs)

How they were obtained:

The three irrational closed forms are evaluated with Arb ball arithmetic. The other eight masses use central-series initialization followed by Taylor continuation in $t=\sqrt{\theta_n}$. Separate runs use 150 and 210 working decimal digits, orders 160 and 210, step caps $1/12$ and $1/18$, and central matching radii $1/2$ and $2/3$. Across all ten finite models, including the analytic controls, the largest relative mass discrepancy is less than $2.97\times10^{-154}$. Only 100 significant digits are stored.

A separate radial-coordinate implementation checks every finite mass and the pressure-integral identity to a relative tolerance of $10^{-40}$. It uses 70 working decimal digits, order 80, exact rational central coefficients, and a leading tail correction after reaching $0<\theta_n<10^{-30}$. The $n=0,1$ analytic models are nonzero controls; deliberately displaced values are rejected. A separate transformed quadrature checks the analytic $n=5$ integral. Readback checks compare the stored decimal intervals with the high-precision computations, test the analytic anchors, repeat the integral identity with the stored masses, and compare the ten radii with the radius table.

These checks do not provide rigorous bounds on ODE truncation, propagation or endpoint errors. The attached generator, checks, argument selection, skill manifest evidence, source hashes and numerical evidence record the reproducible method and its limits.