Moment of inertia $I/(MR^2)$ of polytropic stars

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Numbers

$n$

$I/(MR^2)$

2/5

comment: The incompressible model has constant interior density, so $I/(MR^2)=2/5$.

1/2:

0.3259310823229534719464853993336045443105167882219770751853836902950038575618442312675120714335220737

0.2613819320973155808911488138277561110492315679776804712842305390899741828177249019018295709641793162

comment: The analytic profile $\theta_1(\xi)=\sin(\xi)/\xi$ gives $I/(MR^2)=2/3-4/\pi^2$.

3/2:

0.2045998945746605070652043768951524258597849801408828983706649136983270028195821891945386765202628756

0.1548491833299797436513273920407495539775133666845901937560057552323741781915787541384728977134433185

5/2:

0.1118049209855242369273655988199145244990578627191312536620619176915140576131061617385893780072484024

0.07535763996014873254377092405034916453997994539003603290673525942206637785303632985364500685648686873

7/2:

0.04554837776599655713544578366050906131816355218124753039932280832947497556916702630310876768802866242

0.02257372948677378219084285189762278790604071681480629361867874865366185398525732917669147403534097898

9/2:

0.006895160488497352250471174725496718319925166762305220139435506503538345980048222767880121665337970500

Definition

The dimensionless axial moment of inertia is $I/(MR^2)$, where $I=(8\pi/3)\int_0^R\rho(r)r^4\,dr$ is the moment about a diameter of a spherical Newtonian Lane–Emden polytrope with total mass $M$ and finite radius $R$ [4].

Parameters

$n$

— polytropic index ($0\leq n<5$)

Formulas

(1)

$\dfrac{I}{MR^2}=\dfrac{2J_n}{3\mu_n\xi_1(n)^2}$, where $J_n=\int_0^{\xi_1(n)}\xi^4\theta_n(\xi)^n\,d\xi$, $\xi_1(n)$ is the dimensionless radius, and $\mu_n=\int_0^{\xi_1(n)}\xi^2\theta_n(\xi)^n\,d\xi=-\xi_1(n)^2\theta_n^\prime(\xi_1(n))$ is the dimensionless mass constant [4].

(2)

$R=a\xi_1(n)$, $M=4\pi a^3\rho_c\mu_n$ and $I=(8\pi/3)a^5\rho_cJ_n$, where $r=a\xi$ and $a>0$ is a length scale. Consequently $I/(MR^2)$ is independent of $a$ and $\rho_c$ [2].

(3)

$J_n=\mu_n\xi_1(n)^2-6\int_0^{\xi_1(n)}\xi^2\theta_n(\xi)\,d\xi$. This follows by integrating the Lane–Emden equation twice by parts.

(4)

$\theta_0(\xi)=1-\xi^2/6$ and $\xi_1(0)=\sqrt6$, so $I/(MR^2)=2/5$ at $n=0$.

(5)

$\theta_1(\xi)=\sin(\xi)/\xi$, with its central value defined by continuity, and $\xi_1(1)=\mu_1=$ $\pi$, so $I/(MR^2)=2/3-4/\pi^2$ at $n=1$.

Comments

(6)

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

(7)

For $n>0$, 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 [3].

(8)

The normalization is by $MR^2$. Thus the table also gives the squared radius of gyration divided by $R^2$, where the radius of gyration is $\sqrt{I/M}$. The spherical profile is nonrotating; its moment about a diameter is defined without specifying an angular velocity.

(9)

The indices $n=3/2$ and $n=3$ describe idealized adiabatic stars supported by nonrelativistic and ultrarelativistic pressure, respectively [3]. A convective envelope also motivates $n=3/2$ [4]. Models with $n>3$ are included as mathematical equilibria without a claim of radial dynamical stability.

(10)

The model at $n=5$ has infinite radius and a divergent axial inertia integral. It supplies no finite-radius entry here, although its convergent total mass appears in the mass table [1].

Programs

(P1)

Python

# SageMath, with the attached generate.py and solver.py beside this file.
from generate import PolytropicInertia
g = PolytropicInertia()
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)

[2]

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)

Links

[3]

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

[4]

Wenbin Lu, Polytropic Star, equations (1)–(10)

Similar tables

Radius of Lane–Emden polytropes — gives the surface radius used to normalize the inertia of the same finite models

Mass constants of Lane–Emden polytropes — gives the mass normalization for the same density profiles

Central-to-mean density ratio of Lane–Emden polytropes — describes central concentration for the same models, while axial inertia weights the density by squared distance from the axis

Tidal Love numbers of Newtonian polytropic stars — gives the apsidal-motion response of the same density profiles. The quadrupolar $k_2(n)$ and $I/(MR^2)$ enter the I–Love comparison, whose near-independence of the equation of state for stiff models is approximate [2]

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 analytic-profile cases $0,1$ are intrinsic, and $3/2,3$ are stellar application indices. The remaining six indices are exploratory references for axial inertia in 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 incompressible value is the exact rational $2/5$. The index-one closed form is evaluated with Arb ball arithmetic. Other entries use central-series initialization and Taylor continuation in $t=\sqrt{\theta_n}$, accumulating the axial density moment. Two runs use 150 and 210 working decimal digits, Taylor orders 160 and 210, step caps $1/12$ and $1/18$, and central matching radii $1/2$ and $2/3$. Their worst relative difference over the ten models is less than $3.25\times10^{-154}$; only 100 significant digits are stored.

The exact-profile cases are controls for the numerical integration. A separate implementation in the original radial coordinate checks every selected model, and the two independent moment integrals obey the integration-by-parts identity. A published Padé approximation in [2], Appendix C.4, provides an additional normalization check on $0\leq n\leq3$; its largest observed error on the selected indices is about $0.2142$ percent. The measured error rounds to the paper's $0.21$ percent, which is not used as a strict bound. These checks do not rigorously bound Taylor truncation, propagated ODE error or the radial endpoint approximation. The attached code and numerical evidence record the comparisons and their limitations.