Lane-Emden Stars: Compact Polytropic Models

• Lane-Emden stars are compact, spherically symmetric equilibrium configurations described by a polytropic equation of state and governed by the Lane-Emden equation.
• Exact analytical solutions for specific indices (e.g., n=0,1,5) and numerical methods establish key mass-radius relations and stability criteria.
• Extensions involving quantum corrections, modified gravity, and fractional models enrich the formalism by altering stellar observables and stability thresholds.

Lane-Emden stars are compact, spherically symmetric equilibrium configurations of self-gravitating polytropic fluids governed by the Lane-Emden equation. They constitute canonical models in astrophysics for main-sequence stars, white dwarfs, and gaseous spheres with polytropic equations of state. The Lane-Emden formalism originates from reductions of the hydrostatic equilibrium equations under a polytropic equation of state to an ordinary differential equation with universal profile constants. These objects exhibit a range of stability properties, intricate mass-radius relations, and admit significant generalizations in both quantum plasma and modified gravity contexts.

1. Lane-Emden Equation and Polytropic Stellar Structure

Let $P = K \rho^{1 + 1/n}$ be the polytropic equation of state, with $K$ constant, $n$ the polytropic index, and $\rho$ the fluid density. The stellar structure equations in spherical symmetry reduce to:

$$\frac{1}{r^2} \frac{d}{dr} \left( r^2 \frac{d\theta}{dr} \right) + \theta^n = 0, \qquad \theta(0) = 1,\ \theta'(0) = 0$$

where $\rho(r) = \rho_c \theta^n(r/\alpha)$, $r = \alpha \xi$, and $\alpha^2 = \frac{(n+1)K\rho_c^{1/n - 1}}{4\pi G}$ (Makino, 2012, Wojnar, 2023, Bludman et al., 2010). Exact analytical solutions exist for $n=0$ (incompressible sphere), $n=1$ (sinusoidal), and $n=5$ (infinite support), while general $n$ requires numerical integration.

For $1 < n < 5$ ($6/5 < \gamma < 2$), $\theta(\xi)$ has a finite zero at some $\xi_1$, yielding finite total radius $R = \alpha \xi_1$ and mass $M = 4\pi \rho_c \alpha^3 (-\xi_1^2 \theta'(\xi_1))$ (Bludman et al., 2010, Wojnar, 2023). The $n=3$ polytrope is special: $M$ is independent of $R$ and yields Chandrasekhar's limit for relativistic white dwarfs.

2. Analytical Solutions and Approximate Schemes

For integer polytropic indices,

• $n=0$: $\theta(\xi) = 1 - \xi^2 / 6$, $\xi_1 = \sqrt{6}$.
• $n=1$: $\theta(\xi) = \sin \xi / \xi$, $\xi_1 = \pi$.
• $n=5$: $\theta(\xi) = (1 + \xi^2 / 3)^{-1/2}$, no finite zero.

Approximation schemes improve computational tractability for general $n$. Picard and Padé approximations for $n=3$ provide high-accuracy fits across the stellar interior (Bludman et al., 2010). These functions closely reproduce the numerically integrated Lane-Emden profiles, especially for analytically intractable $n$.

$n$	$\xi_1$	$-\theta'(\xi_1)$
0	$2.449$	$0.816$
1	$3.142$	$0.318$
3	$6.897$	$0.042$
5	$\infty$	$0$

These constants feed directly into the mass-radius relations and stability thresholds (Wojnar, 2023).

3. Physical Profiles, Mass-Radius Relations, and Quantum Corrections

The density and pressure profiles are:

$$\rho(r) = \rho_c \theta\left(\frac{r}{\alpha}\right)^n,\qquad P(r) = K \rho_c^{1 + 1/n} \theta\left(\frac{r}{\alpha}\right)^{n + 1}$$

Near the center, the expansion $\theta(\xi) = 1 - \xi^2 / 6 + \mathcal{O}(\xi^4)$ yields smooth profiles. Near the boundary, $\theta(\xi) \sim C (\xi_1 - \xi)$ so that $\rho(r) \sim (R - r)^n$, $P(r) \sim (R - r)^{n+1}$, capturing the “physical vacuum” (Hölder $C^{1/2}$ continuity of the sound speed across the boundary) (Makino, 2012, Luo et al., 2015, Luo et al., 2015).

Quantum plasma corrections introduce an additional Bohm pressure term, leading to a quantum-modified Lane-Emden equation:

$$\frac{d^2\theta}{dx^2} + \frac{2}{x} \frac{d\theta}{dx} + \theta^n + N_Q \frac{1}{x^2} \frac{d}{dx}(x^2 \frac{d(\theta^{n/2})}{dx}) = 0$$

with $N_Q \ll 1$ for white dwarf parameters. The quantum corrections negligibly affect mass ($\sim 10^{-39}$) and radius, but conceptually they result in slightly reduced Chandrasekhar mass and enlarged radius (Schlickeiser et al., 2012).

4. Generalizations: Modified Gravity, Fractional Models, and Anisotropy

Modified Gravity

In Palatini $f(R)$ theory, Starobinsky corrections $\beta R^2$ alter the Lane-Emden equation, introducing an $\alpha$ parameter that shifts mass and radius predictions:

$$\frac{1}{\xi^2} \frac{d}{d\xi}\left[(1+2\alpha\,\theta^n)\xi^2\theta' + \alpha\xi^3\theta^{2n}\right] = -\theta^n + 3\alpha\theta^{2n}$$

Positive $\alpha$ increases mass and radius, and allows heavier neutron-star models; negative $\alpha$ reduces them. Exact solutions exist for $n=0,1$ (Wojnar, 2018, 1901.10448).

Fractional Lane-Emden Equation

Conformable fractional derivatives generalize the Lane-Emden equation, introducing a parameter $\alpha \in (0, 1]$:

$$x^{2\alpha} D^\alpha(x^{2\alpha} D^\alpha u) + u^n = 0$$

The first zero $x_1(\alpha)$ decreases as $\alpha$ decreases, resulting in more compact, massive stars and altered stability criteria. Accelerated series expansion yields closed-form coefficients for all physical quantities. These models bridge integer-order polytropes to more compact, nonlocal structures (Abd-Elsalam et al., 2019).

Anisotropic Matter

Radial and tangential pressure polytropes yield double-polytrope Lane-Emden equations, introducing an anisotropy parameter $\delta = n_t - n_r$:

$$\frac{1}{\xi^2}\frac{d}{d\xi}\left(\xi^2 \theta'\right) + \varepsilon \theta^{n_r} + \mathcal{F}(\xi, \theta, \Delta, \Delta') = 0$$

Increasing $\delta$ dilutes the central condensation and mass, allowing significant modifications to the Chandrasekhar limit (Abellan et al., 2020).

5. Stability, Instabilities, and Nonlinear Dynamics

Stability analyses reveal that for $4/3 < \gamma < 2$ ($1 < n < 3$), Lane-Emden equilibria are nonlinearly stable under spherically symmetric perturbations; solutions asymptotically converge to equilibrium with explicit decay rates. Physical vacuum regularity is maintained throughout evolution (Luo et al., 2015, Luo et al., 2015).

For $6/5 < \gamma < 4/3$ ($3 < n < 5$), equilibria are nonlinearly unstable. Small perturbations lead to exponential growth away from equilibrium, manifesting as expanding solutions with support tending to infinity (Jang, 2012, Cheng et al., 2023, Cao, 23 Nov 2025, Hao et al., 2023). This strong instability applies not only to inviscid (Euler-Poisson) stars but also viscous cases and remains true for a dense set of initial data near Lane-Emden configurations (Cao, 23 Nov 2025).

In the specific mass-critical case $\gamma=4/3$ for white dwarfs, the Lane-Emden mass coincides with the Chandrasekhar limit; sub-critical configurations do not collapse, establishing the sharp non-collapse criterion (Cheng et al., 2023).

Liquid Lane-Emden stars exhibit central-density-dependent stability in contrast to gaseous stars. For $\gamma<2(d-1)/d$ and large central density, linear instability emerges, dictated by spectral properties of the associated Sturm-Liouville operator; for small central density, stability persists (Lam, 2022).

6. Extensions: Post-Newtonian, Higher-Curvature, Rotation, and Applications

Post-Newtonian corrections ($\epsilon = p_c / (\rho_c c^2)$) have negligible impact for weak-field objects but affect neutron stars, increasing central pressure/temperature up to 60% and lowering central density by 3-5% (Kremer, 2022).

Higher-curvature gravity theories yield a sixth-order gauge-invariant Lane-Emden equation, necessitating additional boundary data from the profile at both center and surface. Analytical solutions exist for $n=0,1$; modifications induce shifts in stellar observables, bounded by constraints on theory parameters (Tonosaki et al., 2023).

Slow rotation and modified gravity enter as perturbations $v$ and $g_{\rm mod}$, respectively, altering the Lane-Emden profiles, mass-radius relations, and minimum main-sequence mass for fully convective stars (Wojnar, 2023).

Astrophysically, Lane-Emden stars provide tractable models for zero-age main-sequence stars, white dwarfs, neutron stars, brown dwarfs, and substellar cooling. Observational constraints on mass and radius feed back into gravity theory parameters and polytropic indices.

7. Summary and Open Problems

Lane-Emden stars encompass the full phenomenology of spherically symmetric, self-gravitating polytropes, including exact, approximate, and numerical solutions; their global stability is dictated by polytropic index and physical vacuum boundary regularity. Quantum, gravitational, and fractional corrections yield a rich variety of modified profiles. The open mathematical challenges include rigorous extension of nonlinear stability/instability theory to fractional and higher-order Lane-Emden equations, non-integer boundary expansions (especially for physically realistic gases), and systematic exploration of the impact of anisotropy, modified gravity, and post-Newtonian effects on observables.

Recent works have resolved key nonlinear instability paradigms (Jang, 2012, Cheng et al., 2023, Cao, 23 Nov 2025, Hao et al., 2023), established precise asymptotic stability for viscous gaseous stars (Luo et al., 2015, Luo et al., 2015), characterized liquid Lane-Emden stability (Lam, 2022), and advanced the modified Lane-Emden formalism in fractional and gravitational frameworks (Abd-Elsalam et al., 2019, Wojnar, 2018, Tonosaki et al., 2023). The Lane-Emden paradigm remains central in theoretical astrophysics, stellar modeling, and the mathematical theory of free-boundary self-gravitating systems.