Fractional Crustal Moment of Inertia

• Fractional crustal moment of inertia is defined as the ratio of the crust's moment to the total moment, crucial for understanding pulsar glitches.
• It is highly sensitive to the core–crust transition pressure, with minor effects from variations in transition density.
• Observational constraints from glitches, such as those from the Vela pulsar, challenge standard EoS models, especially under strong crustal entrainment.

The fractional crustal moment of inertia (ΔI/I) quantifies the proportion of a neutron star’s total moment of inertia residing in the solid crust, a parameter crucial to interpreting pulsar glitches and constraining the equation of state (EoS) of dense matter. ΔI/I is highly sensitive to microphysical properties at the core–crust boundary, notably the transition pressure ($P_t$) and less strongly to the transition density ($\rho_t$). Its astrophysical relevance is anchored in glitch observations—most notably in the Vela pulsar—which require a minimum ΔI/I to explain the observed spin-up events, setting empirical constraints on neutron-star structure.

1. Definition and Formalism

The fractional crustal moment of inertia is defined as:

$$\frac{\Delta I}{I} \equiv \frac{I_{\text{crust}}}{I_{\text{total}}}$$

where $I_{\text{crust}}$ is the moment of inertia of the solid crust and $I_{\text{total}}$ the total moment of inertia of the star. The physical interpretation is direct: ΔI/I gives the maximum fraction of stellar angular momentum storable in the (potentially superfluid) crustal reservoir, which is available for transfer to the observable component during a glitch.

In the slow-rotation Hartle–Thorne approximation, $I_{\text{total}}$ and $I_{\text{crust}}$ are computed from

$$I = \frac{8\pi}{3} \int_{0}^{R} (\varepsilon(r)+P(r))\, e^{-\lambda(r)} \frac{\bar\omega(r)}{\Omega}\, \frac{r^4}{\sqrt{1-2GM(r)/r}} dr$$

where $R$ is the stellar radius, $\varepsilon(r)$ and $P(r)$ are the energy density and pressure, $\omega(r)$ the frame-dragging frequency, and $\bar\omega(r)=\Omega-\omega(r)$. The crustal component is given by restricting the radial integral to $r > R_t$, the crust-core interface.

A widely used analytic approximation expresses ΔI/I in terms of transition properties and global parameters (Atta et al., 2015, Madhuri et al., 2016, Piekarewicz et al., 2014):

$$\frac{\Delta I}{I} \approx \frac{28\pi P_{t} R^{3}}{3Mc^{2}} \frac{1-1.67\xi-0.6\xi^2}{\xi} \left[1 + \frac{2P_{t}}{\rho_{t} m_{b} c^{2}} \frac{(1+7\xi)(1-2\xi)}{\xi^2}\right]^{-1}$$

where $\xi=GM/Rc^2$, $P_t$ and $\rho_t$ are the core–crust transition pressure and density, $m_b$ is the baryon mass, $M$ and $R$ the mass and radius.

2. Determination of Transition Properties and Equations of State

The crust–core transition occurs when uniform β-equilibrated matter becomes unstable to clusterization. Thermodynamic stability requires the curvature matrix of the energy per baryon to be positive definite. Explicitly, the transition is located by solving $V_{\mathrm{thermal}}(\rho_t, x_p)=0$ with

$$V_{\mathrm{thermal}}(\rho, x_p) = \rho^2\left[2\rho\,\epsilon'_{\rho} + \rho^2\,\epsilon''_{\rho\rho} - \rho^2\,\frac{(\epsilon''_{\rho x_p})^2}{\epsilon''_{x_p x_p}}\right]$$

for a given EoS (Atta et al., 2015, Madhuri et al., 2016). The underlying EoS is typically constructed from effective interactions such as the density-dependent M3Y (DDM3Y) (Atta et al., 2015), Skyrme functionals (Madhuri et al., 2016, 2207.13384), or relativistic mean-field (RMF) models (Basu et al., 2018, Dutra et al., 2021).

Representative values for $\rho_t$, $P_t$ (fm$^{-3}$, MeV fm$^{-3}$) appear in the following table for selected models:

Model	$\rho_t$	$P_t$
DDM3Y	0.0938	0.5006
KDE0v1 Skyrme	0.0904	0.5013
NRAPR Skyrme	0.073	0.413

These transition properties are critical inputs to the analytic ΔI/I formula.

3. Sensitivity to Microphysics and Key Dependencies

ΔI/I is extremely sensitive to $P_t$, scaling nearly linearly with this parameter, and also varies as $R^3/M$ (or approximately $R^4/M^2$ in the Newtonian regime). The correction involving $\rho_t$ is subleading, typically contributing $\lesssim5\%$ variation for plausible $\rho_t$ shifts (Atta et al., 2015, Madhuri et al., 2016). Thus $P_t$—and its dependence on the density profile of the symmetry energy—dominates. Uncertainties of $\pm10\%$ in $P_t$ translate to $\pm10\%$ in ΔI/I.

Systematic studies (Zhang et al., 2024, Seif et al., 24 Jul 2025) show that the symmetry energy slope $L$ and skewness $J_\mathrm{sym}$ increase $P_t$, thereby boosting ΔI/I, while larger curvature $K_{\mathrm{sym}}$ or certain higher-order coefficients often reduce it. The incompressibility $K_0$ (isoscalar channel) and still higher-order symmetry terms (e.g., $Q_4$, $K_2$, $I_0$) modulate the core–crust transition properties and thus ΔI/I (Seif et al., 24 Jul 2025).

Meta-model and Bayesian studies spanning chiral EFT constraints predict, for $1.4\,M_\odot$ canonical stars, $\Delta I/I \sim 2$–$6\%$, with the spread reflecting uncertainties in transition inputs and isovector nuclear parameters (Carreau et al., 2018, Lim et al., 2018, Steiner et al., 2014, 2207.13384).

4. Observational Constraints from Glitches and Astrophysical Application

Glitch observations, especially in the Vela pulsar, set empirical lower bounds on ΔI/I. In the standard two-component superfluid model, at least 1.4% of the total moment of inertia must reside in the crust to explain observed glitch amplitudes (Atta et al., 2015, Madhuri et al., 2016, Piekarewicz et al., 2014):

$\frac{\Delta I}{I} > 0.014$

If crustal entrainment is included—where band-structure effects raise the effective mass of superfluid neutrons, reducing the mobile superfluid fraction—the bound increases sharply, up to $\sim7\%$ (Madhuri et al., 2016, 2207.13384, Eya et al., 2018, Piekarewicz et al., 2014), and in some scenarios to 10% or higher. Current microscopic and phenomenological models robustly reach $\Delta I/I \sim 5$–$8\%$ only for low stellar masses or very stiff EoS; the typical $1.4\,M_\odot$ star yields $1$–$6\%$ depending on EoS and entrainment assumptions (Carreau et al., 2018, Lim et al., 2018, Parmar et al., 2021, Atta et al., 2015).

Consequently, matching the observed Vela glitches with crustal superfluid alone is challenging if strong entrainment is realized. This places strong constraints on the EoS and even permits exclusion of certain classes of hybrid stars or soft symmetry-energy models (Li et al., 2014, Hooker et al., 2013). Empirical radius constraints such as

$$R \geq 4.10\,\text{km} + 3.36\,\text{km}\,(M/M_\odot)$$

are derived for DDM3Y, with parallel relations for other EoS models (Atta et al., 2015, Madhuri et al., 2016).

5. Role of Entrainment and Core Participation in Glitch Physics

Crustal entrainment—arising from Bragg scattering of neutrons by the crustal lattice—effectively reduces the available moment of inertia in the inner crustal superfluid by a factor $Y_e<1$. The effective minimum required for the glitch explanation becomes

$$(I_{\rm crust}/I)_{\text{required}} = \frac{0.014}{Y_e}$$

with $Y_e \sim 0.23$–$0.25$ (i.e., entrainment factor $\langle m_n^*/m_n \rangle \approx 4.3$) (Eya et al., 2018, Li et al., 2014); thus, $I_{\rm crust}/I$ must exceed $\sim7\%$.

Large-scale studies show that maximum $\Delta I/I$ for crust-alone models, including realistic entrainment, is rarely above $\sim 3.6$–$5.5\%$ for physically plausible EoS (Madhuri et al., 2016, 2207.13384, Eya et al., 2018). This suggests that either weaker entrainment (possibly due to superfluid pairing) or participation of additional superfluid reservoirs (e.g., core neutrons coupled by vortex–flux-tube pinning) is necessary (2207.13384, Gügercinoğlu et al., 2014, Eya et al., 2018).

6. Systematic EoS and Microphysical Trends

Unified EoS frameworks—Skyrme, RMF, Hartree-Fock, and meta-models—reveal robust microphysical dependencies:

• Symmetry energy ($L$, $K_{\text{sym}}$, $J_{\text{sym}}$): $P_t$ is maximized in models with stiff symmetry energy at sub- and near-saturation densities. Non-monotonic dependence on $L$ allows for models tuned to maximize ΔI/I in the range $\Delta R_{np}(^{208}\text{Pb}) = 0.20$–$0.26$ fm (Piekarewicz et al., 2014, Zhang et al., 2024).
• Macroscopic parameters: ΔI/I decreases with increasing stellar mass (thinner crust) and is maximized for low-mass, large-radius stars. Typical EoS models for $1.4\,M_\odot$ yield $2$–$6\%$ crustal fractions; for $M>1.7\,M_\odot$ the fraction falls below $2\%$ in most models (Carreau et al., 2018, Dutra et al., 2021, Qian et al., 2018).

Incorporation of higher-order isovector and isoscalar EOS parameters (e.g., $K_0$, $Q_0$, $I_0$, $K_4$, $Q_4$) further refines predictions, with stiffer EoS and positive higher-order symmetry coefficients enhancing ΔI/I (Seif et al., 24 Jul 2025).

7. Radius Constraints and Prospects

From the glitch requirement, lower limits on neutron-star radii can be derived using the analytic ΔI/I expression for given masses and microphysical transition inputs. For the DDM3Y EoS, the Vela radius bound is $R \geq 4.10 + 3.36\, M/M_\odot$ km; for the KDE0v1 Skyrme EoS, $R \geq 3.69 + 3.44\, M/M_\odot$ km (Atta et al., 2015, Madhuri et al., 2016). Bayesian and meta-modeling approaches, spanning the experimental uncertainty on nuclear inputs, find that radii $\gtrsim 10$–$12\,$km are necessary for canonical-mass stars to marginally satisfy the strong-entrainment glitch bound (Carreau et al., 2018, Lim et al., 2018, Steiner et al., 2014).

Summary Table: Key Values of ΔI/I for $1.4\,M_\odot$ Star

EoS Model	$P_t$ (MeV fm$^{-3}$)	$\rho_t$ (fm$^{-3}$)	ΔI/I (%) no entrainment	ΔI/I (%) with entrainment
DDM3Y	0.5006	0.0938	$\sim$0.7	$\sim$3.6$\,$*
KDE0v1 Skyrme	0.5013	0.0904	$\sim$0.7	$\sim$3.6
NRAPR Skyrme	0.413	0.073	$\sim$1.0	6.0
BSk24 Skyrme	0.268	0.0808	$\sim$0.8	5.4
RMF (Stiff J,L)	0.54	0.086	1.4	—

*Upper bound set by entrainment arguments (Madhuri et al., 2016, 2207.13384).

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In conclusion, the fractional crustal moment of inertia is a sensitive probe of dense-matter microphysics, encoding constraints on the symmetry energy, higher-order EOS parameters, and neutron star global structure. Meeting glitch-driven constraints, especially under strong crustal entrainment, remains challenging for standard EoS models unless the crust is exceptionally thick or the stellar mass is low. This tension renders ΔI/I a central anchor for multi-messenger neutron-star physics, linking nuclear theory, astrophysical data, and fundamental dense-matter phenomenology (Atta et al., 2015, Madhuri et al., 2016, 2207.13384, Eya et al., 2018, Lim et al., 2018, Carreau et al., 2018, Lim et al., 2018, Li et al., 2014, Dutra et al., 2021, Seif et al., 24 Jul 2025, Zhang et al., 2024).