Numerical calculations of neutron star mountains supported by crustal lattice pressure

This paper presents the first self-consistent numerical calculation of neutron star "mountains" driven by thermal gradients and crustal lattice pressure using realistic Brussels-Montreal equations of state, finding that such mountains are too small to dictate the spin equilibrium of low-mass X-ray binaries.

The Gist

The Big Picture: Listening for Cosmic "Humming"

Imagine the universe is a giant concert hall. For years, we've been listening for the loud, crashing sounds of black holes colliding (like two cymbals smashing together). But now, scientists are trying to hear a much quieter sound: a continuous, steady hum coming from spinning neutron stars.

Neutron stars are the super-dense corpses of dead stars, packed so tightly that a teaspoon of their stuff would weigh a billion tons. If a neutron star spins perfectly round, it's silent. But if it has a "bump" on it—a mountain, even a tiny one—it will wobble as it spins. This wobble creates ripples in space-time called gravitational waves.

The big question this paper asks is: How big can these mountains get? And more importantly, can we actually hear them?

The Problem: The "Perfect" Sphere vs. The "Bumpy" Star

Neutron stars are incredibly strong. Their outer crust is like a solid, super-hard shell made of atomic nuclei. If you try to build a mountain on Earth, gravity pulls it down until it flattens. On a neutron star, gravity is so strong that the crust can only support a mountain about the size of a grain of sand before it cracks.

However, scientists have been looking for a way to build mountains that are just big enough to be detected by our current telescopes (like LIGO). To do this, they need a mechanism to push the crust out of shape without breaking it.

The Old Theory: The "Shifting Floor" Analogy

For a long time, the leading theory was called the "Capture Layer Shift" model.

• The Analogy: Imagine a multi-story building where the floors are made of different materials. If you heat up a specific room on the 5th floor, the floorboards might warp or shift. In a neutron star, as matter falls onto the surface, it gets crushed. At certain depths, the atoms get squeezed so hard they change identity (electrons get captured by protons).
• The Theory: If one side of the star is hotter than the other, these "identity changes" happen at slightly different depths. This creates a wavy, uneven floor inside the star, pushing the crust out of shape.
• The Catch: Recent calculations show that the "floors" in these neutron stars are much more stable than we thought. The identity changes happen at much higher pressures than previously believed, meaning this "wavy floor" effect is much weaker than we hoped. It's like trying to build a sandcastle with wet sand that just won't hold its shape.

The New Theory: The "Hot Lattice" Analogy

This paper proposes a different mechanism. Instead of relying on the shifting floors, the authors look at the crust itself as a giant, hot crystal lattice (like a grid of atoms).

• The Analogy: Think of a metal bridge on a hot day. If one side of the bridge is in the sun and the other is in the shade, the metal expands unevenly. The sun-side expands more, causing the bridge to bend or warp.
• The Mechanism: Neutron stars in binary systems (LMXBs) are often magnetized. These magnetic fields act like a funnel, guiding heat to specific spots on the star. This creates a "hot side" and a "cold side."
  • The "hot side" of the crystal lattice expands slightly.
  • The "cold side" stays contracted.
  • This difference in expansion pushes the crust out of round, creating a "mountain."

The authors call these "Magneto-Thermo-Elastic Mountains." It's a fancy way of saying: Magnetic fields create heat differences, which make the crystal crust expand unevenly, creating a bump.

The Calculation: Doing the Math

The authors didn't just guess; they built a massive, self-consistent computer model.

1. The Star: They used the most realistic "recipe" for neutron star matter available (called BSk equations of state), covering everything from the core to the surface.
2. The Heat: They calculated how heat moves through the star, considering the magnetic fields.
3. The Bump: They calculated exactly how much the crust would bulge out due to the heat.

The Result: A Disappointing (but Honest) Answer

Here is the punchline: The mountains are too small to be heard.

• The Scale: The authors found that even under the most optimistic conditions (a very hot, very fast-spinning star with strong magnetic fields), the resulting "mountain" creates an ellipticity (a measure of how bumpy the star is) of about 0.00000000001 ($10^{-11}$).
• The Goal: To be detected by current telescopes, the star would need to be about 10,000 times bumpier than that (around $10^{-7}$).
• The Comparison: The "Hot Lattice" mountains are roughly the same size as the "Shifting Floor" mountains, but both are far too tiny to explain why some neutron stars stop spinning faster than they do.

Why This Matters

You might think, "If the answer is 'no,' why write a paper?"

1. Ruling Out Options: Science is about elimination. By proving that this specific mechanism creates mountains that are too small, the authors are telling the rest of the community: "Stop looking for gravitational waves from this specific type of bump. It's not going to work."
2. Better Models: They used the most up-to-date physics available. Previous studies used older, less accurate models of neutron star matter. This paper says, "Even with the best modern physics, this mechanism doesn't produce big enough mountains."
3. Future Hope: While these mountains are too small for current detectors, the paper suggests that if we find Ultraluminous X-ray sources (ULXs)—which are neutron stars eating matter at a super-fast rate with super-strong magnetic fields—they might create bigger bumps. If we build better telescopes in the future (like the Einstein Telescope), we might finally hear these cosmic hums.

The Takeaway

The authors tried to find a way to explain how neutron stars get "bumpy" enough to sing a song we can hear. They tested a new theory involving magnetic fields and heat expanding the star's crust like a hot metal bridge.

The verdict? The bridge doesn't bend enough. The mountains are real, but they are microscopic. We need to look for different kinds of stars or build much more sensitive ears to hear them.

Technical Summary

1. Problem Statement

The detection of continuous gravitational waves (CGWs) from rapidly spinning neutron stars (NSs) is a primary goal for current and future observatories like LIGO-Virgo-KAGRA. A leading candidate for such emission is a "mountain"—a mass asymmetry (ellipticity, $\varepsilon$) supported by the star's solid crust.

While theoretical limits suggest the crust can sustain an ellipticity of $\varepsilon_{max} \sim 10^{-7}$, the physical mechanisms capable of generating such deformations in realistic accreting systems (Low-Mass X-ray Binaries, LMXBs) remain uncertain. Previous models for "thermal mountains" relied heavily on the Ushomirsky, Cumming, & Bildsten (UCB) model, which posits that temperature gradients shift the depth of electron capture layers in the crust, creating pressure imbalances. However, recent updates to the Equation of State (EOS) suggest these capture layers are far fewer and have lower threshold energies than previously assumed, potentially rendering the UCB mechanism insufficient to explain observed spin equilibria.

The authors aim to investigate an alternative mechanism: thermal pressure perturbations arising directly from the crustal ionic lattice, rather than electron capture shifts. They seek to perform a fully self-consistent calculation of the entire mountain formation process, from temperature asymmetry generation to elastic crustal readjustment.

2. Methodology

The authors developed a unified framework to calculate the ellipticity of "magneto-thermo-elastic" mountains. The calculation proceeds in four distinct stages:

• Unified Equation of State (EOS):

  • The study utilizes the Brussels–Montreal nuclear energy-density functionals (BSk19, BSk20, BSk21). These provide a thermodynamically consistent, unified description of the NS interior (outer crust, inner crust, and core), overcoming the discontinuities found in older, piecewise EOS models.
  • To handle numerical convergence issues caused by density jumps at nuclear transition layers, the authors fitted the tabulated BSk data with a 23-parameter analytical function (based on Potekhin et al. 2013), achieving an average error of $\approx 0.3\%$.

• Hydrostatic Structure:

  • The background structure is solved using the Tolman-Oppenheimer-Volkoff (TOV) equations.
  • To simplify the elastic and thermal calculations, the authors map the relativistic TOV solution to a Newtonian framework by defining an "effective gravitational acceleration" ($g_{eff}$) that incorporates relativistic corrections, rather than solving the full perturbed Einstein equations.

• Thermal Structure and Asymmetry:

  • Background: Solved using coupled ODEs for energy conservation and Fourier's law, accounting for deep crustal heating (from non-equilibrium reactions) and shallow crustal heating (an unknown mechanism required to match quiescent temperatures).
  • Perturbation: Temperature asymmetries ($\delta T$) are induced by anisotropic heat conduction driven by internal magnetic fields. The authors consider two scenarios:
    1. A crustal toroidal field ($B \sim 10^{12}$ G).
    2. A core toroidal field ($B \sim 10^8$ G), limited by the perturbative assumption $\omega_B \tau \ll 1$.
  • The magnetic field deflects heat flux, creating a latitudinal temperature gradient.

• Elastic Readjustment:

  • The temperature asymmetry induces a pressure perturbation ($\Delta P$). The authors focus on the thermal lattice pressure component ($P_{th}$), which depends on the temperature of the ionic lattice.
  • They solve the linearized elastic equations (Lagrangian displacement field $\xi_i$) to determine how the crust deforms to restore equilibrium.
  • The mass quadrupole moment ($Q_{22}$) and ellipticity ($\varepsilon$) are calculated from the resulting density perturbations.

3. Key Contributions

• Self-Consistent Framework: Unlike previous studies that treated thermal generation and elastic response separately, this paper couples the magnetic-field-induced temperature asymmetry directly to the lattice pressure response within a single model.
• Realistic EOS Integration: This is the first study to utilize the unified BSk19/20/21 EOS for thermal mountain calculations, ensuring thermodynamic consistency across the entire star and avoiding the artificial "bolt-on" of crustal and core models.
• Alternative Mechanism Validation: The paper rigorously tests the crustal lattice pressure mechanism, moving beyond the electron-capture-layer-shift model (UCB) which is now considered less viable due to modern EOS constraints.
• Analytical EOS Fitting: The authors successfully implemented a high-precision analytical fit for accreted EOS data to resolve numerical convergence issues inherent in integrating over sharp density discontinuities.

4. Key Results

• Magnitude of Ellipticity: The calculated ellipticities for magneto-thermo-elastic mountains are extremely small:
  • $\varepsilon \lesssim 10^{-11}$ for all tested configurations.
  • The maximum value found was $\varepsilon_{max} \approx 2.32 \times 10^{-11}$ (BSk21 EOS, high accretion rate $\dot{M} \sim 4.4 \times 10^{-9} M_\odot \text{yr}^{-1}$, high shallow heating $Q_S = 10$ MeV, and crustal field $B=10^{12}$ G).
• Comparison with Torque Balance: The required ellipticity to balance accretion torques via gravitational wave emission (the torque-balance limit) is typically $\varepsilon_{TB} \sim 10^{-9} - 10^{-7}$. The calculated mountains are 2 to 6 orders of magnitude too small to dictate the spin equilibrium of LMXBs.
• Lattice Pressure vs. Capture Layers:
  • When comparing the lattice pressure mechanism to the UCB capture-layer shift mechanism (using modern EOS constraints), the lattice pressure produces ellipticities comparable to or slightly smaller than those from shallow capture layers.
  • However, if capture layers exist in the inner crust (as predicted by older EOS models but not modern ones), they would dominate. With modern EOS (fewer/inner layers), the lattice pressure is a competitive but still sub-dominant source.
• Dependence on Parameters:
  • Ellipticity increases with accretion rate and shallow heating (due to higher crustal temperatures).
  • At very high accretion rates ($\dot{M} > 5 \times 10^{-9} M_\odot \text{yr}^{-1}$), the crust melts at lower densities, limiting the solid region available to support the mountain, causing $\varepsilon$ to plateau or converge.

5. Significance and Conclusion

• Detection Prospects: The study concludes that magneto-thermo-elastic mountains supported by lattice pressure are too small to be detected by current-generation gravitational wave detectors (LIGO/Virgo) or to explain the spin equilibrium of LMXBs. The predicted signal is far below current observational upper limits ($\varepsilon \sim 10^{-7}$).
• Implications for UCB Model: The results reinforce that the traditional UCB capture-layer mechanism, when updated with modern EOS (fewer capture layers, lower $E_{cap}$), also fails to produce mountains large enough to explain observed spin-downs.
• Future Directions:
  • The authors suggest that Ultraluminous X-ray sources (ULXs) might be more promising candidates. Their super-Eddington accretion rates and stronger magnetic fields could generate larger temperature asymmetries and lattice pressures, potentially compensating for their slower spin rates.
  • Future work should extend the formalism to full General Relativity and remove the Cowling approximation to refine the estimates, though the authors expect these corrections to reduce the ellipticity further.

In summary, this paper provides a rigorous, self-consistent rejection of the hypothesis that thermal lattice pressure in standard LMXBs can generate detectable gravitational waves or dictate spin equilibrium, highlighting the need for either more extreme astrophysical environments (like ULXs) or alternative mountain formation mechanisms.