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arxiv: 2605.19082 · v3 · pith:OQGDGYJDnew · submitted 2026-05-18 · 🌌 astro-ph.EP

Dependence on the Equation of State in SPH Simulations of Proto-Uranian Disk Formation from a Giant Impact

Keiya Murashima , Takanori Sasaki This is my paper

Pith reviewed 2026-05-22 09:02 UTC · model grok-4.3

classification 🌌 astro-ph.EP
keywords giant impactUranusequation of stateSPH simulationsdebris disksatellite formationobliquityhydrodynamic modeling
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The pith

For giant impacts matching Uranus's spin, disk mass and size are set by angular momentum while rock fraction varies with the equation of state.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines how different equations of state affect smoothed particle hydrodynamics simulations of a giant impact forming a disk around proto-Uranus. It finds that when impact parameters are chosen to reproduce the planet's current rotation, the resulting disk's mass, size, and the planet's final spin are largely insensitive to the specific EOS or SPH variant used. Instead, these macroscopic features are set by the total angular momentum delivered in the collision. However, the fraction of rock in the disk material changes noticeably with the choice of EOS. This distinction helps explain prior disagreements on disk composition and shows that accurate modeling of the equation of state is needed before disk evolution can be used to test the giant-impact origin of Uranus's satellites.

Core claim

In simulations of a 3 Earth-mass impactor striking proto-Uranus at parameters that reproduce the observed spin state, the post-impact rotation period, the mass of the ejected disk, and the radial extent of that disk remain nearly the same across three different equations of state and two SPH formulations. These outcomes are controlled primarily by the angular momentum of the impact. By contrast, the rock-to-ice ratio in the disk material shows strong dependence on which EOS is adopted.

What carries the argument

Systematic comparison of three distinct EOS models and two SPH schemes (standard and density-independent) in giant-impact simulations tuned to match Uranus's current rotation.

If this is right

  • Disk mass and radial size can be predicted reliably from impact angular momentum alone.
  • Disk rock fraction requires careful EOS selection to constrain possible satellite compositions.
  • Earlier inconsistencies in reported disk compositions are attributable to differences in EOS rather than SPH scheme.
  • The giant-impact scenario for Uranus remains testable once disk-evolution models incorporate the range of rock fractions found here.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Disk-evolution calculations run across the full range of rock fractions reported here could produce bounded predictions for the ice-to-rock ratios of Uranus's regular satellites.
  • The robustness of mass and size to microphysical details suggests that analogous impacts on other ice giants would yield structurally similar disks even if their EOS differs.
  • If multi-phase or temperature-dependent EOS refinements alter the rock fraction by more than the spread seen across the three models tested, satellite-accretion outcomes could shift measurably.

Load-bearing premise

The assumption that a 3-Earth-mass impactor and the explored range of impact parameters are representative of the actual formation event, and that the tested EOS models and SPH schemes capture the dominant physical uncertainties.

What would settle it

Running identical impact simulations with an additional independent EOS model or with a non-SPH hydrodynamics code and checking whether the rock fraction remains strongly EOS-dependent while mass and size stay unchanged.

Figures

Figures reproduced from arXiv: 2605.19082 by Keiya Murashima, Takanori Sasaki.

Figure 1
Figure 1. Figure 1: Particles ejected by collisions simulated with the DISPH method using SESAME/ANEOS EOS (top), Tillotson EOS (middle), and HM80 EOS (bottom), for initial angular momenta Limp = 3.0, 5.0, and 7.0 × 1036 kg m2 s −1 (from left to right). These panels display the state of the system at the final time of the simulation. The dotted circle indicates Uranus’s Roche radius. The components are color-coded as follows:… view at source ↗
Figure 1
Figure 1. Figure 1: Particles ejected by collisions simulated with the DISPH method using SESAME/ANEOS EOS (top), Tillotson EOS (middle), and HM80 EOS (bottom), for initial angular momenta Limp = 3.0, 5.0, and 7.0 × 1036 kg m2 s −1 (from left to right). These panels display the state of the system at the final time of the simulation. The dotted circle indicates Uranus’s Roche radius. The components are color-coded as follows:… view at source ↗
Figure 2
Figure 2. Figure 2: Snapshots of a collision with Limp = 3 × 1036 kg m2 s −1 using the Tillotson EOS. Only particles with z < 0 are shown. Snapshot times are measured in hours from the beginning of the simulation. For collisions with low initial angular momentum (Limp ≲ 3×1036 kg m2 s −1 ) ( [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 2
Figure 2. Figure 2: Snapshots of a collision with Limp = 3 × 1036 kg m2 s −1 using the Tillotson EOS. Only particles with z < 0 are shown. Snapshot times are measured in hours from the beginning of the simulation. cases is greater than that produced in lower-angular-momentum collisions, reaching ∼ 10−2MU in some runs. This trend, consistent with previous studies (e.g., Slattery et al. 1992; Woo et al. 2022), highlights the cr… view at source ↗
Figure 3
Figure 3. Figure 3: Same as [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 3
Figure 3. Figure 3: Same as [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Same as [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 4
Figure 4. Figure 4: Same as [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Median rotation periods of the post-impact planets as a function of the impact angular momentum Limp, for different EOSs and SPH schemes. Panels show (a) SSPH and (b) DISPH. The dashed line indicates the present rotation period of Uranus (17.24 hr; Warwick et al. 1986). excess angular momentum can potentially be shed during the subsequent long-term evolution, such as through disk-planet interactions, atmos… view at source ↗
Figure 5
Figure 5. Figure 5: Median rotation periods of the post-impact planets as a function of the impact angular momentum Limp, for different EOSs and SPH schemes. Panels show (a) SSPH and (b) DISPH. The dashed line indicates the present rotation period of Uranus (17.24 hr; Warwick et al. 1986) [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Disk mass Md,imp as a function of the initial angular momentum, for (a) SSPH and (b) DISPH. Masses are normalized by the total mass of Uranus’s current satellites (Msatellite ∼ 10−4MU). The y-axis is shown on a logarithmic scale. et al. 2020) and to demonstrate that this macroscopic outcome is largely insensitive to the choice of EOS. The weak dependence on EOS or SPH scheme indicates that the gross spin s… view at source ↗
Figure 6
Figure 6. Figure 6: Disk mass Md,imp as a function of the initial angular momentum, for (a) SSPH and (b) DISPH. Masses are normalized by the total mass of Uranus’s current satellites (Msatellite ∼ 10−4MU). The y-axis is shown on a logarithmic scale [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Characteristic disk size ⟨rd,imp⟩ as a function of the initial angular momentum, for (a) SSPH and (b) DISPH. Sizes are normalized by Uranus’s radius. Figures 6 and 7 summarize the dependence of Md,imp and ⟨rd,imp⟩ on Limp. Across all EOSs and both SPH schemes, the resulting disks are systematically more massive than Uranus’s present satellite system. Typical values are Md,imp ∼ 10−3–10−2MU, corresponding t… view at source ↗
Figure 7
Figure 7. Figure 7: Characteristic disk size ⟨rd,imp⟩ as a function of the initial angular momentum, for (a) SSPH and (b) DISPH. Sizes are normalized by Uranus’s radius [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Rock and ice components in the debris disk as a function of Limp for different EOSs in SSPH. The total mass of the rock component is normalized by the present satellite mass Msatellite [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 8
Figure 8. Figure 8: Rock and ice components in the debris disk as a function of Limp for different EOSs in SSPH. The total mass of the rock component is normalized by the present satellite mass Msatellite [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Rock and ice components in the debris disk as a function of Limp for different EOSs in DISPH. The total mass of the rock component is normalized by the present satellite mass Msatellite. properties of the disk appear robust, the composition of the disk—and in particular the rock-to-ice ratio—is much more sensitive to the adopted physical model, as discussed in the following section. 3.4. Rock Abundance in … view at source ↗
Figure 9
Figure 9. Figure 9: Rock and ice components in the debris disk as a function of Limp for different EOSs in DISPH. The total mass of the rock component is normalized by the present satellite mass Msatellite. tidal torques between the disk and forming satellites, or subsequent gravitational interactions (e.g. Crida & Charnoz 2012; Woo et al. 2022). Overall, our results indicate that disk mass and size are governed primarily by … view at source ↗
Figure 10
Figure 10. Figure 10: Snapshots of a collision with a Limp = 3 × 1036 kg m2 s −1 using SESAME and ANEOS EOS with SSPH. Particles with z < 0 are shown. The snapshot times are given in hours from the beginning of the simulation. 4.3. The Role of Initial Internal Structure To understand the origin of the compositional differences in the debris disks across different EOS models, it is essential to consider the initial internal str… view at source ↗
Figure 10
Figure 10. Figure 10: Snapshots of a collision with a Limp = 3 × 1036 kg m2 s −1 using SESAME and ANEOS EOS with SSPH. Particles with z < 0 are shown. The snapshot times are given in hours from the beginning of the simulation. 4.4. Secondary Impacts The differences among EOSs become most pronounced in cases where a secondary collision occurs. Such events arise when material ejected during the first encounter remains marginally… view at source ↗
Figure 11
Figure 11. Figure 11: The same as [PITH_FULL_IMAGE:figures/full_fig_p028_11.png] view at source ↗
Figure 11
Figure 11. Figure 11: The same as [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The same as [PITH_FULL_IMAGE:figures/full_fig_p029_12.png] view at source ↗
Figure 12
Figure 12. Figure 12: The same as [PITH_FULL_IMAGE:figures/full_fig_p018_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Initial density profiles of the target (left column) and the impactor (right column) utilizing the SSPH. From top to bottom, the panels show the profiles generated by the SESAME/ANEOS, Tillotson, and HM80 equations of state. The radial distance is calculated from the center of mass of each respective body at t = 0 [PITH_FULL_IMAGE:figures/full_fig_p030_13.png] view at source ↗
Figure 13
Figure 13. Figure 13: Initial density profiles of the target (left column) and the impactor (right column) utilizing the SSPH. From top to bottom, the panels show the profiles generated by the SESAME/ANEOS, Tillotson, and HM80 equations of state. The radial distance is calculated from the center of mass of each respective body at t = 0. in Section 2.1.3. If this density jump is artificially amplified by the extrapolation, it c… view at source ↗
Figure 14
Figure 14. Figure 14: Initial density profiles of the target (left column) and the impactor (right column) utilizing the DISPH. The layout and the EOS arrangements are identical to those in [PITH_FULL_IMAGE:figures/full_fig_p031_14.png] view at source ↗
Figure 14
Figure 14. Figure 14: Initial density profiles of the target (left column) and the impactor (right column) utilizing the DISPH. The layout and the EOS arrangements are identical to those in [PITH_FULL_IMAGE:figures/full_fig_p020_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Snapshots of the second collision with DISPH and Limp = 7.0 × 1036kg m2 s −1 for SESAME/ANEOS (top), Tillotson EOS (middle), and HM80 EOS (bottom). As shown in [PITH_FULL_IMAGE:figures/full_fig_p035_15.png] view at source ↗
Figure 15
Figure 15. Figure 15: Snapshots of the second collision with DISPH and Limp = 7.0×1036kg m2 s −1 for SESAME/ANEOS (top), Tillotson EOS (middle), and HM80 EOS (bottom). Tillotson model. Future simulations incorporating more advanced multiphase EOS models would be beneficial to further constrain these dynamics. 4.6. Constraints on Formation Scenarios The major Uranian satellites are estimated to have rock-to-ice ratios close to … view at source ↗
Figure 16
Figure 16. Figure 16: The same results as [PITH_FULL_IMAGE:figures/full_fig_p036_16.png] view at source ↗
Figure 16
Figure 16. Figure 16: The same results as [PITH_FULL_IMAGE:figures/full_fig_p022_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Density map of snapshots of second collision for SESAME/ANEOS (left), Tillotson EOS (center), and HM80 EOS (right). mechanisms have been proposed: (i) Impacts involving intrinsically rock-rich impactors, which can enhance the rocky component of the disk (Woo et al. 2022); (ii) subsequent contamination or accre- [PITH_FULL_IMAGE:figures/full_fig_p036_17.png] view at source ↗
Figure 17
Figure 17. Figure 17: Density map of snapshots of second collision for SESAME/ANEOS (left), Tillotson EOS (center), and HM80 EOS (right). The discrepancy between SESAME/ANEOS predictions and the observed satellite compositions suggests that ad￾ditional processes must have operated after the initial impact. Several possible mechanisms have been proposed: (i) Impacts involving intrinsically rock-rich impactors, which can enhance… view at source ↗
Figure 18
Figure 18. Figure 18: Rock and ice components in the disk as a function of Limp for different EOSs in SSPH (Appendix version). The total mass of rock components is normalized by the present satellite mass Msatellite. APPENDIX A. COMPARISON WITH AN ALTERNATIVE CLASSIFICATION OF SPH PARTICLES To investigate the properties of the resulting debris disks, SPH particles in this study were classified into three categories: “planetary… view at source ↗
Figure 18
Figure 18. Figure 18: Rock and ice components in the disk as a function of Limp for different EOSs in SSPH (Appendix version). The total mass of rock components is normalized by the present satellite mass Msatellite [PITH_FULL_IMAGE:figures/full_fig_p025_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Rock and ice components in the disk as a function of Limp for different EOSs in DISPH (Appendix version). The total mass of rock components is normalized by the present satellite mass Msatellite. land C6, N ≈ 105 ), a sensitivity test for the smoothing kernel (Wendland C2, N ≈ 105 ), and a convergence test for spatial resolution (N ≈ 106 ). While minor discrepancies are observed between these cases—such a… view at source ↗
Figure 19
Figure 19. Figure 19: Rock and ice components in the disk as a function of Limp for different EOSs in DISPH (Appendix version). The total mass of rock components is normalized by the present satellite mass Msatellite. Although both definitions yield qualitatively similar trends, the iterative algorithm of Canup & Asphaug (2001) generally produces a smaller disk mass, as it systematically excludes particles that are only weakly… view at source ↗
Figure 20
Figure 20. Figure 20: Rock and ice components in the disk as a function of Limp. Left panel: Baseline simulations utilizing the Wendland C6 kernel and approximately 105 particles. Middle panel: Test for the kernel choice using the Wendland C2 kernel with N ≈ 105 particles. Right panel: Test for spatial resolution using N ≈ 106 particles and the Wendland C6 kernel. Becker, A., Lorenzen, W., Fortney, J. J., et al. 2014, The Astr… view at source ↗
Figure 20
Figure 20. Figure 20: Rock and ice components in the disk as a function of Limp. Left panel: Baseline simulations utilizing the Wendland C6 kernel and approximately 105 particles. Middle panel: Test for the kernel choice using the Wendland C2 kernel with N ≈ 105 particles. Right panel: Test for spatial resolution using N ≈ 106 particles and the Wendland C6 kernel. Barnes, J., & Hut, P. 1986, nature, 324, 446 Becker, A., Lorenz… view at source ↗
read the original abstract

The $98^\circ$ obliquity of Uranus is widely attributed to a giant impact that ejected material and formed a debris disk, which subsequently coalesced into its regular satellites. Previous Smoothed Particle Hydrodynamics (SPH) studies have yielded inconsistent disk compositions, a discrepancy often linked to the variety of numerical and physical modeling assumptions. We address this by presenting SPH simulations that systematically test three distinct EOS models alongside two SPH schemes (standard SPH, and the enhanced density-independent SPH). We utilized a $3M_{\oplus}$ impactor and explored a range of impact parameters which are capable of reproducing Uranus's current spin state. Our primary finding is that for impacts capable of reproducing Uranus's current rotation, the choice of EOS or SPH scheme barely affects macroscopic features such as the post-impact rotation period, disk mass, or disk size; these properties are primarily controlled by the impact's angular momentum. In contrast, the disk's rock fraction is highly EOS-dependent. Our results clarify that while disk mass and size are robust outcomes, the final disk composition is highly model-dependent. Therefore, accurate EOS modeling, integrated with detailed disk evolution studies, is essential to definitively validate the giant impact scenario for Uranus.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript presents SPH simulations of a giant impact on proto-Uranus using a fixed 3 Earth-mass impactor and a range of impact parameters selected to reproduce the planet's observed spin state. It systematically compares three distinct equations of state and two SPH schemes (standard and density-independent), concluding that for spin-matching impacts the choice of EOS or SPH variant has negligible effect on macroscopic disk properties such as post-impact rotation period, disk mass, and disk size, which are instead controlled primarily by the impact angular momentum; in contrast, the disk rock fraction is strongly EOS-dependent. The authors argue that this clarifies prior inconsistencies in the literature and underscores the need for accurate EOS modeling in disk evolution studies.

Significance. If the central result holds, the work provides a useful clarification of why previous SPH studies of Uranian satellite formation have disagreed on disk composition while showing more consistency on bulk properties. By isolating the spin-matching subset of impacts, it demonstrates robustness of mass and size outcomes and isolates composition as the key model-dependent quantity. This has direct implications for subsequent N-body or disk-evolution calculations that rely on the initial disk state. The systematic multi-EOS, multi-scheme comparison is a methodological strength.

major comments (2)
  1. [§2 and abstract] §2 (Simulation Setup) and abstract: the impactor mass is fixed at 3 M_⊕ and impact parameters are chosen specifically to reproduce Uranus's spin. This restricted slice of parameter space may mask EOS sensitivity in disk mass and size; different impactor masses would alter shock heating, vaporization thresholds, and angular-momentum partitioning, potentially producing larger spreads in macroscopic outcomes under the same EOS variations. The manuscript provides no tests at other masses and no explicit argument that 3 M_⊕ is representative, rendering the robustness claim conditional on an untested assumption.
  2. [Results section] Results section (disk property comparisons): while the paper reports that macroscopic features are insensitive to EOS within the explored runs, the quantitative spreads in disk mass and size across EOS models are not tabulated or plotted with error bars that would allow assessment of whether the observed insensitivity is statistically meaningful or merely a consequence of the narrow parameter choice.
minor comments (2)
  1. [Abstract] Abstract: the specific three EOS models and the two SPH schemes should be named explicitly rather than described generically.
  2. [Figures and text] Figure captions and text: ensure consistent notation for rock fraction (e.g., always specify whether it is mass fraction or number fraction) and clarify how particles are classified as belonging to the disk versus the planet.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed review. The comments highlight important aspects of the scope and presentation of our results, which we address point by point below.

read point-by-point responses

  1. Referee: [§2 and abstract] §2 (Simulation Setup) and abstract: the impactor mass is fixed at 3 M_⊕ and impact parameters are chosen specifically to reproduce Uranus's spin. This restricted slice of parameter space may mask EOS sensitivity in disk mass and size; different impactor masses would alter shock heating, vaporization thresholds, and angular-momentum partitioning, potentially producing larger spreads in macroscopic outcomes under the same EOS variations. The manuscript provides no tests at other masses and no explicit argument that 3 M_⊕ is representative, rendering the robustness claim conditional on an untested assumption.

    Authors: We agree that the study is restricted to a 3 M_⊕ impactor and to the subset of impact parameters that reproduce Uranus's observed spin. This choice follows from the scientific goal of examining the giant-impact scenario capable of explaining the planet's obliquity and rotation; prior literature on Uranian satellite formation has commonly adopted similar impactor masses. Within this spin-matching regime our results show that disk mass and size are controlled primarily by angular momentum rather than EOS. We will revise the manuscript to include an explicit statement of this scope, a brief justification for the 3 M_⊕ choice based on existing models, and a clear caveat that sensitivity at other masses remains untested. No new simulations at different masses will be added in this revision. revision: partial

  2. Referee: [Results section] Results section (disk property comparisons): while the paper reports that macroscopic features are insensitive to EOS within the explored runs, the quantitative spreads in disk mass and size across EOS models are not tabulated or plotted with error bars that would allow assessment of whether the observed insensitivity is statistically meaningful or merely a consequence of the narrow parameter choice.

    Authors: We accept this criticism of the presentation. In the revised manuscript we will insert a new table that reports the numerical values of post-impact rotation period, disk mass, disk size, and rock fraction for every EOS–SPH combination, together with the range or standard deviation across the spin-matching impact parameters. We will also add error bars (or shaded ranges) to the relevant comparison figures. These additions will allow readers to judge the magnitude of the spreads directly. revision: yes

Circularity Check

0 steps flagged

No circularity: results from independent numerical variation against external spin target

full rationale

The paper reports direct SPH simulation outcomes for a fixed 3M⊕ impactor across EOS variants and schemes, selecting impact parameters only to match Uranus's observed rotation period as an external benchmark. Macroscopic disk properties are stated to be controlled by angular momentum while rock fraction varies with EOS; these emerge from explicit numerical comparisons rather than any equation reducing a reported quantity to a fitted parameter or self-citation by construction. No self-definitional steps, fitted-input predictions, or load-bearing self-citations appear in the provided text, so the central claims remain independent of the inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim depends on the validity of SPH as a model for giant impacts, the representativeness of the chosen impactor mass and parameter range, and the assumption that the three tested EOS models bracket the relevant material uncertainties. No new particles or forces are introduced.

free parameters (2)
  • Impactor mass
    Fixed at 3 Earth masses to explore impacts that can reproduce the target spin state.
  • Impact parameter range
    Selected to match Uranus's current rotation period; specific values not listed in abstract.
axioms (2)
  • domain assumption SPH and its density-independent variant accurately capture the hydrodynamics and mixing in giant impacts between icy bodies.
    Invoked by using these schemes to generate the disk properties reported.
  • domain assumption The three chosen EOS models adequately sample the range of plausible rock-ice behavior under impact conditions.
    Basis for concluding that rock fraction is highly EOS-dependent.

pith-pipeline@v0.9.0 · 5752 in / 1677 out tokens · 75582 ms · 2026-05-22T09:02:56.711118+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    Our primary finding is that for impacts capable of reproducing Uranus's current rotation, the choice of EOS or SPH scheme barely affects macroscopic features such as the post-impact rotation period, disk mass, or disk size; these properties are primarily controlled by the impact's angular momentum. In contrast, the disk's rock fraction is highly EOS-dependent.

  • IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We employed two Smoothed-Particle Hydrodynamics (SPH) schemes... three distinct EOS models (ANEOS/SESAME, Tillotson EOS, and HM80 EOS)

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