Effects of Rotation on the Gravitational Tug-Boat Mechanism for Neutron-Star Kicks and Implications for Spin-Kick Alignment

Eric G. Blackman; Yiming Lu

arxiv: 2606.17496 · v1 · pith:4YH5IUUTnew · submitted 2026-06-16 · 🌌 astro-ph.HE · astro-ph.SR

Effects of Rotation on the Gravitational Tug-Boat Mechanism for Neutron-Star Kicks and Implications for Spin-Kick Alignment

Yiming Lu , Eric G. Blackman This is my paper

Pith reviewed 2026-06-26 23:57 UTC · model grok-4.3

classification 🌌 astro-ph.HE astro-ph.SR

keywords neutron-star kickstug-boat mechanismspin-kick alignmentcore-collapse supernovaestellar rotationnatal kicksgravitational tug

0 comments

The pith

Rotation of expanding ejecta sets the spin-kick angle in the tug-boat mechanism by the product of expansion-time-to-rotation-period ratio and asymmetry orientation relative to the spin axis.

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

Neutron stars receive natal kicks from the gravitational tug-boat effect of anisotropic ejecta. Prior models omitted initial stellar rotation. This extension incorporates rotation of the asymmetric mass distribution and shows that the resulting spin-kick angle is the product of two factors: one governed by the ratio of shock expansion time to rotation period, and the other by the asymmetry's orientation to the spin axis. Sufficiently rapid rotation averages out non-axisymmetric components, suppressing the perpendicular force component and leaving only a spin-aligned kick. This averaging, however, demands rotation rates that are unrealistic unless magnetic fields efficiently transport angular momentum outward; otherwise alignment is more probable when the mass asymmetry itself is preferentially aligned with the spin axis.

Core claim

We show that the spin-kick angle is determined by the product of two factors, one that depends on the ratio of shock expansion time to the rotation period and the other which depends on the orientation of the asymmetric mass distribution with respect to the spin-axis. For fast enough rotation, the first factor amounts to axially averaging out non-axisymmetry thereby suppressing the perpendicular tug and leaving only a spin-aligned force. However, the rotation speed required for this effect would be unrealistically large unless magnetic fields could transport angular momentum from the core to the outflow efficiently. Otherwise, spin-kick alignment for the tug-boat mechanism would be more like

What carries the argument

The product of the shock-expansion-time-to-rotation-period ratio and the asymmetry-orientation factor relative to the spin axis, which together determine the spin-kick angle.

If this is right

Fast rotation suppresses the perpendicular component of the kick through axial averaging of the asymmetry.
Magnetic angular-momentum transport from core to outflow is required for the averaging mechanism to operate at realistic rotation rates.
Alignment can instead arise when the mass-flux asymmetry is itself preferentially aligned with the spin axis.
The model connects progenitor rotation rate and asymmetry orientation to the statistical distribution of observed spin-kick angles.

Where Pith is reading between the lines

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

Common spin-kick alignments could constrain the efficiency of magnetic angular-momentum transport during the supernova.
If alignments persist at slow rotation rates, the mechanism generating the initial mass asymmetry must itself favor the spin axis.
Full hydrodynamic simulations that include rotation can test whether the minimal model's predictions survive additional physics.
The same two-factor structure may apply to other asymmetric explosion channels that impart kicks.

Load-bearing premise

The derivation assumes a minimalist extension of the tug-boat model that adds only initial rotation and a possible magnetic angular-momentum transport channel while neglecting hydrodynamical back-reaction, neutrino transport, and further instabilities.

What would settle it

A clear observation of strong spin-kick misalignment in a system with independently measured fast core rotation and no evidence for efficient magnetic transport would falsify the averaging pathway as the primary source of alignment.

Figures

Figures reproduced from arXiv: 2606.17496 by Eric G. Blackman, Yiming Lu.

**Figure 1.** Figure 1: Schematic illustration of the non-rotating gravitational tug-boat model. asymmetry. In section 4 we discuss the implications for spin-kick alignment and the possible role of magnetic fields. We present the conclusions, limitations, and targets for further work in section 5. 2. The Gravitational Tug-Boat Mechanism Revisited The basic non-rotating gravitational tug-boat picture can be illustrated with a sim… view at source ↗

**Figure 2.** Figure 2: Time evolution of the NS velocity predicted by Equation (15), including the additional correction factor of 2 as discussed below equation (7). The fiducial parameters are ri = 2000 km, vs = 104 km s−1 , M = 1 M⊙, and ∆m(0) = 2 × 10−2 M⊙. The time origin is shifted to t = 0.2 s, corresponding to the onset of the tug-boat phase. while ∆m(t → ∞) ≈ ∆m(0). (20) Using the representative values discussed in Secti… view at source ↗

**Figure 3.** Figure 3: Schematic illustration of the rotating tug-boat model. (a) Side view in the x-z plane. The effective asymmetric point mass is located in polar angle θ, with the rotation axis aligned with the z-axis. Its radial position increases at a constant velocity vs . (b) Top-down view of the model.φ is the azimuthal angle in the x-y plane and is given by Equation (25). The perturbed mass spirals outward during the e… view at source ↗

**Figure 4.** Figure 4: Probability distribution of the spin-kick angle α for different values of β, calculated from Equation (31). The initial asymmetry direction is assumed to be isotropically distributed, so that p(θ) ∝ sin θ. 0 5 10 15 20 25 30 0.0 0.2 0.4 0.6 0.8 1.0 |sinc( /2)| 2 × 1 2 × 2 2 × 3 2 × 4 2 × 5 |sinc( /2)| = | sin( /2) /2 | [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 5.** Figure 5: Rotational-averaging factor |sinc(β/2)| = |sin(β/2)/(β/2)| as a function of β. The first zero occurs at β = 2π. subtantial spin-kick alignment. However, we must estimate β for real systems. The typical specific angular momentum of the iron core is inferred as j <∼ 1015 cm2 s −1 (Andresen et al. 2019). Neglecting angular momentum loss and magnetic redistribution, [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

read the original abstract

Neutron stars are often born with large recoil velocities, or natal kicks, whose physical origin remains an open question in core-collapse supernova theory. One possible mechanism is the gravitational tug-boat effect, in which anisotropic ejecta gravitationally accelerate the proto-neutron star over a timescale of seconds after shock revival. Observations suggest that the spin-kick angle distribution is not isotropic but skewed toward spin-kick alignment. Previous derivations of the tug-boat mechanism do not include the effect of initial stellar rotation. Here we derive a minimalist extension to assess how rotation of the expanding asymmetric mass distribution influences the spin-kick alignment. We show that the spin-kick angle is determined by the product of two factors, one that depends on the ratio of shock expansion time to the rotation period and the other which depends on the orientation of the asymmetric mass distribution with respect to the spin-axis. For fast enough rotation, the first factor amounts to axially averaging out non-axisymmetry thereby suppressing the perpendicular tug and leaving only a spin-aligned force. However, the rotation speed required for this effect would be unrealistically large unless magnetic fields could transport angular momentum from the core to the outflow efficiently. Otherwise, spin-kick alignment for the tug-boat mechanism would be more likely achieved via the second factor, namely for systems in which the mass flux asymmetry is itself preferentially spin-aligned.

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.

Desk Editor's Note private letter to a colleague

The paper factors spin-kick alignment in the tug-boat model into a rotation-period ratio term and an orientation term, with the first able to average out perpendicular kicks only at unrealistically high spins without magnetic transport.

read the letter

The central result is that the spin-kick angle factors as the product of a term set by the ratio of shock expansion time to rotation period and a term set by the orientation of the mass asymmetry relative to the spin axis. Fast rotation can axially average non-axisymmetry and suppress the perpendicular component, but only if the core spins fast enough, which the paper notes requires efficient magnetic angular-momentum transport to the outflow.

What is new is the explicit two-factor decomposition once rotation is added to the tug-boat setup. Prior work omitted rotation, so this dependence on the time ratio and the alignment of the asymmetry was not derived before. The paper does a clean job of stating the minimalist assumptions up front and flagging that the rotation speeds needed for the averaging effect are unrealistic without additional physics.

The soft spots are the usual ones for an analytic extension: the model neglects hydrodynamical back-reaction, neutrino transport, and non-axisymmetric instabilities beyond the minimal setup. There is no direct comparison to full simulations or to the observed kick-velocity distribution, so the practical weight of the two factors remains to be tested. The derivation itself appears straightforward from the abstract, with no obvious circularity or hidden fitting.

This is for people already working on analytic models of core-collapse kicks and spin-kick correlations. A reader who follows the tug-boat literature will see the new condition on rotation or asymmetry alignment and can use it to constrain future simulations. It is not a first-principles calculation of the kick, but it organizes an observational constraint inside an existing framework.

I would send it to peer review. The claim is modest, the limitations are stated, and the factoring gives modelers a concrete handle to check.

Referee Report

0 major / 2 minor

Summary. The manuscript derives an analytic factorization for the spin-kick angle in an extended gravitational tug-boat model that includes initial stellar rotation. The angle is expressed as the product of (i) a factor depending on the ratio of shock-expansion time to rotation period, which axially averages non-axisymmetric ejecta for sufficiently rapid rotation and thereby suppresses the perpendicular kick component, and (ii) a geometric factor set by the orientation of the asymmetric mass distribution relative to the spin axis. The authors conclude that realistic alignment would require either unrealistically high core rotation rates or efficient magnetic angular-momentum transport, or else a preferentially spin-aligned mass-flux asymmetry; the model is explicitly minimalist and neglects hydrodynamical back-reaction and non-axisymmetric instabilities.

Significance. If the derivation is correct, the result supplies a transparent, parameter-free analytic limit that isolates how rotation modulates the tug-boat kick and spin-kick alignment. The explicit statement of the minimalist assumptions and the unrealistic rotation speeds required without magnetic transport constitute a clear strength, providing a useful benchmark against which more complete simulations can be compared.

minor comments (2)

[Abstract / §3] The abstract states that the first factor 'amounts to axially averaging out non-axisymmetry' for fast rotation; an explicit equation or short derivation of this averaging in the main text would make the factorization immediately verifiable.
[Discussion] The manuscript flags that the required rotation speeds are 'unrealistically large' without magnetic transport; a brief numerical estimate of the critical period (in ms) relative to typical core-collapse timescales would strengthen the claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the supportive summary, recognition of the analytic result as a useful benchmark, and recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper presents an analytical derivation extending the tug-boat model to include rotation, showing the spin-kick angle factors as the product of a shock-expansion-to-rotation-period ratio term and an orientation term. No equations, steps, or claims reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations. The result follows from the stated minimalist assumptions without renaming known results or smuggling ansatzes. This is the normal case of a self-contained first-principles extension.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. No free parameters or invented entities are mentioned. The work rests on standard domain assumptions about the tug-boat timescale and observed spin-kick statistics.

axioms (2)

domain assumption The gravitational tug-boat mechanism operates over a timescale of seconds after shock revival
Stated directly in the abstract as the physical basis for the kick.
domain assumption Observations indicate that the spin-kick angle distribution is skewed toward alignment
Used to motivate the need for an alignment mechanism.

pith-pipeline@v0.9.1-grok · 5784 in / 1482 out tokens · 43130 ms · 2026-06-26T23:57:17.229858+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

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doi: 10.1051/0004-6361: 20064855. Swapnil Shankar, Philipp M ¨osta, Roland Haas, and Erik Schnetter. 3d full- gr simulations of magnetorotational core-collapse supernovae on gpus: a systematic study of rotation rates and magnetic fields.Monthly Notices of the Royal Astronomical Society, 548(2):stag646, 05

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doi: 10.1038/30168. A. Wongwathanarat, H.-Th. Janka, and E. M¨uller. Three-dimensional neutrino- driven supernovae: Neutron star kicks, spins, and asymmetric ejection of nucleosynthesis products.A&A, 552:A126, April

work page doi:10.1038/30168

[1] [1]

doi: 10.1093/mnras/stz990. H. Andresen, R. Glas, and H.-Th Janka. Gravitational-wave signals from 3D supernova simulations with different neutrino-transport methods.MNRAS, 503(3):3552–3567, May

work page doi:10.1093/mnras/stz990

[2] [2]

doi: 10.1093/mnras/stab675. Z. Arzoumanian, D. F. Chernoff, and J. M. Cordes. The Velocity Distribution of Isolated Radio Pulsars.ApJ, 568(1):289–301, March

work page doi:10.1093/mnras/stab675

[3] [3]

doi: 10.1017/pasa.2025.10069. John M. Blondin and Samantha Shaw. Linear Growth of Spiral SASI Modes in Core-Collapse Supernovae.ApJ, 656(1):366–371, February

work page doi:10.1017/pasa.2025.10069 2025

[4] [4]

doi: 10.1086/510614. John M. Blondin, Anthony Mezzacappa, and Christine DeMarino. Stability of Standing Accretion Shocks, with an Eye toward Core-Collapse Supernovae. ApJ, 584(2):971–980, February

work page doi:10.1086/510614

[5] [5]

doi: 10.1086/345812. G. Hobbs, D. R. Lorimer, A. G. Lyne, and M. Kramer. A statistical study of 233 pulsar proper motions.MNRAS, 360(3):974–992, July

work page doi:10.1086/345812

[6] [6]

doi: 10.1111/j.1365-2966.2005.09087.x. H.-T. Janka and E. Mueller. Neutron star recoils from anisotropic supernovae. A&A, 290:496–502, October

work page doi:10.1111/j.1365-2966.2005.09087.x 2005

[7] [7]

2012.21704.x

doi: 10.1111/j.1365-2966. 2005.09669.x. R´emi Kazeroni, J ´erˆome Guilet, and Thierry Foglizzo. Are pulsars spun up or down by SASI spiral modes?MNRAS, 471(1):914–925, October

work page doi:10.1111/j.1365-2966 2005

[8] [8]

doi: 10.1093/mnras/stx1566. Ilya A. Kondratyev, Sergey G. Moiseenko, and Gennady S. Bisnovatyi-Kogan. Magnetorotational neutron star kicks.Phys. Rev. D, 110:083025, Oct

work page doi:10.1093/mnras/stx1566

[9] [9]

URL https://link.aps.org/doi/10.1103/ PhysRevD.110.083025

doi: 10.1103/PhysRevD.110.083025. URL https://link.aps.org/doi/10.1103/ PhysRevD.110.083025. Dong Lai and Yong-Zhong Qian. Neutrino Transport in Strongly Magnetized Proto-Neutron Stars and the Origin of Pulsar Kicks: The Effect of Asymmetric Magnetic Field Topology.ApJ, 505(2):844–853, October

work page doi:10.1103/physrevd.110.083025

[10] [10]

doi: 10.1086/306203. C.-Y . Ng and Roger W. Romani. Birth Kick Distributions and the Spin-Kick Correlation of Young Pulsars.ApJ, 660(2):1357–1374, May

work page doi:10.1086/306203

[11] [11]

doi: 10.1086/513597. J. Nordhaus, T. D. Brandt, A. Burrows, and A. Almgren. The hydrodynamic origin of neutron star kicks.MNRAS, 423(2):1805–1812, June

work page doi:10.1086/513597

[12] [12]

doi: 10.1111/j.1365-2966.2012.21002.x. A. Noutsos, M. Kramer, P. Carr, and S. Johnston. Pulsar spin-velocity align- ment: further results and discussion.MNRAS, 423(3):2736–2752, July

work page doi:10.1111/j.1365-2966.2012.21002.x 2012

[13] [13]

doi: 10.1111/j.1365-2966.2012.20768.x

doi: 10.1111/j.1365-2966.2012.21083.x. Jade Powell, Bernhard M ¨uller, David R. Aguilera-Dena, and Norbert Langer. Three dimensional magnetorotational core-collapse supernova explosions of a 39 solar mass progenitor star.MNRAS, 522(4):6070–6086, July

work page doi:10.1111/j.1365-2966.2012.21083.x 2012

[14] [14]

doi: 10.1093/mnras/stad1292. L. Scheck, K. Kifonidis, H.-Th. Janka, and E. M¨uller. Multidimensional super- nova simulations with approximative neutrino transport. I. Neutron star kicks and the anisotropy of neutrino-driven explosions in two spatial dimen- sions.A&A, 457(3):963–986, October

work page doi:10.1093/mnras/stad1292

[15] [15]

Swapnil Shankar, Philipp M ¨osta, Roland Haas, and Erik Schnetter

doi: 10.1051/0004-6361: 20064855. Swapnil Shankar, Philipp M ¨osta, Roland Haas, and Erik Schnetter. 3d full- gr simulations of magnetorotational core-collapse supernovae on gpus: a systematic study of rotation rates and magnetic fields.Monthly Notices of the Royal Astronomical Society, 548(2):stag646, 05

work page doi:10.1051/0004-6361:

[16] [16]

doi: 10.1093/mnras/stag646

ISSN 0035-8711. doi: 10.1093/mnras/stag646. URL https://doi.org/10.1093/mnras/stag646. H. Spruit and E. S. Phinney. Birth kicks as the origin of pulsar rotation.Nature, 393(6681):139–141, May

work page doi:10.1093/mnras/stag646

[17] [17]

doi: 10.1038/30168. A. Wongwathanarat, H.-Th. Janka, and E. M¨uller. Three-dimensional neutrino- driven supernovae: Neutron star kicks, spins, and asymmetric ejection of nucleosynthesis products.A&A, 552:A126, April

work page doi:10.1038/30168