Spiral Morphology and Radial Migration: Kinematically heating, cooling, and cold

Amy Smock; Karen L. Masters; Kathryne J. Daniel; Lipika Chatur; M. E. Wisz; Rosemary F. G. Wyse

REVIEW 4 major objections 5 minor 165 references

Spiral Morphology and Radial Migration: Kinematically heating, cooling, and cold

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper claims that a rigidly rotating density-wave spiral torques trapped stars progressively farther as its pitch angle increases, that a corotating winding spiral shows the opposite trend, and that this asymmetry makes pitch angle a…

desk verdict The headline cotθ scaling does not follow from the paper's own Equation 32; the simulations are useful but the central analytic claim needs to be reworked. read the letter →

arxiv 2608.10097 v1 pith:DGVUKB3Z submitted 2026-08-10 astro-ph.GA

Kathryne J. Daniel , M. E. Wisz , Karen L. Masters , Rosemary F. G. Wyse , Amy Smock , Lipika Chatur This is my paper · ORCID

classification astro-ph.GA

keywords spiralstructureradialmigrationcoldtorquingcorotationresonancepitchangledensitywavetheorykinematicheatingwinding

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks what actually sets how far transient spiral arms can relocate disk stars through cold torquing, the resonant process that changes a star's orbital radius without heating its random motions. It derives an analytic scaling for the maximum radial excursion of stars trapped at corotation and finds that more open rigidly rotating spirals, those with larger pitch angle, should move stars farther. Tracer-particle simulations in two and three dimensions confirm this trend. The same formalism applied to a corotating winding spiral predicts and shows the opposite: cold torquing becomes more efficient as the pattern shears to smaller pitch angles. A sympathetic reader would care because the result turns pitch angle into an observable test of what spiral arms actually are, and it shows that older bar-based estimates overstate how far a single spiral episode can move stars like the Sun.

What carries the argument

The central object is the logarithmic spiral perturbation $\Phi_{1,s}(R,\phi,t) = \Phi_s \cos[\alpha\ln(R/R_{\rm CR}) + m\Omega_p t - m\phi]$ with $\alpha = m\cot\theta$, whose amplitude $\Phi_s$ depends on pitch angle through the radial wavenumber $k = m\cot\theta/R$. Around corotation, trapped orbits are described by an assumed azimuthal libration $\phi_1(t) = |\phi_1|\cos(\omega t + \delta)$ with maximum amplitude $2\pi/m$; substituting this into the equations of motion yields the radial equation of motion and, at the libration phase $\tau = \pi/(2m)$, the approximate maximum excursion used throughout. That expression carries the argument because it converts spiral morphology, arm number, and pattern speed into a quantitative prediction $\max(\Delta R_g) \propto |\Phi_s(\theta)|\cot\theta$ that the tracer-particle experiments are designed to verify.

What would settle it

Compute the exact libration amplitude and frequency for orbits trapped at corotation in a logarithmic spiral potential by direct numerical integration, then check whether the maximum guiding-centre change follows $\max(\Delta R_g) \propto |\Phi_s(\theta)|\cot\theta$ across pitch angles such as 10, 20, 30, and 40 degrees; a clear departure from that scaling would falsify the central claim.

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Extended reading notes

Core claim

For a rigidly rotating density-wave-like spiral with $m$-fold symmetry and pitch angle $\theta$, stars trapped at corotation librate about the stable maxima of the effective potential in the rotating frame. Solving the linearized equations of motion gives a maximum first-order radial excursion $|R_{1,s}| \approx (\Phi_s/\kappa^2 R_0)[m\cot\theta\,\sin(1/\sqrt{2}m) + 2\sqrt{2}\pi^2 \Omega_0/\omega]$, so the largest change in guiding-centre radius is $\max(\Delta R_g) = 2\max(|R_{1,s}|) \propto |\Phi_s(\theta)|\cot\theta$. More open arms are therefore stronger radial migrators at fixed spiral strength. When the pattern speed instead equals the local circular frequency at every radius, a winding sheared spiral, corotation exists everywhere and the simulations show the rms change in angular momentum growing as the pitch angle decreases, reversing the density-wave trend. The same transient spiral also heats orbits at resonances away from corotation and circularizes a minority of orbits at the inner Lindblad resonance, so a single spiral passage produces cold torquing plus net kinematic heating alongside some resonant cooling.

Load-bearing premise

The entire analytic prediction hangs on the assumed shape of a trapped star's azimuthal libration, a cosine with amplitude $2\pi/m$ and a libration frequency $\omega$ that is not derived from the potential, so if real trapped orbits librate differently, the $\cot\theta$ scaling need not hold.

Editorial extensions

If this is right

For a rigidly rotating spiral of fixed strength, more open patterns produce larger radial changes in the guiding centres of trapped stars, and the 2D and 3D simulations confirm that rms$(\Delta J_\phi)$ grows with pitch angle while kinematic heating stays minimal.
Estimates that use the older bar-symmetry formula $\max(R) \propto \sqrt{|\Phi_b|}$ overstate the maximum radial migration from a transient spiral episode by roughly a factor of two or more for typical pitch angles.
At observed Milky Way spiral amplitudes and pitch angles, cold-torquing excursions are comparable to or smaller than ordinary epicyclic excursions, so a single transient rigid spiral episode moves the Sun at most about 2 kpc rather than the several kiloparsecs a bar formula would suggest.
A winding spiral that corotates everywhere shows the opposite trend: cold torquing becomes more efficient as the pattern shears to smaller pitch angles, with nearly zero kinematic heating.
Spiral morphology alone cannot predict cold-torquing efficiency, so the relation between pitch angle and radial redistribution can distinguish rigid density-wave spirals from winding, corotating spirals in observations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: measuring radial migration strength or azimuthal metallicity variations as a function of pitch angle in galaxy samples could separate density-wave from winding-spiral behaviour, provided an independent constraint on the pattern-speed nature is available.
Editorial inference: if the $\cot\theta$ scaling survives in fully self-consistent spiral simulations, the burden for Milky Way radial migration shifts toward repeated generations of transient patterns rather than one strong episode.
Editorial inference: the assumed cosine form for azimuthal libration with amplitude $2\pi/m$ and an undetermined frequency $\omega$ is the step most worth testing; comparing the analytic maximum against numerically computed libration trajectories would either confirm or falsify the central scaling.
Editorial inference: the resonant-cooling result suggests that old, metal-rich, nearly circular orbits near the Sun should be interpreted not only as accretion relics but also as stars circularised by transient spiral resonances.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's the take: the paper's headline analytic scaling doesn't survive contact with its own equations. I checked the second-pass stress-test and it lands. Equation 32 is the basis for max(ΔRg) ∝ |Φs(θ)| cotθ (Eq. 64), but with Φs ∝ tanθ from Equations 17–18, the first term in Equation 32 is independent of θ and the second is ∝ tanθ. So the red curve in Figure 5 is driven by the amplitude term, not by the geometric cotθ factor. The text also says Equation 32 adopts τ=π/2m, but the coefficients are those of τ=π/4; there are additional factor-of-π inconsistencies between Equation A9 and Equation 30. This is a load-bearing problem, not a typo, because the paper's central result is the cotθ scaling.

That said, there is real value here. The paper is the first to write explicit pitch-angle-dependent expressions for maximum radial excursions for a Lin-Shu spiral, and the reversal for corotating/winding spirals is genuinely new and worth taking seriously. The tracer-particle simulations are competently constructed, the 3D Cox–Gómez implementation is useful, and the discussion of resonant kinematic heating and cooling is careful. The paper is clearly written and engaged with the literature.

The soft spots are proportional to the above. The 2D simulations use the same Lin-Shu potential as the derivation, so they cannot independently validate the cotθ factor; they are equally consistent with the amplitude effect Φs ∝ tanθ. The 3D model is more independent, but there are no error bars and it does not directly test Equation 64. The libration frequency ω in Equation 32 is left unspecified, and §3.2.2 is ambiguous about whether ω is fitted. Minor: the conclusions say the bar-symmetry estimate overestimates by 'many times' and later by 'about 50 percent.'

Recommendation: send to peer review. The numerical trends and the winding-spiral reversal deserve serious referee time, and the analytic issues are fixable, but the authors need to redo the algebra and reframe the scaling claim before this is publishable as is.

Referee Report

4 major / 5 minor

Summary. The paper studies how the morphology of spiral arms—pitch angle, arm number, lifetime, and radial dependence of pattern speed—controls the efficiency of 'cold torquing' at corotation. It derives an analytic expression for the maximum radial excursion of stars trapped at the corotation resonance of a Lin-Shu spiral, Eq. (32), and states in Eq. (64) that max(ΔRg) ∝ |Φs(θ)| cotθ. This leads to the headline prediction that rigidly rotating, density-wave-like spirals torque stars more efficiently when they are more open, while corotating winding spirals show the opposite trend. The paper tests the prediction with 2D and 3D tracer-particle simulations, reports that cold torquing increases with pitch angle for fixed fractional amplitude, and demonstrates a trend reversal for a winding spiral. It also discusses resonant kinematic heating and cooling and observational diagnostics.

Significance. If the central scaling were established, this would be a useful diagnostic for distinguishing density-wave from winding-spiral theories and would correct earlier bar-based estimates of cold-torquing efficiency. The paper has real strengths: it addresses an important question, includes 2D and 3D tracer simulations, uses an independent Cox-Gómez spiral potential in 3D, and explicitly contrasts rigid and winding spiral models. However, the central quantitative claim is undermined by an algebraic inconsistency in Eqs. (32)–(64), and the 2D confirmation shares the same Lin-Shu potential, so the paper's main new quantitative result is not currently supported. The qualitative trend may survive, but the claimed cotθ scaling needs to be re-derived and tested with a controlled numerical experiment.

major comments (4)

[§2.3.4, Eqs. (32) and (64)] The claimed scaling is not what Eq. (32) gives. With the Lin-Shu amplitude, Φs = 2πGΣ εΣ / k and k = m cotθ/R0 (Eqs. 17–18), so Φs ∝ tanθ. Substituting into Eq. (32), the first term in the bracket is ∝ cotθ, making its product with Φs θ-independent, while the second term is proportional to tanθ. Thus |R1,s| behaves as a constant plus a term ∝ tanθ, not as |Φs(θ)| cotθ; the claimed quantity |Φs(θ)| cotθ is actually constant in θ. The increasing red curve in Fig. 5 therefore reflects the amplitude variation Φs ∝ tanθ, not a geometric cotθ enhancement. In addition, the coefficients in Eq. (32) correspond to τ = π/4, not τ = π/(2m) as stated in the text; for m = 4 the stated choice gives cos(π/8) = 0.924 and sin(π/8) = 0.383, not the 1/√2 and 1/√2 used in Eq. (32). This needs to be corrected and the consequences for Eq. (64) and the abstract must be addressed.
[§2.3.3, Eq. (29)] The derivation assumes a specific azimuthal libration φ1(t) = |φ1| cos(ωt + δ) with |φ1| = 2π/m and an unspecified libration frequency ω. This assumption enters directly into Eq. (30) and the final amplitude Eq. (32). The paper gives no derivation of this form from the effective potential near L4/L5 and no check that numerically trapped orbits satisfy it for moderate pitch angles. If the amplitude, phasing, or θ-dependence of ω differs, the predicted scaling changes; since ω is a free parameter, Eq. (32) is not a closed prediction until ω is specified or shown not to matter for the trend.
[§3.2.2] The 2D simulation confirmation is not an independent test of the cotθ dependence. The simulations adopt the same Lin-Shu spiral potential used in the analytic derivation, and they hold εΣ fixed, so Φs ∝ tanθ varies with pitch angle. The measured increase of rms(ΔLz) with θ is therefore consistent with the amplitude scaling alone, without any geometric cotθ effect. The authors should run a control with Φs at R_CR held fixed across pitch angles, or otherwise separate the amplitude contribution from the shape contribution. This is essential because Eq. (64) and the abstract's emphasis on 'more open spiral patterns' are about the geometric dependence.
[§3.2.2, 'best matched'] The statement that the rms(ΔLz) values are 'best matched' to Eq. (32) is not supported by an explicit procedure. No fitting metric, parameter values, or comparison of the θ-dependence is given, and since Eq. (32) contains the unspecified ω, the match is ambiguous. Please provide the quantitative comparison, for example predicted versus measured peak values and slopes, or remove the claim.

minor comments (5)

[§1] The text contains a duplicated word: 'without without kinematically heating a stellar population'.
[Fig. 3 caption] The caption lists pitch angles θ = {5°, 15°, 30°, 45°} while the surrounding text says θ = {5°, 15°, 25°, 35°}; these should be aligned.
[Fig. 4 and §2.3.4] Figure 4 describes the vertical line as purple while §2.3.4 says gray, dotted; the color reference should be consistent.
[§3.4, Eq. (39)] The winding potential wrapper is described only by a phase shift; please state explicitly how the pitch angle evolves with time and what value of ϕ0 is used at t_ref.
[§3.2.1, Eq. (49)] The harmonic indexing is unclear: the text says n = 0 gives the I/OLRs but then refers to n = 1 as the first harmonics; please clarify the indexing convention.

Circularity Check

2 steps flagged · score 4.0 of 10

Claimed cotθ scaling cancels under the paper's own Lin-Shu definitions; 2D confirmation uses the same input potential.

other [Eqs. (17)-(18), (32), (34) in §2.3.4; Eq. (64) in §5]
"The amplitude of the perturbing spiral potential is given by, Φs(R) = 2πGΣ(R)ϵΣ/k ... k = α/R = mcotθ/R0. ... |R1,s| ≈ Φs/κ²R0 ( mcotθ sin(1/(√2m)) + 2√2π² Ω0/ω ). ... max(ΔRg) = 2 max(|R1,s|) ∝ |Φs(θ)|cotθ."

Substituting Φs=2πGΣϵΣ/k and k=mcotθ/R0 into Eq. (32), the first bracket term becomes 2πGΣϵΣ m sin(1/(√2m))/(κ²R0), which is independent of θ, while the second term is proportional to tanθ. Thus Eq. (32) predicts |R1,s| ≈ const + C·tanθ, not ∝|Φs|cotθ; and since Φs∝tanθ, the product |Φs|cotθ is itself constant. The headline scaling Eq. (64) therefore does not follow from the derivation; the monotonic increase of the plotted |R1,s| is carried by the amplitude variation Φs∝tanθ built into the Lin-Shu ansatz, not by the claimed cotθ geometry. In addition, Eq. (32) is said to adopt τ=π/(2m), but its coefficients correspond to τ=π/4, so the 'maximum' expression is not the stated phase.
self definitional [§3.2 and §3.2.2 (Eqs. 16, 37, 38)]
"The adopted underlying axisymmetric potential has the form Φ0=v_c² ln(R/Rp). ... The perturbing potential, Φ1(R,ϕ,t), takes the form described in Equation 37 where the time-independent amplitude for a Lin-Shu spiral potential is given by Equation 16. ... The only condition modified for each simulation within the 2D and 3D suites is the assigned spiral pitch angle, which takes a value θ={10°,20°,30°,40°}."

The 2D tracer experiments are constructed with the same Lin-Shu spiral potential (Eq. 16) and the same amplitude prescription (Eqs. 37-38) used in the analytic derivation, with ϵΣ held fixed. Because in that prescription Φs∝tanθ, the measured increase of rms(ΔLz) with θ inherits the assumed amplitude-pitch-angle dependence; the experiment is a self-consistency check of the algebra rather than an independent validation of the cotθ factor. By construction it cannot falsify the input potential's θ-dependence.

full rationale

The paper's central qualitative contrast — rigidly rotating spirals torque more efficiently when more open, while winding spirals torque more efficiently as they wind — has independent content: the Cox & Gómez 3D potential and the winding-spiral experiment do not assume the Lin-Shu form used in the analytic section, and the trend reversal is a genuine dynamical result. Self-citations (Daniel & Wyse 2015; Daniel et al. 2019; Smock et al. 2026) are contextual and not load-bearing for the central derivation. However, the quantitative prediction Eq. (64) is not actually obtained from Eq. (32): under the paper's own definitions (Eqs. 17-18), the claimed ∝|Φs|cotθ factor cancels to a constant in the first term and leaves only the input amplitude ∝tanθ in the second; the 2D 'confirmation' uses the same Lin-Shu potential and so partly confirms an input assumption. These are partial, construction-level reductions of the central prediction, but not a self-citation chain and not a full equivalence, so the score is 4.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the Lin-Shu density wave model, the weak-perturbation epicyclic framework, and a specific assumed libration waveform with a free frequency ω. No new physical entities are introduced.

free parameters (2)

libration frequency ω = not stated
Introduced in Eq. (30) as the frequency of oscillation about L4/5. It is not derived from the potential or listed for the models. The 'best matched' comparison in §3.2.2 leaves open whether ω was measured from the simulations or treated as a fitting parameter, which affects the quantitative predictive power of Eq. (32).
spiral fractional amplitude ε_Σ = 0.1, 0.2, 0.3 in the 2D model; set by the 3D model parameters
Chosen by hand to span observed spiral arm strengths (Elmegreen et al. 2011; Eilers et al. 2020). It sets the amplitude of Φs in Eq. (17) and is held fixed when varying pitch angle to isolate the morphological dependence. It is an input, not a fitted nuisance.

assumptions (6)

domain assumption The perturbation is weak (|Φ1/Φ0| ≪ 1) and the underlying disk is axisymmetric, so linear epicyclic theory applies.
Invoked in §2.1 following Binney and Tremaine (2008); the entire analytic treatment inherits this from the bar case.
domain assumption The spiral perturbation has the Lin-Shu form (Eq. 16) with radially independent pattern speed Ωp and constant pitch angle θ.
Adopted in §2.3.1; the authors state it is chosen for analytic accessibility and assert that spirals with peak density at corotation should follow similar scalings.
ad hoc to paper Trapped stars librate azimuthally as φ1(t) = |φ1| cos(ωt + δ) with |φ1| = 2π/m.
Stated in Eq. (29). This waveform is the key approximation enabling the closed-form Eq. (30); no justification is given for the amplitude 2π/m.
ad hoc to paper For θ ≲ 30 degrees, the maximum of Eq. (31) occurs at τ = π/2m.
Section 2.3.4 uses this to reduce Eq. (31) to Eq. (32). It is not proved for the full range of pitch angles.
domain assumption The torque determining radial excursions is dominated by the spiral amplitude at corotation.
Section 2.3.1(d) states this as a design choice of the Lin-Shu model.
domain assumption For the winding spiral model, the pattern corotates with the disk at all radii (Ωp(R) = Ω(R)) and winds up due to differential rotation.
Section 3.4; implemented via galpy's CorotatingRotationWrapperPotential (Eq. 39). It is a separate physical model, not a limit of the density wave prescription.

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Cite this review

Pith. "Pith review of Spiral Morphology and Radial Migration: Kinematically heating, cooling, and cold." pith.science (2026). https://pith.science/paper/DGVUKB3Z

@misc{pith2026260810097,
  author       = {Pith},
  title        = {Pith review of: Spiral Morphology and Radial Migration: Kinematically heating, cooling, and cold},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DGVUKB3Z}},
  note         = {Machine review of arXiv:2608.10097}
}

read the original abstract

Transient spiral arms are known to drive radial redistribution of stars and thus could play a central role in shaping disk galaxies, including modifying, over time, the age, chemical, and kinematic (chrono-chemo-dynamic) distributions in the Milky Way. However, the physical factors governing the efficiency of such processes remain poorly understood. This paper investigates how the morphology of spiral arms -- the number, pitch angle, lifetime, and radial dependence of the pattern speed -- influences orbital redistribution through 'cold torquing' at the corotation resonance(s). Analytic expressions are derived for the maximum radial excursion of stars trapped at corotation that explicitly account for spiral morphology, predicting that the efficiency of cold torquing for a density-wave like spiral is greater for more open spiral patterns. Tracer-particle simulations confirm the analytic prediction, in both two- and three-dimensional galactic potentials. In contrast, spirals that have a radially dependent pattern speed such that they corotate with the disk at all radii exhibit the opposite behavior, with cold torquing becoming more efficient as the spiral winds to smaller pitch angles over time. This study further finds that resonant interactions from the same transient spiral causing cold torquing naturally also produces both kinematic heating and cooling of orbits away from corotation. These results demonstrate that spiral morphology alone cannot predict the efficiency of cold torquing and suggest that the relationship between spiral pitch angle and radial redistribution provides a potential diagnostic for distinguishing between competing theories of spiral structure.

Figures

Figures reproduced from arXiv: 2608.10097 by the authors.

**Figure 1.** Illustration showing the effective potential, Φb,eff , of a bar. The bar is horizontally oriented and contours are normalized by the constant circular velocity, vc. The stable local maxima (L4 and L5, circles), unstable saddle points (L1 and L2, plus signs), and stable global minimum (L3, square) are labeled. be generalized to describe motions about the L5, or any equivalent local maximum from a given weak pattern w… view at source ↗

**Figure 2.** The effective potential of a spiral pattern, Φs,eff , with pitch angle θ = 25◦ . A comparison with the effective potential for a bar in [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Illustration showing the invariant curves Φs,eff = Φs,eff (L1) for an m = 4 spiral pattern (Equation 16) in a Mestel disk. Positions ϕ = {π/4, 3π/4, 5π/4, 7π/4} correspond to the unstable L1/2 saddle points in Φs,eff (separatrixes) and are the location of the spiral arms at RCR = 8 kpc. The stable L4/5 local maxima (red dot) have azimuthal positions ϕ = {0, π/2, π, π/2} and are the points around which stars trapped … view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: Comparison of and total for the contributing terms for radial excursions from L4/5 from Equation 30, normalized by the radius of corotation, RCR = R0. These are the term associated with torque from the spiral pattern (light blue, dotted) and the radial distortion term …

**Figure 5.** Figure 5: Various solutions for the maximum radial excursions of a trapped orbit from the radius of corotation, RCR = R0. The model shown is for a Mestel disk with an m = 4 spiral pattern and fractional amplitude ϵΣ. Maximum radial excursions increase with increasing pitch angle…

**Figure 6.** Figure 6: Time evolution of the rms (∆Lz) (top) and rms (∆Enc) (bottom) for populations of orbits meeting resonant criteria with spiral patterns that have pitch angle θ = 10◦ (left), 20◦ (middle), and 30◦ (right). Units are normalized by models values in the underlying axisymmet…

**Figure 7.** Figure 7: Time evolution of rms (∆Lz) from cold torquing and rms (∆Enc) (fractional kinematic heating) for ‘trapped at corotation only’ populations in models with various pitch angles, θ. Units are normalized by models values in the underlying axisymmetric disk at the RCR and th…

**Figure 8.** Figure 8: Time evolution of rms (∆Jϕ) (degree of cold torquing) and rms (∆JR) (kinematic heating) for ‘trapped at corotation only’ populations in models with various pitch angles, θ. Units are normalized by Jϕ,CR, the value of Jϕ at RCR in the underlying axisymmetric disk. The v…

**Figure 9.** Figure 9: Summarizing plot showing the near linear increase in the degree of cold torquing (increasing value for rms (∆Jϕ) with constant rms (∆JR) ∼ 0) for ‘trapped at corotation only’ orbits with increasing pitch angle, θ. 4. DISCUSSION 4.1. Observational Constraints Spiral st…

**Figure 10.** Figure 10: Time evolution of cold torquing from a winding spiral potential. The left panel shows rms (∆Jϕ) and the right shows rms (∆JR). The vertical axes are normalized by the value of Jϕ at RCR in the smooth underlying disk Jϕ,CR and the x-axis is normalized by the orbital pe…

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