Graze-and-Merge Collisions under External Perturbers

Alexandre Emsenhuber; Erik Asphaug

REVIEW 4 major objections 6 minor 26 references

Graze-and-Merge Collisions under External Perturbers

T0 review · 4 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The outcome of a graze-and-merge collision around a star or planet is set by the farthest point of the post-impact orbit, scaled to the Hill radius, with a change in behavior near one-third and three-quarters of that radius.

desk verdict Useful quantitative thresholds for when a third body disrupts graze-and-merge collisions; the Titan application is plausible but sits exactly at the point where their separability assumption is thinnest. read the letter →

arxiv 1908.07557 v1 pith:VUCUKJ64 submitted 2019-08-20 astro-ph.EP

Alexandre Emsenhuber , Erik Asphaug This is my paper · ORCID

classification astro-ph.EP

keywords graze-and-mergecollisionsgiantimpactsHillradiushit-and-runplanetformationsatelliteMonteCarloN-bodysimulationsaccretion

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 when the standard picture of a graze-and-merge collision—two similar-sized bodies collide, the smaller one slows, loops back, and merges after one orbit—still holds when a third massive body such as a star or planet is present. Using large Monte Carlo sets of point-mass orbital evolutions with randomized orientations, it finds that the controlling quantity is the apocenter of the post-impact orbit, measured in units of the Hill radius of the larger remnant. Inside roughly one-third of the Hill radius the usual one-orbit merger picture is preserved; between one-third and three-quarters the second collision becomes increasingly random in angle and velocity and retains little memory of the first impact; beyond about three-quarters a growing fraction of pairs never re-collide and become unbound. The behavior is scale-invariant: the same thresholds apply to planets around a star and satellites around a planet, with only a mild correction for the size of the bodies compared with their Hill sphere. This matters because graze-and-merge collisions are a common accretion channel in giant impacts, and currently used accretion scaling laws treat such pairs as unconditional mergers.

What carries the argument

The object that carries the argument is the scaled apocenter distance $\tilde{r}_{\rm apo}=r_{\rm apo}/r_H$, the farthest separation reached by the two post-impact remnants divided by the radius of the larger body's Hill sphere. The machinery is a staged decomposition of a graze-and-merge collision: the first impact is treated as a hydrodynamical event whose outcome fixes a point-mass orbit, then a hybrid symplectic N-body integrator with direct close-encounter handling evolves that orbit under the central body for 10,000 randomly oriented realisations per configuration, and the second collision is sampled from the resulting encounters. The authors anchor the decomposition by checking one SPH simulation, finding that the relative motion is well described by two-body dynamics once the remnants separate beyond about six mutual radii. A period-ratio estimate, $T_2/T_1\approx\sqrt{\tilde{r}_{\rm apo}^3/24}$, shows that the relative motion of the central body during the loop-back depends only on orientation and $\tilde{r}_{\rm apo}$, which motivates the scale invariance and the use of the Hill radius as the natural yardstick.

What would settle it

Run the paper's Monte Carlo setup with the post-impact apocenter fixed at $\tilde{r}_{\rm apo}=0.5$ but with the target moved from 1 AU to 0.5 AU (or the body radii changed so $\tilde{r}_{\rm coll}$ stays fixed); if the cumulative distribution of delays between the two collisions differs measurably from the paper's $\tilde{r}_{\rm apo}=0.5$ curve, the claimed scale invariance in $\tilde{r}_{\rm apo}$ is wrong. A full SPH simulation with the same first-impact outcome and an explicit third body would also settle whether the point-mass approximation between collisions holds at apocenters in the $0.3$\textendash$0.75$ Hill-radius range.

Watch

Extended reading notes

Core claim

The central claim is that the outcome of a graze-and-merge collision around a third body is governed primarily by the scaled apocenter distance $\tilde{r}_{\rm apo}=r_{\rm apo}/r_H$, the farthest point of the loop-back orbit divided by the Hill radius. The paper's Monte Carlo integrations show that if $\tilde{r}_{\rm apo}\lesssim 0.3$, every realisation returns and collides again after a single orbit, with the second impact nearly aligned with the first. From about one-third to three-quarters of the Hill radius, the central body perturbs the orbit enough that a growing share of realisations miss on the first return, and the distributions of return velocity, impact angle, and orbital-plane alignment drift toward what would be expected for randomly oriented encounters; at $\tilde{r}_{\rm apo}=1.0$ the impact-angle distribution resembles but is not identical to the uniform one, while in the parabolic limit it is statistically indistinguishable from it. Beyond about $\tilde{r}_{\rm apo}\simeq 0.75$, most pairs no longer follow the first impact's geometry and some become unbound, so the process behaves more like a hit-and-run collision than a merger. The paper argues the dependence is scale-free, with a secondary correction encoded by the approximate relation $k\approx \tilde{r}_{\rm apo}-0.2\log_{10}(\tilde{r}_{\rm coll})$, where $\tilde{r}_{\rm coll}=r_{\rm coll}/r_H$, so results obtained for an Earth-like target at 1 AU transfer, within the model's assumptions, to Titan around Saturn and other systems.

Load-bearing premise

The load-bearing premise is that a graze-and-merge collision can be split into independent stages, with the interval between the two collisions governed purely by point-mass gravity; this was checked with only one SPH simulation, and the authors concede it may fail when the Hill sphere is not much larger than the colliding bodies.

Editorial extensions

If this is right

Graze-and-merge collisions with post-impact apocenters inside about $0.3\,r_H$ can be modelled as isolated two-body mergers without meaningful error.
For apocenters between roughly $0.3\,r_H$ and $0.75\,r_H$, the second impact must be treated as a random or distribution-sampled event, not as a deterministic consequence of the first collision.
Accretion scaling laws that count every bound post-impact pair as a merger overestimate growth when the loop-back orbit approaches or exceeds the Hill radius, because a substantial fraction of those pairs either collide only after long chaotic delays or become unbound.
The Titan moon-formation scenario in which Saturn's middle-sized moons are produced during Titan's accretion is the most affected application: the modelled grazing collisions reach about half of Titan's Hill radius, so Saturn's tides make them hit-and-run events rather than mergers, which reshapes the predicted moon-forming mass budget.
N-body planet-formation codes should reinsert the bound but widely ranging remnants of a first impact into the integration, treating the graze-and-merge to hit-and-run transition as smooth rather than as a yes/no accretion criterion.

Reading between the lines

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

Because the paper fixes the first impact angle at $60^\circ$, a natural extension is to vary impact angle and check whether the outcome map indeed shifts with $\tilde{r}_{\rm coll}$ as the eccentricity argument suggests: more grazing angles should behave like effectively larger $\tilde{r}_{\rm coll}$, and more head-on angles like smaller ones.
The two-variable scaling $k\approx \tilde{r}_{\rm apo}-0.2\log_{10}(\tilde{r}_{\rm coll})$ is presented as a rough fit; a higher-resolution survey of the $\tilde{r}_{\rm apo}$–$\tilde{r}_{\rm coll}$ plane could test whether a single contour of $k$ really collapses the return-time, impact-angle, and alignment distributions, or whether the fit underestimates early returns at low $\tilde{r}_{\rm apo}
The same Hill-scaled apocenter criterion could be applied to satellite-accretion and binary-asteroid settings, since the scale-invariance argument is general, but the validation rests on a single SPH simulation, so the transfer of thresholds to settings where the Hill sphere is only a few body radii wide remains to be verified.
A practical follow-up would be to convert the return-time and impact-property distributions into a simple parametrized return-probability function of $\tilde{r}_{\rm apo}$ and $\tilde{r}_{\rm coll}$ for use in giant-impact population synthesis.

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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

Two things to know. The paper establishes practical boundaries for when an external body breaks a graze-and-merge collision: with the post-impact orbit's apocenter within about 0.3 Hill radius, the usual single-orbit return and the geometry of the second collision survive; beyond about 0.75 Hill radius, single-orbit returns become a minority and the second impact angle and plane orientation trend toward random; some pairs become unbound. Those are new numbers, and they come with an approximate scaling relation, k ≈ r_apo/r_H − 0.2 log10(r_H/r_coll), that lets you map the Earth-at-1-AU results onto other body sizes and distances. For anyone treating late-stage accretion with N-body codes that cannot afford resolved hydrodynamics for every encounter, this is immediately useful.

The work is straightforward and mostly careful. The Monte Carlo N-body scheme is described in detail, they vary r_H/r_coll independently from r_apo/r_H, they test the scale-invariance claim with a Saturn-mass system, and the cumulative distributions in Figures 5–9 actually support the claimed transitions. The authors are also honest that the second collision's velocity is barely affected — the action is in timing and geometry, not energy. The dependence on EA19 is appropriate; this is a direct sequel, and the citation is not padding.

Now the soft spots, in proportion. The staged-separability assumption — hydro first encounter, point-mass orbit, hydro second encounter — is validated by exactly one SPH run, and that run shows two-body behavior only beyond about six mutual radii, while the N-body integrations start at two mutual radii. Section 7.4 explicitly concedes the decomposition may fail when the Hill radius is not much larger than the bodies. This is not a problem for the Earth-at-1-AU suite, where r_H/r_coll is around 150. It is a problem for their headline application: Titan's Hill radius is about 20 Titan radii, and the Asphaug-Reufer apocenters are about 0.5 r_H, which for a 0.2 mass ratio is roughly 6 r_coll — just at the edge where their own SPH test says hydrodynamics stops. So the critique of the Titan formation scenario rests on treating the entire intermediate orbit as point-mass, in the regime where their validation is thinnest. One or two more SPH cases at wider separation or different impact angles would substantially harden the claim.

Two smaller issues. The return impact angle is fixed at 60°, and Section 6.2 admits the thresholds should shift with impact angle but does not quantify it. The abstract says "usually escape" at 3/4 Hill radius; the body text says a minority of realizations return in one orbit and some pairs become unbound. That is an overstatement, but a fixable one.

Bottom line: this deserves normal peer review. The central threshold result is supported by the N-body experiments and is worth having in the literature. I would send it out and ask for more SPH validation before final acceptance.

Referee Report

4 major / 6 minor

Summary. The paper studies the intermediate orbital phase of graze-and-merge collisions (GMCs) when the colliding pair orbits a third, more massive body. The authors replace the hydrodynamical post-impact evolution with point-mass N-body integrations in the Mercury code, drawing initial conditions from the expected outcome of the first collision: two remnants separated by twice their mutual radii, with the unperturbed return impact angle fixed at 60 degrees and the unperturbed apocenter scaled by the Hill radius. A Monte Carlo procedure samples 10,000 orientations of the post-impact orbit for each apocenter value. The main results are that the standard single-orbit GMC picture holds only for apocenters within about 0.3 r_H, that between roughly 1/3 and 3/4 r_H the return geometry is progressively randomized while the pair still generally re-collides, and that beyond about 3/4 r_H most realizations no longer return on the first orbit and some become unbound. The paper also presents a scaling relation in Eq. (6) involving the apocenter-to-Hill-radius ratio and the body-size-to-Hill-radius ratio, and applies the results to argue that the Asphaug and Reufer (2013) Titan formation scenario is significantly affected by Saturn's presence.

Significance. If the thresholds and scaling relation hold, the paper makes a useful contribution to planet and satellite formation: it shows that GMCs cannot always be treated as isolated two-body mergers, identifies a smooth transition between GMC and hit-and-run regimes, and provides a cheap N-body methodology for exploring the intermediate orbit. The Monte Carlo strategy is efficient, the scale-invariance argument in Section 3 is clearly motivated, and the main thresholds are supported by the cumulative distributions in Figures 5-9. The paper also offers falsifiable predictions: the 0.3 r_H and 0.75 r_H transitions could be tested with dedicated hydrodynamical simulations. However, the current validation of the central separability assumption is thin, and the paper's strongest astrophysical conclusion about the Titan scenario rests on that assumption in a regime where it is least tested.

major comments (4)

[Sections 2 and 4; Section 7.4] The separability assumption is validated by only one SPH simulation, and that validation does not cover the initial conditions actually used in the main suite. Section 2 finds that two-body behavior begins only after the remnants separate by about six mutual radii, yet Section 4 starts the N-body integrations with the bodies separated by twice their mutual radii. The interval from 2 to 6 r_coll is therefore treated as point-mass dynamics without direct support. This is not a purely academic concern: for the Titan application in Section 7.3, r_H/r_coll is about 12.6 and r_apo ~ 0.5 r_H corresponds to only about 6.3 r_coll, so the entire modeled intermediate orbit lies at the edge of the validated regime. Section 7.4 itself concedes that separability 'might not be entirely correct when the Hill radius is not much larger than the body sizes.' I ask the authors to either add SPH validation for other impact angles, mass ratios, and smaller r_H/r_coll values, or clearly restrict the claims to the validated regime and soften the Titan conclusion accordingly.
[Section 4 and Section 6.2] The main suite fixes the unperturbed return impact angle at 60 degrees, and the paper provides no N-body series with different impact angles. Section 6.2 states that the results 'slightly depend on the choice of the impact angle' and even predicts the direction of the shift for grazing versus more head-on collisions, but no simulations are shown. Since the impact angle controls the pericenter-to-rcoll relationship and hence the orbital eccentricity, the central thresholds at 0.3 r_H and 0.75 r_H, as well as the scaling relation in Eq. (6), are established for a single geometry. The generality claims in Section 7.1 therefore go beyond what is demonstrated. A small number of additional series with, for example, 30 and 80 degree return angles would make the parameter dependence quantitative.
[Abstract versus Section 5.1 and Section 7.1] The abstract states that when the loop-back orbit reaches about 3/4 of the Hill radius, the smaller body 'will usually escape the target.' This is not what the simulations show. Section 5.1 reports that for r_apo/r_H = 1.0 more than 90% of realizations collide within 1e6 T1, and even the parabolic limiting case has a return rate above 80% within 1e6 T1. Section 7.1 more carefully states only that 'some of the pairs get unbound.' The abstract should be reworded to match the actual results, for example by saying that most realizations no longer return after a single orbit and a minority become unbound.
[Section 7.3] The claim that the specific Asphaug and Reufer (2013) Titan collisions 'are not, in fact, accretionary, but should be counted as hit and run' is stronger than the presented simulations justify. The N-body runs use a 60 degree return angle and a mass ratio gamma = 0.1, which are not shown to match the A&R collision parameters. Moreover, at r_apo/r_H ~ 0.5, the paper's own Figure 6 indicates that a substantial fraction of realizations still return after a single orbit, and Section 5.1 shows that all pairs at this apocenter collide within 100 T1. The conclusion may be correct, but as written it overreaches; rephrasing to say that the A&R scenario is 'likely to be significantly perturbed' or 'needs to be re-evaluated' would be more appropriate unless A&R-like initial conditions are explicitly simulated.

minor comments (6)

[Table 1] There are typos in Table 1: 'Merucry' should be 'Mercury,' and the table header 'T able 1' has an extra space.
[Figure 4 caption] The caption contains a duplicated word: 'the plane denotes denotes the orbital planet of m1 about m0'; also 'planet' should presumably be 'plane.'
[Section 5.2.2] The legend text 'Note that the colors are for different sets than in Figures 5 or 5' should refer to the correct companion figure (probably Figure 7), not repeat Figure 5.
[Section 6.1] In the discussion of rtilde_coll = 0.1, the text refers to 'the lower boundary on Figure 6,' but the relevant panel is Figure 14, which plots the r_H/r_coll axis; Figure 6 has r_apo/r_H on the horizontal axis.
[Section 5.2.1] The word 'apoceneters' is a typo for 'apocenters.'
[Section 7.2] There is a duplicated 'where' in the sentence beginning 'For instance, in the case where where rtilde_coll = 0.1...'

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the 1/3 and 3/4 Hill-radius thresholds emerge from N-body integrations, not from the fitted scaling law or from self-citations.

full rationale

The paper's central claims are the return-time, impact-angle, and alignment transitions near 1/3 and 3/4 of the Hill radius. These are outputs of 10,000-realisation Mercury N-body integrations with apocenter distance as the controlled input and uniformly random orbital orientations; they are read off cumulative distributions in Figures 5, 6, 8, and 9, and are not imposed by any equation. The initial conditions deliberately fix the unperturbed return angle at 60 degrees, so the low-apocenter agreement with 60 degrees is a consistency check of the numerical setup, not a derived prediction; the physically meaningful result is the progressive randomisation at larger apocenter, which is not in the initial conditions. Equation (6) is explicitly introduced as a rough fit to reproduce Figure 6, and is then tested on independent N-body sets with different r_H/r_coll values in Section 6.2 (Figure 15), so it is a fitted scaling relation with validation rather than a fitted input renamed as a prediction. Self-citations to EA19 are used for one SPH validation run, the bound-remnant search, the Monte Carlo procedure, and contextual comparison of return rates; none of these is load-bearing for the main thresholds. The staged-separability assumption is stated openly and its limitation is conceded in Section 7.4, which is an acknowledged scope restriction rather than a circular argument. Overall, the derivation chain is self-contained with respect to the central claim; the minor non-load-bearing self-citations do not make it circular.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central thresholds rest on the point-mass separability of GMC stages (validated by one SPH run), a specific initial-orbit construction with a fixed 60-degree return angle and mass ratio 0.1, a circular target orbit, and the Mercury integrator's accuracy. Equation (6) adds one fitted coefficient. No new physical entities are introduced.

free parameters (3)

Reference return impact angle = 60 degrees
The post-collision orbit is built so that, without a perturber, the second impact would occur at 60 degrees. Section 6.2 states that the results shift for other impact angles, so the thresholds are conditional on this choice.
Remnant mass ratio m2/m1 = 0.1
The main simulation suite uses 1 Earth mass and 0.1 Earth mass, and the Saturn-like suite keeps the same ratio. The scale-independence argument holds only for a fixed mass ratio, which is not varied.
Scaling coefficient in Eq. (6) = 0.2
The relation k = rapo/rH - 0.2 log10(rcoll/rH) is described as a rough fit to the simulation grid in Figure 14, so this coefficient is calibrated on the same data it is later used to predict.

assumptions (5)

ad hoc to paper A GMC can be divided into independent stages, with point-mass two-body dynamics applying once the remnants are separated by about six mutual radii.
Section 1 states the assumption; Section 2 validates it with one SPH simulation (0.9 Earth-mass target, 0.2 Earth-mass projectile, v/v_esc=1.1, angle 52.5 degrees). A single validation does not cover the parameter space used later.
domain assumption The post-encounter orbit is fully described by an apocenter set to a fraction of the Hill radius and a pericenter that would produce a 60-degree impact angle in the unperturbed case, with the orientation distributed uniformly on a sphere.
Section 4 describes the pericenter iteration and uniform sampling of phi, theta, psi. Actual GMC remnants from hydrodynamical simulations may have a different spread of orbits, which is not explored.
domain assumption The largest remnant moves on a circular orbit around the central body so that the Hill radius is constant in time.
Section 3: 'If we assume that the largest remnant is on an almost circular orbit around the central body, then Hill radius remains almost constant in time.' Eccentric target orbits are not covered.
domain assumption Mercury's hybrid symplectic and Bulirsch-Stoer integration accurately tracks close encounters over up to 10^6 orbital periods.
Section 4 states the integration algorithm but reports no convergence tests or energy-error statistics, leaving the long-delay tails of the distributions dependent on integrator reliability.
standard math Keplerian two-body dynamics and energy conservation give the correct unperturbed mapping between impact velocity, apocenter, and period ratio.
Equations (1) and (3) are used to convert velocities to apocenters and to estimate period ratios. These are standard background results, not contributions of this paper.

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Cite this review

Pith. "Pith review of Graze-and-Merge Collisions under External Perturbers." pith.science (2026). https://pith.science/paper/VUCUKJ64

@misc{pith2026190807557,
  author       = {Pith},
  title        = {Pith review of: Graze-and-Merge Collisions under External Perturbers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VUCUKJ64}},
  note         = {Machine review of arXiv:1908.07557}
}

read the original abstract

Graze-and-merge collisions (GMCs) are common multi-step mergers occurring in low-velocity off-axis impacts between similar sized planetary bodies. The first impact happens at somewhat faster than the mutual escape velocity; for typical impact angles this does not result in immediate accretion, but the smaller body is slowed down so that it loops back around and collides again, ultimately accreting. The scenario changes in the presence of a third major body, i.e. planets accreting around a star, or satellites around a planet. We find that when the loop-back orbit remains inside roughly 1/3 of the Hill radius from the target, then the overall process is not strongly affected. As the loop-back orbit increases in radius, the return velocity and angle of the second collision become increasingly random, with no record of the first collision's orientation. When the loop-back orbit gets to about 3/4 of the Hill radius, the path of smaller body is disturbed up to the point that it will usually escape the target.

Figures

Figures reproduced from arXiv: 1908.07557 by the authors.

**Figure 1.** Dynamics of a graze and merge collision, as seen from the target’s reference frame. The projectile, coming in on a hyperbolic orbit from the bottom, shown with a dashed red line, whose reference axis is shown with the dotted red line collides with the target at a shallow impact angle (point 1). After the initial collision, a second body (composed mostly from the projectile) remains on an elliptical orbit (point 2), … view at source ↗

**Figure 2.** Properties of relative motion after an initial encounter where the two bodies remain bound; i.e. a GMC. The left panel shows the impact velocity, calculated assuming energy conservation, and the right panel the impact angle, calculated assuming angular momentum conservation. The corresponding time since initial encounter is show at the top. ties as function of separation of the two main remnants, given in terms of t… view at source ↗

**Figure 3.** Impact velocity required to achieve a certain apocenter, given in terms of the collision radius. The blue line is computed following the same method described in Section 4 while the dotted black line denotes the maximum reachable distance assuming energy conservation (equivalent to a perfect head-on case). and a theoretical calculation based solely on energy conservation (which would be valid for head-on orbits), a… view at source ↗

Figures from the paper (12 more)

**Figure 4.** Figure 4: Definition of the different angles. The plane denotes denotes the orbital planet of m1 about m0; φ is measured in that plane. star, we perform a series of 10 000 dynamical evolution runs. The largest remnant is assumed to be on a circular orbit around the central star,…

**Figure 6.** Figure 6: Fraction of time delays as function of the scaled apocenter distance ˜rapo = rapo/rH for the case of an Earthlike target at 1 AU from a Sun-mass star, without any other bodies present. located at the same distance than the Hill radius, then only about 10% of realisati…

**Figure 7.** Figure 7: Distribution of impact velocities for the different dynamical evolution series varying as a function of the scaled apocenter distance ˜rapo as given in the legend. The expected value for each case if there were no perturbations is given by the vertical dashed lines, wh…

**Figure 8.** Figure 8: Distribution of impact angle for the different dynamical evolution series varying the as function of the scaled apocenter distance ˜rapo, as given in the legend. The initial conditions are such that without perturbation, the return impact angle would be 60◦ . Note tha…

**Figure 9.** Figure 9: Distribution of angle between symmetry planes of the successive collisions for the different dynamical evolution series varying the as function of the scaled apocenter distance r˜apo, as given in the legend. Note that the colors are for different sets than in [PITH_F…

**Figure 10.** Figure 10: Distribution of angle between symmetry planes of the successive collisions for the dynamical evolution series with rapo/rH = 0.80, and broken down by time delays between the collisions. Colors are identical to the categories provided in [PITH_FULL_IMAGE:figures/full…

**Figure 11.** Figure 11: Time delay between successive collisions as function of the scaled apocenter distance ˜rapo = rapo/rH from the two-body problem. Colors are the same as in [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]

**Figure 12.** Figure 12: 2-D histogram of the orientation of the second collisions for the set with rapo/rH = 0.8. The values of φ and θ are determined from the orbital motion at the moment the second collision is detected. The position of the bin edges were selected so that each bin has the …

**Figure 13.** Figure 13: Fraction of time delays (∆T) as function of the inverse of the scaled physical radius ˜rcoll, for ˜rapo = 0.5. The colors are the same as in [PITH_FULL_IMAGE:figures/full_fig_p013_13.png]

**Figure 14.** Figure 14: Fraction of runners that return within 1.5T2 for each set of dynamical evolution. The dashed lines represent a rough fit (see text); the values for the colors of each of the lines are 1.00, 0.87, 0.70, 0.57, 0.44, 0.23 and 0.13, from the bottom left to the top right. …

**Figure 15.** Figure 15: Cumulative distributions of impact angles (left panels), offset angle between successive collisions (center panels) and time delays between successive collisions (right panels) for dynamical evolution sets have to have similar relationships between apocenter-to-Hill r…

**Figure 16.** Figure 16: Comparison of the masses for the largest body (left panel) and second largest (right panel) for one graze and merge collision modeled using Smooted Particle Hydrodynamics (SPH) using two different methods (see text): simple friends-of-friends (FoF) search (dashed line…

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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, " * write output.state after.block = add.period write newline

ENTRY address archivePrefix author booktitle chapter doi edition editor eprint howpublished institution journal key month number organization pages publisher school series title misctitle type volume year version url label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts ...

[2] [2]

write newline

" write newline "" before.all 'output.state := FUNCTION format.url url empty "" new.block "" url * "" * if FUNCTION format.eprint eprint empty "" archivePrefix empty "" archivePrefix "arXiv" = new.block " " eprint * " " * new.block " " eprint * " " * if if if FUNCTION format.doi doi empty "" " " doi * " " * if FUNCTION format.pid doi empty eprint empty ""...

[3] [3]

sj c ; HR3c= cg 96m nI] XPHٛڤ ^QFXv z|e

thebibliography [1] 20pt to REFERENCES 6pt =0pt 10pt plus 3pt =0pt =0pt =1pt plus 1pt =0pt =0pt -12pt =13pt plus 1pt =20pt =13pt plus 1pt \@M =10000 =-1.0em =0pt =0pt 0pt =0pt =1.0em @enumiv\@empty 10000 10000 `\.\@m \@noitemerr \@latex@warning Empty `thebibliography' environment \@ifnextchar \@reference \@latexerr Missing key on reference command Each re...

2017

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