"Low-speed impacts between rubble piles modeled as collections of
polyhedra, 2", D. G. Korycansky, E. Asphaug, submitted to *Icarus*.

We present the results of additional calculations involving the collisions of rubble piles. In our new work, the calculations were made using the Open Dynamics Engine (ODE). ODE is an open-source library for the simulation of rigid-body dynamics and incorporates a high-level collision-detection and resolution routine. We have found that using ODE as a ``physics engine'' for our calculations results in a speed-up of approximately a factor of 30 compared with our previous code, allowing calculations with greatly increased numbers of objects and also many more calculations be performed. In this paper we report on the results of almost 1200 separate runs, the bulk of which were carried out with 1000 to 2000 elements. We carried out calculations with three different combinations of the coefficients of friction $\mu$ and (normal) restitution epsilon: low (mu=0, epsilon=0.8), medium (mu=0, epsilon=0.5), and high (\mu=0.5, epsilon=0.5) dissipation. In general, head-on collisions between equal mass objects have values of Q*D, the kinetic energy per unit mass such that the mass of largest fragment equals half the target mass m_1=0.5m_T. We found values of $\qd$ ranging from ~4 to 14 J/kg$^{-1} for low to high dissipation cases; monodisperse objects disrupted somewhat more easily than power-law objects in general. For the low-dissipation case, Q*D was roughly constant with decreasing mass ratio, whereas for the medium and high-dissipation cases, Q*D increased with decreasing mass ratio of impactor m_i to target mass m_T over the range 0.001 < m_i/m_T < 1 up to sim 80 J/kg for mu=0.5 and epsilon=0.5. For oblique collisions of equal-mass objects, mildly off-center collisions (b/b_0=0.5) seemed to be as efficient or possibly more efficient at collisional disruption as head-on collisions. More oblique collisions were less efficient and the most oblique collisions we tried (b/b_0=0.866) required up to Q*D 200 J/kg for high-dissipation power-law objects. For calculations with smaller numbers of elements (total impactor n_i+target n_T=20 or 200 elements) we found that collisions were more efficient for smaller numbers of more massive elements, with Q*D values as low as 1.2 J/kg or the low-dissipation case and n_i+n_T=20. We also analyzed our results in terms of the relations proposed by Stewart Leinhardt and where m_1/(m_i+m_T) = 1-Q_R/2Q^*RD where Q_R is the impact kinetic energy per unit total mass m_i+m_T and Q^*{RD} is the corresponding disruption energy. Although there is a significant amount of scatter, our results generally bear out the suggested relation.

"Predictions for the LCROSS mission", D. G. Korycansky, C. S. Plesko,
M. Jutzi, E. Asphaug, and A. Colaprete,
*Meteoritics and Planetary Science,* submitted.

We describe the results of a variety of model calculations for predictions of observable results of the LCROSS mission to be launched in 2009. Several models covering different aspects of the event are described along with their results. We start with a brief discussion of crater scaling laws as applied to the impact of the EDUS second stage and resulting estimated crater diameter and ejecta mass. Next we describe results from the RAGE hydrocode as applied to modeling the short-timescale (t<0.1 s) thermal plume that is expected to occur immediately after the impact. We present results from several large-scale smooth-particle hydrodynamics (SPH) calculations, along with results from a ZEUSMP hydrocode model of the crater formation and ejecta mass-velocity distribution. We finish with two semi-analytic models, the first being a Monte Carlo model of the distribution of expected ejecta, based on scaling models using a plausible range of crater and ejecta parameters, and the second being a simple model of observational predictions for the shepherding spacecraft (S-S/C) that will follow the impact for several minutes until its own impact into the lunar surface.

For the initial thermal plume we predict an initial expansion velocity of
~7 km/s, and a maximum temperature of ~1200 K. Scaling
laws for crater formation and the SPH calculation predict a crater with
a diameter of ~15 m, a total ejecta mass of ~10^{6}
kg, with
~10^{4} kg reaching an altitude of 2 km above the target. Both the
SPH and ZEUSMP calculations predict a maximum ejecta velocity of ~1
km/s. The semi-analytic Monte Carlo calculations produce more
conservative estimates (by a factor of ~5) for ejecta at 2 km, but
with a large dispersion in possible results.

"Implications of an impact origin of for the martian hemispheric dichotomy",
F. Nimmo, S. D. Hart, D. G. Korycansky, and C. B. Agnor,
*Nature*, v. 453, 1220-1223.

The observation that one hemisphere of Mars is lower and has a thinner crust than the other (the "hemispheric dichotomy") [1-3] has been a puzzle for thirty years. The dichotomy may have arisen due to internal mechanisms such as convection [4,5]. Alternatively, it may have been caused by one [6] or several [7] giant impacts, but quantitative tests of the impact hypothesis have not been published. Here we use a high-resolution 2D axisymmetric hydrocode [8,9] to model vertical impacts over a range of parameters appropriate to early Mars. We propose that the impact model, in addition to excavating a crustal cavity of the correct size, explains two additional observations. First, crustal disruption [e.g. 10] at the impact antipode is likely responsible for the observed antipodal decline in magnetic field strength [11]. Second, the impact-generated melt forming the northern lowlands crust is predicted to derive from a deep, depleted mantle source. This prediction is consistent with characteristics of the Martian shergottite meteorites [12,13] and suggests a dichotomy formation time ~100 Myr after Martian accretion [13], comparable to that of the Moon-forming impact on Earth [14].

We report on calculations of the on-shore runup of waves that might be
generated by the impact of sub-km asteroids into the deep ocean.
The calculations were done with the COULWAVE code, which models the propagation
and shore-interaction of non-linear moderate- to long-wavelength waves
(kh less than pi) using the extended Boussinesq
approximation. We carried out runup calculations for several different
situations:

1) laboratory-scale monochromatic wavetrains onto simple slopes,
2) 10-100 metre monochromatic wavetrains onto simple slopes,
3) 10-100 metre monochromatic wavetrains
onto a compound slope representing a typical
bathymetric profile of the Pacific coast of North America,
4) time-variable scaled
trains generated by the collapse of an impact cavity in deep water onto
simple slopes, and
5) full-amplitude trains onto the Pacific coast profile.
For the last case, we also investigated the effects of bottom friction
on the runup.
For all cases,
we compare our results with the so-called ``Irribaren scaling'': The relative
runup R/H_0=xi=s(H_0/L_0)^{-1/2}, where the runup is R, H_0 is the
deep-water waveheight, L_0 is the deep-water wavelength, s is the
slope and xi is a dimensionless quantitity known as the Irribaren number.
Our results suggest that Irribaren scaling breaks down
for shallow slopes s less than 0.01 when xi less than 0.1-0.2, below which
R/H_0 is approximately constant. This regime corresponds to steep waves
and very shallow slopes, which are the most relevant for impact tsunami, but
also the most difficult to accesss experimentally.

We have run high-resolution, three-dimensional, hydrodynamic simulations
of the impact of comet Shoemaker-Levy 9 into the atmosphere of Jupiter. We
find that the energy deposition profile is largely similar to the previous
two-dimensional calculations of Mac Low and Zahnle though
perhaps somewhat broader in the range of height over which the energy is
deposited. As with similar calculations for impacts into the Venusian
atmosphere, there is considerable sensitivity in the results to small changes
in the initial conditions, indicating dynamical chaos. We calculated the
median depth of energy deposition (the height z at which 50% of the
bolide's energy has been released) per run. The mean value among runs is
~ 70 km below the 1-bar level, for a 1-km diameter impactor of porous
ice of density rho=0.6 g cm^{-3}. The standard deviation among these
runs is 14 km. We find little evidence of a trend in these results with the
resolution of the calculations (up to 57 cells across the impactor, or 8.8-m
resolution), suggesting that resolutions as low as 16 grid cells across the
radius of the bolide yield good results for this particular quantity.

Visualization of the bolide breakup shows that the ice impactors were shredded and/or compressed in a complicated manner but evidently did not fragment into separate, coherent masses, unlike a calculation for a basaltic impactor. The processes that destroy the impactor take place at significantly shallower levels in the atmosphere (~ -40 km for a 1-km diameter bolide) but the shredded remains have enough inertia to carry them down another scale height or more before they lose their kinetic energy.

Models of impactors covering a ~ 600-fold range of mass ^{1.2}.

"Polyhedron models of asteroid rubble piles in collision"
D. G. Korycansky, E. Asphaug,
*Icarus*, v. 181, 605-617

We present results of modeling rubble-pile asteroids as collections of polyhedra. We solved the equations of rigid-body dynamics, including frictional/inelastic collisions, for collections with as many as several hundred elements. As a demonstration of the methods and to compare with previous work by other researchers, we simulated low-speed collisions between km-scale bodies with the same overall parameters as those simulated by Leinhardt et al. (2000). High-speed collisions appropriate to present-day asteroid encounters require additional treatment of shock effects and fragmentation and are the subject of future work; here we study regimes appropriate to planetesimal accretion and re-accretion in the aftermath of catastrophic events.

Collisions between equal-mass objects at low speeds (<10^{3} cm
s^{-1}) were
simulated for both head-on and off-center collisions between rubble piles
made of a power-law mass spectrum of sub-elements. Very low-speed head-on
collisions produce single objects from the coalescence of the the impactors.
For slightly higher speeds, extensive disruption occurs, but re-accretion
produces a single object with most of the total mass. For increasingly
higher speeds, the re-accreted object has smaller mass, finally resulting in
complete catastrophic disruption with all sub-elements on escape trajectories
and only small amounts of mass in re-accreted bodies. Off-center collisions
at moderately low speeds produce two re-accreted objects of approximately equal
mass, separating at greater than escape speed. At high speed, the complete
disruption occurs as with the high-speed head-on collisions. Low to moderate
speed head-on collisions result in objects of mostly oblate shape, while
higher speed collisions produce mostly prolate objects, as do off-center
collisions at moderate and high speeds. Collisions carried out with the same
dissipative coefficients (coefficient of restitution epsilon_{n}=0.8,
zero friction) as used by Leinhardt et al. (2000).
result in a value for specific energy for
disruption Q_{D}^{*}~ 1.4 J/kg, somewhat lower than the value
of 2 J/kg found by them, while collisions with friction and a lower coefficient
of restitution [epsilon_{n} f_{t}=1, mu=0.5, used by
Michel et al. (2004) for SPH+N-body calculations]
yield Q_{D}^{*} ~ 4.5 J/kg.

"Offshore breaking of impact tsunami: the Van Dorn effect re-visited",
D. G. Korycansky, Patrick J. Lynett,
*Geophysical Research Letters* , **32**, 10 L10805

We report on calculations of the shoaling and off-shore breaking of typical wavetrains from sub-km impactors into the deep ocean. We use the wave propagation code COULWAVE to compute the propagation of waves through simple bathymetry profiles typical of the North American Pacific coast and the Gulf of Mexico. Numerical results are consistent with those predicted by nonlinear shoaling theory. Our primary result is that large long-period waves of the type considered should indeed break far offshore, as suggested by W. G. Van Dorn for similar waves generated by underwater explosions. Typical breaking distances range from ~3-17 km for the Pacific coast, and up to ~ 200 km for the Gulf coast. The inclusion of bottom friction affects the results; for very gentle slopes like the Gulf coast, a modest amount of bottom friction supplies enough dissipation to suppress wave breaking.

Index Terms: 4255 Oceanography: General: Numerical modeling (0545, 0560); 4564 Oceanography: Physical: Tsunamis and storm surges; 5420 Planetary Sciences: Solid Surface Planets: Impact phenomena, cratering (6022, 8136).

"Modeling Crater Populations on Venus and Titan",
D. G. Korycansky, K. Zahnle,
*Planetary and Space Science* **53**, 695-710

We describe a model for crater populations on planets and satellites with
dense atmospheres, like those of Venus and Titan. The model takes into
account ablation (or mass shedding), pancaking, and fragmentation.
Fragmentation is assumed to occur due to the hydrodynamic instabilities
promoted by the impactors' deceleration in the atmosphere. Fragments that
survive to hit the ground make craters or groups thereof. Crater sizes are
estimated using standard laws in the gravity regime, modified to take
into account impactor disruption. We use Monte Carlo methods to pick
parameters from appropriate distributions of
impactor mass, zenith angle, and velocity.
Good fits to the Venus crater populations (including
multiple crater fields) can be found with reasonable values of model
parameters. An important aspect of the model is that it reproduces the
dearth of small craters on Venus: this is due to a cutoff
on crater formation we impose, when the expected crater would be smaller than
than the (dispersed) object that would make it. Hydrodynamic effects alone
(ablation, pancaking, fragmentation) due to the passage of impactors through
the atmosphere are insufficient to explain the lack of small craters.
In our favored model, the observed number of craters (940) is
produced by ~5500 impactors with masses > 10^{15} gm,
yielding an age of 730+\- 110 Myr (1-sigma uncertainty) for the venusian
surface.

We apply the model with the same parameter values to Titan
to predict crater populations under differing assumptions of impactor
populations.
We assume that the impactors (comets) are made of 50%
porous ice.
Predicted crater production rates are ~190 craters per 10^{9} year.
The smallest craters on Titan are
predicted to be ~2 km in diameter.
If the impactors are composed of solid ice (density 0.92 gm cm^{-3}),
crater production rates increase by ~70% and the smallest crater
is predicted to be ~1.6 km in diameter.
We give cratering rates for denser comets and atmospheres 0.1 and 10 times
as thick as Titan's current atmosphere. We also explicitly address
leading-trailing hemisphere asymmetries that might be seen if Titan's
rotation rate were strictly synchronous over astronomical timescales:
if that is the case, the ratio of crater production on
the leading hemisphere to that on the trailing hemisphere is ~ 4:1.

keywords: Impact processes, Craters, Venus, Titan

"Astroengineering, or how to save the Earth in only one billion years",
D. G. Korycansky,
*Revista Mexicana Astronomia y Astrophysica, Conf. Ser.*, **22**,
117-120.

Korycansky et al. (2001) have presented a scheme for altering planetary orbits in the solar system, in particular that of the Earth as a means to escape (for a period) the consequences of the secular brightening of the Sun over the next few billion years. In this paper I discuss that work, present background information, and attempt to understand consequences of the ideas involved.

"Simulations of impact ejecta and regolith accumulation on
asteroid Eros", D. G. Korycansky, E. Asphaug,
*Icarus*, **171**, 110-119.

We have carried out a set of Monte Carlo simulations of the placement of
impact ejecta on asteroid 433 Eros, with the aim of understanding the
distribution and accumulation of regolith. The simulations consisted of two
stages: (1) random distribution of primary impact sites derived from a uniform
isotropic flux of impactors, and (2) integration of the orbits of test
particle ejecta launched from primary impact points until their re-impact or
escape. We integrated the orbits of a large number of test particles
(typically 10^{6} per individual case). For those particles that did
not escape we collected the location of their re-impact points to build up a
distribution on the asteroid surface.

We find that secondary impact density is mostly controlled by the overall topography of the asteroid. A gray-scale image of the density of secondary ejecta impact points looks, in general, like a reduced-scale negative of the topography of the asteroid's surface. In other words, regolith migration tends to fill in the topography of Eros over time, whereas topographic highs are denuded of free material. Thus, the irregular shape of Eros is not a steady-state configuration, but the result of larger stochastic events.

keywords: asteroids, dynamics

"Atmospheric Impacts, Fragmentation, and Small Craters on Venus",
D. G. Korycansky, K. Zahnle,
*Icarus*, **169**, 287-299.

We use high-resolution three-dimensional numerical models of aerodynamically disrupted asteroids to predict the characteristic properties of small impact craters on Venus. We map the mass and kinetic energy of the impactor passing though a plane near the surface for each simulation, and find that the typical result is that mass and energy sort themselves into one to several strongly peaked regions, which we interpret as more-or-less discrete fragments. The fragments are sufficiently well separated as to imply the formation of irregular or multiple craters that are quite similar to those found on Venus. We estimate the diameters of the resulting craters using a scaling law derived from the experiments of Schultz and Gault (1985) of dispersed impactors into targets. We compare the spacings and sizes of our estimated craters with measured diameters tabulated in a Venus crater database (Herrick and Phillips, 1994; Herrick et al. 1997; Herrick 2003) and find quite satisfactory agreement, despite the uncertainty in our crater diameter estimates. The comparison of the observed crater characteristics with the numerical results is an after-the-fact test of our model, namely the fluid-dynamical treatment of large impacts, which the model appears to pass successfully.

keywords: impacts, craters, Venus

"Orbital dynamics for rigid bodies", D. G. Korycansky

In this paper I describe a method for calculating the motions of a collection of self-gravitating rigid bodies. The bodies are described by polyhedra of general shapes with triangular faces. The gravitational potential of such objects can be calculated via the "polyhedron gravity" routines of Werner (1994) and Werner and Scheeres (1996), and the resulting mutual forces and torques calculated via surface integration. Additional components of the overall scheme include updating the spin vector and orientation of the bodies, and their positions, velocities, and angular momenta after each timestep. Collisions are allowed, and treated via the impulse approximation. Inelastic or frictional collisions can be handled via coefficients of restitution. After the description of the scheme, I present some results that verify global conservation of momentum and energy during sample calculations.

keywords: dynamics

"Impact evolution of asteroid shapes. 1. Random mass redistribution" D. G. Korycansky, E. Asphaug,

We explore whether the cumulative effect of small-scale meteoroid bombardment can drive asteroids into non-axisymmetric shapes comparable to those of known objects (elongated prolate forms, twin-lobed binaries, etc). We simulate impact cratering as an excavation followed by the launch, orbit, and re-impact of ejecta. Orbits are determined by the gravity and rotation of the evolving asteroid, whose shape and spin change as cratering occurs repeatedly. For simplicity we consider an end-member evolution where impactors are all much smaller than the asteroid, and where all ejecta remain bound. Given those assumptions, we find that cumulative small impacts on rotating asteroids lead to oblate shapes, irrespective of the chosen value for angle of repose or for initial angular momentum. The more rapidly a body is spinning, the more flattened the outcome, but oblateness prevails. Most actual asteroids, by contrast, appear spherical to prolate. We also evaluate the timescale for reshaping by small impacts and compare it to the timescale for catastrophic disruption. For all but the steepest size distributions of impactors, reshaping from small impacts takes more than an order of magnitude longer than catastrophic disruption. We conclude that small scale cratering is probably not dominant in shaping asteroids, unless our assumptions are naive. We believe we have ruled out the end-member scenario; future modeling shall include angular momentum evolution from impacts, mass loss in the strength regime, and craters with diameters up to the disruption threshold. The ultimate goal is to find out how asteroids get their shapes and spins, and whether tidal encounters in fact play a dominant role.

keywords: asteroids, dynamics

"High-resolution simulations of the impacts of asteroids into the venusian atmosphere III: further 3D models", D. G. Korycansky, K. J. Zahnle,

We report on high-resolution three-dimensional calculations of oblique impacts into planetary atmospheres, specifically the atmosphere of Venus, extending the results of Korycansky et al. (2000, Icarus 146, 387-403; 2002, Icarus 157, 1-23). We have made calculations for impacts at 0degrees, 45degrees, and 60 degrees from the vertical, different impactor velocities (10, 20, and 40 km /s), and different impactor masses and orientations. We present results for porous impactors using a simple model of porosity. We have investigated the sensitivity to initial conditions of the calculations [as a follow-up to the results found in Korycansky et al. (2002)] and resolution effects. For use in cratering calculations, we fit simple functions to the numerical results for mass and momentum that penetrate to a given altitude (column mass) and investigate the behavior of the fit coefficients as functions of impactor parameters such as mass, velocity, and impact angle. Generally speaking, the mass and momentum (and hence resulting crater diameters) depend primarily on impactor mass and mass of atmosphere encountered and weakly or not at all on other parameters such as impactor velocity, impact angle, or porosity. The column mass to which the last portion of the impactor penetrates is approximately equal to the mass of impactor at the top of the atmosphere before the impact takes place. Finally, we present the beginnings of a simplified but physically based model for the impactor and its fragments to reproduce the mass and momentum fluxes as a function of height during the impact.

keywords: impact processes

"High-resolution simulations of the impacts of asteroids into the Venusian atmosphere II: 3D models", D. G. Korycansky, K. J. Zahnle, M.-M. Mac Low,

We compare high-resolution 2D and 3D numerical hydrocode simulations of asteroids striking the atmosphere of Venus. Our focus is on aerobraking and its effect on the size of impact craters. We consider impacts both by spheres and by the real asteroid 4769 Castalia, a severely nonspherical. body in a Venus-crossing orbit. We compute mass and momentum fluxes as functions of altitude as global measures of the asteroid's progress. We find that, on average, the 2D and 3D simulations are in broad agreement over how quickly an asteroid slows down, but that the scatter about the average is much larger for the 2D models than for the 3D models. The 2D models appear to be rather strongly susceptible to the "butterfly effect," in which tiny changes in initial conditions (e.g., 0.05% change in the impact velocity) produce quite different chaotic evolutions. By contrast, the global properties of the 3D models appear more reproducible despite seemingly large differences in initial conditions. We argue that this difference between 2D and 3D models has its root in the greater geometrical constraints present in any 2D model, and in particular in the global conservation of enstrophy in 2D that forces energy to pool in large-scale structures. It is the interaction of these artificial large-scale structures that causes slightly different 2D models to diverge so greatly. These constraints do not apply in 3D and large scale structures are not observed to form. A one-parameter modified pancake model reproduces the expected crater diameters of the 3D Castalias reasonably well.

keywords: impact processes, cratering, near Earth asteroids, Venus

"Non-linear dynamics of the corotation torque", N. J. Balmforth, D. G. Korycansky,

The excitation of spiral waves by an external perturbation in a disc deposits angular momentum in the vicinity of the corotation resonance (the radius where the speed of a rotating pattern matches the local rotation rate). We use matched asymptotic expansions to derive a reduced model that captures non-linear dynamics of the resulting torque and fluid motions. The model is similar to that derived for forced Rossby wave critical layers in geophysical fluid dynamics. Using the model we explore the saturation of the corotation torque, which occurs when the background potential (specific) vorticity is redistributed by the disturbance. We also consider the effects of dissipation. If there is a radial transport of potential vorticity, the corotation torque does not saturate. The main application is to the creation, growth and migration of protoplanets within discs like the primordial solar nebula. The disturbance also nucleates vortices in the vicinity of corotation, which may spark further epochs of planet formation.

keywords: accretion, accretion discs, hydrodynamics, methods : analytical

"Astronomical engineering: A strategy for modifying planetary orbits"
D. G. Korycansky, G. Laughlin, F. C. Adams,
*Astrophysics and Space Science*

The Sun's gradual brightening will seriously compromise the Earth's biosphere
within ~ 10^{9} years. If Earth's orbit migrates outward, however, the
biosphere could remain intact over the entire main-sequence lifetime of the Sun.
In this paper, we explore the feasibility of engineering such a migration over
a long time period. The basic mechanism uses gravitational assists to (in
effect) transfer orbital energy from Jupiter to the Earth, and thereby enlarges
the orbital radius of Earth. This transfer is accomplished by a suitable
intermediate body, either a Kuiper Belt object or a main belt asteroid. The
object first encounters Earth during an inward pass on its initial highly
elliptical orbit of large (~ to 300 AU) semimajor axis. The encounter transfers
energy from the object to the Earth in standard gravity-assist fashion by
passing close to the leading limb of the planet. The resulting outbound
trajectory of the object must cross the orbit of Jupiter; with proper timing,
the outbound object encounters Jupiter and picks up the energy it lost to Earth.
With small corrections to the trajectory, or additional planetary encounters
(e.g., with Saturn), the object can repeat this process over many encounters.
To maintain its present flux of solar energy, the Earth must experience roughly
one encounter every 6000 years (for an object mass of 10^{22} g). We
develop the details of this scheme and discuss its ramifications.

keywords: solar system, earth

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* 12 January 2009 *