www.sciencemag.org/cgi/content/full/science.aaa4747/DC1

Supplementary Materials for

The shape and structure of cometary nuclei as a result of low-velocity

accretion

M. Jutzi* and E. Asphaug

*Corresponding author. E-mail: martin.jutzi@space.unibe.ch

Published 28 May 2015 on Science Express

DOI: 10.1126/science.aaa4747

This PDF file includes:

Materials and Methods

Figs. S1 to S7

Table S1

Full Reference List

Captions for Data S1 to S7

Other Supplementary Material for this manuscript includes the following:

(available at www.sciencemag.org/cgi/content/full/science.aaa4747/DC1)

Data S1 to S7 as zipped archives

Materials and Methods

We use a smooth particle hydrodynamics (SPH) impact code (20-22). This parallel (35) code includes

self-gravity and is specially suited to modeling planetesimal collisions. To avoid numerical rotational

instabilities, the scheme suggested by Speith (2006) is used (36).

A pressure dependent shear strength (friction) is included by using a standard Drucker-Prager yield

criterion (22). As shown in (22,26), granular flow problems (of cohesionless material) are well

reproduced using this method.

For the simulations with cohesion, we use Y0 > 0 and include a tensile fracture model (20), using

parameters that lead to an average tensile strength of ~ 100 Pa. Porosity is modeled using a P-alpha

model (21) with a simple quadratic crush curve and parameters Pe, Ps as indicated in Table S1. Note

that σc = Ps/2 is used as an estimate of the compressive strength. The initial density of both target and

impactor is ~ 0.5 g/cm3.

We use a modified Tillotson equation of state (EOS; c.f. 37) with parameters for water ice. It is

adequate for modelling the meters-per-second collisions considered here, where the most important

response is the solid compressibility. As long as there is porosity, the compressibility is limited not by

the EOS but by the crush curve of the P-alpha model; as noted, we find that despite the substantial

deformation, only a minor volume fraction of the colliding material is substantially compactified, so a

detailed EOS is not important.

The elastic wave speed ce is not well represented by the EOS in any case, for a weak aggregate body.

For compact geologic ice ce~1 km/s is typical, and according to the Tillotson EOS parameters, ce~3

km/s. Porous ice can have ce~0.1 km/s. So, we apply a reduced bulk modulus (leading term in the

Tillotson EOS; see Table S1), which leads to more realistic wave speeds. The approach has the

additional major advantage that the time-steps becomes large enough to carry out the simulations over

many dynamical timescales. Note that the ce >> vimp still applies for the impact velocities considered in

this study, so the materials do not artificially shock up. A similar approach was applied by (38) to study

planetesimal collisions.

The relevant material parameters used for the simulations are indicated in Table S1.

Both the target and the impactor are evolved to hydrostatic equilibrium in the simulation before the

collision. Our study does not include initial rotation. We use Nt = 4x105 particles for the target and the

same number per unit mass for the impactor for our nominal simulations. The cases with cohesion were

studied using a lower resolution (Nt ~ 1x105).

Two cometesimals are separated by a distance rinit = 3(RT + RP), and an initial relative velocity vinit is

applied to achieve a specific impact velocity vimp and impact angle θ at contact. Initial velocities are

varied so that vesc < vimp < 2 vesc where vesc = [2G(MT+MP)/(RT+RP)]0.5

and MP and MT are the colliding

(spherical) masses of radii RT > RP, and G is the gravitational constant.

The simulations are carried out up to a few days of simulation time, depending on the type of collision

result. The self-gravity timescale is hours, and the final rotation period is around twice that in typical

simulations. While the final configurations in the splat type collisions are reached relatively quickly

(several hours), the bi-loped forming collisions take much longer to reach the final state (Figure 1).

Overall, ~100 simulations were performed (see Figure 2 and Figures S2-S6). Each simulation takes one

to several weeks to complete, depending on the collision type.

Figure S1: Estimation of the critical radius Rc at which the overburden pressure leads to compaction,

plotted as a function of the radius Rt of the body. The layers below Rc are significantly compacted. In

this simplified model, Rc is approximately computed for a given object with radius Rt by equating the

crushing pressure to the overburden pressure at the Radius Rc: σc = Poverburden(R=Rc), assuming a

homogenous density of ρ = 500 kg/m3. This is for dust-aggregate compaction pressures, many orders of

magnitude smaller than previously considered (32).

Figure S2: Results for Mt/Mp = 4 and cohesionless bodies.

Figure S3: Results for Mt/Mp = 8 and cohesionless bodies.

Figure S4: Results for Mt/Mp = 2 and bodies with cohesion.

Figure S5: Results for Mt/Mp = 4 and bodies with cohesion.

Figure S6: Results for Mt/Mp = 8 and bodies with cohesion.

Figure S7: Same run as shown in Figure 4, right (with tensile strength). Materials whose prescribed

material tensile strength was exceeded are plotted here, with dark being undamaged regions. Two

different viewing angles are shown.

Extended Data Figure 6: Same run as shown in Figure 3, right (with cohesion). Materials whose pre-

scribed material cohesion was exceeded are plotted here, with dark being undamaged regions. Two differ-

ent viewing angles are shown.

type Pe (Pa) Pe (Pa) ρs0 (kg/m3) ρ0 (kg/m

3) α0 A (Pa) μ Y0 (Pa) YT (Pa)

No cohesion 101 10

3 910 440 2.08 2.67 10

4 0.55 0 0

With cohesion 102 10

4 910 440 2.08 2.67 10

4 0.55 10

3 10

2

Table S1: Material parameters used. Crush curve parameters Pe and Ps (c.f. 21), density of matrix

material ρs0, initial bulk density ρ0, initial distention α0, bulk modulus A, friction coefficient μ, cohesion

Y0, average tensile strength YT.

Captions for Supplementary Data Files:

Data S1 to S6 (same caption for files named aaa4747_SupportingFile_Other_seq1,

aaa4747_SupportingFile_Other_seq2, aaa4747_SupportingFile_Other_seq3,

aaa4747_SupportingFile_Other_seq4, aaa4747_SupportingFile_Other_seq5,

aaa4747_SupportingFile_Other_seq6):

SPH data for simulations presented in Figure 2. The file names indicate the velocity (vimp/vesc),

impact angle and mass ratio, respectively. The first 3 columns correspond to the coordinates of

the SPH particles (in cm), the last column is the body number (1: target, 2: impactor). The single

.gz files are ascii files compressed using gzip.

Data S7 (file named aaa4747_SupportingFile_Other_seq7_v2.gz):

SPH data for simulations presented in Figure 3. The first 3 columns correspond to the

coordinates of the SPH particles (in cm), the last column is the body number (1: target, 2: first

impactor, 3: second impactor). The single .gz files are ascii files compressed using gzip.

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