Formation of bi-lobed shapes by sub-catastrophic collisions: A late origin of comet 67P/C-G's structure
AAstronomy & Astrophysics manuscript no. Jutzi_and_Benz_final c (cid:13)
ESO 2018August 27, 2018
Formation of bi-lobed shapes by sub-catastrophic collisions
A late origin of comet 67P/C-G’s structure
M. Jutzi, W. Benz
Physics Institute, University of Bern, NCCR PlanetS, Sidlerstrasse 5, 3012 Bern, Switzerlande-mail: [email protected]; [email protected]
Received – ; accepted –
ABSTRACT
Context.
The origin of the particular shape of a small body like comet 67P / Churyumov-Gerasimenko (hereafter 67P / C-G) is a topicof active research. How and when it acquired its peculiar characteristics has distinct implications on the origin of the solar system andits dynamics
Aims.
We investigate how shapes like the one of comet 67P / C-G can result from a new type of low-energy, sub-catastrophic im-pacts involving elongated, rotating bodies. We focus on parameters potentially leading to bi-lobed structures. We also estimate theprobability for such structures to survive subsequent impacts.
Methods.
We use a smooth particle hydrodynamics (hereafter SPH) shock physics code to model the impacts, the subsequent reaccu-mulation of material and the reconfiguration into a stable final shape. The energy increase as well as the degree of compaction of theresulting bodies are tracked in the simulations.
Results.
Our modelling results suggest that the formation of bi-lobed structures like 67P / C-G is a natural outcome of the low energy,sub-catastrophic collisions considered here.
Conclusions.
Sub-catastrophic impacts have the potential to alter the shape of a small body significantly, without leading to majorheating or compaction. The currently observed shapes of cometary nuclei, such as 67P / C-G, maybe a result of such a last major shapeforming impact.
Key words.
Comets: general – Comets: individual: 67P / C-G – Kuiper belt: general – Planets and satellites: formation
1. Introduction
Whether cometary nuclei structures as observed today are pris-tine and preserve a record of their original accumulation, or area result of later collisional or other evolutionary processes is stillmuch debated (e.g. Weissman et al. 2004; Mumma et al. 1993;Sierks et al. 2015; Rickman et al. 2015; Morbidelli & Rickman2015; Davidsson et al. 2016).Based on data from the European Space Agency’s Rosettarendezvous mission (Sierks et al. 2015), it was suggested that theparticular bi-lobe structure of comet 67P / C-G was formed duringthe early stages of the Solar System (Massironi et al. 2015), pos-sibly by low velocity accretionary collisions (Jutzi & Asphaug2015) and therefore should be considered as a primordial body.On the other hand, Morbidelli & Rickman (2015) show that inthe "standard scenario" of the early dynamical evolution of theSolar system, an object of the size of comet 67P / C-G would haveexperienced a high number of catastrophic collisions and thuscould not have survived. This study has been improved and adetailed analysis of the survival probability of a 67P / C-G-likeobject is presented in a companion paper (Jutzi et al., 2016, sub-mitted; hereafter Paper I). It is found that even in the scenariowithout a long-lasting primordial disc, comet 67P / C-G couldnot have conserved its primordial shape and a large number ofshape-changing collisions should have occurred during its life-time. This conclusion holds also true for generic bi-lobe struc-tures, which might evolve further through a fission-merging cy-cle as recently suggested by Scheeres et al. (2016). These results strongly indicate that neither the current shape of comet 67P / C-G nor its two individual lobes can actually be primordial.Alternatively, 67P / C-G-like bi-lobe structures could be theresult of collisional disruptions of somewhat larger bodies takingplace at a later stage in the history of the solar system (e.g. Rick-man et al. 2015; Morbidelli & Rickman 2015; Marchi et al. 2015,Paper I). During these later stages, typical relative velocities be-tween small bodies have grown much larger than their mutualescape velocity ( V >> V esc ≈ / s) of kilometer-sized bod-ies and therefore direct bi-lobe formation by collisional merg-ers of similar sized bodies (Jutzi & Asphaug 2015) is no longera viable mechanism. However, suitable low relative velocitiescould be found again between fragments / aggregates following ahigher-velocity collisional disruption of a larger parent body. Inthis secondary bi-lobe formation scenario, the dispersed materialfollowing a catastrophic collision re-accumulates in two gravita-tionally bound bodies which subsequently collide at low speed.Re-accumulation has been extensively studied for asteroids andresults show that such behaviour is indeed occurring (e.g. Michelet al. 2001, 2003; Michel & Richardson 2013). A study of large-scale disruption and sub-sequent reaccumulation in the case ofcometary parent bodies is currently in progress (Schwartz et al.,2016, in prep.).However, if the currently observed 67P / C-G structure didform as a result of a late catastrophic disruption of a larger parentbody, the resulting shape must then survive until today. In otherwords, from the time of formation until present day it cannothave experienced a single subsequent shape-changing collision.
Article number, page 1 of 10 a r X i v : . [ a s t r o - ph . E P ] N ov & A proofs: manuscript no. Jutzi_and_Benz_final
In this context, the di ff erence between the specific impact energy Q required to catastrophically disrupt the parent body of 67P / C-G and the energy su ffi cient to change its shape Q reshape is a cru-cial quantity, as it determines the relative number of such events.On average, the larger the specific energy di ff erences, the largerthe number of shape changing impacts compared to the numberof disruption events. Catastrophic disruptions are usually char-acterised by the specific impact energy Q ∗ D , which leads to theescape of half of the initial mass involved in the collision. As itis shown in Paper I, the ratio Q ∗ D / Q reshape is very large, of thethe order of 10 . Hence, there are on average many more shapechanging collisions than disruptive collisions and it is unlikelythat 67P / C-G could have conserved its shape, unless the disrup-tion of the parent body has occurred very recently.As we will show in this paper, 67P / C-G-like bi-lobe struc-tures could also emerge from low energy, sub-catastrophic im-pacts on parent bodies of cometary size. It turns out that the spe-cific energy Q sub for these types of impacts is much closer tothe specific re-shape energy Q reshape and hence the collision fre-quency di ff erence between the two events is much smaller. Thistranslates directly in a larger survival rate and therefore a higherprobability to observe a shape like 67P / C-G.As suggested recently (Scheeres et al. 2016), comets withtwo-component shapes might enter a fission-merging cycle, oncethey enter the inner solar system and experience changes in theirspin rate. Hence, its is possible that bi-lobe structures undergoepisodic shape changes and the present day observed shape issimply the result of the last of such episodes. Even though thisscenario adds the possibility for a time dependent macroscopicshape change, it is important to realise that a two-componentobject of cometary size must exist to begin with. Hence, evenin this case the question of the long-enough survival of bi-lobeshapes is a central question (Paper I).Observational results, such as the abundance of super-volatiles (CO, CO , N ) (e.g. Hässig et al. 2015; Le Roy et al.2015), the detection of primordial molecules (Bieler et al. 2015),and the evidence for a low formation temperature (Rubin et al.2015) suggest that comet 67P / C-G cannot have experienced anysubstantial, global scale heating after its formation. Further con-straints include the high porosity and the observed homogene-ity of the nucleus, which appears to be constant in density on aglobal scale without large voids (Pätzold et al. 2016). As a con-sequence, any proposed formation mechanism of bi-lobe shapesmust be able to operate within very tight constraints on energyinput and compaction of porous material.In this paper we investigate the final shapes resulting from anew type of low-energy, sub-catastrophic impacts on elongated,rotating bodies that meets the constraints mentioned above. Wecarry out a set of 3D smooth particle hydrodynamics simula-tions of impacts to investigate the possibility to forming bi-lobestructures reminiscent of those objected for cometary nuclei. Insection 2 we present our model approach and describe the setupand initial conditions. Representative bi-lobe forming collisionare presented in section 3.1. The results (final shapes) of ourparameter space exploration are shown in section 3.2; heatingand compaction e ff ects are discussed in section 3.3. In section4 we investigate the probabilities for a 67P / C-G-like structuresformed in such a manner to avoid destruction by subsequentshape changing collisions.Finally we discuss our bi-lobe formation model in the con-text of the question of how primordial comets are (section 5).
2. Bi-lobe formation by sub-catastrophic impacts
Remote observations of cometary nuclei suggest that a largefraction of these objects have elongated rather than sphere-likeshapes (Lamy et al. 2004). This is interesting as it turns out thatelongated bodies are naturally easier to ’split’ in two componentsthan spherically symmetric bodies. This is even more true whenthey are rotating around their short axis, and centrifugal forcesact in opposite directions at each end of the body. Impacts onsuch elongated rotation bodies might therefore act as a splittingmechanism leading first to two distinct bodies which can poten-tially form a binary system or eventually merge together forminga bi-lobed body.To study under which conditions such splitting can indeedproduce such bi-lobed structures, we investigate the e ff ects ofimpacts on rotating ellipsoids. We use axis ratios and rotationperiods which are consistent with the observed range of val-ues typical for cometary nuclei (Lamy et al. 2004). We considerimpacts in the sub-catastrophic regime for which most of themass remains bound and accumulates on the main body (possi-bly made of two components). The impact scenario investigatedhere is very di ff erent from the case of a catastrophic break-up ofa large parent body (Schwartz et al., 2016, in prep.) where thelargest re-accumulated remnants contain only a small fraction ofthe initial mass. However, such remnants of catastrophic disrup-tions might have properties (elongated and rotating) similar tothe targets considered here. The modelling approach used here is the same as in Paper I. Weuse a parallel smooth particle hydrodynamics (SPH) impact code(Benz & Asphaug 1995; Ny ff eler 2004; Jutzi et al. 2008; Jutzi2015) which includes self-gravity as well as material strengthmodels. To avoid numerical rotational instabilities, the schemesuggested by Speith (2006) has been implemented.In our modelling, we include an initial cohesion Y > Y T ∼ P e , P s , ρ , ρ s and α . Following the approachin Paper I, we also introduce the density of the compacted ma-terial as ρ compact = / m to define the initial distention α = ρ compact /ρ = . − /α ∼ c e ∼ / s.The relevant material parameters used in the simulations areprovided in Table 1. There is an infinite number of combinations of impact parame-ters in the case of non-spherical, rotating targets. Therefore, forpractical reasons, we have to limit the size of the initial param-eter space that can be investigated. The aim of this paper being
Article number, page 2 of 10. Jutzi, W. Benz: Formation of bi-lobed shapes by sub-catastrophic collisions
Table 1.
Material parameters used in the simulations. Crush curve parameters P e and P s (Jutzi et al. 2008), density of matrix material ρ s , initialbulk density ρ , density of the compacted material ρ compact , initial distention α , bulk modulus A , friction coe ffi cient µ , cohesion Y , average tensilestrength Y T . P e (Pa) P s (Pa) ρ s (kg / m3) ρ (kg / m3) ρ compact (kg / m3) α A (Pa) µ Y (Pa) Y T (Pa)10
910 440 1980 4.5 2.67 × center xy + dx- dx + dy z (out of plane) Fig. 1.
Impact geometries. We use an ellipsoid with a constant length of L = ff erent axis ratios (0.4 and 0.7). The ellipsoidsare rotating with a rotation axis either along the y or z direction. Variousimpact points are investigated, as illustrated in the plot. The distance | dx | = | dy | =
670 m. more a demonstration of principle than a complete investigationof all possibilities, we limit ourselves to a few promising cases ofrelatively central collisions. However, a larger sample of variousrotation rates, orientations as well as a range of impact locationshave been investigated. The parameters defining the impact ge-ometries used in this study are illustrated in Figure 1.Target and impactor have both the same initial material prop-erties including their initial bulk density which is set to ρ ∼ / m .We use two di ff erent ellipsoidal targets both of length L = .
04 km but di ff ering from each other by their axis ratios (0.4 and0.7). With the assumed initial density, the two ellipsoids have aninitial masses of 4.7 × kg and 1.45 × kg, respectively.The impactor size is R p =
100 m (in collisions involving thetarget with an axis ratio of 0.4) and R p =
200 m (in collisionsinvolving the target with an axis ratio of 0.7). The correspondingprojectile masses are M p = × kg and M p = × kg,respectively. The impact velocities are chosen in the range of200-300 m / s. We note that the sizes of the impactors as well asthe collision velocities are motivated mainly by numerical ratherthan physical reasons. Smaller impactors at higher speeds wouldnot be well resolved (spatially) in our simulations and would alsorequire smaller timesteps to avoid unphysical oscillations. Evenat a relatively modest resolution ( ∼ SPH particles), thesesimulations are quite challenging even for a parallelised code.This is because the simulations have to extend over a very largenumber of dynamical timescales (and hence involve a very largenumber of timesteps) before the final structure of the resultingobject can be determined (typically one day real time). We note,however, that the impact velocities considered here are super-sonic (sound speed ∼
100 m / s) and the projectile small enoughfor the impact to be taking place in the so-called point sourceregime. Hence, our results can be scaled to larger impact speedsusing appropriate scaling laws.
3. Results
We present snapshots from the simulation of two representativecases of bi-lobe forming collisions in Figures 2 and 3. Due to theelongated nature of the targets, the immediate post-impact massdistribution is concentrated at two locations. As a result, sub-sequent re-accumulation due to gravity leads to the formationof two distinct bodies. Centrifugal forces due to target rotationenhance the initial separation of these two masses. Material re-accumulating at low relative velocity during this process mightalso lead to the formation of layered structures on each individ-ual body, such as observed on 67P / C-G (Massironi et al. 2015)and computed by Jutzi & Asphaug (2015). Finally, the two maingravitationally bound bodies eventually collide with each otherat low velocity (determined by their mutual gravity and hence oforder ∼ / s) within ∼ one day forming a 67P / C-G-like bi-lobeshape.We note that the formation of a bi-lobe structure in such lowvelocity collisions ( V ∼ V esc ) of two gravitationally bound ob-jects is consistent with previous results (Jutzi & Asphaug 2015).While these studies considered these low velocity impacts to oc-cur during the early days of the solar system before small bodiesbecame scattered by growing planets, the impact scenario pre-sented here provides additional possibilities for such low veloc-ity collisions to occur much later in the history of the solar sys-tem.Due to numerical limitations (section 2.3), the velocities con-sidered are at the lower limit of the expected average velocities inthe initial left-over planetesimal disc and are significantly lowerthan average velocity after disc dispersal (Morbidelli & Rickman2015). However, the critical energies Q sub for sub-catastrophicbi-lobe formation for di ff erent impact velocities (and the cor-responding projectile sizes) can be obtained using appropriatescaling (section 3.3 below). It is important to point out that thepost-impact focusing of the mass into two distinct locations,which has not been observed in previous simulations, is not dueto the low velocity of the impacts, but rather the result of the spe-cific target properties, namely their elongated shapes, the initialrotation and the relatively high porosity. All these properties aretypical for real comets (Lamy et al. 2004), but were not consid-ered in previous impact studies. Moreover, the collision energiesconsidered here are between the cratering and the catastrophicregime, an area that has not been well explored before. A summary of our simulations, using various rotation rates, ori-entations as well as a range of impact points, is presented inFigs. 4 and 5, which show the final shapes resulting from thecollisions considered. We find that for the conditions investi-gated here, there is a reasonably large fraction of shapes thathave 67P / C-G-like bi-lobe structures. While this is true for thetwo di ff erent targets (di ff erent axis ratio) considered here, bodieswith two distinct lobes are more probable outcomes in collisionsinvolving the more elongated target. Article number, page 3 of 10 & A proofs: manuscript no. Jutzi_and_Benz_final
27 h17 h11 h8.3 h 5.6 h2.8 h1.1 h0.0 h
Fig. 2.
Comet 67P / C-G shape formation by sub-catastrophic collisions. Shown is an example of an SPH calculation of an impact on a rotatingellipsoid. After the initial disruption, subsequent re-accumulation leads to the formation of two lobes. This processes may include the possibleformation of layers. The two lobes are gravitationally bound and collide with each other within ∼ one day forming a bi-lobed structure. The finalshape is also shown in Figure 4. Initial conditions: o ff axis (impact point + dy ; see Figure 1) impact of a R p =
200 m impactor with a velocity of V =
300 m / s on a target with axis ratio 0.7 (mass M t = × kg) and rotation period of T = y -axis (see Figure1). Fig. 3.
Same as Figure 2 but showing a case with a more elongated target, di ff erent rotation axis and impactor properties. The final shape is shownin Figure 5. Initial conditions: o ff axis (impact point + dx ; see Figure 1) impact of a R p =
100 m impactor with a velocity of V =
250 m / s on atarget with axis ratio 0.4 (mass M t = × kg) and rotation period of T = y axis (see Figure 1). The specific impact energy is defined as Q = . µ r V / ( M t + M p ) (1)where µ r = M p M t / ( M t + M p ) is the reduced mass. We note that µ r (cid:39) M p for M p << M t , as it is the case here. In Figure 6,we compare the specific energies Q sub of the sub-catastrophicimpacts considered in this study to the specific impact energiesfor catastrophic collisions ( Q ∗ D ) as well as for shape changingimpacts on comet 67P / C-G ( Q reshape ) (see Paper I). Following the scaling law already applied for Q ∗ D and Q reshape , we can write Q sub = aR µ V − µ (2)with R = µ = a isdetermined by the impact conditions used for the two di ff erenttargets (section 2.3). For the case with the axis ratio of 0.4 withuse the impact velocity of 250 m / s. Table 2 lists the values ofparameter a for the various cases. Article number, page 4 of 10. Jutzi, W. Benz: Formation of bi-lobed shapes by sub-catastrophic collisions
Rotation axis (period) + z (12h) + y (12h) + y (6h) I m pa c t po i n t center+ dx+dy + z (9h) + y (9 h)- dx (not finished) Fig. 4.
Shapes resulting from sub-catastrophic disruptions of rotation ellipsoids. Shown are the results for di ff erent impact positions, rotation axisand periods (see Figure 1 for the impact geometries). The impact velocity is V =
300 m / s and the impactor size R p =
200 m. The initial target massis 1.45 × kg and the target axis ratio is 0.7. The mass of the final bodies is of the order of ∼ + y (6h), + dy ]. Rotation axis (period) + z (12h)+ y (12h) I m pa c t po i n t / v e l o c i t y centerv = 300 m/s+ dxv = 200 m/s+ dxv = 250 m/s - y (12h) Fig. 5.
Shapes resulting from sub-catastrophic disruptions of rotation ellipsoids. Shown are the results for di ff erent impact positions, velocitiesand rotation periods and axis (see Figure 1). The impact velocity is labeled in the plot; the impactor size is R p =
100 m. The initial target mass is4.7 × kg and the target axis ratio is 0.4. The mass of the final bodies is of the order of ∼ y (12h), + dx with v = / ms]. Article number, page 5 of 10 & A proofs: manuscript no. Jutzi_and_Benz_final
Table 2.
Parameters (SI units) for the scaling law Q crit = aR µ V − µ ,where R is the target radius and V the impact velocity. The scaling for Q reshape only holds for a fixed size, corresponding to comet 67P / C-G ( R = Scaling µ aQ ∗ D Q reshape (average) 0.42 2.50e-6 Q sub (0.4) 0.42 1.66e-5 Q sub (0.7) 0.42 4.90e-5 Also shown in Figure 6 is the maximal global temperature in-crease dT resulting from the impacts. To estimate an upper limitfor the global dT , we assumed that all kinetic impact energy isconverted into internal energy: dT = Q / c p for which a constantheat capacity c p =
100 J / kg / K has been adopted (see Paper I).This rough estimate already indicates that for impacts with en-ergies comparable to Q = Q sub , the maximal global temperatureincrease must remain relatively small. The actual temperaturedistribution depends on how much of the available kinetic en-ergy is converted into heating and what fraction of the heatedmaterial remains on the body. Moreover, due to the highly dis-sipative characteristics of porous material, the compressed andtherefore heated region remains very localised to the vicinity ofthe impact and therefore such an impact a ff ects little the bulkcontent of volatile elements and the bulk porosity.We demonstrate this by plotting in Figure 7 the fraction ofmaterial that actually experienced a temperature increase largerthan a certain dT for the two cases presented in Figures 2 and3. As it can be seen, only (cid:46)
1% of the mass in the final bod-ies experienced a temperature increase larger than a few K. Ac-cording to the scaling laws (Figure 6) higher impact velocitieswould require slightly higher specific impact energies in orderto result in a similar final bi-lobed configuration. For km / s im-pact velocities, we would therefore expect the curves in figure 7to be slightly shifted towards higher dT . However, as discussedabove, the heating would remain localised and on a global scale, dT remains limited to relatively small values even at high im-pact speeds. We note that the bulk of the dT values found hereare generally much smaller than the sublimation temperatures ofthe observed super-volatiles, which are are typically T s >
20 K(Yamamoto 1985; Meech & Svoren 2004).
Using the same procedure as in Paper I, we compute the cumula-tive distribution of the porosity in the final body, which takes intoaccount compaction as well as the addition of macroporosity byreaccumulation of ejected material (Figure 8).Only a small fraction of the target mass ( (cid:46) (cid:38) F o r m a t i o n b y s u b - c a t a s t r o p h i c c o lli s i o n s S h a p e c h a n g i n g F o r m a t i o n b y c a t a s t r o p h i c d i s r u p t i o n s R = . k m R = k m A x i s r a t i o . A x i s r a t i o . -1
100 1000 10 -3 -2 -1 Q c r i t ( J / K g ) M a x . g l oba l t e m pe r a t u r e i n c r ea s e d T m a x ( K ) v (m/s) Q *D (R = 5 km)Q *D (R = 1.8 km)Q sub (axis ratio 0.4)Q sub (axis ratio 0.7)Q reshape , Y T = 10 PaQ reshape , Y T = 100 PaQ reshape , Y T = 1000 Pa Fig. 6.
Critical specific impact energies (adapted from Paper I): Shownare specific energies for catastrophic disruptions ( Q ∗ D ), shape changingcollisions ( Q reshape ) and sub-catastrophic impacts ( Q sub ). For the latter,a target radius R = V =
300 m / s (axisratio 0.7) and V =
250 m / s (axis ratio 0.4) have been assumed. -4 -3 -2 -1 -3 -2 -1 T e m pe r a t u r e i n c r ea s e d T ( K ) Mass fraction Axis ratio 0.7Axis ratio 0.4
Fig. 7.
Fraction of material in the final body that experienced a temper-ature increase larger than a certain value dT . The curves correspond tothe cases shown in Figure 2 (axis ratio 0.7) and Figure 3 (axis ratio 0.4),respectively. depth. On smaller scales, layers of varying porosities are ob-served, suggesting that stratification such as observed on 67P / C-G (Massironi et al. 2015), may be produced by the reaccumula-tion of material on each single lobe (see also Figures 2 and 3).In some locations, porosity decreases with increasing depths, assuggested by CONSERT measurements of the first ∼
100 m ofthe subsurface (Ciarletti et al. 2015). However, we stress that thespatial resolution of our simulations (of the order of ∼
100 m)does not allow to directly compare our results with these mea-surements. It also prevents us from making predictions regardingthe size distribution of reaccumulated boulders.
Article number, page 6 of 10. Jutzi, W. Benz: Formation of bi-lobed shapes by sub-catastrophic collisions P o r o s i t y Mass fractionAxis ratio 0.7, including macro-porosityAxis ratio 0.7, compaction onlyAxis ratio 0.4, including macro-porosityAxis ratio 0.4, compaction onlyInitial porosity
Fig. 8.
Cumulative distribution of the porosity in the final body. Thecurves correspond to the cases shown in Figure 2 (axis ratio 0.7) andFigure 3 (axis ratio 0.4), respectively. The porosity calculation takesinto account compaction as well as the increase of macorporosity. Forcomparison, the porosity distributions resulting from compaction onlyare shown as well. The final average porosity (compaction plus addi-tion of macroporosity by reaccumulation) is 85.1% (axis ratio 0.7) and82.3% (axis ratio 0.4), while the initial porosity was 77.8%.
Fig. 9.
Cross-sections showing the distribution of the porosity in thefinal body. The plots correspond to the cases shown in Figure 2 (top)and Figure 3 (bottom).
4. Survival probabilities
If the currently observed 67P / C-G structure did indeed form as aresult of a collision event as shown above, for it to be observedtoday implies that it has not su ff ered any shape-changing colli-sions since the time of its formation.Assuming that the formation of a bi-lobed structure of thesize of 67P / C-G requires at least a specific impact energy of Q f orm , we can estimate the average number of subsequent shape- changing collisions which have the minimum energy Q reshape (asdetermined in Paper I), as well as the probability to avoid all ofthese collisions. To allow for this, we calculate from Equation 1the minimal projectile radius delivering a specific impact energy Q ≥ Q min as R min = (2 Q min V ) / R t / V (3)where we assume that target and impactor have the same den-sity and that M p << M t (for impact velocities of a few hun-dred m / s, this is true even for Q min = Q ∗ D ). The number of im-pacts on a target of size R t by projectiles within in the size range R min ≤ R t ≤ R max during a time interval δ t , can be written as (e.g.Morbidelli & Rickman 2015): N = P i δ t (cid:90) R max R min π ( R t + R p ) N p ( R p ) dR p (4)where P i is the average intrinsic collision probability. Follow-ing Morbidelli & Rickman (2015) we use R max =
50 km. N p ( R p )represents the number of projectiles with a radius between R p and R p + dR p . This distribution is not known precisely but inagreement with other studies of small body size distribution,we assume a di ff erential size distribution of bodies given by dN / dr ∼ r q . For the exponent q we use the same values as incalculations performed in Paper I: q = − . , − . − . N ( Q min = Q f orm ) within the time interval δ t having aminimal specific impact energy Q f orm allowing the formation of67P / C-G-like structure. In addition to these collisions, during thesame time interval a number of reshaping collisions N ( Q min = Q reshape ) will take place. Given that Q f orm is larger than Q reshape , N ( Q min = Q reshape ) will be larger than N ( Q min = Q f orm ). In otherwords, for one formation event ( N ( Q min = Q f orm ) =
1) therewill be on average N rs , norm reshaping collisions. This can be ex-pressed mathematically as: N rs , norm = N ( Q min = Q reshape ) − N ( Q min = Q f orm ) N ( Q min = Q f orm ) , (5)or rewritten: N rs , norm = N ( Q min = Q reshape ) N ( Q min = Q f orm ) − N rs , norm for the scenario of the formation of the67P / C-G shape as a result of either a catastrophic break-up event(where we assume that a specific energy of at least Q f orm = Q ∗ D is required) or a sub-catastrophic formation event as illustratedabove (where Q f orm = Q sub ). The numbers obtained are shownin Table 3 for two di ff erent projectile velocities V =
500 m / s and V = / s, which are representative for the relative velocitiesof small bodies within in the planetesimal disc before and afterdispersal, respectively (Morbidelli & Rickman 2015). To compute the survival probability of a 67P / C-G-like struc-ture formed by a collision event of a certain type, we can usethe number of reshaping collisions per unit number of forma-tion collisions N rs , norm derived in the previous section. In thefollowing, we estimate this survival probability P survival for thedi ff erent formation scenarios by assuming that a 67P / C-G struc-ture was formed as a result of the last possible collision. Thismeans the last collision involving the required specific energy
Article number, page 7 of 10 & A proofs: manuscript no. Jutzi_and_Benz_final
Table 3.
Average number of shape-changing collisions N rs , norm for oneformation event [ N ( Q min = Q form ) = N rs , norm is computed for thecatastrophic break-up with Q form = Q ∗ D (with R = Q form = Q sub and two axis ratios (0.4 and0.7, respectively). V Formation type q = -2.5 q = -3 q = -3.50.5 Catastrophic 4.60 23.1 90.9km / s Sub-catastrophic (0.7) 2.09 4.93 9.2Sub-catastrophic (0.4) 1.12 2.18 3.462.0 Catastrophic 6.58 30.1 112km / s Sub-catastrophic (0.7) 2.57 5.41 9.9Sub-catastrophic (0.4) 1.34 2.30 3.61 Table 4.
Survival probabilities in the di ff erent scenarios. Computed isthe bi-lobe shape survival probability, P survival , against any subsequentshape changing collision. These survival probabilities are computed forthree di ff erent scenarios a) a catastrophic disruption of a target with R t = R t = / C-G sized bodies with R t ∼ c n = q = -2.5, c n = q = -3 and c n = q = -3.5, respectively). Type q = -2.5 q = -3 q = -3.5Catastrophic (3 km) 2.00E-02 1.07E-03 7.85E-05Including c n T disc =
400 Myrs < < < T disc = Q f orm = Q ∗ D or Q f orm = Q sub ). We then take into accountthat on average, only a fraction f of the shape changing colli-sions N rs , norm take place after the structure formed: n f = f N rs , norm (7)We further assume that these n f events follow a Poisson dis-tribution. In this case, the probability that all subsequent shapechanging collisions are avoided is given by P ( f ) = e − n f = e − f N rs , norm (8)To obtain the total survival probability P survival we integrate overall possible orders in which the collisions may take place P survival = (cid:90) P ( f ) N rs , norm + d f (9)where we use the weight factor 1 / ( N rs , norm + P survival = N rs , norm = are given in Table 4. Forcomparison, we also show in Table 4 the survival probabilitiesin the case of a primordial formation of a 67P / C-G-like bi-lobedstructure (Paper I). For the standard scenario with a giant planetinstability taking place at ∼
400 Myrs, we adapt the number of Note that the calculations presented here do not take into accountthe probability of the formation of a bi-lobe shape in a specific impactevent with a given energy. As discussed in section 3.2, this probabilityis reasonably high, but is not possible to quantify. disruptive collisions computed in Morbidelli & Rickman (2015)to the number of shape-changing collisions using Equation 4and Q reshape (for a tensile strength of 100 Pa; Paper I). The sur-vival probability then corresponds to the probability of missingall shape-changing collisions. For the scenario with no transientdisc, the survival probabilities are computed in Paper I.The results vary significantly between the di ff erent scenar-ios, and also strongly depend upon the size distribution of avail-able projectiles (characterised by q ). Preliminary results fromthe New Horizons mission to Pluto and Charon based on thecrater size frequency distribution suggest that q ∼ − . q ≥
3, we find that P survival is of the order of 1-10 % in the framework of a formation fol-lowing a sub-catastrophic collision while P survival < P survival < − , even in the conservative assumption thatthere was no initial massive transient planetesimal disc. We notethat alternative models of the dynamical evolution (Davidssonet al. 2016) (see discussion in Paper I) predict a much smallercollisional evolution and prefer shallower slopes of the size dis-tribution, which would generally increase the survival probabil-ities computed here. An overview of the various scenarios con-sidered in our study is presented in Figure 10.Note that so far we have not taken into account the pos-sibility of a final shape evolution via a fission-merging cycle(Scheeres et al. 2016). In this scenario, the bi-lobe forming colli-sions presented here would not form 67P / C-G directly, but rathera two-component structure with the right size ratio. This two-component body would then later evolve into the final shapevia (one or several) fission-merging cycle, once the comet entersthe inner solar system. In this scenario, the survival probabilitiesmight be somewhat larger than those computed above since gen-eral two-component structures require a slightly higher impactenergy to be ’destroyed’ (Paper I).It is likely that the parent body from which the 67P / C-G-like structure formed is not primordial itself (as indicated by thesurvival probabilities in Figure 10) and has already experiencedsome collisional evolution or maybe a result of a collisional cas-cade.
5. Discussion and Conclusions
The analysis of the survival of the global structure of comet67P / C-G shape (Paper I) strongly suggests that such a shape can-not be primordial. It must have formed as a result of a collisionat a subsequent time (most probably within the last Gyr). At thistime, the relative velocities between cometary-sized bodies aresuch ( V >> V esc ) that the formation mechanism invoked previ-ously, namely the collisional mergers at low velocity ( V ≈ V esc )of similar-sized bodies (Jutzi & Asphaug 2015), cannot work di-rectly anymore.In this paper, we present an alternative scenario. We inves-tigate the final shapes resulting from a new type of low-energy,sub-catastrophic impacts on elongated, rotating bodies, using a3D SPH shock physics code. Our modelling results suggest thatsuch collisions result in "splitting" events which frequently leadto formation of bi-lobe 67P / C-G-like shapes. This mechanismmight not only explain the bi-lobe shape of some cometary nu-clei but could potentially also provide an explanation for struc-
Article number, page 8 of 10. Jutzi, W. Benz: Formation of bi-lobed shapes by sub-catastrophic collisions (formed by low velocity collision) For T disk = 400 Myrs , Y T = 100 Pa: P survival (-2.5) < ; P survival (-3.0) < ; P survival (-3.5) < formation by catastrophic break-up P survival ( q ): probability to miss all subsequent shape-changing or disruptive collisions for different values of q; N reshape : number of shape-changing collisions; N rs,norm number of shape-changing collisions for one formation event; f : fraction of collisions taking place after bi-lope formation: N disrupt : number of disruptive collisionssome collisional evolutionsome collisional evolutionprimordial body formation by sub-catastrophic collision todaytodaytoday P survival (-2.5) ~ ; P survival (-3.0) ~ ; P survival (-3.5) ~ P survival (-2.5) ~ P survival (-3.0) ~ P survival (-3.5) ~ For T disk = 400 Myrs: P survival (-2.5) < ; P survival (-3.0) < ; P survival (-3.5) < today primordial bodyprimordial bodyprimordial body N rs,norm * fN rs,norm * fN reshape N disrupt massive transient disk T disk ~ 400 Myrs (nominal) or T disk ~ 0 Myrs (hypothetical) disk dispersal phase / scattered disk (~4 Gyrs)planet formation phase (a few Myrs) For T disk = 0 Myrs, Y T = 100 Pa: P survival (-2.5) ~ ; P survival (-3.0) ~ ; P survival (-3.5) ~ For T disk = 0 Myrs: P survival (-2.5) ~ ; P survival (-3.0) ~ ; P survival (-3.5) ~ (1)(2)(4)(5) formation by catastrophic break-upprimordial body today P survival (-2.5) ~ ; P survival (-3.0) ~ ; P survival (-3.5) ~ N rs,norm * f (3) For T disk = 400 Myrs: P survival (-2.5) < ; P survival (-3.0) < ; P survival (-3.5) < For T disk = 0 Myrs: P survival (-2.5) ~ ; P survival (-3.0) ~ ; P survival (-3.5) ~ N disrupt N rs,norm * f Fig. 10.
Scenarios for the formation of a 67P / C-G-like bi-lobe structure. For each scenario, the probability to avoid all subsequent shape-changingcollisions is shown. For the cases of a late formation by a collision, we also indicate the probability that the parent body did not experience anyprior catastrophic disruption. tures observed in the asteroid population, as for instance the par-ticular forms of the asteroids 25143 Itokawa or 4179 Toutatis.According to our model, comets are not primordial in thesense that their shape and structure formed during the initialstages of the formation of the Solar System. Rather, the finalstructure is the result of the last major shape-forming impact.The sub-catastrophic collisions investigated here provide a pos-sibility of bi-lobe formation with small impact energies. Suchsmall-scale impacts are much more frequent than catastrophicdisruptions and the probability for such a shape-forming eventto occur without a subsequent shape-destroying event occurringuntil today is estimated to be reasonably high. We note that thetwo-component structure resulting from the type of collisionsinvestigated here might further evolve by fission-merging cycleonce the comet enters the inner Solar system, as suggested re-cently (Scheeres et al. 2016).Although the individual collisions considered in this workcan alter the global shape, their respective energy is smallenough not to lead individually to any substantial global scaleheating or compaction. In this sense, our formation model isconsistent with the observed "pristinity" of 67P / C-G (e.g. Ru-bin et al. 2015; Bieler et al. 2015). However, it is likely that theparent body from which the 67P / C-G-like structure ultimatelyformed must have undergone significant collisional evolution(Figure 10; see also Paper I), or is itself the result of a collisional disruption of a larger parent body. Several paths through the col-lisional cascade being able to lead to the similar-sized bodies,the cumulative e ff ects of impact heating and compaction experi-enced during the 4.6 Gyrs of evolution by the material compo-nents that eventually form the comets observed today are di ffi -cult to establish with certainty. This formation degeneracy im-plies that it is not possible to reconstruct uniquely the detailedcollision history nor the number and sizes of the parent bodies.Nevertheless, an upper limit for the size of parent bodies is givenby the fact that larger bodies are subjected to internal heating byshort-lived radionuclides (e.g. Prialnik et al. 2008) that will sig-nificantly alter the pristine nature of the material and therefore beincompatible with observations of porosity and content in highlyvolatile elements in comets. Interestingly, this upper limit of thesize of the parent body coupled with the requirement that the cu-mulative e ff ects of impacts in terms of compaction and volatilelosses can only be very moderate provide strong constraints forthe duration and intensity of the collisional bombardment.Whether these constraints are compatible with a scenario ofa massive planetesimal disc phase existing for 450 Myr, as pro-posed by the Nice model (Tsiganis et al. 2005; Gomes et al.2005; Morbidelli et al. 2012), remains to be analysed in detail.We note that Davidsson et al. (2016) suggest that the number ofobjects in the disc was much smaller, leading to less collisions(however, this model has other issues; see discussion in Paper I). Article number, page 9 of 10 & A proofs: manuscript no. Jutzi_and_Benz_final
On the other hand, our analysis of heating and porosity evolutionin the impacts considered here as well the regimes investigatedin Paper I (shape-changing impacts as well as catastrophic dis-ruptions) indicates that collisionally processed objects may stilllook "primitive". It is found that such bodies can still have ahigh porosity, and could have retained their volatiles, since thesecollisions generally do not lead to large-scale heating of the ma-terial bound in the largest remnant. A more detailed study of theoutcome of large-scale catastrophic disruptions is currently inprogress (Schwartz et al., in prep., 2016).In any case, given our current understanding of the dynam-ics of the small bodies in the outer solar system, it is unlikelythat the currently observed shape of comet 67P / C-G is primor-dial (even in the hypothetical scenario in which no initial massiveplanetesimal disc was existing; Paper I). According to the calcu-lations presented here, it may have formed as a result of the lastmajor shape-forming impact. Nevertheless, should future inves-tigations show that the collisional cascade does not preserve thepristine nature of cometary material, we would be facing the con-clusion that the current knowledge of the dynamics of small bod-ies in the outer regions of the solar system is seriously flawed.In this sense, comets provide invaluable tools to probe the originand evolution of our solar system.
Acknowledgements.
M.J. and W.B. acknowledge support from the Swiss NCCRPlanetS. We thank the referees B. Davidsson and N. Movshovitz for their thor-ough review which helped to improve the paper substantially.
References