1. #1488 Small Bodies Physics Posters. Thu, p.m., Poster location #139 ANGULAR MOMENTA OF COLLIDED RAREFIED PREPLANETESIMALS S. I. Ipatov 1,2,* 1 Space Research Institute, Moscow, Russia; 2 Department of Physics, Catholic University of America, Washington DC, 20064, USA; * Present address: Alsubai Establishment for Scientific Studies, Doha, Qatar; siipatov@hotmail.com, http://faculty.cua.edu/ipatov 1

2. Introduction The binary fractions in the minor planet population are about 2 % for main-belt asteroids, 22 % for cold classical trans-Neptunian objects (TNOs), and 5.5 % for all other TNOs (Noll 2006). There are several hypotheses of the formation of binaries for a model of solid objects. In recent years, new arguments in favor of the model of rarefied preplanetesimals - clumps have been found (e.g. Makalkin and Ziglina 2004, Johansen et al. 2007, 2011, Cuzzi et al. 2008, Lyra et al. 2008). These clumps could include meter sized boulders in contrast to dust condensations earlier considered. Sizes of preplanetesimals could be up to their Hill radii. Our studies of formation of binaries (see Ipatov 2009, 2010a-b) testify in favor of existence of rarefied preplanetesimals and can allow one to estimate their sizes at the time of their mutual collision. In the studies presented in the poster, the angular velocities used by Nesvorny et al. (AJ, 2010, 140, 785-793) as initial data for simulations of contraction of rarefied preplanetesimals are compared with the angular velocities obtained at a collision of two such preplanetesimals moving in circular orbits. The comparison showed that the angular momenta of rarefied preplanetesimals needed for formation of small- body binaries can be obtained at collisions of preplanetesimals. Trans-Neptunian objects, including trans-Neptunian binaries, could be formed from contracting rarefied preplanetesimals. 2

3. Angular momentum at a collision of two preplanetesimals moving in circular heliocentric orbits Ipatov (2010a) obtained that the angular momentum of two collided RPPs (rarefied preplanetesimals with radii r 1 and r 2 and masses m 1 and m 2 ) moved in circular heliocentric orbits equals K s =k Θ ∙(G∙M S ) 1/2 ∙(r 1 +r 2 ) 2 ∙m 1 ∙m 2 ∙(m 1 +m 2 ) -1 ∙a -3/2, where G is the gravitational constant, M S is the mass of the Sun, and a difference in semimajor axes a of RPPs equals Θ∙(r 1 +r 2 ). At r a =(r 1 +r 2 )/a<<Θ and r a <<1, one can obtain k Θ ≈(1-1.5∙Θ 2 ). The mean value of |k Θ | equals to 0.6. It is equal to 2/3 for mean positive values of k Θ and to 0.24 for mean negative values of k Θ. The resulting momentum is positive at 0<Θ<(2/3) 1/2 ≈0.8165 and is negative at 0.8165<Θ<1. The minimum value of k Θ is -0.5. In the case of uniform distribution of Θ, the probability to get a reverse rotation at a single collision is about 1/5. 3

4. Angular velocity at a collision of two preplanetesimals The angular velocity ω of the RPP of radius r=(r 1 3 +r 2 3 ) 1/3 formed as a result of a collision equals K s /J s. Momentum of inertia of a RPP is equal to J s =0.4∙χ∙(m 1 +m 2 )∙r 2, where χ=1 for a homogeneous sphere considered by Nesvorny et al. It was obtained that ω=k Θ (0.4χ) -1 ∙(r 1 +r 2 ) 2 ∙r -2 ∙m 1 ∙m 2 ∙(m 1 +m 2 ) -2 Ω, where the angular velocity of the motion of a preplanetesimal around the Sun Ω=(G∙M S ) 1/2 a -3/2. As K s /J s is proportional to (r 1 +r 2 ) 2 ∙r -2, then ω does not depend on r 1, r 2, and r, if (r 1 +r 2 )/r=const. Therefore, ω will be the same at different values of k r if we consider RPPs with radius equal to k r r Hi, where r Hi ≈a∙m i 1/3 (3M S ) -1/3 is the radius of the Hill sphere of mass m i (m 1, m 2, or m=m 1 +m 2 ). However, if at some moment of time after a collision of uniform spheres with radii r 1 and r 2, the radius r c of a compressed sphere equals k rc ∙r, then (at χ =1) the angular velocity of the compressed RPP is ω c =ω∙k rc -2. Below we consider r 1 =r 2, r 3 =2r 1 3, m 1 =m 2 =m/2, and χ=1. In this case, we have ω=1.25∙2 1/3 k Θ Ω≈1.575k Θ Ω, e.g., ω≈0.945Ω at k Θ =0.6. 4

5. Comparison of angular momenta obtained at collisions of preplanetesimals with those used by Nesvorny et al. as initial data Nesvorny et al. (2010) made computer simulations of the contraction of preplanetesimals in the trans-Neptunian region. They considered initial angular velocities of preplanetesimals equal to ω o =k ω Ω o, where Ω o =(G∙m) 1/2 r -3/2, k ω =0.5, 0.75, 1, and 1.25, m=m 1 +m 2. In most of their runs, r=0.6r H, where r H is the Hill radius for mass m S. Also r=0.4r H and r=0.8r H were used. Note that Ω o /Ω=3 1/2 (r H /r) 3/2 ≈1.73(r H /r) 3/2, e.g., Ω o ≈1.73Ω at r= r H. Considering ω=ω o, in the case of Hill spheres, we have k ω =1.25∙2 1/3 3 -1/2 k Θ χ -1 ≈0.909k Θ χ -1. This relationship shows that for k Θ =χ=1 at collisions of RPPs, it is possible to obtain the values of ω=ω o corresponding to k ω up to 0.909. In the case of a collision of Hill spheres and the subsequent contraction of the RPP to radius r c, the obtained angular velocity is ω rc =ω H (r H /r c ) 2, where ω H ≈1.575k Θ Ω. For this RPP with radius r c, ω o =(r H /r c ) 3/2 Ω oH (where Ω oH is the value of Ω o for the Hill sphere) and ω rc /ω o is proportional to (r H /r c ) 1/2. At r c =0.6r H, the collision of Hill spheres can produce K s corresponding to k ω up to 0.909∙0.6 -1/2 ≈1.17. Nesvorny et al. (2010) obtained binaries or triples only at k ω equal to 0.5 or 0.75 (not to 1 or 1.25). Therefore, one can coclude that the initial angular velocities of RPPs used by Nesvorny et al. (2010) can be obtained at collisions of RPPs. Note that the values of ω at the moment of a collision are the same at collisions at different values of k r, but ω rc is proportional to r 2 in the case of contraction of a RPP from r to r c. 5

6. Frequency of collisions of preplanetesimals The number of collisions of RPPs depends on the number of RPPs in the considered region, on their initial sizes, and on the time dependences of radii of collapsing RPPs. Cuzzi et al. (2008) obtained the «sedimentation» timescale for RPPs to be roughly 30-300 orbit periods at 2.5 AU for 300 μm radius chondrules. Both smaller and greater times of contraction of RPPs were considered by other authors. According to Lyra et al. (2009), the time of growth of Mars-size planetesimals from preplanetesimals consisted of boulders took place in five orbits. It may be possible that a greater fraction of RPPs had not collided with other RPPs before they contracted to solid bodies. For an object with mass equal to m o = 6∙10 17 kg ≈ 10 -7 M E (where M E is the mass of the Earth), e.g., for a solid object with diameter d=100 km and density ρ≈1.15 g cm -3, its Hill radius equals r Ho ≈4.6∙10 -5 a. For circular orbits separated by this Hill radius, the ratio of periods of motion of two RPPs around the Sun is about 1+1.5r Ha ≈1+7∙10 -5, where r Ha =r Ho /a. In this case, the angle with a vertex in the Sun between the directions to the two RPPs will change by 2π∙1.5r Ha ∙n r ≈0.044 radians during n r =100 revolutions of RPPs around the Sun. 6

7. Frequency of collisions of preplanetesimals Let us consider a planar disk consisted of N identical RPPs with radii equal to their Hill radii r Ho and masses m o =6∙10 17 kg. The ratio of the distances from the Sun to the edges of the disk is supposed to be equal to a rat =1.67 (e.g., for a disk from 30 to 50 AU), N=10 7 and M Σ =m o N=M E. A RPP can collide with another RPP when their semimajor axes differ by not more than 2r Ho. If the number of RPPs depends linearly on a or depends on a 2, then the mean number N m of RPPs which can collide with the RPP is ≈2N∙r Ha (a rat +1)∙(a rat -1) -1 ≈10 7 ∙4.6∙10 -5 ∙8≈3.7∙10 3 or ≈4N∙r Ha (a rat 2 +a rat +1)∙(a rat 2 -1) -1 ∙3 -1 ≈1.9∙10 3, respectively. The mean number N c of collisions of the RPP during n r revolutions around the Sun can be estimated as (1.5r Ha ∙n r )∙N m. At N m ≈3∙10 3 and r Ha ≈4.6∙10 -5, we have N c ≈0.2n r. N c is proportional to N∙r Ha 2, to N∙m o 2/3, and to M Σ ∙m o -1/3. Therefore, for N=10 5 and d=1000 km (i.e., for M Σ =10M E ), N c is also 0.2n r. Some collisions were tangent and did not result in a merger. RPPs contracted with time. Therefore, the real number of mergers can be much smaller than that for the above estimates. We suppose that the fraction of RPPs collided with other RPPs during their contraction can be about the fraction of small bodies with satellites, i.e., it can be about 1/4 in the trans-Neptunian belt. 7

8. Mergers and contraction of preplanetesimals For a primary of mass m p and a much smaller object, both in circular heliocentric orbits, the ratio of tangential velocity v τ of encounter to the escape velocity v esc-pr on the edge of the Hill sphere of the primary is v τ /v esc-pr =k Θ ∙3 -1/6 ∙(M Sun /m p ) 1/3 ∙a -1. This ratio is smaller for greater a and m p. Therefore, the capture was easier for more massive preplanetesimals and for a greater distance from the Sun (e.g., it was easier in the trans-Neptunian region than in the asteroid belt). Densities of RPPs can be very low, but their relative velocities v rel at collisions were also very small. The velocities v rel were smaller than the escape velocities on the edge of the Hill sphere of the primary (Ipatov 2010a). If collided RPPs are much smaller than their Hill spheres, and if their heliocentric orbits are almost circular before the encounter, then the velocity of the collision does not differ much from the parabolic velocity v par at the surface of the primary RPP (with radius r pc ). Indeed, v par is proportional to r pc -0.5. Therefore, collisions of RPPs could result in a merger (followed by possible formation of satellites) at any r pc <r H. Johansen et al. (2007) determined that the mean free path of a boulder inside a cluster - preplanetesimal is shorter than the size of the cluster. This result supports the picture of mergers of RPPs. In calculations made by Johansen et al. (2011), collided RPPs merged. 8

9. The period of axial rotation of a solid planetesimal formed after contraction of a rarefied preplanetesimal If a RPP got its angular momentum at a collision of two RPPs at a=1 AU, k Θ =0.6, k r =1, and m 1 =m 2, then the period T S1 of rotation of the planetesimal of density ρ=1 g cm -3 formed from the RPP with intial radius k r r H is ≈0.5 h, i.e., is smaller than the periods (3.3 and 2.3 h) at which velocity of a particle on a surface of a rotating spherical object at the equator is equal to the circular and escape velocities, respectively. The period of axial rotation T S1 of the solid planetesimal is proportional to a -1/2 ρ -2/3. Therefore, T S1 and the fraction of the mass of the RPP that could contract to a solid core are smaller for greater a. For those collided RPPs that were smaller than their Hill spheres and/or differ in masses, K s was smaller (and T S1 was greater) than for the above case with k r =1 and m 1 =m 2. 9

10. Formation of trans-Neptunian objects and their satellites The above discussion could explain why a larger fraction of binaries are found at greater distances from the Sun, and why the typical mass ratio (secondary to primary) is greater for trans-Neptunian objects (TNOs) than for asteroids. The binary fractions in the minor planet population are about 2% for main-belt asteroids, 22% for cold classical TNOs, and 5.5% for all other TNOs (Noll 2006). Note that TNOs moving in eccentric orbits (the third category) are thought to have been formed near the giant planets, closer to the Sun than classical TNOs (e.g., Ipatov 1987; Levison & Stern 2001; Gomes 2003, 2009). Some present asteroids (especially those with diameter d<10 km) can be debris of larger solid bodies. Most of rarefied preasteroids could turn into solid asteroids before they collided with other preasteroids. In the considered model of binary formation, two colliding RPPs originate at almost the same distance from the Sun. This point agrees with the correlation between the colors of primaries and secondaries obtained by Benecchi et al. (2009) for trans-Neptunian binaries. In addition, the material within the RPPs could have been mixed before the binary components formed. 10

11. Prograde and retrograde rotation of small bodies In their models Nesvorny et al. (2010) supposed that the angular velocities they used were produced during the formation of RPPs without collisions, and they had problems to explain a retrograde rotation of a collapsing clump. About 20% of collisions of RPPs moving in almost circular heliocentric orbits lead to retrograde rotation. Note that if all RPPs got their angular momenta at their formation without mutual collisions, then the angular momenta of minor bodies without satellites and those with satellites could be similar (but actually they differ considerably). In our opinion, those RPPs that formed TNOs with satellites got most of their angular momenta at collisions. The formation of classical TNOs from RPPs could have taken place for a small total mass of RPPs in the trans-Neptunian region, even given the present total mass of TNOs. Models of formation of TNOs from solid planetesimals (e.g., Stern 1995, Kenyon & Luu 1998, 1999) require a massive primordial belt and small (~0.001) eccentricities during the process of accumulation. However, the gravitational interactions between planetesimals during this stage could have increased the eccentricities to values far greater than those mentioned above (e.g., Ipatov 2007). This increase testifies in favor of formation of TNOs from rarefied RPPs. 11

12. Conclusions The angular velocities used by Nesvorny et al. (AJ, 2010, 140, 785- 793) as initial data for simulations of contraction of rarefied preplanetesimals are compared with the angular velocities obtained at a collision of two such preplanetesimals moving in circular orbits. The comparison showed that the angular momenta of rarefied preplanetesimals needed for formation of small-body binaries can be obtained at collisions of preplanetesimals. Trans-Neptunian objects, including trans-Neptunian binaries, could be formed from contracting rarefied preplanetesimals. The time of contraction of preplanetesimals in the trans-Neptunian region probably did not exceed a hundred of revolutions around the Sun. The fraction of preplanetesimals collided with other preplanetesimals during their contraction can be about the fraction of small bodies with satellites, i.e., it can be about 1/4 in the trans- Neptunian belt. 12

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