PD3/EX4. P159. EPSC2013-290

ANGULAR MOMENTA OF COLLIDED RAREFIED PREPLANETESIMALS

Sergei I. Ipatov 1,2 1 Space Research Institute, Moscow, Russia;

2 Department of Physics, Catholic University of America, Washington DC, 20064, USA; siipatov@hotmail.com

Introduction 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. (2010) as initial data for simulations of contraction of rarefied preplanetesimals are compared with the angular velocities obtained at a collision of two rarefied 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.

Angular momentum and angular velocity 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 r1 and r2 and masses m1 and m2) moved in circular heliocentric orbits equals

Ks=kΘ∙(G∙MS)1/2∙(r1+r2)2∙m1∙m2∙(m1+m2)-1∙a-3/2, where G is the gravitational constant, MS is the mass of the

Sun, and a difference in semimajor axes a of RPPs equals Θ∙(r1+r2). At ra=(r1+r2)/a<<Θ and ra<<1, one can obtain kΘ≈(1-1.5∙Θ2). The mean value of |kΘ| equals 0.6. It is equal to 2/3 for mean positive values of kΘ and to 0.24 for mean negative values of kΘ. 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 kΘ 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. The angular velocity ω of the RPP of radius r=(r1

3+r23)1/3 formed as a result of a collision equals Ks/Js.

Momentum of inertia of a RPP is equal to Js=0.4∙χ∙(m1+m2)∙r2, where χ=1 for a homogeneous sphere considered by Nesvorny et al. It was obtained that ω=kΘ(0.4χ)-1∙(r1+r2)2∙r-2∙m1∙m2∙(m1+m2)-2Ω, where the angular velocity of the motion of a preplanetesimal around the Sun Ω=(G∙MS)1/2a-3/2. As Ks/Js is proportional to (r1+r2)2∙r-2, then ω does not depend on r1, r2, and r, if (r1+r2)/r=const. Therefore, ω will be the same at different values of kr if we consider RPPs with radius equal to krrHi, where rHi≈a∙mi

1/3(3MS)-1/3 is the radius of the Hill sphere of mass mi (m1, m2, or m=m1+m2). However, if at some moment of time after a collision of uniform spheres with radii r1 and r2, the radius rc of a compressed sphere equals krc∙r, then (at χ =1) the angular velocity of the compressed RPP is ωc=ω∙krc

-2. Below we consider r1=r2, r3=2r13, m1=m2=m/2, and χ=1. In this case, we have

ω=1.25∙21/3kΘΩ≈1.575kΘΩ, e.g., ω≈0.945Ω at kΘ=0.6.

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 ωo=kω Ωo, where Ωo=(G∙m)1/2r-3/2, kω=0.5, 0.75, 1, and 1.25, m=m1+m2 . In most of their runs, r=0.6rH, where rH is the Hill radius for mass m. Also r=0.4rH and r=0.8rH were used. Note that Ωo/Ω=31/2(rH/r)3/2≈1.73(rH/r)3/2, e.g., Ωo≈1.73Ω at r= rH. Considering ω=ωo, in the case of Hill spheres, we have kω=1.25∙21/33-1/2kΘχ-1≈0.909kΘχ-1. This relationship shows that for kΘ=χ=1 at collisions of RPPs, it is possible to obtain the values of ω=ωo corresponding to kω up to 0.909. In the case of a collision of Hill spheres and the subsequent contraction of the RPP to radius rc, the obtained angular velocity is ωrc=ωH(rH/rc)2, where ωH≈1.575kΘΩ. For this RPP with radius rc, ωo=(rH/rc)3/2ΩoH (where ΩoH is the value of Ωo for the Hill sphere) and ωrc/ωo is proportional to (rH/rc)1/2. At rc=0.6rH, the collision of Hill spheres can produce Ks corresponding to kω up to 0.909∙0.6-1/2≈1.17. Nesvorny et al. (2010) obtained binaries or triples only at kω 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 kr=r/rHi, but ωrc is proportional to r2 in the case of contraction of a RPP from r to rc .

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 mo = 6∙1017 kg ≈ 10-7 ME (where ME 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 rHo≈4.6∙10-5a. For circular orbits separated by this Hill radius, the ratio of periods of motion of two RPPs around the Sun is about 1+1.5rHa≈1+7∙10-5, where rHa=rHo/a. In this case, the angle with a vertex in the Sun between the directions to the two RPPs will change by 2π∙1.5rHa∙nr≈0.044 radians during nr=100 revolutions of RPPs around the Sun. Let us consider a planar disk consisted of N identical RPPs with radii equal to their Hill radii rHo and masses mo=6∙1017 kg. The ratio of the distances from the Sun to the edges of the disk is supposed to be equal to arat=1.67 (e.g., for a disk from 30 to 50 AU), N=107 and MΣ=moN=ME. A RPP can collide with another RPP when their semimajor axes differ by not more than 2rHo. If the number of RPPs depends linearly on a or depends on a2, then the mean number Nm of RPPs which can collide with the RPP is ≈2N∙rHa(arat+1)∙(arat-1)-1 ≈107∙4.6∙10-5∙8≈3.7∙103 or ≈4N∙rHa(arat

2+arat+1)∙(arat2-1)-1∙3-1≈1.9∙103, respectively. The mean number Nc of collisions of the RPP during nr

revolutions around the Sun can be estimated as (1.5rHa∙nr)∙Nm. At Nm≈3∙103 and rHa≈4.6∙10-5, we have Nc≈0.2nr. Nc is proportional to N∙rHa

2, to N∙mo2/3, and to MΣ∙mo

-1/3. Therefore, for N=105 and d=1000 km (i.e., for MΣ=10ME), Nc is also 0.2nr. 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 0.3 in the trans-Neptunian belt.

Mergers and contraction of preplanetesimals For a primary of mass mp and a much smaller object, both in circular heliocentric orbits, the ratio of tangential velocity vτ of encounter to the escape velocity vesc-pr on the edge of the Hill sphere of the primary is vτ/vesc-pr=kΘ∙3-1/6∙(MSun/mp)1/3∙a-1. This ratio is smaller for greater a and mp. 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 vrel at collisions were also very small. The velocities vrel 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 vpar at the surface of the primary RPP (with radius rpc). Indeed, vpar is proportional to rpc

-0.5. Therefore, collisions of RPPs could result in a merger (followed by possible formation of satellites) at any rpc<rH. 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.

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, kΘ=0.6, kr=1, and m1=m2, then the period TS1 of rotation of the planetesimal of density ρ=1 g cm-3 formed from the RPP with intial radius krrH 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 TS1 of the solid planetesimal is proportional to a-1/2ρ-2/3. Therefore, TS1 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, Ks was smaller (and TS1 was greater) than for the above case with kr=1 and m1=m2.

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 0.3 for cold classical TNOs and 0.1 for all other TNOs (Noll et al. 2008). 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.

Prograde and retrograde rotation of small bodies and binaries In their models Nesvorny et al. (2010) supposed that the angular velocities of RPPs 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.

The angular momentum of the preplanetesimal formed by accumulation of small objects As it was noted by Ipatov (2010a), the period of axial rotation of the RPP – Hill sphere that grew by collisions with smaller objects equals Ts≈7∙χ-1∙a3/2∙(G∙MS)-1/2∙ΔK (the typical tangential velocity of collisions was supposed to be equal to vτ=0.6∙vc∙rH∙a-1), where ΔK is the difference between the fraction of positive increments of angular momentum and the fraction of negative increments, vc is the velocity of motion of a RPP. After the contraction of the RPP to radius r=krH∙rH, its angular velocity equals ω≈0.9∙χ-1∙a-3/2∙(G∙MS)1/2∙ΔK∙krH

-2. Considering ω=ωo=kωΩo and rH≈a(m/3MS)1/3, we obtain kω≈0.9∙χ-1∙3-1/2ΔK∙krH

-2. At ΔK=0.9 and χ=1, kω≈0.47krH-2. As kω≈1.3 at krH=0.6, then the RPP formed by accumulation of

smaller objects moved in almost circular heliocentric orbits could get the angular velocity equal to that used by Nesvorny et al. (2010) as initial data. However, for such a model, all TNOs could formed with satellites, and the angular momentum of the formed systems would be positive. The model of growth of a RPP by accumulation of small objects moved in almost circular orbits does not explain a large difference between angular momenta of single and binary small bodies. For such model, all trans-Neptunian objects formed from such preplanetesimals would have satellites because the angular momentum of the preplanetesimal would be much greater than that for a single trans-Neptunian object.

Conclusions The angular velocities used by Nesvorny et al. (2010) as initial data for simulations of contraction of rarefied preplanetesimals are compared with the angular velocities obtained at a collision of two 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 0.3 in the trans-Neptunian belt.

ReferencesBenecchi S.D., Noll K.S., Grundy W.M., Buie M.W., Stephens D.C., Levison H.F., 2009, Icarus 200, 292-303Cuzzi J.N., Hogan R.C., Shariff K., 2008, Astrophys. J. 687, 1432-1447Goldreich P., Ward W.R., 1973, Astrophys. J. 183, 1051-1061Gomes R.S., 2003, Icarus 161, 404-418Gomes R.S., 2009, Celestial Mechanics 104, 39-51Ipatov S.I., 1987, Earth, Moon, & Planets 39, 101-128 Ipatov S.I., 2007, abstract, Lunar. Planet. Sci. XXXVIII, #1260 Ipatov S.I., 2009, abstract, Lunar. Planet. Sci. XL, #1021 Ipatov S.I., 2010a, MNRAS, 403, 405-414, http://arxiv.org/abs/0904.3529 Ipatov S.I., 2010b, Proc. IAU Symp 263, IAU v. 5, Cambridge Univ. Press, pp. 37-40 Ipatov, S.I., 2013, Proc. IAU Symp. No. 293, in press. Johansen A., Oishi J. S., Mac Low M.-M., Klahr H., Henning T., Youdin A., 2007, Nature 448, 1022-1025Johansen A., Klahr H., & Henning T., 2011, Astron. Astrophys. 529, A62, 16 pagesKenyon S.J., Luu J.X., 1998. Astron. J. 115, 2136-2160Kenyon S.J., Luu J.X., 1999. Astron. J. 118, 1101-1119Levison H.F., Stern S.A., 2001, Astron. J. 121, 1730-1735Lyra W., Johansen A., Klahr H., Piskunov N., 2008. Astron. Astrophys 491, L41-L44Makalkin A.B., Ziglina I.N., 2004, Sol. Syst. Research, 38, 288-300Nesvorny D., Youdin A.N., Richardson D.C., 2010, Astron. J. 140, 785-793Noll, K.S., Grundy, W.M., Stephens, D.C., Levison, H.F., Kern, S.D., 2008, Icarus 194, 758-768. Stern S.A., 1995, Astron. J., 110, 856-868

The file with this 1-page poster is on http://faculty.cua.edu/ipatov/epsc2013-planetesimals.ppt . The source file of the poster (several A4 pages) is on http://faculty.cua.edu/ipatov/epsc2013s-planetesimals.ppt