1. Моменты количества движения сталкивающихся разреженных препланетезималей С.И. Ипатов siipatov@hotmail.com, http://faculty.cua.edu/ipatov Рассматривается формирование планетезималей из разреженных препланетезималей. Показано, что моменты количества движения препланетезималей, необходимые для формирования двойных малых тел Солнечной системы (особенно в занептунной области), могли быть получены при столкновениях разреженных препланетезималей. В 2012 г. сравнивались моменты количества движения сталкивающихся разреженных препланетезималей с моментами количества движения, использовавшимися в [Nesvorny et al., AJ, 2010, 140, 785-793] в качестве начальных данных при исследовании формирования двойных малых тел при сжатии разреженных препланетезималей. A file with an older version of the presentation can be found on http://faculty.cua.edu/ipatov/iau2012planetesimals.ppt Main results of these studies were published in Mon. Not. R. Astron. Soc., 2010, v. 403, 405-414 (http://arxiv.org/abs/0904.3529).http://arxiv.org/abs/0904.3529 1

2. Introduction 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). There are several hypotheses of the formation of binaries for a model of solid objects. For example, Goldreich et al. (2002) considered the capture of a secondary component inside Hill sphere due to dynamical friction from surrounding small bodies, or through the gravitational scattering of a third large body. Weidenschilling (2002) studied collision of two planetesimals within the sphere of influence of a third body. Funato et al. (2004) considered a model for which the low mass secondary component is ejected and replaced by the third body in a wide but eccentric orbit. Studies by Astakhov et al. (2005) were based on four-body simulations and included solar tidal effects. Gorkavyi (2008) proposed multi- impact model. Ćuk, M. (2007), Pravec et al. (2007) and Walsh et al. (2008) concluded that the main mechanism of formation of binaries with a small primary (such as near-Earth objects) could be rotational breakup of ‘rubble piles’. More references can be found in the papers by Richardson and Walsh (2006), Petit et al. (2008), and Scheeres (2009). 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, 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 presented below (see also 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. 2 2

3. Scenarios of formation of binaries at the stage of rarefied preplanetesimals Application of previous solid-body scenarios to preplanetesimals. The models of binary formation due to the gravitational interactions or collisions of future binary components with an object (or objects) that were inside their Hill sphere, which were considered by several authors for solid objects, could be more effective for rarefied preplanetesimals. For example, due to almost circular heliocentric orbits, duration of the motion of preplanetesimals inside the Hill sphere could be longer and the minimum distance between centers of masses of preplanetesimals could be smaller than for solid bodies, which usually moved in more eccentric orbits. Two centers of contraction. Some collided rarefied preplanetesimals had a greater density at distances closer to their centers, and sometimes there could be two centers of contraction inside the preplanetesimal formed as a result of a collision of two rarefied preplanetesimals. For such model, binaries with close masses separated by a large distance (up to a radius of a Hill sphere) and with any value of the eccentricity of the orbit of the secondary component relative to the primary component could be formed. The observed separation distance can characterize sizes of encountered preplanetesimals. Most of rarefied preasteroids could contract into solid asteroids before they collided with other preasteroids. 3 3

4. Scenarios of formation of binaries at the stage of rarefied preplanetesimals Excessive angular momentum. Formation of some binaries could be caused by that the angular momentum that they obtained at the stage of rarefied preplanetesimals was greater than that could exist for solid bodies. During contraction of a rotating rarefied preplanetesimal, some material could form a cloud (that transformed into a disk) of material moved around the primary. One or several satellites of the primary could be formed from this cloud. The angular momentum of any discovered trans-Neptunian binary is smaller than the typical angular momentum of two identical rarefied preplanetesimals having the same total mass and encountering up to the Hill sphere from circular heliocentric orbits. Hybrid scenario. Both above scenarios could work at the same time. In this case, it is possible that besides massive primary and secondary components, there could be smaller satellites moving around the primary (and/or the secondary) at smaller distances. For binaries formed in such a way, separation distance between main components can be different (e.g. large or small). 4 4

5. Data presented in the Table For six binaries, the angular momentum K scm of the present primary and secondary components (with diameters d p and d s and masses m p and m s ), the momentum K s06ps =K sΘ =v τ ∙(r p +r s )∙m p ∙m s /(m p +m s )=k Θ ∙(G∙M Sun ) 1/2 ∙(r p +r s ) 2 ∙m p ∙m s ∙(m p +m s ) -1 ∙a -3/2 (v τ is the tangential component of velocity v col of collision) of two collided Hill spheres -preplanetesimals with masses m p and m s moved in circular heliocentric orbits at k Θ ≈(1-1.5∙Θ 2 )=0.6 (this value of |k Θ | characterizes the mean momentum; the difference in semimajor axes equaled to Θ∙(r p +r s ), r p +r s was the sum of radii of the spheres), and the momentum K s06eq of two identical collided preplanetesimals with masses equal to a half of the total mass of the binary components (i.e. to 0.5m ps, where m ps =m p +m s ) at k Θ =0.6 are presented in the Table. All these three momenta are considered relative to the center of mass of the system. The resulting momentum of two colliding spheres is positive at 0<Θ<(2/3) 1/2 ≈0.8165 and is negative at 0.8165<Θ<1. Formulas and other details of calculation of momenta can be found in Ipatov (2010a). K spin =0.2π∙χ∙m p ∙d p 2 ∙T sp -1 is the spin momentum of the primary (χ=1 for a homogeneous sphere; T sp is the period of spin rotation of the primary). L is the distance between the primary and the secondary. In this Table we also present the values of 2L/d p and L/r Htm, where r Htm is the radius of the Hill sphere for the total mass m ps of the binary. Three velocities are presented in the last lines of the Table, where v τpr06 is the tangential velocity v τ of encounter of Hill spheres at present masses of components of the binary, v τeq06 is the value of v τ for encounter of Hill spheres at masses equal to 0.5m ps each, and v esc-pr is the escape velocity on the edge of the Hill sphere of the primary. 5 5

6. binaryPluto (90842) Orcus 2000 CF 105 2001 QW 322 (87) Sylvia (90) Antiope a, AU39.4839.343.843.943.493.156 d p, km2340950170108?28688 d s, km1212260120108?1884 m p, kg 1.3×10 22 7.5×10 20 2.6×10 18 ?6.5×10 17 ?1.478×10 19 4.5×10 17 m s, kg 1.52×10 21 1.4×10 19 for ρ=1.5 g cm -3 9×10 19 ?6.5×10 17 ? 3×10 15 for ρ=1 g cm -3 3.8×10 17 L, km19,570870023,000120,0001356171 L/r Htm 0.00250.00290.040.30.0190.007 2L/d p 16.918.327122009.53.9 T sp, h153.3105.1816.5 K scm, kg∙km 2 ∙s -1 6×10 24 9×10 21 5×10 19 3.3×10 19 10 17 6.4×10 17 K spin, kg∙km 2 ∙s -1 10 23 10 22 1.6×10 18 at T s =8 h 2×10 17 at T s =8 h 4×10 19 3.6×10 16 K s06ps, kg∙km 2 ∙s -1 8.4×10 25 9×10 22 1.5×10 20 5.2×10 19 3×10 17 6.6×10 18 K s06eq, kg∙km 2 ∙s -1 2.8×10 26 2×10 24 2.7×10 20 5.2×10 19 8×10 20 6.6×10 18 (K scm +K spin )/K s06eq 0.020.010.20.630.050.1 v τeq06, m∙s -1 6.12.20.360.262.00.82 v τpr06, m∙s -1 5.51.80.30.261.30.82 v esc-pr, m∙s -1 15.05.80.80.535.31.7 Table. Angular momenta of several small-body binaries 6

7. Comparison of angular momenta of present binaries with model angular momenta For the binaries presented in the Table, the ratio r K =(K scm +K spin )/K s06eq (i.e., the ratio of the angular momentum of the present binary to the typical angular momentum of two colliding preplanetesimals – Hill spheres moving in circular heliocentric orbits) does not exceed 1. For most of observed binaries, this ratio is smaller than for the binaries considered in the Table. Small values of r K for most discovered binaries can be due to that preplanetesimals had already been partly compressed at the moment of collision (could be smaller than their Hill spheres and/or could be denser for distances closer to the center of a preplanetesimal). Petit et al. (2008) noted that most other models of formation of binaries cannot explain the formation of the trans-Neptunian binary 2001 QW 322. For this binary we obtained that the equality K sΘ =K scm is fulfilled at k Θ ≈0.4 and v τ ≈0.16 m/s. Therefore in our approach this binary can be explained even for circular heliocentric orbits of two collided preplanetesimals. 7

8. Formation of axial rotation of Pluto and inclined mutual orbits of components Pluto has four satellites, but the contribution of three satellites (other than Charon) to the total angular momentum of the system is small. To explain Pluto’s tilt of 120 o and inclined mutual orbit of 2001 QW 322 components (124 o to ecliptic), we need to consider that thickness of a disk of preplanetesimals was at least of the order of sizes of preplanetesimals that formed these systems. Inclined mutual orbits of many trans-Neptunian binaries testify in favor of that momenta of such binaries were acquired mainly at single collisions of rarefied preplanetesimals, but not due to accretion of much smaller objects (else primordial inclinations of mutual orbits relative to the ecliptic would be small). It is not possible to obtain reverse rotation if the angular momentum was caused by a great number of collisions of small objects with a larger preplanetesimal (for such model, the angular momentum K s and period T s of axial rotation of the formed preplanetesimal were studied by Ipatov 1981a-b, 2000). 8 8

9. -- Sizes of preplanetesimals In the considered model, sizes of preplanetesimals comparable with their Hill spheres are needed only for the formation of binaries at a separation distance L close to the radius r Htm of the Hill sphere (such as 2001 QW 322 ). For other binaries presented in the Table (and for most discovered binaries), the ratio L/r Htm does not exceed 0.04. To form such binaries, sizes of preplanetesimals much smaller (at least by an order of magnitude) than the Hill radius r Htm are enough. The observed separation distance L can characterize the sizes of contracted preplanetesimals. 9

10. Capture of colliding 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 (designations are presented on page 5 in “Data presented in the Table”). 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 for preplanetesimals in the trans-Neptunian region than in the asteroid belt). The ratio of the time needed for contraction of preplanetesimals to the period of rotation around the Sun, and/or the total mass of preplanetesimals could be greater for the trans-Neptunian region than for the initial asteroid belt. It may be one of the reasons of a larger fraction of trans-Neptunian binaries than of binaries in the main asteroid belt. At greater eccentricities of heliocentric orbits, the probability of that the encountering objects form a new object is smaller (as collision velocity and the minimum distance between centers of mass are greater and the time of motion inside the Hill sphere is smaller) and the typical angular momentum of encounter up to the Hill sphere is greater. 10 10

11. Angular momentum at a collision of two preplanetesimals moving in circular heliocentric orbits Ipatov (2010a) obtained that the angular momentum of two collided RPPs (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. 11

12. 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 (0.4 χ) -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 radii k r r Hi, where r Hi ≈a m i 1/3 (3 M S ) -1/3 is the radius of the Hill sphere of mass m i (m 1, m 2, or m). 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 =2 r 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 ≈1.575 k Θ Ω, e.g., ω≈0.945 Ω at k Θ =0.6. 12

13. Comparison of angular momenta obtained at collisions 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 S ) 1/2 r -3/2, k ω = 0.5, 0.75, 1, and 1.25, m S = m 1 + m 2. In most of their runs, r=0.6 r H, where r H is the Hill radius for mass m S. Also r=0.4 r 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 RRPs, 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. 13

14. 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∙1.5 r Ha n r ≈0.044 radians during n r =100 revolutions of RPPs around the Sun. 14

15. 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). For 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 2 r Ho, i.e., the mean number of RPPs which can collide with the RPP is ~N ∙4 r Ho /(a rat -1)≈10 7 ∙4∙4.6∙10 -5 /0.67≈2.7∙10 3. The mean number N c of collisions of the RPP during n r revolutions around the Sun can be estimated as (N ∙4 r Ha /(a rat -1)) (2 ∙1.5 r Ha n r ) ≈2.7 ∙10 3 ∙1.4 ∙10 -4 n r ≈0.4 n 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 0.4n 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. 15

16. Mergers and contraction of preplanetesimals 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 < 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.. 16

17. 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. 17

18. 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. 18

19. 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 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 my 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, Davis & Farinella 1997, 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. 19

20. Conclusions The models of binary formation due to the gravitational interactions or collisions of future binary components with an object (or objects) that were inside their Hill sphere, which were considered by several authors for solid objects, could be more effective for rarefied preplanetesimals. Some collided rarefied preplanetesimals had a greater density at distances closer to their centers, and sometimes there could be two centers of contraction inside the rotating preplanetesimal formed as a result of a collision of two rarefied preplanetesimals. In particular, binaries with close masses separated by a large distance and with any value of the eccentricity of the orbit of the secondary component relative to the primary component could be formed. The observed separation distance can characterize the sizes of contracted preplanetesimals. Most of rarefied preasteroids could contract into solid asteroids before they collided with other preasteroids. Formation of some binaries could have resulted because the angular momentum that they obtained at the stage of rarefied preplanetesimals was greater than that could exist for solid bodies. During contraction of a rotating rarefied preplanetesimal, some material could form a disk of material moving around the primary. One or several satellites of the primary could be formed from this cloud. 20 20

21. Conclusions Both above scenarios could take place at the same time. In this case, it is possible that besides massive primary and secondary components, there could be smaller satellites moving around the primary (and/or secondary). The angular momentum of any discovered trans-Neptunian binary is smaller than the typical angular momentum of two identical rarefied preplanetesimals having the same total mass and encountering up to the Hill sphere from circular heliocentric orbits. This difference and the separation distances, which are usually much smaller than radii of Hill spheres, testify in favor of that most of preplanetesimals had already been partly compressed at the moment of a mutual collision. 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. 21

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