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215 th AAS meeting (January 5, 2010, Washington). Presentation number 344.01.

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### Calculations of the characteristic time elapsed up to the encounter of two objects to radius of sphere rs

215th AAS meeting (January 5, 2010, Washington). Presentation number 344.01.

Probabilities of Collisions of Migrating Small Bodies and Dust Particles with Planets

Sergei Ipatov

Catholic University of America, USA

- The file with this presentation can be found on http://faculty.cua.edu/ipatov/present.htm
- (http://faculty.cua.edu/ipatov/aas2010-probabilities.ppt )

Initial data. Methods of calculations

The orbital evolution of >30,000 bodies with initial orbits close to those of Jupiter-family comets (JFCs), Halley-type comets, long-period comets, and asteroids in the resonances 3/1 and 5/2 with Jupiter, and also of >20,000 dust particles produced by these small bodies was integrated a few years ago. Gravitational influence of 7 planets (Venus-Neptune) was taken into account. For dust particles, we also consider radiation pressure, Poynting-Robertson drag and solar wind drag.

First series (n1): Orbital elements were the same as those of 20 real comets (with numbers 7, 9, 10, 11, 14, 16, 17, 19, 22, 26 30, 44, 47, 51, 57, 61, 65, 71, 73, 75) with 5<P<9 yr, but different values of the mean anomaly were considered.

Second series (n2): Orbital elements were the same as those of 10 real comets (with numbers 77, 81, 82, 88, 90, 94, 96, 97, 110, 113) with 5<P<15 yr, but different values of the mean anomaly were considered.

In each of other series, initial orbits were close to the orbit of one comet (2P, 9P, 10P, 22P, 28P, 39P, or 44P).For runs with Comet 2P,we considered also Mercury.

Bodies initially located atthe 3:1 and 5:2 resonances with Jupiter.

Methods of integration. We used the SWIFT package by Levison and Duncan (Icarus, 1994, v. 108, 18-36). Evolution of N orbits was calculated using the Bulirsh-Stoer method (BULSTO) with the error per integration step less than =10-9, or =10-8, or some value between these two values. Also =10-12 and =10-13 were used. We also used a symplectic method with an integration step 3≤ds≤10, or ds=30 days (RMVS3).

The considered time interval in one run usually was equal to the largest dynamical lifetime of bodies in the run (until all bodies reached 2000 AU or collided with the Sun). Sometimes it reached several hundreds Myr.

Based on a set of orbital elements during evolution, recently we studied the probabilities of collisions of migrating particles and bodies (during their dynamical lifetimes) with all planets.

T2=6.28·kp·Ts ·R ·kv/(rs·kfi) - planar model,

T3=T2·Δi·R/rs - spatial motel;

R is the distance of encounter from the Sun, kfi is the sum of angles (in radians) with apices in the Sun, within which the distance between orbits is less than rs, Ts is the synodic period of revolution, kp=P2/P1, where P2>P1, Pi is a period of revolution of the i-th body around the Sun.

In order to take into account that velocity at distance R from the Sun differs from the mean velocity, we used coefficient kv=sqrt{2a/R-1}.

In contrast to the approach used by Opik (1951) and Arnold (1965), T depends on orbital orientations and on a synodic period.

Ipatov, S.I., Evolution times for disks of planetesimals, Soviet Astronomy, v. 32 (65), 560-566 (1988).

Ipatov, S.I. and Mather, J.C., Comet and asteroid hazard to the terrestrial planets, Advances in Space Research, v. 33, 1524-1533 (2004).

Motion of JCOs in NEO orbits for a long time

- The motion of several former Jupiter-crossing objects (JCOs) inside Jupiter’s orbit for a long time was obtained both for integration with the use of BULSTO and with a symplectic method.
- A few JCOs got orbits located inside Jupiter’s orbit and moved in such orbits for millions or even hundreds of millions of years. The probability of a collision of such object with a terrestrial planet can be greater than the total probability of thousands of other JCOs. Actually, comets split into mini-comets during many millions of years.
- For one object (from 10P runs with BULSTO) its probability of collisions with Earth and Venus was0.3 and 0.7, respectively.
- For another object (from 2P runs) during its lifetime (352 Myr) its probability of collisions with Earth, Venus and Mars was 0.172, 0.224, and 0.065, respectively. For all 12000 other objects, with BULSTO such probability was 0.2, 0.18, and 0.04, respectively.
- For series 2P, at ds=3 days (i.e., for a symplectic method), a lifetime of one object was 400 Myr, and it moved on Inner-Earth, Aten, Apollo, and Amor orbits during 2.5, 2.2, 44.9, and 80.8 Myr, respectively. At t=6.5 Myr this object got an orbit with e=0.03 and a=1.3 AU, and then until 370 Myr the eccentricity was less than 0.4 and often was even less than 0.2. The probability of a collision of this object with the Earth was about 1, and it was greater than that for all other 99 objects in that run by two orders of magnitude, i.e. greater by four orders of magnitude than the mean probability for one 2P object.
- Levison et al. (Icarus, 2006, 182, 161) argued that our obtained orbits with a~1 AU were due to the fact that collisions with terrestrial planets were not taken into account in our runs. Based on the orbital elements obtained in our runs, we can conclude that probabilities of collisions of migrating bodies in our runs before bodies got orbits with a<2 AU were very small and the reason of the transformations of orbits was not in close encounters with the terrestrial planets. Real probabilities of collisions of bodies moving in orbits with a<2 AU were only after bodies had already got such orbits and moved in them for hundreds of Myr.

Distribution of all migrating objects in runs with BULSTO(time in Myr during which a was in interval with a width of 0.005 AU (a-b) or 0.1 AU (c-d))

Probabilities P=10-6Pr of collisions of bodies with the terrestrial planets

V V E E E M M -

, ds N Pr TrPr Tr TcPr T r Td

n1 10-9 1900 2.4 4.2 4.5 7.9 1760 6.1 30.0 0.7 20

n1 10d 1200 25.4 13.8 40.1 24.0 600 2.48 35.2 3.0 25.7

n1 10d 1199* 7.88 9.70 4.76 12.6 2650 0.76 16.8 2.8 10.3

n2 10-9 4000 9.9 24.4 11.6 35.6 3060 2.12 56.3 2.7 7.7

n2 10d10000 14.7 24.8 14.9 36.1 2420 2.88 56.1 3.1 90.1

2P 10-9 501*141 3451103973610 10.5430 18. 249

9P 10-9800 1.31.83.7 4.11100 0.7 9.7 1.2 2.6

10P 10-92149*28.3 41.3 35.6 71 1970 10.3 16.4 1.6 107

22P 10-910001.44 2.98 1.76 4.87 2770 0.74 11.0 1.6 1.5

28P10-97501.7 21.8 1.9 34.7 18260 0.44 68.9 1.9 0.1

39P10-97501.06 1.72 1.19 3.03 2550 0.31 6.82 1.6 2.7

44P 10-95002.58 15.8 4.01 24.9 6210 0.75 46.3 2.0 8.6

3:1 10-9 288 1286 1886 1889 2747 1450 488 4173 2.7 5169

5:2 10-9288 101 173 318 371 1160 209 1455 0.5 1634

Tr(the mean time in a planet-crossing orbit) and Td are in Kyr, Tc=Tr/Pr in Myr, Pr =106P=106P/N for Venus=V, Earth=E, Mars=M. Td is the mean time spent in orbits with Q=a(1+e)<4.2 AU. r is the ratio of times spent in Apollo and Amor orbits. For one object (from 10P runs), its probability of collisions with Earth and Venus was0.3 and 0.7, respectively. For another object (from 2P runs) during its lifetime (352 Myr), its probability of collisions with Earth, Venus and Mars was 0.172, 0.224, and 0.065, respectively. For 12,000 other objects with BULSTO such probability was 0.2, 0.18, and 0.04, respectively. Results with the BULSTO code at 10-910-8 are marked as 10-9, those at 10-12 are marked as 10-12, and those with the RMVS3 code are at integration step ds. For the lines which do not include all bodies in a series of runs, the number of objects N is marked by *.

Probabilities of collisions of migrating particles with the Earth

Fig. 1. The probability P of collisions of dust particles and bodies (during their dynamical lifetimes) with the Earth versus β (the ratio between the radiation pressure force and the gravitational force) for particles launched from asteroids (ast), trans-Neptunian objects (tno), Comet 2P/Encke at perihelion (2P per), Comet 2P/Encke at aphelion (2P aph), Comet 2P/Encke in the middle between perihelion and aphelion (2P m), Comet 10P/Tempel 2 (10P), Comet 39P/Oterma (39P), long-period comets (lp) at e=0.995 and q=0.9 AU, and Halley-type comets (ht) at e=0.975 and q=0.5 AU (for lp and ht runs, initial inclinations were from 0 to 180o). If there are two points for the same β, then a plot is drawn via their mean value. Probabilities presented at β~10-5 are for small bodies (β=0). Probabilities presented only for bodies were calculated for initial orbits close to orbits of Comets 9P/Tempel 1 (9P), 22P/Kopff (22P), 28P/Neujmin (28P), 44P/Reinmuth 2 (44P), and test asteroids from resonances 3:1 and 5:2 with Jupiter at e=0.15 and i=10o (‘ast 3:1’ and ‘ast 5:2’). For ‘n1’ and ‘n2’, initial orbits of bodies were close to 10-20 different Jupiter-family comets.

Probabilities of collisions of migrating particles with Venus and MarsThe ratios of probabilities of collisions of JFCs with Venus, Mars, and Mercury to the mass of a planet usually were not smaller than those for Earth. Probabilities of collisions of considered particles with Venus were of the same order as those for Earth, and those for Mars were about an order of magnitude smaller.

Fig. 2. The probability P of collisions of dust particles and bodies (during their dynamical lifetimes) with Venus (left) and Mars (right) versus β (the ratio between the radiation pressure force and the gravitational force). Designations are the same as those for Fig. 1.

Probabilities of collisions of migrating particles with MercuryDepending on a source of dust, probabilities of collisions of considered particles with Mercury can be smaller or greater than for Mars.

Fig. 3. The probability P of collisions of dust particles and bodies (during their dynamical lifetimes) with Mercury versus β (the ratio between the radiation pressure force and the gravitational force). Designations are the same as those for Fig. 1.

Probabilities of Collisions of Migrating Bodies and Dust Particles with the Earth

- The probability of a collision of Comet 10P with the Earth during the dynamical lifetime of the comet was PE≈1.4∙10-4, but 80% of this mean probability was due only to one body among 2600 considered bodies with initial orbits close to that of Comet 10P. For runs for Comet 2P, PE≈(1-5)∙10-4. For most other considered JFCs, 10-6<PE<10-5. For Comets 22P/Kopff and 39P/Oterma, PE≈(1-2)∙10-6; and for Comets 9P/Tempel 1, 28P/Neujmin 1 and 44P, PE≈(2-5)∙10-6.
- For all considered JFCs, PE>4∙10-6 even if we exclude a few bodies for which the probability of a collision of one body with the Earth could be greater than the sum of probabilities for thousands of other bodies. The Bulirsh-Stoer method of integration and a symplectic method gave similar results.
- For dust particles produced by comets and asteroids, PE was found to have a maximum (~0.001-0.005) at 0.002≤β≤0.01, i.e., at d~100 μm (this value of d is in accordance with observational data). These maximum values of PE were usually (exclusive for Comet 2P) greater at least by an order of magnitude than the values for parent comets.

Probabilities of collisions of migrating particles with Jupiter and SaturnProbabilities of collisions of considered particles and bodies with Jupiter during their dynamical lifetimes are smaller than 0.1. They can reach 0.01-0.1 for bodies and particles initially moved beyond Jupiter’s orbit. For bodies and particles initially moved inside Jupiter’s orbit, the probabilities are usually smaller than the above range and can be equal to zero.

Fig. 4. The probability P of collisions of dust particles and bodies (during their dynamical lifetimes) with Jupiter (left) and Saturn (right) versus β (the ratio between the radiation pressure force and the gravitational force). Designations are the same as those for Fig. 1.

Probabilities of collisions of migrating particles with Uranus and NeptuneProbabilities of collisions of migrating particles (exclusive for trans-Neptunian particles) with other giant planets were usually smaller than those with Jupiter. The total probability of collisions of any typical considered body or particle with all planets didn’t exceed 0.2.

Fig. 5. The probability P of collisions of dust particles and bodies (during their dynamical lifetimes) with Uranus (left) and Neptune (right) versus β (the ratio between the radiation pressure force and the gravitational force). Designations are the same as those for Fig. 1.

Delivery of water to the terrestrial planets during the formation of the giant planets

- The total mass of water delivered to the Earth during the formation of the giant planets is Mw=MJPJEki, where MJ is the total mass of planetesimals from the feeding zones of these planets that got Jupiter-crossing orbits during evolution, PJE is the probability P of a collision of a JCO with the Earth during its lifetime, and ki is the portion of water ices in planetesimals.
- For MJ=100m (where m is the mass of the Earth), ki=0.5, and PJE=410-6(this value is even less than those in series n1 and n2), we obtainedMw=210-4m. This value is aboutthe mass of the Earth’s oceans.
- The larger value of P for Earth we have calculated compared to those argued by Morbidelli et al. [2000] (P (1-3)10-6)and Levison et al. [2001] (P = 410-7 ) is caused by the fact that in our runs we considered a larger number of Jupiter-crossing objects and the main portion of the probability of collisions was caused by a small fraction (0.01-0.001)of bodies, each of which moved at least several hundreds of thousands years in the Earth-crossing orbits with Q<4.7 AU. We also considered that the mass supply from the Uranus-Neptune region is about 100m , in contrast to (20-30)mestimated by other authors, which increase the volatiles delivery. The total mass of water delivered to Venus can be of the same order of magnitude and that delivered to Mars can be less by a factor of 3 or 4 than that for Earth.Ancient Venus and Mars could have large oceans.

References

[1] Ipatov, S.I. (1987) Accumulation and migration of the bodies from the zones of giant planets, Earth, Moon, and Planets, v. 39, N 2, pp. 101-128.

[2] Ipatov, S.I. (1993) Migration of bodies in the accretion of planets, Solar System Research (translated from Astronomicheskii Vestnik), v. 27, N 1. pp. 65-79.

[3] Ipatov, S.I. (2001) Comet hazard to the Earth, Adv. in Space Research, v. 28, N 8, pp. 1107-1116.

[4] Ipatov, S.I. and Mather, J.C. (2003) Migration of trans-Neptunian objects to the terrestrial planets, Earth, Moon, and Planets, v. 92, 89-98.

[5] Ipatov, S.I. and Mather, J.C. (2004) Migration of Jupiter-family comets and resonant asteroids to near-Earth space, Annals of the New York Academy of Sciences, v. 1017, pp. 46-65.

[6] Ipatov, S.I. and Mather, J.C. (2004) Comet and asteroid hazard to the terrestrial planets, Adv. in Space Research, v. 33, N 9, 1524-1533.

[7] Ipatov, S.I. and Mather, J.C. (2006) Migration of small bodies and dust to near-Earth space, Adv. in Space Research, v. 37, N 1, 126-137.

[8] Ipatov, S.I. and Mather, J.C. (2007) Migration of comets to the terrestrial planets, Proc. of the IAU Symposium No. 236 “Near-Earth Objects, Our Celestial Neighbors: Opportunity and Risk” (14-18 August 2006, Prague, Czech Republic), ed. by A. Milani, G.B. Valsecchi & D. Vokrouhlický, pp. 55-64.

[9] Ipatov, S.I., Kutyrev, A., Madsen, G.J., Mather, J.C., Moseley, S.H., Reynolds, R.J. (2008) Dynamical zodiacal cloud models constrained by high resolution spectroscopy of the zodiacal light, Icarus, v. 194, N. 2, 769-788

[10] Marov, M. Ya. and Ipatov, S.I. (2005) Migration of dust particles and volatiles delivery to the terrestrial planets, Solar System Research, v. 39, N 5, 374-380

[11] Ipatov, S.I., (2010) Collision probabilities of migrating small bodies and dust particles with planets, Proceedings of the IAU Symposium 263 "Icy bodies in the Solar System" (Rio de Janeiro, Brazil, 3-7 August, 2009), ed. by D. Lazzaro, D. Prialnik, R. Schulz, J.A. Fernandez, in press. http://arxiv.org/abs/0910.3017

- Files with the above papers are available on http://faculty.cua.edu/ipatov/list-publications.htm

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