Transcript Probabilities of Collisions of Migrating Bodies and Dust Particles

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/aas2010probabilities.ppt )
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Initial data. Methods of calculations
The orbital evolution of >30,000 bodies with initial orbits close to those of Jupiterfamily 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 at the 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
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(during their dynamical lifetimes) with all planets.
Calculations of the characteristic time elapsed up to the
encounter of two objects to radius of sphere rs
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).
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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 was 0.3 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 of4Myr.
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))
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Probabilities P=10-6Pr of collisions of bodies with the terrestrial planets
, ds
n1 10-9
n1 10d
n1 10d
n2 10-9
n2 10d
2P 10-9
9P 10-9
10P 10-9
22P 10-9
28P 10-9
39P 10-9
44P 10-9
3:1 10-9
5:2 10-9
N
1900
1200
1199*
4000
10000
501*
800
2149*
1000
750
750
500
288
288
V
Pr
2.4
25.4
7.88
9.9
14.7
141
1.3
28.3
1.44
1.7
1.06
2.58
1286
101
V
Tr
4.2
13.8
9.70
24.4
24.8
345
1.8
41.3
2.98
21.8
1.72
15.8
1886
173
E
Pr
4.5
40.1
4.76
11.6
14.9
110
3.7
35.6
1.76
1.9
1.19
4.01
1889
318
E
Tr
7.9
24.0
12.6
35.6
36.1
397
4.1
71
4.87
34.7
3.03
24.9
2747
371
E
Tc
1760
600
2650
3060
2420
3610
1100
1970
2770
18260
2550
6210
1450
1160
M
Pr
6.1
2.48
0.76
2.12
2.88
10.5
0.7
10.3
0.74
0.44
0.31
0.75
488
209
M
T
r
30.0 0.7
35.2 3.0
16.8 2.8
56.3 2.7
56.1 3.1
430 18.
9.7 1.2
16.4 1.6
11.0 1.6
68.9 1.9
6.82 1.6
46.3 2.0
4173 2.7
1455 0.5
Td
20
25.7
10.3
7.7
90.1
249
2.6
107
1.5
0.1
2.7
8.6
5169
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 was 0.3 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 10-910-8 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
6 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
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‘n2’, initial orbits of bodies were close to 10-20 different Jupiter-family comets.
Probabilities of collisions of migrating particles
with Venus and Mars
The 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
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pressure force and the gravitational force). Designations are the same as those for Fig. 1.
Probabilities of collisions of migrating particles
with Mercury
Depending 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.
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Probabilities of collisions of migrating particles with
Jupiter and Saturn
Probabilities 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.
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Probabilities of collisions of migrating particles
with Uranus and Neptune
Probabilities 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 12
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=410-6 (this
value is even less than those in series n1 and n2), we obtained Mw=210-4m. This
value is about the 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 = 410-7 ) 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)m estimated 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.
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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/list14
publications.htm