UDC 523.52

Migration of Small Bodies to Earth

(Short title: Migration of Small Bodies)

S. I. Ipatov

Keldysh Institute of Applied Mathematics, Russian Academy
of Sciences, Miusskaya pl. 4, Moscow, 125047 Russia

@Abstract - The present migration of bodies to the Earth from the
asteroid and trans-Neptune belts is investigated. Computer
simulations of the evolution of disks that originally consisted of
planets and hundreds of bodies located in various regions of the
Solar System were carried out. It was found that either perihelia
or aphelia of orbits of bodies that collide the Earth lie mainly
near the Earth's orbit. The orbit of a near-Earth asteroid (NEA)
varies significantly before the collision of an NEA with the
planet. Most of the asteroids of the Amor group should have come from
the asteroid belt. Mean lifetimes of the main-belt asteroids with
diameter less than 100 km are shorter than the age of the Solar
System. A certain fraction of bodies migrating to the Earth
from various regions of the Solar System, has replenished the family
of asteroids with orbits entirely inside Earth's orbit, and
several of them lie inside the orbit of Venus. The number of
asteroids of this family may be large. If the orbital elements of two
celestial bodies vary before their collision, the characteristic
time elapsed up to this collision may be several times less than
that for fixed average inclinations and eccentricities. The mean
time between collisions of NEAs with diameters greater than 1 km
with Earth may be less than 100 000 yr. The number of NEAs
that are ejected into hyperbolic orbits is about ten times as
large as the number of objects colliding with Earth. For most bodies
that collide with Earth, the time interval between settling into an
orbit intersecting with the Earth's orbit and colliding with Earth does
not exceed 10 Myr.

Received November 10, 1994

INTRODUCTION

     A large number of papers devoted to the migration of small
bodies in the Solar System were published during last years. The
bodies from the internal part of the asteroid belt, in particular,
from the secular resonances @[nu]@5, @[nu]@6, and @[nu]@16
(Morbidelli @et al.@, 1994; Ferraz-Mello, 1994; Froeschl@e and
Morbidelli, 1994; Froeschl@e and Scholl, 1988, 1989; Froeschl@e
@et al.@, 1991; Morbidelli @et al.@, 1994; Scholl and Froeschl@e,
1990, 1991), the Kirkwood gaps (Wetherill, 1985, 1987, 1988,
1989a,b, 1991, 1992; Wetherill and Chapman, 1988; Wisdom, 1985),
the trans-Neptune belts (E@neev, 1980; Wetherill, 1991), and the
Hills and Oort clouds (Marochnick @et al.@, 1989) are considered
to be the present sources of replenishment of near-Earth asteroids
(NEAs), as well as meteorites colliding the Earth. The above
sources are listed in order of increasing distances from the Sun.
The asteroid belt and the trans-Neptune belt are considered to be
the main sources.
     The regions of values of semimajor axes @a of orbits, which
correspond to minima in the distribution of asteroids with @a, are
called the Kirkwood gaps. For these gaps the ratio of periods of
revolution of an asteroid and Jupiter around the Sun is close to
3:1, 5:2, 2:1, or 7:3. At the resonances @[nu]@5 (@g = g@5) and
@[nu]@6 (@g = g@6) the mean precessional rate @g@[omega]@ of the
longitude of perihelion of an asteroidal orbit is equal to the
mean precessional rate of the longitude of perihelion of Jupiter
(@g@5@[omega]@J@ 4".30 / yr) and Saturn (@g@6@[omega]@S@ 27".77 /
yr), respectively. At the resonance @[nu]@16, we have the relation
@[Omega]@[Omega]@J@ -25".73 / yr, where @[Omega]@ and @[Omega]@J@
are the mean precessional rates of the longitude of ascending node
of orbits of an asteroid and Jupiter, respectively (Froeschl@e and
Morbidelli, 1994; Froeschl@e and Scholl, 1989).
     NEAs are conventionally divided into the following groups
(Sokolsky, 1992): asteroids of the Apollo group (@a > 1 AU, @q =
@a(1-e) < 1.02 AU), asteroids of the Amor group (1.02 < @q < 1.33
AU), asteroids of the Aten group (@a < 1 AU), asteroids of the
group @X (@Q = a(1+e) < 1.02 AU), as well as short-period comets
and meteor streams with large fragments. Here @a is the semimajor
axis of an asteroid orbit, @e is its eccentricity, @q and @Q are
the perihelion and aphelion distances, respectively.
     According to Wetherill (1988, 1989a) and Weissman @et al@.
(1989), it is difficult to explain the number of objects of the
Apollo and Amor groups and features of their orbits (for example,
their mean inclinations, which are larger than those in the main
asteroid belt), if one considers only asteroidal sources. Therefore,
a considerable part of NEAs can belong to extinct comets. The
results of investigations of transformation of small bodies'
orbits under the influence of planets, submitted by Andreev @et
al.@ (1991), also testify in favor of the comet origin of some
part of NEAs and meteorites, which was suggested by O@pik (1963).
The orbit of the asteroid 3200 almost coincides with the orbit of
the meteorite stream Gemenid, this is an indication of cometary
origin of this asteroid (Babadzhanov, 1987). Reviews of papers on
the origin of meteorites were published by Sobotovich and
Semenenko (1985) and Shor (1973). According to Babadzhanov (1987),
about 98 - 99% of meteor bodies with masses less than 100 g in the
vicinity of the Earth have cometary origin. Britt @et al.@ (1992)
have shown that spectral variations in the collection of
meteorites are considerably wider than those of asteroids. These
authors have not found direct meteorite analogies among the
asteroids of B-, D-, F-, P-, and T-types in the external part of
the asteroid belt.
     Several large objects were found in the zone of the giant
planets. The diameter of Hidalgo (@a = 5.79 AU, @e = 0.71,
inclination @i = 42.5@[degree]) is considered to be equal to 39-60
km (Olsson-Steel, 1987), and that of Chiron (@a = 13.67 AU, @e =
0.38, @i = 6.9@[degree]) is equal to 180 km (Hartmann @et al.@,
1990) or @300 km (Luu, 1994). Diameters of objects 5145 (Pholus)
and 1993 HA@2 , which, as well as Chiron, came from the
trans-Neptune belt, are equal to 190 and 60 km, respectively.
Dimensions of the object 1991 DA (5335 Damocles; @a = 11.87 AU, @e
= 0.867, @i = 61.89@[degree], @q = a(1-e) = 1.58 AU) are
considerably less and are about 5 - 8 km (Bailey and Hahn, 1992).
The object 1994 TA (@a = 15.1 AU, @e = 0.0 and @i = 6.4@[degree])
was discovered recently. Bailey (1992) considers that certain
comets belonging to the Jupiter family were formed 10@4 years ago
as a result of disintegration of a hypothetical comet as large as
Chiron, because the number of comets of the Jupiter family cannot
be explained if only the capture of near-parabolic comets is taken
into account.
     According to Whipple (1964), Fernandez (1980), and E@neev
(1980), distances of bodies of the trans-Neptune belt (the Kuiper
belt) from the Sun basically lie in the range from 40 to 60 AU,
and the total mass of the bodies of this belt @M@[Sigma]@10@m@,
where @m@ is the mass of the Earth. Hamid @et al.@ (1968)
supposed that @M@[Sigma]@1.3m@ at @a@50 AU. Marov (1994) evaluated
the total number of comet-like bodies in the Kuiper belt to be ~
10@8 and their total mass @M@[Sigma]@0.002m@. The first object
(1992 QB1) of the trans-Neptune belt was found in 1992 at a
distance of 42 AU (Cowen, 1992). By the end of 1994, 17 objects
were discovered in the trans-Neptune belt. Semimajor axes of
orbits of six trans-Neptune objects presented by Luu (1994) lie in
interval 32.3 - 43.8 AU, @e@0.07, @i < 8@[degree], and their
diameters are located in the interval from 100 to 280 km. As these
objects were detected in the region of about 1.5 sq. degree of the
sky, Luu (1994) considers that there are more than 2@10@4 such
objects in the trans-Neptune belt, and their total mass @M@0.03m@.
Stern (1991) considers that diameters of the largest objects can
exceed 1000 km.
     According to Bailey (1990), the planar Hills cloud is located
at the distance 10@3@a@2 - 3@10@4 AU and the spherically symmetric
Oort cloud lies at the distance 2 - 3@10@4@a@10@5 AU from the Sun.
Hills (1981) believed the border of between the Oort and Hills
clouds to be equal to 1 - 2@10@4 AU and considered the mass of the
Hills cloud to be by two orders greater than that of the Oort
cloud. Tsitsin @et al.@ (1993) think that the ensemble of relic
planetesimals begins with the asteroid belt and extends up to the
external part of the Hills cloud.
     Yeates (1989), Frank @et al.@ (1986a,b) suppose that a large
number of small snow comets exists near the Earth. However,
Dessler (1991) put forward serious objections to the existence of
a large number of such comets.
     The number of bodies in the asteroid belt with diameters @d
greater than a certain value @D is considered to be proportional to
@D@[-[alfa]]. For asteroids with radius equal to several
kilometers, @[alfa] is in the interval (2, 2.5), and for particles
colliding Gaspra and having diameters from 20 to 60 m it is close
to 3.3 (Farinella @et al.@, 1994); i.e., the values of @[alfa] are
different for various @D. Bailey (1994) supposes that @[alfa] = 2
for comets. For objects crossing the orbit of the Earth, @[alfa]@2
(Farinella @et al.@, 1994). Canavan (1993) has shown that for NEAs
with 40 m < @d < 2 km Morrison's data indicate that the impact
frequency is characterized by a power law with @[alfa] = 2. At @d
< 40 ¬ he obtained @[alfa] = 3.
     The space age of stony meteorites of various groups is ranged
from 0.1 to 365 Myr, and most of data varies within the limits of
2 - 25 Myr (Sobotovich and Semenenko, 1985). The tentative estimates
of sizes of meteorite parent bodies were about tens of kilometers
according to Bronsht@en (1987) and did not exceed hundreds of
meters corresponding to data by Sobotovich and Semenenko (1985).
Padev@et and Jake@s (1993) think that chondrites probably
originated in bodies larger than comets are today. According to
Bronsht@en (1987), parent bodies repeatedly collided other bodies,
and smaller age of stony meteorites as compaired with iron meteorites
is explained by smaller strength of stony meteorites, which could
not survive until the age of 10@8 yr.
     The lifetime of asteroids crossing only the orbit of Mars was
evaluated by Arnold (1964, 1965) to be ~ (1 - 3)@10@9 yr,  nd the
dynamical lifetime (until ejection into a hyperbolic orbit or a
collision with a planet) of Hidalgo is ~ 3@10@5 yr. Wetherill
(1975) and Olsson-Steel (1988) supposed that the dynamical
lifetime of a Jupiter-crossing NEA is ~ 10@6 yr. Wetherill (1991)
obtained that the mean dynamical lifetime of an Earth-crossing
asteroid is ~ 10@7 - 10@8 yr. An Earth-crossing object is ejected
into a hyperbolic orbit in ~ 10@4 - 10@5 yr, and the terrestrial
planets capture only 1 of 500 such objects. Under taking into
account the resonances with Jupiter, which prevent close
encounters with Jupiter, the fraction of objects captured by the
terrestrial planets could be 10 and even 100 times greater. The
time of "decoupling" of such objects with Jupiter was appreciated
by Wetherill (1991) to be equal to 10@5 - 10@6 yr. The objects
interacting with planets can spend part of time in resonances with
these planets (Zausaev and Pushkarev, 1993; Ipatov, 1979, 1981a;
Krezak, 1994; Milani @et al.@, 1989; Whipple and Shelus, 1993).
For example, the object 1985 WA moves during the larger part of
time in the resonances 5:2, 8:3, and 13:5 with Jupiter (Milani @et
al.@, 1989). Levison and Duncan (1994) obtained that the median
dynamical lifetime of short-period comets equals to 4.5@10@5 yr,
that is, for a half of the comets the lifetime is greater and for
another half is smaller than 4.5@10@5 yr. The median lifetime of
the comet P/Encke (@a = 2.2 AU, @e = 0.85, @i = 12@[degree], @Q =
a (1 + e) = 4.1 AU, @d@5- 10 km) is obtained to be equal to 10@5
yr.
     Wetherill (1991) considered that NEAs should come from the
Kuiper belt, but not from the Oort cloud in order to supply
present inclinations of orbits of NEAs. The time of active life of
the comet P/Enke is appraised by Wetherill (1988) to be equal to
10@3 -10@4 yr. Therefore, a capture of one Enke-type comet in 10@4
yr is required for supplying the Apollo group only by comets.
Kazimirchack-Polonskaya (1971) was the first who investigated
migration of comets from the orbit of Neptune inwards the Solar
System. The sizes of large comet nuclei are presented by Marov
(1994) and Mc-Fadden (1994). For example, sizes of the Halley
comet are equal to 16.0@8.2@8.4 km.
     Hahn and Bailey (1990) considered evolution of the orbit of
Chiron and 82 pseudobodies moving in orbits close to the orbit of
Chiron for interval of @+@10@5 yr. For the considered time span,
variations in @a reached 20 - 30 AU, but most of the objects
stayed in elliptical orbits. In one of variants of runs, the orbit
of pseudoChiron crossed the orbit of the Earth during 4@10@3 yr.
On the basis of the result that 4 or 5 objects (of 83) were
ejected into parabolic orbits within 10@4 years, it is assumed
that a half of the objects will be ejected within 1.1 - 1.4@10@6
yr.
     Nakamura and Yoshikawa (1993) investigated the evolution of
orbits of Chiron, Hidalgo, and objects P/Schwassmann-Wachmann 1
(SW1) and 1992 AD (5145), as well as 12 close orbits for each of
these objects. From analysis of the obtained plots of dependence
of semimajor axis @a on time, it follows, in particular, that the
orbit of Hidalgo crosses the orbit of the Earth during more than
10% of time.
     Evolution of the orbit of object 1991 DA was investigated by
Bailey and Hahn (1992) by numerical integration of equations of
motion for @2@10@5 yr. It was obtained that this object avoids
close encounters with Jupiter and Saturn. The time up to ejection
of the object into hyperbolic orbit was evaluated to be equal to
~1 - 3@10@6 yr. Investigating the evolution of 100 test bodies in
orbits close to the orbit of object 1991 DA, Bailey and Hahn
(1992) used a Monte-Carlo simulation based on the O@pik's scheme
and obtained an ejection half-life of about 1.4@10@6 yr. The time
during which the orbit of this object crosses the orbit of the
Earth is obtained by Steel and Asher (1992) to be equal to 1% and
by Steel (1993) to be equal to 10% of its lifetime.
     The characteristic times between collisions of asteroids and
times up to captures of asteroids in resonances were investigated
by Bottke @et al.@ (1994) and Farinella @et al.@ (1993, 1994).
Migration of asteroids (including migration to the orbit of the
Earth) during planet formation was considered by Wetherill (1992),
Ipatov (1993a), Safronov and Ziglina (1991).
     The problem of migration of bodies from the asteroid and
trans-Neptune belts to the Earth can be divided on some
subproblems: migration to orbits of planets, migration under the
influence of planets to the orbit of the Earth, evolution of
orbits of bodies near the orbit of the Earth, and probability of
their collisions with the Earth. In the present paper, we consider
several stages of the process of migration of small bodies to the
Earth. In contrast to previous works, we pay a main attention to
the investigation of migration of asteroid debris through the
Kirkwood gap 5:2, as well as to migration of bodies of the
trans-Neptune belt. Migration of asteroids from other gaps and
resonances was studied earlier by Wetherill (1985, 1987, 1988,
1989a,b, 1991, 1992), Yoshikawa (1990, 1991), Morbidelli @et al.@
(1994), Wisdom (1982), Ferraz-Mello (1994), Froeschl@e and
Morbidelli (1994), Froeschl@e and Scholl (1988, 1989), Froeschl@e
@et al.@ (1991), Scholl (1992), Scholl and Froeschl@e (1990,
1991), and other researchers and is not considered here. At
investigation of probabilities of collisions of asteroids between
themselves and with the Earth, together with the usually
considered model of fixed values of semimajor axes,
eccentricities, and inclinations, we also study the case when
elements of orbits of asteroids vary up to collisions. The present
paper is a summary of the results presented by Ipatov (1992b,
1993d,e,f, 1994b,c,e,f) on several conferences and in a preprint
(Ipatov, 1994d).

USED ALGORITHMS AND FORMULAS

@Algorithms of Simulation@

    Our investigations of migration of bodies were based on a
series of analytical estimates, on results of numerical
integration of the equations of motion for the three- and four-body
problem obtained by using a Bulirsh and Stoer (1966) code BULSTO,
as well as on results of simulation of evolution of orbits of
small bodies obtained with the help of the method of spheres of
action. While taking into account the gravitational influence of
objects (planets and bodies) by the method of spheres of action,
we suppose that objects move in unperturbed Keplerian orbits
outside the sphere of action, and we treated the relative motion
of objects in the two-body problem inside the sphere. The radius
of the sphere is equal to @r@s = R@[mu]@[2 / 5], where @R is the
distance of two encounting objects from the Sun, and @[mu] is a
ratio of the sum of masses of these objects to the mass of the
Sun. A brief description of the used algorithm of the method of
spheres is published by Ipatov (1978, 1991). A detailed
description of the algorithm of simulation for the evolution of
orbits of gravitating bodies moving around the Sun is given by
Ipatov in the Report @N-1211 of the Institute of Applied
Mathematics, USSR Academy of Sciences, for 1985. In contrast to
other authors, I used another algorithm for calculations of the
probability @p@ij of encounter of two objects up to @r@s (see
below). In particular, in my algorithm @p@ij depended on the
synodic period of revolution.
    The probability and deterministic methods were used for
choosing the pairs of encounting bodies (up to the radius of
corresponding sphere of action). A brief description of these
methods is presented by Ipatov (1991, 1992c, 1993b), and a
detailed description of used algorithms is given by Ipatov
(1993c). When the probability method is used, encounters of objects
in a disk are considered as a random process and pairs of objects
encounting up to @r@s are chosen with the help of encounter
probability matrix. When the determistic method is used, the
instant @[tau]@ij = @[tau]@ij@ + @[Delta]@[tau]@ij of the isolated
encounter of these objects is considered instead of the
probability @p@ij of encounter of two (@ith and @jth) objects,
(where @[tau]@ij@ = max {t@i, t@j}, @t@i is the instant when @ith
object encountered last time, @[Delta]@[tau]@ij@1/p@ij) and the
minimum value of @[tau]@ij is determined, where @i = 1,...,@N, @j
= @i+1,...,@N, @N is the number of objects in the disk. The exact
determination of @[Delta]@[tau]@ij requires a large volume of
calculations; therefore, I used the approximate formulas for
calculation of @[Delta]@[tau]@ij (see below).
     The results of numerical simulation show (Ipatov, 1993c),
that if the number of bodies is not small, then the time of
evolution of the disk is on the order of magnitude higher when the
probability approach is used than in the case of the deterministic
method. Other characteristics of the evolution of disks, as well
as times up to collisions of individual bodies with planets, are
approximately identical with both methods. To my opinion, the
deterministic approach is more physically substantiated.
     The evolution of spatial disks was investigated by Ipatov
(1982, 1987a) by an appropriate reduction of the spatial
(three-dimensional) evolution problem to the planar
(two-dimensional) one on the basis of the computer simulation
results of the evolution of planar disks of high density
fictitious bodies. The density of the fictitious bodies was chosen
so that the changes in orbital eccentricities between successive
collisions of the fictitious bodies were approximately the same as
in the considered spatial disks of solid bodies. The results of
investigations of accumulation of the terrestrial planets obtained
by Ipatov (1987a, 1993a) with this method are approximately the
same as the results obtained by Wetherill (1980) by direct
simulation of the evolution of spatial disks.

@Characteristic Time up to a Collision of Two Objects@

     In the present section for several models, the characteristic
time up to a collision of two objects orbiting the Sun is
considered. This time is designated above by @[Delta]@[tau]@ij.
The formulas of calculation the time used by us are based on the
results of presented by Ipatov (1978, 1988c). At derivation of
these formulas, the gravitational influence of objects is taken
into account by the method of spheres of action.
     For the planar model at fixed orbits, the time elapsed up to
a collision of two objects is obtained to be equal to

  @. (1)

Here @k@[teta] = 1 + (v@p / v@r)@2, @k@[fi] = @[Delta]@[fi] /
r@s@, @[Delta]@[fi] = @[Delta]@[fi]@1 + @[Delta]@[fi]@2 is the sum
of the angles (in radians) with vertex in the Sun within which the
distance (along the radial line with vertex at the Sun) between
the orbits is less than the radius @r@s of sphere of action (Fig.
1); @r@s@ = r@s / @R, @R is the distance of objects from the Sun,
@r@[Sigma] is the sum of their radii, @v@p / @v@r is the ratio of
parabolic and relative velocities of objects, @T@s = @PT@s@r is
the sinodic period of revolution, @P is the period of revolution
of a more massive object around the Sun; @r@[Sigma]@k@[Teta] /
@r@s is the probability of a collision of the considered objects
encounting each other at the distance @r@s. The lower index in
@T@2 indicates that the planar (two-dimensional) model is
considered. Depending on the considered model, @[xi] varies from
0.5 to 1. If one consider the isolated interaction of two objects,
then the value of @[xi] depends on the longitudes of perihelia and
on the initial (at the moment of calculation of @T@2) angle
@[Delta]@u@0 with vertex at the Sun between directions to objects
(from the fast object to the slow one). In the specific case, when
orbits are circular and do not vary, it is possible to show that
if @[Delta]@u@0 = 2@[pi]@[xi] radian, then after the time
@[xi]@T@s the objects will be on one radial line with vertex at
the Sun. The probability of the fact that this radial line will
lie inside of angles @[Delta]@[fi]@ (i.e., the probability of
encounter up to @r@s during the time @[xi]@T@s) is equal to
@[Delta]@[fi] / 2@[pi]. The mean value of @[xi] equals 0.5 in this
case. Successive catches (locations of objects in one radial line
with vertex at the Sun) will take place in the time T@s, i.e.,
@[xi]@1. This condition is not fulfiled, if each object can
interact with several objects. In this case for the time interval
@T@s calculated at some moment of time @t@0 for a pair of objects,
their orbits and arguments of latitude @u can change under the
influence of other objects. Furthermore, the elements of orbits of
real objects vary also with their motion outside spheres of
action.
     If the angle between planes of orbits of two objects is equal
to @[Delta]@i, then for other equal conditions the probability of
encounter up to @r@s is @k@i = @[pi][sin][Delta]@i / r@s@ times
greater than that for the planar case (Ipatov, 1978). Deriving
this relation for @k@i, I considered that encounters up to r@s are
possible only at @h@(u) = r@2@[sin]u[sin][Delta]@i@r@s, where
h@(u) is the length of perpendicular droped from the second object
on the plane of the orbit of the first object, @r@2 is the
distance from the Sun to the second object, @u is the angle
between one of two radial lines, which make the line of
intersection of planes of orbits of objects, and the direction to
the second object. We designate by @[Delta]u the sum of angles @u
for which this inequality is satisfied. We consider the case when
max{@h(u)} > r@s and sin@[Delta]u@[Delta]u (the values of
@[Delta]u are taken in radians). The mean size of projection of
r@s on the plane of the first orbit is equal to @r@s@' = r@s / 2
(Ipatov, 1978). Therefore, we obtain @k@i = (2@[pi] / @[Delta]u)
(r@s / r@s@') = @[pi][sin][Delta]i / r@s@.
    The probability of a collision of objects encounting up to the
distance @r@s at @[Delta]i > r@s@ is equal to @k@[teta]@
(r@[Sigma] / r@s)@2. Here and below in formulas, the values of
@[Delta]i are taken in radians. Therefore, using the formula (1),
we obtain the following formula for calculation of a
characteristic time @T@3 up to a collision of objects in the case
when their semimajor axes @a, eccentricities @e, and the angle
@[Delta]i (@[Delta]i > r@s@) between planes of orbits almost does
not vary before their collisions:

  @, (2)

where @k@[alfa] = 2@[pi]@2[xi]P(R/r@[Sigma])@2. Deriving this
formula, I considered that the values of @[omega] (the argument of
perihelion) and @[Omega] (the longitude of ascending node)
considerably vary before a collision of the objects. At the
assumptions mentioned above, the probability of a collision of two
objects at the time @[Delta]@t is equal to @[Delta]@t / T@3.
Another approach to the calculation of @T@3 was suggested by O@pik
(1951). One of further developments of this approach is presented
by Kramer and Shestaka (1987). Bottke and Greenberg (1993)
compared the values of @P@i = 1 / T@3 obtained by using various
modification's of the O@pik's formula. In this formula, as well as
in formula (2), @T@3 is proportional to sin@[Delta]@ but, in
contrast to (2), depends on the mean relative velocity of
encounter up to @r@s and does not depend on @T@s and @k@[fi]. In
reality, the values of @T@3 can vary considerably for the pairs of
bodies with various values of @T@s (i.e., with various ratio of
semimajor axes of orbits) and other identical orbital elements.
     We consider the model in which, for the time elapsed up to
encounter of objects up to @r@s, the angle @[Delta]i is uniformly
distributed from @[Delta]i@min to @[Delta]i@max (@[Delta]i@max >
r@s@). In this case, Ipatov (1988c) obtained that at
sin@[Delta]i@[Delta]i the ratio of probability of encounter up to
@r@s in the planar and spatial models @k@i@. At @[Delta]i@min = 0
we have @k@i@, where @[nu]@i@. The ratio of this value of @k@i to
the value of @k@i obtained for the fixed value of @[Delta]i equals
to @[Delta]i@max@. Thus, if @[Delta]i varies from 0 to
@[Delta]i@max (@[Delta]i@max@ > r@s@) during evolution, then the
time elapsed up to a collision of two objects

 @. (3)

       In the models studied in the present paper, considered time
intervals considerably exceed @T@s and variations of orbits at
time @T@s usually are relatively small. Therefore, if it is not
mention specially, in Tables and below in the text, our obtained
characteristic times elapsed up to collisions of bodies are
presented at @[xi] = 1. Ipatov (1994d) considered the case @[xi] =
0.5. The changes in orbital elements of bodies during the
evolution that are not taken into account by me usually reduce the
values of these times. Therefore, the values of characteristic
times between collisions of bodies obtained at @[xi] = 1 can be
considered as upper estimates. In formulas (1) - (3) the values of
@T@2, @T@3, and @T@3@ depend on @k@[xi]@ = @[xi]@ and are
identical at various @[xi], @T@s, and @k@[fi] for which @[xi] =
const.
     In the report N-1211 of the Institute of Applied Mathematics,
USSR Academy of Sciences, for 1985 for some values of orbital
elements and masses @m@1 and @m@2 of objects, I investigated the
dependences of @[Delta]@[fi] / r@s@ on orbital elements. It was
obtained that if orbits are eccentrical, then @[Delta]@[fi] can
considerably depend on the initial orientations of orbits. At
other identical data, @[Delta]@[fi] is maximum, if the perihelion
of one orbit is close to the aphelion or perihelion of another
orbit. The results of computer runs showed that at @m@1 >> m@2,
@m@1 ~ 0.01 - 10 m@, @a@2 / a@1 = 0.99, @e@1 = 0, and 0.05 < @e@2
< 0.4 in the correlation @[Delta]@[fi] ~ e@2@[-@[gamma]], the
value of @[gamma] is close to one and @[Delta]@[fi] / r@s@20 at
@e@2 = 0.2.
     If the orbit of a body intersects the orbit of the Earth and
other things are equal, the values of @[Delta]@[fi] are greater
for the values of @q = a (1 - e) which are more close to 1 AU. The
same statements are true, if for definition of @[Delta]@[fi] =
@[Delta]@[fi]@v we consider not the radius of sphere of action but
any other radius @r@v considerably smaller than @R. We can observe
small bodies encounting the Earth only at small distances @r@v.
Therefore, the probability of detection of these bodies is
proportional to @[Delta]@[fi]@v and is greater at smaller
inclinations @[Delta]@i. The search of NEAs in the vicinity of the
Earth-Moon system with the help of 0.9 m telescope began in
January 1991. Rabinowitz @et al.@ (1993) presented elements of
orbits of 15 NEAs with diameter @d@50 m. A half of these NEAs has
@e < 0.2. For such small NEAs we have @r@v ~ 0.005 - 0.05 AU. For
11 of 15 NEAs considered above, the relation 0.93@q@1.02 AU is
valid. The mean inclination of NEAs with @d < 50 m equals to
10@[degree]; that is, in 1.5 time smaller than that for large NEAs.
Rabinowitz @et al.@ (1993) assumed that small asteroids with small
eccentricities form the belt near the orbit of the Earth. However,
it is possible to explain the features of the distribution of
detected small NEAs on @e, @q, and @i on the base of dependences
of @[Delta]@[fi] considered above taking into account that we can
observe mainly those small NEAs for which @[Delta]@[fi] is small.
Therefore, the distribution of small NEAs in their orbital
elements may be the same as that for greater NEAs.

MIGRATION OF BODIES TO THE ORBITS OF PLANETS

@Migration of Bodies into the Kirkwood Gaps@

     At present the bodies can get into the Kirkwood gaps and in
the resonances @[nu]@5 and @[nu]@16 due to mutual collisions and
gravitational influence of large asteroids. The results of
computer simulation of the evolution of an asteroid orbit under
the influence of Ceres obtained by the method of spheres of action
show (Ipatov, 1978) that in the time of existence of the Solar
System the variations in a semimajor axis of the orbit of the
asteroid could reach several hundredths AU. If close encounters of
Ceres and an asteroid can take place, then for greater
variations in the difference of their semimajor axes,
the variations in the semimajor axis of this asteroid are greater.
According to the results presented by Samoilova-Yakhontova (1973),
the mass of Ceres equals to ~ 20% of the mass of the asteroid
belt. Therefore, Ceres make a significant contribution in
variations of asteroidal orbital elements. Part @k@r of the
asteroids that due to gravitational interactions got into the
resonances, which move asteroids to the Earth at the time @t@ss of
the existance of the Solar System, most likely does not exceed
0.1.
    We designate by @N@ast the total number of main-belt asteroids
with diameter greater than some value of @D@0 and by 1 / @k@e the
ratio of the number of asteroids that got into the main-belt
resonances to the number of these asteroids that became NEAs. Then
after time interval @[Delta]@t, @N@+ = k@ asteroids will become
NEAs. The decrease of the number of NEAs, due to collisions with
planets and their ejections into hyperbolic orbits during this
time interval, equals to @N@- = N@NEO@, where N@NEO is the number
of NEAs with diameters greater than @d@0, @T@d is the mean
dynamical lifetime of NEAs. At @D@0 = 1 km, Bintsel @et al.@
(1991) obtained @N@NEO ~ 10@6, and @N@ast@10@3 according to
Morrison (1992). Taking into account that @T@d ~ 10@7 yr and
k@r@0.1, we obtain that @N@+@; that is, a part of NEAs that get
into the resonances under the influence of the largest asteroids
made, probably, several percents.
     Crushings are more effective suppliers of bodies into the
Kirkwood gaps; i.e., the bodies coming from the asteroid belt are
presented mainly by debris of asteroids. Let us use the formulas
(2) - (3) for investigations of a characteristic time between
collisions of asteroids.
     Even at fixed inclinations of orbits of two asteroids, the
angle @[Delta]i between the planes of their orbits varies with
time due to variations in the longitude of ascending node.
Therefore, at estimates of the characteristic time between
collisions of asteroids, it is better to consider the case of
variable @[Delta]i and to use formula (3). We may consider
approximately that @k@[teta]@1 and @k@[fi]@5 for asteroids. In
this case we have

 @, (4)

where @a = 1 AU, @P is the period of revolution of the Earth
around the Sun. If @[Delta]i@, then @T@3@. At @r@[Sigma] equal to
1 m, 1 km, and 100 km, the values of @[eta]i are equal to 23, 15,
and 9, respectively. Therefore, for a model of variable @[Delta]i,
the characteristic time between collisions of asteroids is 5 - 10
times smaller than that at @[Delta]i = const. I designate by
@T@3@[teta] the characteristic time elapsed up to a collision of
two objects obtained by using the O@pik's formula. As well as
@T@3, this time is determined at fixed @[Delta]i. As the
dependences of @T@3 and T@3@[teta] on @[Delta]i are identical,
then averaging @T@3@[teta] by @[Delta]i, we obtain that
@T@3@[teta]@* = T@.
    Bottke @et al.@ obtained that the mean intrinsic collision
probability @P@i calculated at @r@[Sigma] =1 km) equals to
2.86@10@[-18] km@[-2]@yr@[-1] for main-belt asteroids. Using
formula (2), at @R = 2.8 AU, @[Delta]i =10@[degree], @T@ =10
years, @k@[teta]@1, @k@[fi]@5, and [xi] = 1, one obtains @P@i = 1
/ T@3 = 8.3@10@-19 km@[-2]@yr@[-1]. This value is 3.4 time smaller
than the value of @P@i obtained by Bottke @et al.@ (1994), and
Farinella and Davis (1992). The larger values of @P@i presented by
these authors are caused by the fact that in these papers @P@i is
calculated not for mean values of @e and @i, but as a ratio of the
sum of probabilities of collisions of various pairs of asteroids
with diameter @d@50 km to the number of such pairs. The values of
@P@i@* = 1 / T@3@* are @[eta] / 2 times greater than the values of
1 / @T@3 and depend on diameters of asteroids. They exceed the
value of @P@i obtained by Bottke @et al.@ (1994) and for
considered values of @R, @T@s, @k@[teta], and @k@[fi] at
@[Delta]i@max@* = 20@[degree] and @[xi] =1 are equal to 6@10@[-18]
and 3.7@10@[-18] km@[-2]@yr@[-1] for 1 km and 100 km asteroids,
respectively. If we will take into account the variations in
eccentricities of asteroidal orbits, as well as the distribution
of asteroids on semimajor axes of orbits, then we will obtain
greater values of @P@i@*.
     Let us consider the model in which each asteroid can collide
@k@N@N asteroids with the same radius @r and the values of
@T@3@* are identical for all pairs of asteroids and are designated
by @T@3e@*, where @N is the total number of asteroids with radius
@r; @k@N = 1, if each asteroid can collide with any other
asteroid. In this case the time elapsed up to the first collision
of some asteroid with some other asteroid equals to @T@cN = T@ and
the mean time elapsed up to the moment of the first collision of
some asteroids with radius @r between themselves equals to @T@cF =
T@.
     In the case of collisions of asteroids of radius @r with
asteroids of the smaller radius @r@', the times considered above
are equal to @T@cN@' = T@ and T@cF@' = T@, where @T@3@' is the
value of @T@3@* for asteroids with radii @r and @r@' (i.e., the
mean time elapsed up to the moment of first collision of these two
asteroids), @k@N@'@N@' is the number of asteroids @r@' which can
collide the asteroid @r; @N and @N@' are the total number of
bodies @r and @r@', respectively. According to Farinella and Davis
(1992), @k@N = 0.88.
     Let us designate by @s@0 the maximal ratio of masses of
asteroids, at which the catastrophic destruction of the greater
asteroid takes place. Piotrowsky (1952) showed that @s@0@100 for
the asteroid belt. The larger values of @s@0 were obtained at
later investigations. For example, Galt (1969) considered that
@s@0 can reach 10@3 -10@4 and in some cases even 10@6 due to
additional destructions connected with the effects arising owing
to the waves reflecting from restricting surfaces. Dohnanyi (1971)
supposed that @s@0@250v@c@2, where @v@c is the collision velocity
in km s@[-1]. The ratio of diameters of colliding asteroids, at
which the catastrophic destruction of both asteroids takes place,
decreases with the increase of diameter of the greater asteroid
(Davis @et al.@, 1979 ). Petit (1993) considers that @s@0 ~ 10@4 and
large bodies (@d@100 km) reaccrete and nearly do not lose mass at
collisions. The value of @s@0 = 10@4 was used also by Williams and
Wetherill (1994). The result of a collision depends also on that,
the impact is frontal or sliding. At @s@0 = 10@3 and @s@0 = 10@4,
the asteroid of radius @r@' = 0.1r and @r@' = 0.056r,
respectively, destroys the asteroid of radius r.
     Piotrowsky (1952) considered that @s@o = v@ct@2 / (2@k@e)
(where @v@ct is the velocity of collision) and the energy per unit
of mass, which is necessary for destruction of the target, @k@e ~
10@7 - 10@9 egr g@[-1], and the smallest of these numbers
corresponds to earthly rocks. Zharkov and Kozenko (1985) considered
that the density of energy, which is necessary for destruction of
Fobos, is equal to 3@10@7 erg cm@[-3]. It is followed from the
experiments by Galt that 10@6 - 10@7 erg g@[-1] are enough for the
total destruction of a body. Values of @k@e considerably depend on
the composition of the target. It is more difficult to destroy an
iron celestial body than a stony and, furthermore, an icy body.
Housen @et al.@ (1991) obtained that @k@e is minimum (smaller than
10@6 erg g@[-1]) at the target radius @r@* of several kilometers
and increased rather quickly with increasing of @r at @r > @r@*
and slowly with decreasing of @r at @r < @r@*. Barge and Pellat
(1993) investigated the laws of destruction of planetesimals
taking into account velocities of collisions. These authors
obtained that energy which is necessary for the destruction of
the target depends on @v@ct (@v@ct@[0.35]). The more wide review
of papers on high-velocity collisions of solid bodies is presented
by Ipatov (1981b) in Appendix 5.
     When @r = 0.5 km (i.e., at diameter d=1 km), [Delta]i@max@* =
20@[degree], @[pi]@2@[xi] / (k@[fi]@k@[teta]) = 2, @T@s@r = 2, @R
= 2.8 AU, @N = 10@6, @N@' = 10@8 (i.e., at @r@' = 0.1r and
@N@'@r'@[-2]), and @k@N = @k@N@' = 0.88, the values of @T@3e@*,
@T@cN, @T@cF, @T@3@', @T@cN', and @T@cF@' are equal to 1.6@10@17,
2@10@11, 3.4@10@5, 4.8@10@17, 5.4@10@9, and 5.4@10@3 years,
respectively. At smaller values of @r@' (i.e., at larger values of
@s@0) the values of @N@' are greater and the values of @T@cN@' and
@T@cF@' are smaller. For example, at @r@' = 0.056r and
@N@'@r'@[-2], the number of bodies that can destruct the asteroids
@r@' is by a factor of 10{2/3}@4.64 greater and the lifetime of an
asteroid @r is by a factor of 4.64 smaller than at @r@' = 0.1r.
Times @T@cN@' and @T@cF@' are determined above for the auxiliary
model of two types of bodies. If we consider that @N and @N@' is
the number of asteroids with radii greater than @r and @r@',
respectively, then the values of @T@cN, @T@cF, @T@cN@', and
@T@cF@' are smaller than those for this auxiliary model. For more
exact estimates it is necessary to take into account the
distribution of asteroids on masses, semimajor axes and other
orbital elements and to consider more strict laws of destruction
of asteroids.
     Assuming @N@r@[-2] we obtain that formulas for calculation of
@T@cN and @T@cN@' do not depend directly on @r and @T@cF@r@2. The
values of @[eta]@i are greater for smaller masses. Therefore, the
values of @T@cN are almost the same for bodies @r and @r@", if
@N/N"@r@[-[alfa]], where @[alfa] = 1.93 for @r = 1 km and @[alfa]
= 1.89 for @r = 100 km, @N@" is the number of bodies @r@".
     The values of @T@cN@' submitted above indicate that
collisional lifetime of majority of small main-belt asteroids does
not exceed the age of the Solar System and most of small asteroids
are products of destruction of larger asteroids. The review of
papers that considered the reconstruction of the initial
distribution of masses in the ring of small planets is presented
by Shor (1973).
     At @d = 1 km for the considered above auxiliary model of two
types of bodies, one has @T@cF@' = 6@10@3@[xi] years, and,
therefore, small bodies often can get into the Kirkwood gaps.
According to Wetherill (1989b), there are 10@4 bodies with
diameter greater than 10 km in the asteroid belt. At @N@d@[-2] the
values of T@cD@' for @d equal to 10 and 100 km are greater by a
factor of 100 and 10@4, respectively, than those for @d = 1 km.
Debris can get into the gaps not at each collision. Therefore, the
time interval, in which asteroids of diameter of some kilometers
get into the gaps, can exceed 10@6 yr. The ancient age of iron
meteorites (@10@8 yr) may be caused by a larger time interval
between the collisions of large asteroids at which the destruction
of their iron nuclei takes place. Iron debris of asteroids are
stable against collisions and live longer than stony one.
     Many bodies of the Apollo - Amor groups penetrate far in the
main asteroid belt. Therefore, collisional lifetimes of these
bodies are not differed considerably from the appropriate values
for main-belt bodies. Above for the auxiliary model of two types
of bodies, these times are designated by @T@cN@'. Times @T@cN and
@T@cN@' are proportional to @r, when @N@r@[-3]. The number of
bodies, which diameter @d exceeds some value of @D, is considered
to be proportional to @D@[-[alfa]]. It is noted in the
Introduction that @[alfa]@3 at @D@40 m. Therefore, small bodies of
the main belt of asteroids, as well as of the Apollo - Amor
groups, are crushed much more often (probably, by orders) than
asteroids with @d ~ 1 km. For example, for @s@0 = 10@4 supposing
@N@D@[-2] at @D > 40 m and @N@D@[-3] at @D@40 m, one obtains that
the number of bodies with diameter @d@"@0.056 m, which are able to
destruct a body with @d = 1 m, are by a factor of 40 / 0.056@700
greater, and a collisional lifetime is by a factor of 700 smaller
than in the case, when @N@D@[-2] at any @D. For the model of two
types of bodies with diameter @d = 1 ¬ and @d' = 0.056 ¬ at
@N@D@[-2], one has @T@cN@' = 1.4@10@9@[xi] years. Therefore, if,
as it is mentioned above, @[alfa] varies at @D = 40 m, then the
lifetime of a body with @d = 1 m does not exceed 2@10@6 years.
     The number of NEAs of diameter @d > 1 km is considered to be
approximately by a factor of 10@3 smaller than the number of
asteroids of the same size in the main asteroid belt. Therefore,
at consideration of collisions between NEAs, the values of @T@cN
and @T@cN@' are approximately 10@3 times and the values of @T@cF
and @T@cF@' are approximately 10@6 times greater than the values
obtained for main-belt asteroids, and the probability of such
collisions is comparatively small. Those NEAs, which have come
from the trans-Neptune belt and, probably, on the whole are icy,
can be destructed at collisions with smaller bodies than stony or
iron bodies. Such icy objects could be destructed during all their
way from the trans-Neptune belt.
       The estimates of the frequency of collisions of asteroids
presented above are obtained for the probability method for
choosing pairs of encounting objects. For the deterministic
method, the frequency of collisions of objects can be several
times greater and characteristic times elapsed up to catastrophic
destructions of bodies can be several times smaller than the
estimates obtained with the help of the probability method. The
ratio of these characteristic times obtained with the help of the
deterministic and probability methods can be smaller than 0.1, if
one uses formula (2) obtained at fixed values of @a, @e, and
@Delta]i. The above estimates of characteristic times elapsed up
to destructions of bodies are based on the values of @T@3@*, for
which definition the variations of @[Delta]i elapsed up to a
collision are taken into account. Therefore, in this case the
ratio of times will be more close to one than that at fixed values
of @[Delta]i. At @s@0 = 10@4 the values of @T@cN@' are about 5
times smaller than the corresponding values of @T@cN@' obtained
above at s@0 = 10@3, i.e., at @r@' = 0.1r. Therefore, the
lifetimes of asteroids with diameter @d ~ 1 km, for which the
energy necessary for destruction of 1 g of substance is minimum
(Housen @et al.@, 1991), can be by one order (or even tens times)
smaller than the values of @T@cN@' obtained at @r@' = 0.1r.
     For finding out, which (the probability or deterministic)
method for choosing pairs of colliding asteroids in the disk
better simulates the real evolution, it is necessary to made
numerical integration of equations of motion of the @N-body problem
at large @N in large time intervals and to compare the obtained
characteristic times between consecutive collisions of bodies with
the similar times obtained with the use of the probability and
deterministic methods. For simplicity, at first it is possible to
consider the model with fictitious bodies of large sizes moving in
orbits not varying with time.
     For the model of two types of bodies considered above at @d =
1 km, I obtained that @T@cN@'@6@10@9@[xi] yr. The account of the
listed above factors can decrease the estimate of the lifetime of
such asteroid to ~ 10@8 yr. Depending on the accepted laws of
destruction, Bottke @et al.@ (1994) obtained the lifetime of such
asteroid a little smaller than 10@8 yr (see Fig. 4a in the paper
by Bottke @et al.@ (1994)) or ~ 3@10@6 yr (Fig. 5a in the same
paper).
     Farinella @et al.@ (1993) obtained that the lifetime of a
body with @D = 50 km equals to 2@10@9 yr, if @N@' = 3@10@5 bodies
capable to destroy this body are present in the asteroid belt. At
such value of @N@' similarly to my calculations of the values of
@T@cN@' presented above for @d = 1 km, I receive the lifetime of
50-km body equal to 1.2@10@9@[xi] yr.

@Migration of Resonant Asteroids to Orbits of Planets@

     The results of runs carried out by Yoshikawa (1990, 1991),
Ipatov (1980a, 1988a, 1989a, 1992a,d), Morbidelli @et al.@ (1994),
Wisdom (1982, 1985), Froeschl@e and Morbidelli (1994), Froeschl@e
and Scholl (1988, 1989), Froeschl@e @et al.@ (1991), Scholl
(1992), and Scholl and Froeschl@e (1990, 1991) show that
eccentricities of orbits of some bodies from the Kirkwood gaps
3:1, 5:2, and 2:1 and also from the resonances @[nu]@5, @[nu]@6,
and @[nu]@16 can increase to such values that orbits of these
bodies begin to cross the orbits of the terrestrial planets. At
the resonance @[nu]@6, @a = 2.4 AU, @e = 0.14, and @i =
16@[degree], the characteristic time elapsed up to the intersection
the orbit of the Earth by a body is obtained to be 10@6 years.
Wetherill (1985, 1987, 1988) and Wisdom (1985) considered the gaps
3:1 and 5:2 as possible sources of filling up the Apollo and Amor
groups. However, to my opinion, the number of Earth-crossing
asteroids is greater for the 5:2 gap than that for these gaps
because, in contrast to other resonances, many Mars-crossing
asteroids are also Earth-crossers for the 5:2 resonance (Ipatov,
1989a, 1992a,d).
     By numerical integration of motion equations for the problem
of three bodies (Sun - Jupiter - asteroid), Ipatov (1989a,
1992a,d) investigated the evolution of orbits of 500 fictitious
asteroids located in vicinity of the resonance 5:2. It was
obtained that at this resonance orbital eccentricities of
asteroids usually vary with a period @P@10@5 years. For the planar
case at initial eccentricity @e@0 = 0.15 and the exact resonance
5:2, perihelia of orbits of about one third of 36 considered
fictitious asteroids with various orbital orientations laied
inside the orbit of the Earth (Ipatov, 1992a,d). Another third of
asteroids reached only the orbit of Mars. For the spatial model,
maximal eccentricities of resonant orbits on the average were
greater than those for the planar model. At initial inclination
@i@0 = 10@[degree] and @e = 0.15, four among six considered
resonant orbits with various orientations (i.e., two thirds)
crossed the orbit of the Earth during the evolution. The regions
of initial values of @a, at which the orbit of the Earth is
reached, are different at various initial orientations of orbits
and on the average make about a half of the width of the Kirkwood
gap 5:2 reaching this width at some orientations of orbits.
Therefore, at @e@0 =0.15, about 1 / 6 and up to 1 / 3 of
asteroidal debris got into the 5:2 Kirkwood gap can reach the
orbit of the Earth for the planar and spatial models,
respectively. For the planar model, the part of the bodies, which
got into the 5:2 gap and reach only the orbit of Mars, is almost
the same, as the part of bodies moving in Earth-crossing orbits.
At @i@0 = 10@[degree] it can be smaller.
     The part of bodies reaching the orbit of the Earth from the
3:1 gap in ~ 10@5 years is considerably smaller than that from the
5:2 gap (Yoshikawa, 1991). The average time of migration of bodies
to the orbit of the Earth from other regions of the Solar System
are greater than that from the 5:2 gap.
     The borders of regions of initial values of semimajor axes
@a@0 and eccentricities e@0 of orbits of fictitious asteroids, in
which, for some initial orientations of orbits, the orbit of Mars
is reached in ~ 10@5 years, are close to the borders of the 5:2
Kirkwood gap (Ipatov, 1989a, 1992a,d). At @e@0.2 for some initial
orientations of asteroidal orbits, the orbit of the Earth is
reached almost for the same regions of values of @a@0 and @e@0, as
the orbit of Mars. Considered fictitious asteroids reached the
orbits of these planets for specific types of relationships
between variations in asteroidal eccentricity and the difference
of longitudes of perihelia of orbits of an asteroid and Jupiter
(Ipatov, 1989a, 1992a,d).
     Hahn @et al.@ (1991) supposed that the role of the 5:2 gap in
migration of bodies to the Earth is smaller than that of the 3:1
gap because of encounters of asteroids with Jupiter. However, the
results obtained by me indicate that close encounters of asteroids
from the 5:2 gap with Jupiter almost do not take place even at large
eccentricities of orbits of these asteroids.
     Evolution of orbits of 26 objects of the Aten - Apollo - Amor
groups in time interval ~ 10@5 years was considered by Hahn and
Lagerkvist (1988). Among these objects, three objects (1985 WA,
1986 DA, and 1986 JK) were in the resonance 5:2 for some time, two
(887 and 1995) were in the resonance 3:1, and one (1986 RA) was in
the resonance 2:1; i.e., a fraction of the bodies located in the
resonance 5:2 was maximum.
     Investigating spectral characteristics of meteorites,
Golubeva and Shestopalov (1992) obtained that @S-asteroids with
spectra similar to those of meteorites are concentrated mainly
among objects from dynamical Apollo - Amor groups, as well as in a
range of heliocentric distances 2.8 - 3.0 AU. These authors came
to conclusion that astrophysical data on @S-asteroids testify in
favor of the hypothesis by Ipatov (1989a) that the resonance 5:2
(2.824 AU) is a probable mechanism of moving of matter from the
asteroid belt to the vicinity of the orbit of the Earth.
     Some bodies from the trans-Neptune belt can get into the
asteroid belt, but probability of this phenomenon is very small.
One can suppose the following mechanism of evolution. Some of such
bodies reach the orbit of Jupiter, and then, due to their close
encounters or collisions with asteroids, their orbits can get
inside the orbit of Jupiter. However, a decrease of aphelia of
these orbits is small on the whole. Small perturbations of
resonant orbits can cause the changes in the type of variations in
orbital elements (in particular, the orbit can become unresonant)
and the large variations in maximum eccentricity of a body
(Ipatov, 1989a, 1992a). Therefore, if some bodies from the
trans-Neptune belt rarely got into the asteroid belt, then more
often such movement took place near the Kirkwood gaps.

@Migration of Bodies from the Trans-Neptune Belt to the Orbit of
Neptune@

     My investigations show that gravitational influence of the
largest objects of the trans-Neptune belt can be of one of the
causes of migration of bodies from this belt to the orbit of
Neptune. Considering the planar four-body problem (the Sun and
three objects), Ipatov (1980b, 1988b, 1994a, 1995) investigated
the evolution of orbits of three gravitationally interacting
objects. The masses of the objects equalled to the mass of Pluto,
initial values of semimajor axes of orbits of objects were close
to 50 AU, and initial orbits were almost circular. The results of
computer integration of the equations of motion for the problem of
four bodies showed that the maximum eccentricity increased up to
0.05 after 3@10@5 revolutions of objects around the Sun. Further
evolution was investigated by the method of spheres of action.
Ipatov (1980b) obtained that the perihelion distance of one of the
objects can decrease by one third after 10% of the lifetime of the
Solar System.
      The real trans-Neptune belt is not planar. However, the
number of objects constituting it can be large. Basing on the
presented above results of computer runs for three objects, one
can obtain that, for @N ~ 100 such objects at 40@a@60 AU and
the mean inclination @i@av = 10@[degree], the mean eccentricity
@e@av grew from 0.05 to 0.2 during the age of the Solar System.
For a disk of @N identical objects at @e@av@50@h@1, Ipatov (1994a,
1995) obtained that velocity of variations in @e@av is
proportional to @N@[mu]@1@, where @[mu]@1 is the mass of an
object, @M@[Sigma] = N@[mu]@1, @[eta]@i@, @h@1@, 50@h@1@0.07 at
@[mu]@1 = 7.5@10@[-9]@M@[odot], and @M@[odot] is the mass of the
Sun. Therefore, at simultaneous increase of @M@[Sigma] by a factor
of 10 and decrease of @[mu]@1 by a factor of 10@3, one obtains
almost the same variations in @e@av. If we consider evolution of
the disk constituting by identical bodies with masses @[mu]@1 ~
5@10@[-12]@M@[odot] (sizes of such bodies are close to sizes of
the discovered bodies of the trans-Neptune belt), then the growth
of @e@av up to 0.1 and @i@av up to 4@[degree] takes place during
the lifetime of the Solar System for the mass of the disk
@M@[Sigma] equal to several masses of the Earth. If there are
larger bodies in the disk, then such growth of @e@av and @i@av
takes place at smaller values of @M@[Sigma].
     During the accumulation of planets under the gravitational
influence of planetesimals from the zones of the giant planets,
orbits of some objects of the trans-Neptune belt could obtain
eccentricities @e@0.1 (Ipatov, 1987b) and could preserve them
until today. This makes easy the present migration of these
objects to the orbit of Neptune. My results show that if there are
some bodies with masses about the mass of Pluto (or a larger number
of smaller bodies) in the trans-Neptune belt, then, due to their
gravitational influence, some bodies of this belt can migrate to
Neptune and then further inside the Solar System.
    Investigating migration of bodies during the formation of the
giant planets, Ipatov (1993a) obtained that semimajor axes of the
embryos of Uranus and Neptune could increase considerably under
the gravitational influence of planetesimals from the zones of
Uranus and Neptune migrated inside the Solar System.
Eccentricities of orbits of these embryos remained small at that
time. For the trans-Neptune belt, a similar mechanism
of exchange of angular momentum by migrating bodies can cause that
the largest bodies moving initially in slightly eccentrical orbits
in the inner edge of the trans-Neptune belt increased their
semimajor axes during the age of the Solar System. In this
case the largest bodies must be located in the middle of the
trans-Neptune belt.
     Duncan and Quinn (1993), Torbett and Smoluchowski (1990)
investigated the gravitational influence of the giant planets as
another possible mechanism of migration of bodies from the
trans-Neptune belt to the orbit of Neptune. Torbett and
Smoluchowski (1990) showed that some bodies with @a@32 AU and @e@0
can cross the orbit of Neptune in ~ 10@7 years. Due to the mutual
gravitational influence of bodies of the trans-Neptune belt
considered by me, the migration of some of these bodies to the orbit
of Neptune can be possible at larger initial values of @a and
smaller initial values of @e than at those obtained by Torbett and
Smoluchowski (1990). Mutual gravitational influence of bodies
located in the middle or in the external part of the trans-Neptune
belt can cause migration of some of these bodies into the internal
part of the belt, from which they can quickly migrate to the orbit
of Neptune due to gravitational influence of the giant planets. Ip
and Fernandez (1991) suggested the hypothesis, according to which
comets are ejected from the trans-Neptune belt under the influence
of bodies, as large as Mars or Earth, presented in the belt.

RESULTS OF COMPUTER SIMULATION OF MIGRATION OF BODIES TO THE ORBIT
OF THE EARTH UNDER INFLUENCE OF PLANETS

@Variants of Computer Runs@

    My investigations of migration of bodies to the Earth from
various regions of the Solar System are based on results of
computer simulation of evolution of orbits of @N@0 = 500 bodies
orbiting the Sun under the perturbing influence of planets and
falling on them at collisions. The gravitational influence of
planets was taken into account by the method of spheres of action.
The deterministic method for choosing pairs of encounting objects
was used in most computer runs. Time between isolated encounters
of pairs of objects up to the radius @r@s of the corresponding
sphere of action was determined by using a formula similar to
formula (2) (at @r@[Sigma] = @r@s and @k@[teta] =1). Using the
probability method, I obtained that orbital evolution is
approximately the same, but the time of evolution is greater.
Interactions of bodies between themselves were not considered.
     In each run, initial values of semimajor axes and
eccentricities of orbits were identical for all bodies and were
equal to @a@0 and @e@0, respectively. If is not mentioned
especially, then below I speak about the spatial
(three-dimensional) model (at mean inclination @i@av@15@[degree]).
The planar model was considered as a limited case of small
inclinations and for a greater statistics of collisions. Initial
orientations of orbits of bodies were various and were chosen with
the help of pseudorandom numbers. Some pairs of considered
values of @a@0 and @e@0 are presented in Table 1. Initial orbits
of bodies crossed an orbit of at least one planet. The value of
@T@h in Table 1 designates the time intervals (obtained at @[xi]
= 1) during which the number of bodies @N decreases by a factor of
two (from @N@0 to @N@0 / 2), @k@[oplus] = N@[oplus] / N@0, where
@N@[oplus] is the number of bodies collided the Earth, and @k@h is
the ratio of the number of bodies ejected during evolution into
hyperbolic orbits to the number of bodies collided planets.
     The pair (@a@0 = 1.7 AU and e@0 = 0.5) corresponds to a
typical orbit of an asteroid of the Apollo group. Computer
investigations of the three-body problem showed that some
fictitious asteroids from the 3:1 gap under the influence of
Jupiter increase their orbital eccentricities from 0.15 to 0.4 in
@10@5 years (Yoshikawa, 1990; Ipatov, 1980a, 1988a; Wisdom, 1982)
and begin to cross the orbit of Mars not reaching the orbit of the
Earth. The pair (@a@0 = 2.5 AU and @e@0 = 0.4) corresponds to such
case. As it was mentioned above, among the bodies getting into the
5:2 Kirkwood gap, a fraction of bodies reached the orbit of the Earth
in 10@5 years, probably, is not smaller than 1 / 6. Ipatov (1989a)
obtained the typical maximum eccentricity of orbits of bodies
reached the orbit of the Earth at the 5:2 resonance to be equal to
0.7. Therefore, I considered that the pair (@a@0 = 2.82 AU and
@e@0 = 0.7) corresponds to the characteristic orbits reaching the
orbit of the Earth from the 5:2 resonance. Evolution of orbits of
those bodies from the 5:2 gap that cross only the orbit Mars is
more similar to the evolution of orbits for the pair (@a@0 = 2.5
AU and @e@0 =0.4). Orbits of some bodies from the 3:1 gap can
cross the orbit of the Earth under the influence of Jupiter, and
evolution of their orbits is similar to the evolution of orbits
when @a@0 = 2.82 AU and @e@0 = 0.7. Under the influence of Jupiter
in ~ 10@4 - 10@5 years, eccentricities of orbits of resonant
asteroids can vary by several fold, and these orbits reach the
orbits of Mars and Earth only during a part of period of
variations in eccentricity. Therefore, real times of motion of
asteroids in resonances are greater by several fold than those for
the considered model.
     The pair (@a@0 = 25 AU and @e@ = 0.5) corresponds to objects
with orbits crossing the orbits of Neptune and Uranus. The pair
(@a@0 = 7.0 AU and @e@0 = 0.5) characterizes some bodies moving
from the zone Neptune to the Earth. If objects cross only the
orbit of Neptune, then time @T@h of twofold increase of number of
bodies constituting the disk can be considerably larger than that
for @a@0 = 25 AU and @e@0 = 0.5.
     Ipatov (1987b, 1989b) considered migration of bodies with
various initial values of semimajor axes located in the zones of
Uranus and Neptune under the influence of planets. It was obtained
that more than 1% of bodies from these zones could reach the orbit
of the Earth.

@Ejection of Bodies into Hyperbolic Orbits and Their Collisions
with Planets@

    In all variants of runs presented in Table 1, it was obtained
that the mass of bodies ejected into hyperbolic orbits for the
spatial model is approximately one order of magnitude more than
the mass of bodies collided the planets. Even for the planar model,
not less than a third of bodies was ejected into hyperbolic orbits.
Therefore, dynamic lifetimes of such bodies (up to their ejections
into hyperbolic orbit) are one order of magnitude less than
characteristic times up to their collisions with planets obtained
by using formulas (2) - (3) without account of ejection of bodies.
    The relative values of number of bodies ejected into
hyperbolic orbits and eccentricities @e that lie in the indicated
intervals are presented in Table 2 for a series of runs. It was
shown that 1.01@e@2 for 94 - 99% of bodies. Small (@e < 1.0001)
eccentricities were obtained mainly at @a@0 = 1.7 AU and @e@0 =
0.5. Investigating migration of bodies during the last stages of
accumulation of the giant planets, together with the distribution
in @e of bodies ejected in hyperbolic orbits, Ipatov (1989b)
considered also the distribution of these bodies in specific
angular momentum @c@' = @a (1 - e@2).
     For pairs (@a@0 = 2.5 AU and @e@0 = 0.4) and (@a@0 = 2.82 AU
and @e@0 = 0.7) a fraction of initial bodies collided the Earth
during evolution is obtained to be about 0.06 and 0.01,
respectively. By numerical integration of equations of motion,
Levison and Duncan (1994) obtained that 0.5% of short-period
comets during their lifetimes cross the orbit of the Earth in
@[Delta]t = 10@3 - 10@4 years. Assuming @[Delta]t = 10@4 years and
considering the characteristic time elapsed up to a collision of an
Earth-crossing comet with the Earth to be equal to 5@10@7 years
(see below the values of @T@*), one obtains a fraction of comets
collided the Earth to be equal to @[zeta] = 10@4@ = 10@[-6]. The
total mass @M@[Sigma] of planetesimals in the feeding zones of
Uranus and Neptune could be about 100 @m@[oplus], where @m@[oplus]
is the mass of the Earth (Ipatov, 1987b); that is,
@M@[Sigma]@[zeta] ~ 10@[-4]@m@[oplus]. Therefore, a large amount
of water could be delivered to the Earth during the time of
accumulation of Uranus and Neptune. Note for comparison that the
mass of water in oceans is ~ 2@10@[-4]@m@[oplus]. Masses of bodies
migrated to the Earth from the trans-Neptune belt could reach the
mass of Chiron. The concrete estimates of migration of the matter
to the Earth from the trans-Neptune belt depend on the
distribution of bodies of the belt in their masses and elements of
orbits, which is not almost known at present.
     Semimajor axes and eccentricities of orbits of bodies
collided the Earth are presented in Figs. 2a - 2h for a series of
pairs of @a@0 and @e@0. It was obtained that perihelia or aphelia
of orbits of bodies collided the Earth were located mainly near
the orbit of the Earth. For the spatial model, aphelia of orbits
of many bodies collided the Earth were close to 1 AU even at @a@0
> 1 AU. Eccentricities of these orbits usually were small. We note
that perihelia of two among four known orbits of meteorites were
close to 1 AU: 0.97 and 0.99 AU (Bronsht@en, 1987, 1994).
     As well as in the paper by Wetherill (1989), the number of
objects collided Venus was close to the number of bodies collided
the Earth and sometimes exceeded this number a little. A fraction of
bodies collided Mercury and Mars considerably depended on the
values of @a@0 and @e@0. For @a@0 = 2.5 AU and @e@0 = 0.4, a fraction
of bodies collided Mercury was two times smaller than that
collided the Earth and, at @a@0 = 1.7 AU and @e@0 = 0.5, was a
little less than that for the Earth. For other initial data
presented in Table 1, this fraction was of the same order as that
for the Earth. For @a@0 = 2.5 AU and @e@0 = 0.4, the fraction of
bodies collided Mars was three times greater than that for the
Earth. It was considerably less at other runs.

@Migration of Bodies in the Solar System@

     The distribution of semimajor axes and eccentricities of
orbits of bodies at some stages of evolution of considered disks
(when the number of bodies in disks @N = 250 and 100) is presented
in Figs. 3 - 5. The circle marks the values of @a@0 and @e@0. As
in Fig. 2, rhombuses and crosses mark the values of @a and @e for
spatial and planar models, respectively. Dashed lines mark the
values of @a and @e, for which aphelia or perihelia are equal to
semimajor axes of orbits of planets. For comparison in Fig. 6, the
values of @a and @e for real asteroids with perihelia of orbits
less than 1.33 AU, Troyans, and comets are presented.
     In all considered runs, the semimajor axes of orbits of some
bodies increased and those of other bodies decreased during
evolution. Bodies tried to penetrate in all regions of the Solar
System. Some bodies from the Apollo group penetrated beyond the
Neptune's orbit (Fig. 5a), and some bodies from the zone of
Neptune changed the orbits in such a way that they wholly got
inside the orbit of Venus (Figs. 3h and 4h).
     Among bodies that came from zones of the giant planets, a
fraction of Earth-crossing bodies is by one order of magnitude more
than a fraction of bodies which are only Mars-crossers, and usually
@e > 0.6 (Fig. 3g - 3h and 4g - 4h). These results show that
Mars captures a small number of bodies coming from the
trans-Neptune belt. Taking into account that the number of
asteroids of the Amor group is greater than that of the Apollo
group (according to data by Helin and Shoemarker (1979), 1500 and
800 objects brighter than 18 in the absolute magnitude,
respectively) and most of asteroids of the Amor group could not
come from the Apollo group, one obtains that most of objects of
the Amor group should have the asteroidal origin.
     Because of the limitation of the method of spheres of action,
in the considered runs the number of Trojans with large (> 0.2)
eccentricities of orbits was obtained to be greater than the real
one. Actually, such objects are quickly ejected by Jupiter. The
number of real Trojans of radius @r > 1 km is estimated by
Fernandez (1994) to be ~ 10@5 and with diameters @d > 15 km to be
equal to several thousands.
     At @a@0 = 25 AU, @e@0 = 0.5, and @N = 100, we obtained gaps
in a distribution of perihelia of orbits of bodies near the orbits
of the giant planets (Fig. 5d).
    At the last stages of evolution of all considered disks (even
at @a@0 = 25 AU), orbits of some bodies wholly were located inside
the orbits of the Earth and Venus, and for some disks the fraction of
bodies, which orbits wholly lay inside the orbit of the Earth, was
significant (Fig. 4a, 4c, 4e). Ipatov (1993b) suggested that the
number of such bodies in the Solar System can be large and some of
these bodies can migrate to the orbit of the Earth. At @a@0 = 0.3
AU and @e@0 = 0.4 (bodies which initially were only
Mercury-crossers), about 3% of initial bodies collided the Earth
in the course of disk evolution.

@Times of Evolution of Disks of Bodies@

     At @a@0 = 1.7 AU and @e@0 = 0.5, as well as at @a@0 = 2.82 AU
and @e@0 = 0.7, @T@h (the time of twofold decrease of number of
bodies in a disk) is two orders of magnitude less than the mean
times @T@3@2.6 - 2.8@10@8 years elapsed up to collisions of bodies
with the Earth obtained with the help of formula (2) for these
values of @a@0 and @e@0 at @[Delta]i = 15@[degree]. For the planar
model, the values of @T@h is one order of magnitude less than the
values of @T@2 (compare Table 1 and Table 3). These differences
of the values of @T@h and @T@3 (or @T@2) are caused by that in the
first place those bodies approach with planets for which times
elapsed up to encounters with planets are minimum (less than the
mean time @T@3). In the spatial case, one more order of magnitude
in the mentioned differences of times is caused by ejection of
majority of bodies into hyperbolic orbits. The method of spheres
of action used by me does not take into account that bodies that
are in resonances with Jupiter, as a rule, do not encounter up to
the radius of sphere of action of Jupiter. The account of
resonances with planets increases times elapsed up to ejection
of bodies into hyperbolic orbits and times elapsed up to their
collisions with planets.
     For @a@0 = 2.82 AU and @e@0 = 0.7 (initial orbits of bodies
cross the orbits of Mars and Earth) at @N = N@0 / 2, it was
obtained that orbits of 0.1@N@0 of bodies are Earth-crossers, but
are not Jupiter-crossers, and times of their further evolution can
by orders of magnitude exceed @T@h. Taking into account a fraction of
Earth-crossing asteroids and that part of time when their
perihelia lie inside the orbit of the Earth, I obtain that more
than 10% of bodies from the 5:2 resonance that collide the Earth
collide it in less than 10@6 years. Times elapsed up to last
collisions can exceed 10@8 years for all considered initial data.
     At @a@0 = 2.5 AU and @e@0 = 0.4 (a typical orbit for the 3:1
resonance crossing only the orbit of Mars (Ipatov, 1980a)) the
time @T@h is one order of magnitude more than that at @a@0 = 1.7
AU and @e@0 = 0.5 (a typical orbit of an asteroid of the Apollo
group) and by several tens of times greater than that at @a@0 =
2.82 AU and @e@0 = 0.7 (a typical orbit for the 5:2 resonance
crossing the orbits of Mars and Earth (Ipatov, 1980a, 1989a)).
These distinctions are connected with that the bodies originally
encountered only with Mars is rather slowly go into Earth-crossing
orbits. In the first two series of runs mentioned above,
eccentricities of orbits of bodies crossing the orbit of the Earth
mainly are smaller than those at @a@0 = 2.82 AU and @e@0 = 0.7,
and, hence, their dynamical lifetimes and probabilities of their
collisions with the Earth are larger. Therefore, though number of
Earth-crossing bodies is greater for the 5:2 gap than that for the
3:1 gap, a fraction of bodies collided the Earth from the 5:2 gap can
be smaller than that from the 3:1 gap.
     For @a@0 = 7 AU and @e@0 = 0.5, the values of @T@h are one
order of magnitude less than the values of lifetimes of real
Jupiter-crossing objects. It is connected with a limitation of the
method of spheres of action and also with that we watch mainly the
objects with large lifetimes, and other objects already were
ejected into hyperbolic orbits or collided planets.
     Small bodies interacting with planets are in resonances with
these planets during part of time of evolution (Zausaev and
Pushkarev, 1993; Ipatov, 1979, 1981a; Kresak, 1994; Milani @et
al.@, 1989; Whipple and Shelus, 1993). Therefore, the real times
elapsed up to collisions of bodies with planets or up to their
ejection into hyperbolic orbits can be larger than those
obtained with the help of the method of spheres of action.

CHARACTERISTIC TIMES ELAPSED UP TO COLLISIONS OF BODIES WITH
PLANETS

@Characteristic Times Elapsed up to Collisions of Asteroids with
the Earth@

    Using formula (2) at @[xi] = 1, we calculated the values of
mean times @T'@k = T@3@(k) (where @k = 1,...,93) elapsed up to
collisions with the Earth for 93 Earth-crossing asteroids (ECAs)
discovered by the middle of 1991 (only one of these ECAs has
diameter @d < 50 m). Mean inclination of orbits of the ECAs is
close to 15@[degree], and their mean eccentricity is equal to
0.54. It is obtained that @T'@k@4@10@6 years for two ECAs (with
numbers 4015 and 4179). Such small values of @T@'@k for the
asteroid @N 4015 are connected with a large (@76) value of ratio
of @k@[fi] = @[Delta]@[fi] / r@s@* and with a small (2.9@[degree])
inclination @[Delta]i, and for the asteroid @N 4179 we have
@[Delta]i = 0.4@[degree]. For asteroids @N 1566 (Icarus) and @N
3753, values of @T'@k were obtained to be equal to 8@10@9 and
8@10@10 years. For the last asteroid, @a = 0.998 AU and a
synodical period of revolution @T@s = 294 years.
      The values of @k@[fi] lie in interval (5, 20) for 80% ECAs,
it is obtained that @k@[fi] < 5 for 9%, and we have @k@[fi]@20 for
11%. The mean value of @k@[fi] is obtained to be equal to 12.7.
For most of real ECAs, the values of @k@[teta] are located in the
range from 1 to 2, and @k@[teta] > 3 only for two ECAs.
     Orbital elements of real asteroids can considerably vary
before their collisions with planets, and times elapsed up to
these collisions can be strongly differed from the values of
@T'@k obtained at fixed orbital elements. The results of
numerical integration of equations of motion for the planar
three-body problem (Sun - Earth - asteroid) show that a semimajor
axis of an asteroidal orbit can decrease from 1.7 to 1.1 AU in
10@4 years. Therefore, lifetime of a NEA does not depend much on
its initial orbit.
     We calculated also the values of @T@3 for various values of
@a, @e, and @[Delta]i. It is obtained that the values of @T@3 grow
with increasing of @e and @[Delta]i and with decreasing of ³@a - 1
AU³. For example, for @a = 1.1 AU, we have @T@3 = 4@10@6@[xi]
years at @e = 0.1 and @[Delta]i = 5@[degree], and @T@3 =
1.2@10@10@[xi] years at @e = 0.6 and @[Delta]i = 60@[degree]. For
@a = 1.25 AU at @[Delta]i = 80@[degree] and @e@0.7, and also at @e
= 0.9 and @[Delta]i@50@[degree], the values of @T@3 exceed the age
of the Solar System; i.e., a probability of collisions with the
Earth of bodies permanently orbiting in highly inclined and
eccentrical orbits is small. However, due to variations in orbital
elements of bodies during evolution and ejection of most of them
into hyperbolic orbits, it is very improbable that bodies could be
preserved near the orbit of the Earth during the time of its
accumulation even at large initial values of @e and @[Delta]i.
      The values of @T@2, @T@3, @k@[teta], and @T@3@* presented in
Table 3 are obtained for @e = 0.5, @k@[fi] = 10, @[xi]T@s@r = 2,
and (v@r)@2 = (e@2 + @[Delta]i@2)(v@c)@2, where @v@c is the
circular velocity of a planet around the Sun, and the values of
@[Delta]@i are taken in radians. The cases when @[Delta]i =
10@[degree] and @[Delta]i@*@max =20@[degree], as well as when
@[Delta]i=15@[degree] and @[Delta]i@*@max =30@[degree] are
considered.
     For the giant planets, the values of @T (@T@2, @T@3, and
@T@3@*) can be a little less than the values of @T presented in
Table 3 since for these planets at @e = 0.5 the values of @k@[fi]
can exceed 10, and @T@1/k@[fi]. However, the time of motion of
bodies in resonances with the giant planets is usually greater
than that with the terrestrial planets.
     The collisions occured more often at inclinations @[Delta]i <
@[Delta]i@*@max / 2 than at @[Delta]i > @[Delta]i@*@max / 2.
Therefore, the values of @T@3@* are several times (2.5 for the
Earth) smaller than the values of @T@3 obtained at @[Delta]@i =
@[Delta]i@*@max / 2. The value of @T@3 obtained for the Earth at
@[Delta]@i = 15@[degree] and @[xi]@T@s = 1 year is by a factor of
2 greater than 10@8 years, i.e., the characteristic time elapsed
up to a collision with the Earth obtained by Wetherill (1988). At
@[xi]T@s = 2 years this difference is some more in two times
larger.
     In general case, the probability of a collision is smaller at
smaller eccentricities of orbits of collided objects. For the
model, at which eccentricity of an orbit of a NEA varies from
@e@min to @e@max, values of a characteristic time @T elapsed up to
a collision will be less than appropriate values of @T obtained at
fixed @e = (e@min + e@max ) / 2.
    For @[xi]=1, Table 3 presents also the values of the mean time
elapsed up to a collision of ECA with the Earth calculated by
formula

@, (5)

Here @T@3@(k) is the value of @T@3 for @kth real ECA, and @N is
the total number of known ECAs. The same formula for calculation
of @T@* was used for other planets, but values of @T@3@(k) were
calculated for the objects, which semimajor axes of orbits are by
a factor of @a@p greater than those for @kth ECA, where @a@p is a
ratio of semimajor axes of orbits of a planet and the Earth.
     For the Earth, the value of @T@* is 5 times as small as the
value of @T@3 obtained for mean values of @e and @[Delta]i for
ECAs at @T@s =2 years. As at calculation of @T@*, a distribution
of bodies in their orbital elements is taken into account, then
@T@* more exactly than @T@3 and @T@3@* characterizes the mean time
elapsed up to a collision of an ECA with the Earth. Bottke @et
al.@ (1994) obtained that the mean time elapsed up to a collision
of an ECA with the Earth similar to @T@* is equal to 1.34@10@8
years. This value exceeds the values of @T@* (3.8@10@7 and
7.6@10@7 years) obtained by me at @[xi] = 0.5 and @[xi] = 1 by a
factor of 3.5 and 1.7, respectively. A little more than a half of
NEAs cross the orbit of the Earth. Therefore, the mean time
elapsed up to a collision of a NEA with the Earth is two times
greater than that for an ECA.

@Migration of Bodies and Meteorite Ages@

     A probability of ejection of a NEA into a hyperbolic¥á orbit
is by one order of magnitude less than a probability of its
collision with the Earth, and most of NEAs leave from the Solar
System in time t@10@7 years. These asteroids can collide not only
the Earth but also Venus and other planets. Therefore, for
majority of bodies collided the Earth, time interval between
obtaining an Earth-crossing orbit and colliding the Earth is in
several times smaller than @T@* and does not exceed 10@7 years.
     Let us consider the following model. There were @N@0 ECAs at
the initial moment of time. We investigate variation in the number
of these ECAs with time @t. Assuming the number of ECAs collided
the Earth in time @[Delta]t to be equal to @[Delta]t@N / T@* and
the number of ECAs ejected into hyperbolic orbits or collided
other planets or the Sun to be in @k times greater than this
number, we obtain

@. (6)

Denoting @[beta] = N / N@0, from (6) we obtain @t = -T@*@ln@[beta]
/ (k + 1) = -T@d@ln@[beta], where @T@d = @T@* / (k + 1) is the
dynamical lifetime of an ECA. At 1 + @k = 10 for @[beta] = 0.5,
0.1, 0.01, and @4.54@10@[-5], we have @t = @T@**@0.07@T@*,
@0.23T@*, @0.46T@*, and = @T@*, respectively, where the value of
@t at @[beta] = 0.5 equals to @T@**@5.3@10@6 years at @[xi] = 1.
Thus, a half of all ECAs that collided the Earth collide it in
time t@5@10@6 years after these asteroids began to cross the orbit
of the Earth. Number of ECAs will be constant and equal to @N@0,
if together with accounting collisions and ejections into
hyperbolic of orbits mentioned above, one supposes that
@N@0@[Delta]t / (k+1)T@* new ECAs come at time @[Delta]t from
other regions of the Solar System.
      Collisional lifetime of a body with diameter equal to 1 m in
the asteroid belt is obtained above to be equal to @T@cN ~ 10@6
years. Earth-crossing bodies frequently penetrate rather far in
the asteroid belt. Therefore, times @T@cN elapsed up to their
destruction are not considerably smaller than those for bodies in
the asteroid belt, and they are several times smaller than @T@**.
These estimates agree with the fact that usually stony meteorites
are the result of several of destructions (Bronsht@en, 1994;
Chapman, 1990).
     The mean lifetime of @H-, @L- and @LL-chondrites after
separation from parent bodies is found by Alekseev (1993) to be
equal to ~ 3@10@7 years. Therefore, majority of chondrites can
spend a considerable part of their life in their way to the orbit
of the Earth.
     A sharp peak is displayed in the region of 6-7@10@6 years in
distribution of ages of @H-chondrites (Alekseev, 1993). Some
asteroids from the 5:2 and 3:1 Kirkwood gaps can reach the orbit
of Mars and sometimes that of the Earth in ~ 10@5 years
(Yoshikawa, 1990, 1991; Ipatov, 1980a, 1988a, 1989a; Wisdom,
1982). The characteristic time @T@h of twofold decrease of number
of such asteroids does not exceed 10@7 years (Table 2). These
results show that the part @H-chondrites with age ~ 6-7@10 years
could have come from the 5:2 and 3:1 gaps.
     Small number of @LL-chondrites with age @t < 8@10@6 years can
be caused by a long way (for example, from the trans-Neptune belt)
of @LL-chondrites to the orbit of the Earth. A decrease of number
of chondrites with age @t with increase of @t at @t > 10@7 years
(Alekseev, 1993), probably, is caused mainly by ejection of NEAs
into hyperbolic orbits.
      Ancient age of iron meteorites (> 10@8 years) can be caused
by that the last collision of large asteroids occured more than
10@8 years ago, and iron debris formed thus almost are not crushed
before their collisions with the Earth.
     Probably, planetesimals formed in the zone of the giant
planets are one of sources of @L-chondrites. During formation of
the giant planets, some of these planetesimals penetrated in the
asteroid belt (Ipatov, 1993a; Safronov and Ziglina, 1991) and
could collide asteroids. Part of debris of these planetesimals
could stay in the asteroid belt (mainly in its external part).
Considerably larger number of such planetesimals could get into
the trans-Neptune belt (Ipatov, 1987b). Many bodies of the
trans-Neptune belt could be formed directly in the belt (Safronov,
1994).

@Frequency of Collisions of Bodies of Various Masses with the
Earth@

     Morrison (1992) considers that the number of NEAs with
diameter @d > 1 km is about 1000 - 2000. A little more than a half
of known NEAs with perihelion @q@1.33 AU cross the orbit of the
Earth. Therefore, the number N@* of ECAs with diameter @d > 1 km
is considered below to be equal to 500 or 1000. Assuming @T@* =
5@10@7 years and @N@* = 500, one obtains that a body with diameter
@d@1 km collide the Earth on the average one time in @T@* / @N@*
=10@5 years. The time elapsed up to such collision is by a factor
of 2 smaller at @N@* = 1000. About 10 NEAs with diameter @d@5 km
are found, and @D@20 km for two of them (Eros and Ganymed).
Similar estimates show that NEAs with such diameters can collide
the Earth in 10@7 and 5@10@7 years, respectively. At @T@* =
7.6@10@7 years, these estimates are by a factor of 1.5 greater. As
this value of @T@* is an upper estimate and it is considered that
1000 is a more probable value of @N@*, then one obtains that a
body with diameter greater than 1 km collides the Earth on the
average not less often than once in 10@5 years. However, times
elapsed up to collisions of NEAs with the Earth can increase
several times (most likely, less than 1.5 times), if the time of
moving of NEAs in resonances with the Earth are taken into
account. In future it is desirable to estimate characteristic
times elapsed up to collisions of NEAs with the Earth by numerical
integration of equations of motion at large time intervals.
Characteristic times elapsed up to collisions of bodies with sizes
mentioned above with the Earth are smaller, if the real number of
such NEAs is larger. Earlier Morrison (1993) obtained a few larger
times between collisions with the Earth of bodies with diameter
@d@1 km. According to his estimates, bodies with diameter @d@150
m, @1 km, and @5 km collide the Earth in 5@10@3, 3@10@5 and 10@7 -
3@10@7 years, respectively, and a diameter of crater exceeds @d by
a factor of 10 - 15.
    Geophisical and paleontological data show (Morrison, 1993)
that a body with diameter 10 or 15 km collided the Earth 6.5@10@7
years ago. As a result, majority of species of animal and plants
were lost. Nine deathes of organisms with average time interval
equal to 3@10@7 years took place during the last 2.5@10@8 years
(Bronsht@en, 1987). Among the impact structures found on Russian
territories, the following astroblems have the largest sizes:
Popigaiskaya with diameter @D@100 km and age @T@3.6 - 3.7@10@7
years, Karskaya (@D@65 km and @T@7@10@7 years), and
Puchezh-Katunskaya (@D@80 km and @T@1.75@10@8 years) (Masaitis @et
al.@, 1994).
     Steel and Asher (1992) obtained that a body similar to Chiron
crosses the orbit of the Earth during ~ 0.01 of time interval. The
orbit of Hidalgo crosses the orbit of the Earth during ~ 0.1 of
the considered time interval (see Fig. 2 of the paper by Nakamura
and Yoshikawa (1993)). Supposing that such bodies existed
permanently and assuming the characteristic time elapsed up to a
collision of an ECA with the Earth @T@*8*10@7 years, one obtains
that a celestial body of size as that of Hidalgo collide the Earth
at least 10 times during the lifetime of the Solar System and,
probably, a body such as Chiron could collide the Earth. A
probability of collisions increases, if one takes into account
that earlier the number of ECAs could be larger.
     Let us consider that on the average one body with diameter
@d@0 collides the Earth in time @T@0 and number of bodies with
diameter greater than @D is proportional to (@D / d@0)@[alfa].
Let @N@* be the number of ECAs with diameter @d@1 km (see above).
Then the folowing relation is true:

@. (7)

According to Ivanov (1992), @T@0 = 2 years at @d@0 = 1 m. In this
case at @T@* / N@* = 2@10@5, @T@* / N@* = 8@10@4, and @T@* / N@* =
4@10@4 years from (7), one obtains @[alfa] = 1.67, @[alfa] = 1.53,
and @[alfa] = 1.43, respectively. According to estimates by
Lebedinets (1993), on the average 1.6 iron or stony bodies with
mass larger than 100 tons collide the Earth during a year. In this
case at @T@* / N@* = 8@10@4 years and @[rho] = 3 g sm@[-3] (i.e.,
for a body with diameter @d@*4 m), time elapsed up to a collision
of the Earth with a body with diameter larger than @D is equal to
@T(D)@1.6(D/d@*)@2@0.1(D/d@0)@2 years. Probably, this estimate of
@[alfa] is more close to the real one than the estimates based on
the results by Ivanov (1992), because, as it is noted in
Introduction, @[alfa]@2 for comets and asteroids. Ceplecha (1992)
suppose that values of @[alfa] differ much for various values of
@D and the main mass influx onto the Earth belongs to stony bodies
with mass of 10@12 - 10@15 kg and to inactive comets with mass of
10@4 -10@7 kg.
     More exact estimates of probabilities of collisions of NEAs
of various masses with the Earth require the improvement of our
knowledge about distributions of NEAs on masses and orbital
elements. Together with observations, investigations of sources of
filling up NEAs will allow to make more accurate estimates of this
distribution.

@Times Elapsed up to Collisions of Bodies with Various PLanets@

     At calculation of values of times @T@3, @T@3@*, and @T@*
presented in Table 3, it was considered for various planets that
ratios of semimajor axes of orbits of bodies encountering with a
planet to the semimajor axis of the orbit of this planet, as well
as other orbital elements of bodies are the same for all planets.
The values of these times presented in Table 3 are by a factor of
2.4 - 2.5 for Venus and by a factor of 2.7 - 3.6 for Mercury
smaller than corresponding values for the Earth. For Mars these
times are by a factor of 16 - 20 greater than those for Earth. The
ratio of a semimajor axis of an orbit of a body crossing (for some
values of longitude of perihelion and longitude of ascending unit)
orbits of several planets to a semimajor axis of an orbit of a
planet is various for various planets, and orbits of bodies, as a
rule, cross the orbits of not all terrestrial planets. Therefore,
fractions of bodies collided different planets (see above) differ
from relations between values of 1 / @T@3 for these planets.
     For Jupiter, values of @T@3, @T@3@* and @T@* (i.e.,
collisional lifetimes of Jupiter-crossing bodies) are by a factor
of 10 - 20 smaller than those for the Earth. Therefore, if the
orbit of a body crosses the orbits of Jupiter and Earth, then in
general case the probability of its collision with Jupiter is
10-20 times larger than that with the Earth. However, such body
can be in the resonance with Jupiter for a long time (Milani @et
al.@, 1989) and thus it can collide only the Earth and some other
terrestrial planets. Therefore, the relative probability of its
collision with the Earth can be larger than the above estimates.
     Characteristic times @T@3, @T@3@*, and @T@* elapsed up to
collisions of bodies with Neptune presented in Table 3 exceed the
lifetime of the Solar System. Therefore, some planetesimals
crossing only the orbits of Neptune and Pluto and orbiting in
inclined orbits could be preserved from the time of formation of
the Solar System. As orbits of real planetesimals varied in the
course of evolution, the overwhelming majority of other
planetesimals migrated inside the Solar System and was ejected
into hyperbolic orbits or collided planets (Ipatov, 1993a).

CONCLUSIONS

     Migration of celestial bodies to the Earth from various
regions of the Solar System and sources of filling up groups of
near-Earth asteroids (NEAs) are considered. In contrast to papers
by other authors, I used the other algorithms of calculation of
probabilities of collisions of bodies and for the first time
investigated migration of bodies to the Earth from the 5:2
Kirkwood gap.
    By numerical integration of equations of motion for the
three-body problem (Sun - Jupiter - asteroid) the time variations
in orbital elements of 500 test asteroids are investigated in the
vicinity of the 5:2 resonance with the motion of Jupiter. The
approximate estimates show that more than 1 / 6 of debris of
asteroids, which got into the 5:2 Kirkwood gap at eccentricity @e
= 0.15, can reach the orbit of the Earth in 100 thousand years.
These debris can make an appreciable part of those chondrites of
the group @H, which age is less than 10 Myr. Mean times of
migration of bodies to the orbit of the Earth from other regions
of the Solar System are larger than those from the 5:2 gap.
      Results of numerical investigations of the four-body problem
(the Sun and three bodies) show that due to mutual gravitational
influence some bodies of the trans-Neptune belt can migrate to the
orbit of Neptune. Part of these bodies subsequently can migrate
inside the Solar System and some bodies can reach the orbit of the
Earth.
     The investigations of evolution of disks originally
consisting of planets and hundreds of bodies located in various
regions of the Solar System were carried out. Gravitational
influence of planets was taken into account by the method of
spheres of action. The results of investigations show that about
1% of bodies from the 5:2 gap could collide the Earth. Perihelia
or aphelia of orbits of bodies collided the Earth mainly lie near
the orbit of the Earth. An orbit of a NEA strongly varies before
its collision with a planet. A large amount of water could be
delivered to the Earth during time accumulation of Uranus and
Neptune.
     Among the bodies which came from the zones of the giant
planets, a fraction of bodies moving in Earth-crossing orbits is one
order of magnitude more than a fraction of bodies that are only
Mars-crossers, and usually e>0.6. These results indicate that
majority of asteroids of the Amor group should have come from the
asteroid belt.
     Part of bodies migrated to the Earth from various regions of
the Solar System subsequently filled up the family of asteroids,
which orbits entirely lied inside the orbit of the Earth (and some
of them inside the orbit of Venus). The number of objects of this
family can be large. These objects are dangerous because they come
from the Sun direction and it is difficult to predict their
appearance.
     In contast to papers by other authors, I considered that
values of time elapsed up to a collision of two celestial objects
depend on their synodical period of revolution and for a series of
models also take into account variations in orbital elements in
the course of evolution. The orbit of a real NEA can strongly vary
before its collision with the Earth, and the characteristic time
elapsed up to this collision is less than the value of time
obtained at fixed mean values of eccentricities and
inclinations of orbits. For example, for variations of inclination
@i of a NEA from 0 to @i@max = 30@[degree] and for other equal
conditions, the time elapsed up to a collision of a NEA with the
Earth is approximately 2.5 times smaller than that at fixed @i =
15@[degree]. At consideration of collisions of bodies with
diameter equal to 1 m in the asteroid belt, the account of
variations in @i decrease the value of characteristic time elapsed
up to a collision of these bodies in more than 10 times.
     Our obtained times elapsed up to collisions of bodies with
the Earth mainly are a few smaller than the values obtained by
other authors. The characteristic time elapsed up to a collision
of a NEA with the Earth is about 100 Myr. The probability of
ejection of a NEA into a hyperbolic orbit is by one order of
magnitude more than the probability of its collision with the
Earth. For most of bodies collided the Earth, a time interval
between obtaining an Earth-crossing orbit and colliding the Earth
does not exceed 10 Myr. A NEA with diameter larger than 1 km
collides the Earth on the average not less often than once in 100
thousand years.
     Asteroids, which got into the Kirkwood gaps due to
gravitational influence of the largest asteroids, can make only
several percents of NEAs. A larger number of bodies can get into
the gaps as a result of mutual collisions of asteroids. Therefore,
NEAs that come from the asteroid belt mainly are products of
high-velocity collisions. It was obtained that a mean time elapsed
up to the moment of the first collision of some asteroids with
diameters @d = 1 km and @d@'@0.1@d is smaller than 10 thousand
years. The mean lifetime of small (< 100 km) asteroids of the main
belt does not exceed the lifetime of the Solar System, and the
lifetime of a body with diameter equal to 1 m in the asteroid belt
can be about a million years.
     @H-chondrites can come to the Earth through the resonances in
the asteroid belt. A small number of @LL-chondrites, which age
does not exceed 8@10@6 years, can be caused by a long way of
@LL-chondrites from the trans-Neptune belt. Ancient age of iron
meteorites, probably, is caused by a large time interval between
collisions of large asteroids.
     I would like to thank ’.Œ. E@neev for attention to the work
and V.P. Volkov, V.N. Zharkov, and I.N. Ziglina for a series of
useful remarks.
     This work was supported by the Russian Foundation for
Fundamental Research, project no 93-02-17035.

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TABLES

Table 1. Several characteristics of evolution of a disk
constituting by 500 bodies under influence of planets

1. @a@0 (AU)
2. planar model
3. @T@h (years)
4. spatial model
5. Designations: @a@0 is the initial value of semimajor axes of
orbits of bodies in AU, @e@0 is the initial eccentricity of orbits
of bodies, @T@h is the time during which number of bodies decrease
by a factor of two, @k@[oplus] is a fraction of bodies collided the
Earth during evolution, @k@h is a ratio of number of bodies
ejected into hyperbolic orbits to number of bodies collided
planets

Table 2. Fraction of bodies (in percents) ejected into hyperbolic
orbits with eccentricities located in various intervals

Table 3. Characteristic times elapsed up to collisions with
planets obtained for several models

1. Planet
2. at
3. Mercury
   Venus
   Earth
   Mars
   Jupiter
   Saturn
   Uranus
   Neptune
4. Note: Times @T@2, @T@3, @T@3@*, and @T@* are presented in
terrestrial years and obtained by using formulas (1) - (3) and
(5), respectively. Values of @T@2, @T@3, and @T@3@* are presented
at @[xi]@T@s@r / k@[fi] = 0.2, @e = 0.5, and @[Delta]i@*@m =
@[Delta]i@*@max, and values of @T@* are presented at @[xi] = 1.


FIGURE CAPTIONS

Fig. 1. The example of angles (@ASB and @CSD) with vertex at the
Sun (S) within which the distance @RR@' along the radial line @SR
between orbits is less than @r@s (@AA' = BB' = CC' = DD' = r@s).

Fig. 2. Semimajor axes @a and eccentricities @e of orbits of
bodies collided the Earth. The centre of a circle corresponds to
initial values of @a and @e. Dashed lines designate the values of
@a and @e, at which aphelia or perihelia are equal to 1 AU.
Rhombuses correspond to the spatial model and crosses correspond
to the planar one. Starting data: @a@0 =1.7 AU and @e@0 = 0.5
(a), @a@0 = 2 AU and @e@0 = 0.4 (b), @a@0 = 2.5 AU and @e@0 =
0.4 (c), @a@0 = 2.82 AU and @e@0 = 0.7 (d), @a@0 = 3.1 AU and
@e@0 = 0.7 (e), @a@0 = 7 AU and @e@0 = 0.5 (f), @a@0 = 25 AU and
@e@0 = 0.5 (g - h).

In Figs. 2a - 2h legends at the vertical axes (eigth times
repeated):
1. @e
and at the horizontal axes:
2. @a, AU.

3.  (a)
4.  (b)
5.  (c)
6.  (d)
7.  (e)
8.  (f)
9.  (g)
10. (h)

Fig. 3. Semimajor axes @a and eccentricities @e of orbits of
bodies at @N@0 = 500 and @N = 250. Starting data: @a@0 =1.7 AU and
@e@0 = 0.5 (a - b), @a@0 = 2.5 AU and @e@0 = 0.4 (c - d), @a@0
= 2.82 AU and @e@0 = 0.7 (e - f), @a@0 = 7 AU and @e@0 = 0.5
(g), @a@0 =25 AU and @e@0 = 0.5 (h). Designations are the same
as in Fig. 2.

In Figs. 3a - 3h legends at the vertical axes (eigth times
repeated):
1. @e
and at the horizontal axes:
2. @a, AU.

3.  (a)
4.  (b)
5.  (c)
6.  (d)
7.  (e)
8.  (f)
9.  (g)
10. (h)

Fig. 4. Semimajor axes @a and eccentricities @e of orbits of
bodies at @N@0 = 500 and @N = 100. Initial data are the same as in
Fig. 3. Designations are the same as in Fig. 2.

In Figs. 4a - 4h legends at the vertical axes (eigth times
repeated):
1. @e
and at the horizontal axes:
2. @a, AU.

3.  (a)
4.  (b)
5.  (c)
6.  (d)
7.  (e)
8.  (f)
9.  (g)
10. (h)

Fig. 5. Semimajor axes @a and eccentricities @e of orbits of
bodies at @N@0 = 500 and @N = 100 (a - d, g), and at N@0 = 100 and
@N = 50 (e - f), and the values of @a and @e collided the Earth
(h). Starting data: @a@0 =1.7 AU and @e@0 = 0.5 (a), @a@0 = 2.82
AU and @e@0 = 0.7 (b), @a@0 =7 AU and @e@0 = 0.5 (c), @a@0 = 25 AU
and @e@0 = 0.5 (d), @a@0 =0.3 AU and @e@0 = 0.4 (e), @a@0 = 0.6 AU
and @e@0 = 0.4 (f - h). Designations are the same as in Fig. 2.

In Figs. 5a - 5h legends at the vertical axes (eigth times
repeated):
1. @e
and at the horizontal axes:
2. @a, AU.

3.  (a)
4.  (b)
5.  (c)
6.  (d)
7.  (e)
8.  (f)
9.  (g)
10. (h)

Fig. 6. Semimajor axes @a and eccentricities @e of orbits of
real asteroids with perihelia are less than 1.33 AU
(rhombuses), Trojans (crosses), and comets (squares).

In Figs. 6a - 6b legends at the vertical axes (two times
repeated):
1. @e
and at the horizontal axes:
2. @a, AU.

3.  (a)
4.  (b)