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Stony meteoroid space erosion and drag: Effect on cosmic ray exposure ages
Accepted Manuscript
Stony Meteoroid Space Erosion and Drag: Effect on Cosmic Ray Exposure Ages David Parry Rubincam PII: DOI: Reference:
S0019-1035(16)30106-3 10.1016/j.icarus.2017.04.003 YICAR 12428
To appear in:
Icarus
Received date: Revised date: Accepted date:
26 April 2016 3 April 2017 5 April 2017
Please cite this article as: David Parry Rubincam , Stony Meteoroid Space Erosion and Drag: Effect on Cosmic Ray Exposure Ages, Icarus (2017), doi: 10.1016/j.icarus.2017.04.003
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Highlights Particles impacting a meteoroid cause space erosion in addition to a drag force The meteoroid’s shrinking size quickens the drift to a resonance Cosmic ray exposure ages of stony meteoroids depend on drift time and erosion The upper limits on CRE ages agree well with observed ages
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Stony Meteoroid Space Erosion and Drag: Effect on Cosmic Ray Exposure Ages David Parry Rubincam
Geodesy and Geophysics Laboratory, Code 61A Earth Sciences Division NASA Goddard
voice: 301-614-6464 fax: 301-614-6522
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Space Flight Center Building 34, Room S280 Greenbelt, MD 20771
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email: [email protected]
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Abstract
Collisions with dust particles in retrograde orbits cause space erosion on stony
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meteoroids in addition to the particle drag which causes drift toward resonances. The spacing between resonances determines the maximum drift time and sets upper limits on
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the neon-21 cosmic ray exposure (CRE) ages for meteoroids less than ~1 m in radius,
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while space erosion controls the limit for radii greater than ~1 m; the limits accord well
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with the measured CRE ages of stony meteorites.
1. Introduction
Currently the Yarkovsky effects (Öpik, 1951; Rubincam, 1987, 1995) are thought to be the main mechanism for delivering meteoroids to the resonances and explaining
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their cosmic ray exposure (CRE) ages (Peterson, 1976, p. 106; Rubincam, 1995, p. 1592; Afonso et al., 1998; Farinella et al., 1998; Bottke et al., 2000, 2002, 2006). However, Rubincam (2015, p. 117-118) has proposed that space erosion and Wiegert (2015) that drag, both resulting from the collisions with small particles, may also be important for
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CRE ages and resonance delivery.
Wiegert (2015) investigated drag from cometary dust in retrograde orbits. He found that high energy retrograde impacts on meteoroids in prograde orbits produce drag that could be competitive with the Yarkovsky effects for delivering meteoroids in the
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asteroid belt to the resonances.
Rubincam (2015) investigated the effect on meteoroid CRE ages from space erosion caused by dust impacts. However, only impacts from Divine’s (1993) core and
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asteroidal dust populations were considered; these populations have no retrograde orbits. Retrograde dust impacts have a much more profound effect on meteoroid CRE
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ages than Divine’s populations. Accordingly, Wiegert’s (2015) retrograde collision scenario is investigated here for its effect on CRE ages.
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The present paper puts upper limits on
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Ne CRE ages, as determined by space
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erosion and as well as drift time to resonances. The upper limit on neon-21 CRE ages is determined not just by the maximum drift time, which depends on the spacing between
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resonances, but also depends on how quickly meteoroids are eroded. It is found that drift time controls the upper limit on CRE age for stony meteoroids with Rres ~1 m, where Rres is a meteoroid’s radius when it reaches a resonance, while space erosion controls the limit when Rres ~1 m. The limits agree well with observed CRE ages of stony meteoroids, which are ~150 106 y (e.g., McSween, 1999; Eugster et al., 2006).
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Only stony meteoroids originating in the asteroid belt are investigated. Meteoroid sources closer to the Earth, such as Near-Earth Objects (NEOs) (Morbidelli et al., 2006) are not dealt with here. Iron meteoroids are much less susceptible to space erosion and are only briefly mentioned. No Yarkovsky effects are considered here; only drag from
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collisions with small particles.
2. Terminology
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Wiegert (2015) uses the term “meteoroids” for the small objects that hit a much larger object. He calls the larger object an “asteroid”. The terminology here instead follows Rubincam (2015), using “dust” or “particles” to describe the small objects, while
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“meteoroid” describes the large object. The “dust” can be larger than what is normally considered to be dust-sized (Shirley, 1997), perhaps ranging up to a few grams in mass,
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while a “meteoroid” might range in size from several centimeters up to tens of meters.
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“Asteroid” is reserved for bodies larger than meteoroids.
3. Assumptions
Consider a meteoroid in a circular prograde ecliptic orbit about the Sun in the plane of the Solar System, with orbital semimajor axis a and mean motion
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n = (GMS/a3)1/2
(1)
where G is the universal constant of gravitation and MS is the Sun’s mass. The meteoroid
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is taken to be a sphere, with radius R and density , so that its mass M is
M = 4R3/3 .
(2)
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The meteoroid is assumed to suffer only head-on collisions (see Fig. 1), meaning that the dust is in a circular orbit with the same semimajor axis as the meteoroid, but in the retrograde orbit, and is presumably cometary dust spiraling in toward the Sun from
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Poynting-Robertson drag (Wiegert, 2015). Also, the meteoroid is assumed to be regolith-
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free.
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4. Single impact on a meteoroid
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Two things happen when a dust particle of mass mD impacts the meteoroid. One is that the impact creates a microcrater, so that fragments with a total mass Mej are ejected
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from the crater and leave the meteoroid. Thus the meteoroid undergoes space erosion and becomes less massive and smaller in size. The other is that momentum is transferred to the meteoroid, altering its orbit. The momentum transfer comes from two sources: the momentum of the impacting particle and the momenta of the ejecta. By action-reaction,
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the departing ejecta kick the meteoroid in the opposite direction; see Housen and Holsapple (2011) and Wiegert (2015). The total momentum from the ejecta is estimated as follows. Nakamura et al.
dN ej dm
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(1994) find that an impact on basalt ejects fragments according to a power law
= N ej0 m-a
(3)
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where dNej is the number of fragments ejected in the range m + dm, where m is the mass of a fragment, and Nej0 and are constants. Also, they find that the fragments are ejected with speeds
(4)
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Vej = Vej0 m-g
where Vej0 and are also constants.
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The total amount of mass Mej ejected from a single collision will then be by (3)
ò
m2
m1
æ N0 ö mej (dN ej / dmej )dmej = çç ej ÷÷ ( m22-a - m12-a ) è 2 -a ø
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M ej =
(5)
where m2 and m1 are the maximum and minimum masses of the ejecta and 2 0. The total amount of kinetic energy carried away by the ejecta by (3) and (4) is
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ò
1 Eej = 2
m2 m1
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mej (Vej ) (dN ej / dmej )dmej = 2
N ej0 (Vej0 )
2
(m 2(2 - a - 2g )
2-a -2g 2
- m12-a -2g )
(6)
N ej0 =
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where 2 2 0, so that
2 -a M ej m - m12-a
(7)
2-a 2
(V )
=
2(2 - a - 2g ) æ m22-a - m12-a öæ Eej ö ÷ . ç 2-a -2g ÷ç 2 -a - m12-a -2g øçè M ej ÷ø è m2
(8)
M
0 2 ej
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and
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The fragments will be assumed to be ejected uniformly over a hemisphere centered on the microcrater, no matter how grazing the impact. The component pej of the
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ejecta’s momentum normal to the meteoroid’s surface will then be
1 2
ò
m2
m1
mejVej (dN ej / dmej )dmej =
N ej0 Vej0 2(2 - a - g )
(m
2-a -g 2
- m12-a -g )
(9)
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pej =
from all fragments, where the factor of ½ comes from integrating over the hemisphere and 2 0. Thus by (5)-(9)
pej = P0 ( M ej Eej )
1/2
,
(10)
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where
2 ù 1/2 é m22-a -g - m12-a -g ) é (2 - a )(2 - a - 2g ) ù ê ( ú P0 = ê ú ê 2-a 2 2-a 2-a -2g 2-a -2g ú 2(2 a g ) m m m m ë û ( 2 )( 2 )û 1 1 ë
1/2
2 ù 1/2 é 1- f 2-a -g ) é (2 - a )(2 - a - 2g ) ù ê ( ú P0 = ê ú ê 2 2-a 2-a -2g ú 2(2 a g ) ë û (1- f ) (1- f )û ë
(11)
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Writing m1 = fm2 in the above equation, where f < 1, yields
.
1/2
,
(12)
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so that P0 depends only on the ratio of the largest and smallest masses, and not the actual
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values of the masses.
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Gault et al. (1972) find for polycrystalline rocks that
1+2z
cos2 Q ,
(13)
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M ej » K éëmD (Dv)2 / 2ùû
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where is colatitude on the meteoroid as shown in Fig. 1, K is a constant, and mD(v2)/2 is the kinetic energy of the impacting dust particle, with v = 2na being the relative speed between the meteoroid and the dust particle. Gault et al. find = 0.065. Also, K = Kstone = 7 10-6 for stony meteoroids, where Mej is measured in kilograms and the kinetic energy mD(v2)/2 is measured in joules (Rubincam, 2015, p. 114).
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Not all of the kinetic energy of the dust particle will be converted into the kinetic energy of the ejecta (Housen and Holsapple, 2011; Wiegert, 2015, p. 24). Thus the total kinetic energy of the ejecta will be
Eej = e éëmD (Dv)2 / 2ùû
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(14)
(M
1+z
ej
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where is an efficiency factor such that 1. By (13) and (14)
1/2 1/2 Eej ) = (e K ) éëmD (Dv)2 / 2ùû cosQ .
(15)
The dust will pepper the meteoroid’s leading hemisphere. It is reasonable to assume that
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the impacts will be spread uniformly over the R2 cross-sectional area seen by the dust
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particles (Fig. 1). The impacts on an annulus of radius Rsin and width Rcos d and averaged over the leading hemisphere give a factor of ½, so that by (13) the average
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amount of ejected mass M ej is (Rubincam, 2015, p. 114)
1+2z
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M ej = R2 éëmD (Dv)2 / 2ùû
1+2z æK ö . = ç ÷éëmD (Dv)2 / 2ùû è2ø
2p
p /2
0
0
ò ò
cos3Q sinQ dQ dF / p R2
(16)
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Similarly, by (10) and (16) the average amount of momentum carried off in the meteoroid’s along-track direction will be
=
2p
p /2
0
0
ò ò
cos3 Q sinQ dQdF / p R2
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1+z
1/2 pej = R2 P0 (e K ) éëmD (Dv)2 / 2ùû
1+z P0 (e K)1/2 é 2 ù m (Dv) / 2 D ë û 2
(17)
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where one factor of cos in the integral comes from (15), another from the annulus, and another from the component of momentum in the along-track direction.
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5. Time to reach a resonance
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Section 4 considered the average effect of a single dust particle impact on the meteoroid. It is now time to consider the evolution of the meteoroid’s orbit from multiple
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impacts. A meteoroid shrinks in size and mass from space erosion (Whipple and Fireman,
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1959; Fisher, 1961; Whipple, 1962; Schaeffer et al., 1981; Hughes, 1982; Wieler and Graf, 2001, p. 227; Welten et al., 2001; Rubincam, 2015), altering the magnitude of the
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forces on it from the size change. The shrinkage affects the drift time to the resonances (Wiegert, 2015), as shown below. Let ND be the number density of the dust particles in the mass range mD to mD +
dmD. Here ND is assumed to be high enough that it is as though the meteoroid is
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immersed in a gas and has a constant drag force on it. The total density retro of the dust particles by adding up all the masses mD is
rretro = ò N D mD dmD .
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(18)
Let Pej be the sum of all the ejecta momentum in the meteoroid’s along-track
dPej dt
= - ò (mD Dv + pej )(N D Dv)p R 2 dmD
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direction. The force on the meteoroid in the along-track direction will then be by (17)
(19)
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where t is time. In (19) the mDv comes from the inelastic collision with the dust particle, while the pej comes from the ejecta momenta.
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If a meteoroid is in the asteroid belt and the resonances are a distance a apart,
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then how long does it take the meteoroid to reach a resonance? The rate of change of the meteoroid’s orbital semimajor axis a with time for a circular orbit will be given by (e.g.,
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Blanco and McCuskey, 1960, p. 178)
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0 0 3Falong 3Falong da 2S 2Falong = = ==dt n nM 2nr R 2nr (R0 - R& t)
(20)
where S is the along-track acceleration, n is given by (1), M is given by (2), and MS = 0 0 Falong = R2 Falong , with Falong being the along-track force and Falong is positive and
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independent of R. Also in (20), the meteoroid’s radius shrinks from abrasive space erosion (Gault et al., 1972) with time according to
R = R0 - R& t ,
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(21)
where R0 is the meteoroid’s radius at time t = 0 and R& is the rate at which the radius changes with time. It turns out that R& is constant so long as the particle environment
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does not change (Rubincam, 2015). If it is assumed that a is a short enough distance so that every quantity on the right-hand side of (20) can be considered constant except t,
(22)
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0 æ ö R& 3Falong ç Da = ln 1- Tdrift ÷ . ÷ 2nr R& çè R0 ø
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then (20) can be integrated to give
R0 1- e- L ) , ( & R
(23)
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Tdrift =
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Solving for Tdrift, the time to reach a resonance, yields
where
L=
2nr Da R& 0 3Falong
.
(24)
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Using (21) to eliminate R0 in (23) yields
Rres L (e -1) , R&
(25)
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Tdrift =
where Rres is the meteoroid’s radius when it reaches the resonance.
Equations (20)-(25) are general for a meteoroid in a circular orbit; they hold
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regardless of which directions the impacting particles bombard the meteoroid. For the particular case considered here of dust particles in the retrograde circular orbit
1+z 1 2N D mD (Dv)2 + P0 (e K)1/2 (N D Dv)éëmD (Dv)2 / 2ùû dmD ò 2
(26)
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0 Falong,retro =
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The rate of change in radius R& retro can be calculated via (2) and (16):
(27)
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dM dR = 4pr R 2 = - ò M ej (N D Dv)p R 2 dmD dt dt retro
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so that
dR K R& retro = =dt 8r
ò (N
1+2z
D
Dv)éëmD (Dv)2 / 2ùû
dmD
(28)
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and
ò (N
1+2z
D
Dv)éëmD (Dv)2 / 2ùû
dmD 1+z
ò 12N D mD (Dv)2 + 6P0 (e K ) (N DDv)éëmD (Dv)2 / 2ùû dmD 1/2
.
(29)
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Lretro =
nK Da
6. Stony meteoroids
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For simplicity it will be assumed for stony meteoroids that = 0 in (16), so that mass loss is proportional to energy. However, K will be replaced with K1, such that
1+2z
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K1 éëmD (Dv)2 / 2ùû = K éëmD (Dv)2 / 2ùû
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for mD = 1.5 10-8 kg, for which ND peaks at the Earth (Love and Brownlee, 1993), and v = 34.470 103 m, as is appropriate for the asteroid belt at 3 AU. For K = 7 10-6,
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these assumptions give K1 = 9.33 10-6 s2 m-2 and (26) becomes
{
}
1 2 rretro (Dv)2 + P0 (e K1 )1/2 Dv éërretro (Dv)2 / 2ùû 2
(30)
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0 Falong,retro »
where retro is given by (18), while (28) becomes
K R& retro » 1 Dv éërretro (Dv)2 / 2ùû . 8r
(31)
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Using (30) and (31) in (25) yields
Rres Lretro (e -1) , R& retro
(32)
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Tdrift,retro »
Lretro »
nK1 Da Dv 24 + 6P0 (e K1 ) Dv 1/2
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where now
,
(33)
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2 æ 3AU ö æ Da ö 184.78 ç ÷ ÷ ç è a ø è 0.1AU ø » é P ùæ e ö1/2 æ 3AU ö1/2 24 + 297.82 ê 0 úç ÷ ç ÷ ë (2 / 3) ûè 0.5 ø è a ø
(34)
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Lretro
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ND. This can be written
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so that retro is independent of retro and thus does not depend on the functional form of
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where a is now in AU. Writing retro for stony meteoroids in the form of (34) makes the quantities in
parentheses and square brackets equal to 1 for the chosen nominal values. The resonances in the middle of the main asteroid belt at 3 AU are ~0.1 AU apart (Nesvorný et al., 2002).
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The value for is taken to be 0.5 as assumed by Wiegert (2015), for the reasons he gives on his page 25. Finding a nominal value for P0 is harder work. The experimental data of Nakamura and Fujiwara (1991) find that 5/3; a value of = 5/3 exactly is adopted
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here. Giblin (1998) finds that ~0 ~1/6, with an average value of 1/13. It is assumed here that the range is exactly 0 1/6. When < 1/6, (12) can be used to find P0. However, if = 1/6 exactly, then 2 2 = 0, and P0 can be obtained via a limit
2 ù 1/2 é é (2 - a )x ù ê (1- f 2-a -g ) ú P0 = ê 2ú ë 2(2 - a - g ) û ê (1- f 2-a ) (1- f x ) ú ë û
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process as follows. Set x = 2 2 in (12) so that
1/2
M
.
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Using
(x ln f )2 (x ln f )3 f =1+ x ln f + + +... 2! 3!
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x
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(Selby, 1974, p. 472) in the equation and letting x 0 yields
2 ù 1/2 é é (2 - a ) ù ê (1- f 2-a -g ) ú P0 = ê 2ú ë 2(2 - a - g ) û ê (1- f 2-a ) ( -ln f ) ú ë û
for 2 2 = 0.
1/2
(35)
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The range 10-3 f 10-5 is assumed here, as seems appropriate from Nakamura and Fujiwara (1991) and Nakamura et al. (1994), where (3) appears to start failing at the smaller masses. Using the adopted ranges and values of , , and f in (12) and (35) yields the tight range of values 0.623 P0 0.699 (Fig. 2). Values of < 5/3 lead to P0 0.704.
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Hence a mid-range value of P0 = 2/3 appears to be quite reasonable.
R& retro as given by (31) can also be put into a form similar to (34). Love and Brownlee (1998) find that the Earth accumulates dust at the rate E = ~4 107 kg y-1 =
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1.27 kg s-1. This is the total rate at the Earth for dust in all orbits. Also, some of the observed rate E is due to gravitational focusing by the Earth. Hence to find the rate in free space for dust in circular retrograde orbits, E must be adjusted downward to sE, where s 1 (Wiegert (2015).
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The total retrograde dust density at the Earth retro,Earth can be found from
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RE2retro,EarthvE = sE, where vE = 60000 m s-1 and RE = 6.371 106 m is the Earth’s
smE æ 3AU ö -3 -20 æ 3AU ö ç ÷ = 5.53´10 s ç ÷ kg m . 2 è a ø 3p RE DvE è a ø
(36)
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rretro =
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radius. The value for retro extrapolated to the asteroid belt is then (Wiegert, 2015, p. 25):
Here the nominal value of s is taken to be 0.5 as assumed by Wiegert (2015). A density of
= 2800 kg m-3 is typical for stones. Thus (31) with the aid of (36) can be written:
æ s öæ 2800 kg m -3 R& retro = 0.0074 ç ÷ç è 0.5 øè r
öæ 3AU ö5/2 ÷ ÷ç øè a ø
m My-1 .
(37)
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Eq. (37) can also be written
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æ 1m ö R& retro » çç ÷÷ è Tmetre,retro ø
(Rubincam, 2015, p. 115), so that
öæ a ö æ 0.5 öæ r Tmetre,retro =134 ç ÷ç ÷ç ÷ è s øè 2800 kg m -3 øè 3AU ø
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5/2
My .
(38)
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Here Tmetre,retro is the time it takes to wear down a meteoroid’s radius by 1 m from the dust in the retrograde orbit.
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For the nominal values of a = 3 AU, a = 0.1 AU, P0 = 2/3, = 0.5, s = 0.5, and
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= 2800 kg m-3, (34) works out to be
retro 0.574 ,
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(39)
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while (38) is
Tmetre,retro (40)
=
134
106
y
.
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Hence for the chosen parameters
Tdrift,retro » ( eLretro -1) Tmetre,retro (Rres /1m) = (104 ´10 6 y)(Rres /1m)
(41)
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where Rres is in meters. The diamonds in Fig. 3 show Tdrift,retro/(Rres/1 m) as a function of a.
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7. Stony cosmic ray exposure ages
For meteoroids traveling to the resonances from the Yarkovsky effects or dust impacts, but with no space erosion, Tdrift would tend to roughly coincide with the CRE
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age. In the Yarkovsky scenario, the material that becomes a meteoroid is most likely deep inside an asteroid and shielded from cosmic rays. It is then liberated by a catastrophic
ED
impact. Thereafter the meteoroid spends most of its time journeying to the resonance
PT
exposed to the cosmic rays. Once the meteoroid reaches the resonance, relatively little time is spent within the resonance before hitting the Earth (Gladman et al., 1997). Hence
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the CRE age is essentially the drift time to the resonance. In the collision scenario, dust impacts cause space erosion, which affects the CRE
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age. Though the surface of any object is exposed to cosmic rays, the surface gets worn away, taking its cosmic ray products with it. Thus a meteoroid has an erosion CRE age , which is younger than the length of time the meteoroid has existed as an independent body (Rubincam, 2015).
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Fig. 4 (solid curve) shows the
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Ne erosion age Ne,retro as a function of Rres for
meteoroids at a = 3 AU for Tmetre,retro = 134 106 y as given by (40). Here the erosion age is computed for a meteoroid with an initial radius of 5 m that shrinks steadily down by space erosion from retrograde particle impacts, all the while being bombarded by cosmic
The computation of the
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rays; the drift time Tdrift,retro does not enter into the calculation.
Ne erosion age Ne,retro follows Rubincam (2015). The
neon production at various depths in a meteoroid of radius R is parameterized according
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to equations (11) – (13) of that paper. The amount of neon at depth is computed as the meteoroid’s radius shrinks from space erosion until the meteoroid reaches its final radius. The erosion age is then chosen to fit the neon concentration inside the meteoroid at half its final radius as though the meteoroid always had its final radius and never eroded. It
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was found that
(42)
PT
ED
Tmetre .
to a high degree of approximation.
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Note in Fig. 4 that Ne,retro 0.41Tmetre for Rres ~1 m, regardless of how long it
takes the meteoroid to reach a resonance. (How 0.41Tmetre varies with a is shown as the
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open circles in Fig. 3.) But once the radius shrinks below ~1 m, Ne,retro rises sharply. Of interest in Fig. 4 is the shaded region under the curves. For Rres ~0.7 m, the
CRE age of a meteoroid journeying to a resonance via collisions is less than or equal to the time Tdrift,retro to travel between resonances spaced a distance a= 0.1 AU apart. But for Rres ~0.7 m, it is space erosion which controls the upper limit on the CRE age,
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so that ages have to be less than or equal to the erosion age Ne,retro. Therefore a meteoroid must have an age somewhere in the shaded region under the dashed line or solid curve for the assumed travel distance. The maximum CRE age at 3 AU is ~75 106 y for Rres 0.7 m, which is where Tdrift,retro = Ne,retro.
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Fig. 5 is similar to Fig. 4, but with Divine’s (1993) core and asteroidal populations added to the retrograde population. Divine’s populations, which are prograde, do not overlap with the retrograde population. Divine’s populations are
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assumed to cause space erosion but not drag. At 3 AU, Tmetre,Divine 666 106 y (Rubincam, 2015, Fig. 8), so that Tmetre, retro+Divine 112 106 y. The maximum CRE age then lowers to ~69 106 y at 3 AU when Rres = ~0.6 m, and the erosion age for Rres > ~1 m drops below 50 106 y. (Because (42) holds, the erosion age curve in Fig. 5 suffers
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only a change in the vertical scale. However, Tdrift, retro+Divine suffers more than a change in
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vertical scale because the increased R& in (24) changes , so that the dashed line meets the curve at Rres = ~0.6 m instead of ~0.7 m.)
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Eugster et al. (2006) show the CRE ages of many stony meteorites in their Fig.s 4, 5, and 6 compiled from various papers. The CRE ages are ~100 106 y for chondrites
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and achondrites except for the aubrites, which range up to ~150 106 y.
8. Iron meteoroids
For iron meteoroids, it was assumed in Rubincam (2015, p. 116) that K = Kiron 7.8 10-8, a value estimated from the experimental results of Matsui and Schultz (1984), who chilled iron targets down to 107 K. However, the value of Kiron may be higher.
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Assume that Mej/m = Kiron(v)2/2, so that the ejecta mass is proportional to energy. According to Table 1 of Matsui and Schultz, the Gibeon meteorite yielded Mej/m = 2.55 when hit by a basaltic projectile with speed 5000 m s-1, giving Kiron = 2 10-7. Using this
2 æ 3AU ö æ Da ö 3.96 ç ÷ ÷ ç è a ø è 0.1AU ø » é P ùæ e ö1/2 æ 3AU ö1/2 24 + 43.60 ê 0 úç ÷ ç ÷ ë (2 / 3) ûè 0.5 ø è a ø
(43)
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Liron,retro
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value in (33) and (31) results in
and
öæ a ö æ 0.5 öæ riron Tmetre,retro =17618ç ÷ç ÷ -3 ÷ ç è s øè 7870 kg m øè 3AU ø
My .
(44)
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M
5/2
where the nominal density of iron is taken to be 7870 kg m-3. This and the other nominal
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values at 3 AU yield iron = 0.0585 and
(
Liron,retro
)
-1 Tmetre,retro (Rres /1m) =1057´10 6 (Rres /1m)y
(45)
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Tdrift,retro » e
for an iron meteoroid traveling 0.1 AU. On the other hand, the Toluca meteorite yielded Mej/m = 23.3 when struck by a steel projectile traveling at 3380 m s-1 because the meteorite had oxide shales that spalled on impact. This case gives Kiron = 4 10-6, iron = 0.362, and Tmetre,retro = 384 106 y. These numbers are in rough accord with the CRE
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ages of irons: iron meteorites tend to have CRE ages in the range 108 - 109 y (e.g., McSween, 1999; Eugster et al. 2006). So it may be that drift time controls the CRE ages for irons. However, the parameters are uncertain, and no attempt is made here to estimate
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erosion ages of irons meteoroids.
9. Discussion
In the collision scenario investigated here, da/dt is negative and meteoroids drift
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sunward toward a resonance. Drag from collisions with dust particles inevitably implies space erosion. If the drag on a stony meteoroid is mainly due to the ejecta, as in the case of particles in the retrograde orbit as investigated by Wiegert (2015), then space erosion
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will also be substantial. Space erosion can profoundly affect CRE ages of meteoroids. Space erosion, rather than drift time Tdrift to a resonance, controls the upper limit of CRE Ne ages for meteoroids with R ~1 m, as shown in Fig.s 4 and 5.
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The upper limit on allowed ages in Fig.s 4-5 fall in the range ~70 106 y for
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stony meteoroids at 3 AU. This accords well with the observed CRE ages, most of which
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range from ~106 y to ~100 106 y (e.g., McSween, 1999; Eugster et al., 2006). The exception is the aubrites. The aubrites are physically weak, but yet have ages
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which range up to ~150 106 y. They may come from asteroids in high-inclination orbits (McSween, 1999, p. 247; Eugster et al., 2006, p. 841; Ćuk et al., 2014). If so, they may have suffered fewer collisions than they would have in ecliptic orbits and thus have longer CRE ages.
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The results depend upon the chosen values of parameters such as , f, and P0, which are not certain. If s or are larger or smaller than the nominal values, then the drift times and the erosion ages will go up or down. The main dependence of P0 appears to be on f (Fig. 2), and f is not well determined. Also, the approximation = 0 was used get
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(34) and (38), avoiding needing to know ND as a function of mD. More exact calculations would use = 0.065, but necessitate knowing the functional form of ND in order to perform the required numerical integrations.
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The results obtained here assume that the meteoroids are regolith-free. A regolith would complicate calculating drift time and erosion age, and is beyond the scope of this paper. The effect of multiple large collisions on CRE ages is not considered here (Vokrouhlický and Farinella, 2000), nor is the catastrophic destruction of meter-sized
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objects (Bottke et al., 2005).
All the calculations assume that dust particle concentrations have remained steady
Shaeffer et al., 1981).
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for hundreds of millions of years. This may not be the case (e.g., Farley, 2001, p. 1194;
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It was implicitly assumed here that Rres, the meteoroid’s radius when it reaches a
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resonance, is the same radius it had just before entering the Earth’s atmosphere. In other words, the time between injection into the resonance and falling on Earth is so short that
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not much erosion takes place. However, the case of meteoroids pumped up into highly elliptical Earth-crossing orbits and hit by retrograde-orbit impactors should be investigated to see how high or low the erosion might be. The Yarkovsky effects give CRE ages similar to those found here (e.g., Bottke et al., 2006; Ćuk et al., 2014). Hence a question arises: what percentage of CRE ages is due
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to Yarkovsky and what percentage is due to collisional drift/space erosion? Could something interfere with the operation of the Yarkovsky effects, such as the YarkovskyO’Keefe-Radzievskii-Paddack (YORP) effect (Paddack, 1969; Paddack and Rhee, 1975; Rubincam, 2000; Bottke et al., 2006; Vokrouhlický et al., 2007) or collisions (Wiegert,
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2015)? It may be that the Yarkovsky effects play a smaller role in determining CRE ages than currently thought.
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Acknowledgments
I thank Susan Fricke and Susan Poulose for excellent programming support. I thank Richard Durisen and an anonymous referee, both of whom corrected errors and
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made excellent suggestions which greatly improved the paper. This paper was inspired by
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Wiegert (2015). I thank Herbert V. Frey for being supportive of my research.
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Öpik, E. J., 1951. Collision probabilities with the planets and the distribution of interplanetary matter. Proc. R. Irish Acad. 54A, 165-199.
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Rubincam, D. P., 1995. Asteroid orbit evolution due to thermal drag. J. Geophys. Res. 100, 1585-1594. doi: 10.1029/94JE02411.
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Rubincam, D. P., 2000. Radiative spin-up and spin-down of small asteroids. Icarus 148, 2-11. doi: 10.1006/icar.2000.6485. Rubincam, D. P., 2015. Space erosion and cosmic ray exposure ages of stony meteorites. Icarus 245, 112-121. http://dx.doi.org/10.1016/j.icarus.2014.09.005.
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Shaeffer, O. A., Nagel, K., Fechtig, H., Neukum, G., 1981. Space erosion of meteorites and the secular variation of cosmic rays (over 109 years). Planet. Space Sci. 29, 11091118. doi: 10.1016/0032-0633(81)90010-6.
Shirley, J. H., 1997. Dust. In: Shirley, J. H., Fairbridge R. W. (Ed.s), Encyclopedia of
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Selby, S. M., 1974. CRC Standard Mathematical Tables, 22nd ed. CRC Press, Cleveland, OH.
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Vokrouhlický, D., Farinella, P., 2000. Efficient delivery of meteorites to the Earth from a wide range of asteroid parent bodies. Nature 407, 606-608.
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Vokrouhlický, D., Beiter, S., Nesvorný, D., Bottke, W. F., 2007. Generalized YORP evolution: Onset of tumbling and new asymptotic states. Icarus 191, 636-650. doi:
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10.1016/j.icarus.2007.06.002.
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Welten, K. C., Nishiizumi, K., Caffee, M. W., Schultz, L., 2001. Update on exposure ages of diogenites: The impact and evidence of space erosion and/or collisional
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disruption of stony meteoroids. Meteoritics & Planet. Sci. 36, supplement, A223. Wiegert, P. A., 2015. Meteoroid impacts on asteroids: A competitor for Yarkovsky and YORP. Icarus 252, 22-31. https://dx.doi.org/10.1016/j.icarus.2014.12.022.
Whipple, F. L., 1962. Meteorite erosion in space. Astron. J. 67, 285. doi: 10.1086/108864.
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Whipple, F. L., Fireman, E. L., 1959. Calculation of erosion in space from the cosmic-ray exposure ages of meteorites. Nature 183, 1315. doi:10.1038/1831315a0. Wieler, R., Graf, T., 2001. Cosmic ray exposure: History of meteorites. In: PeuckerEhrenbrink, B., Schmitz, B. (Ed.s), Accretion of Extraterrestrial Matter Throughout
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Earth’s History. Kluwer Academic/Plenum Publishers, New York, pp. 221-240.
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Fig. 1 Geometry of dust particles impacting on a spherical meteoroid. The along track direction is shown by the thick arrow. Dust particles (black dots) traveling in the
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retrograde direction collide head-on with the meteoroid. The particles are spread evenly
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over the cross-sectional area R2 seen by the particles, where R is the meteoroid’s radius.
is the longitude on the meteoroid, while is colatitude. A particle impacting on the
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annulus of radius Rsin and width Rcos d ejects mass which is proportional to cos2
. The meteoroid is assumed to randomly orient itself in space, so that it always remains an eroding sphere.
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Fig. 2 P0 computed for = 5/3 and = 0, 1/12, and 1/6 when f = 10-3, 10-4, or 10-5. The symbols are joined with straight lines for clarity. A nominal value of P0 = 2/3 is assumed
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in this paper.
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Tdrift,retro/(Rres/1 m) (diamonds) and = 0.41Tmetre (open circles) for stony
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Fig. 3
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Rubincam
meteoroids as a function of semimajor axis a for the nominal values a = 0.1 AU, P0 =
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2/3, = 0.5, s = 0.5, and = 2800 kg m-3. Rres is the radius when the meteoroid reaches the resonance. Erosion age = 0.41Tmetre is the age for meteoroids when Rres > ~1 m. The
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straight lines joining the data points are for clarity.
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Rubincam
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Fig. 4 Drift time Tdrift, retro (dashed line) to a resonance and 21Ne erosion age Ne,retro (solid curve) plotted as a function of Rres, which is the meteoroid’s radius when it reaches the
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resonance, for the nominal values of the parameters. The meteoroid is at a = 3 AU in a prograde circular orbit in the ecliptic and drifts inward to a resonance. The distance
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between resonances is 0.1 AU. The shaded region gives the allowable CRE ages in the collision scenario for meteoroids between the resonances. Here Tmetre = 134 106 y.
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Fig. 5. Plot similar to Fig. 4, but for the combined retrograde and Divine populations. Divine’s populations, which are prograde, are assumed to cause space erosion but not
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drag. The addition of Divine’s populations shortens Tmetre to 112 106 y.
Stony Meteoroid Space Erosion and Drag: Effect on Cosmic Ray Exposure Ages David Parry Rubincam PII: DOI: Reference:
S0019-1035(16)30106-3 10.1016/j.icarus.2017.04.003 YICAR 12428
To appear in:
Icarus
Received date: Revised date: Accepted date:
26 April 2016 3 April 2017 5 April 2017
Please cite this article as: David Parry Rubincam , Stony Meteoroid Space Erosion and Drag: Effect on Cosmic Ray Exposure Ages, Icarus (2017), doi: 10.1016/j.icarus.2017.04.003
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Highlights Particles impacting a meteoroid cause space erosion in addition to a drag force The meteoroid’s shrinking size quickens the drift to a resonance Cosmic ray exposure ages of stony meteoroids depend on drift time and erosion The upper limits on CRE ages agree well with observed ages
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Stony Meteoroid Space Erosion and Drag: Effect on Cosmic Ray Exposure Ages David Parry Rubincam
Geodesy and Geophysics Laboratory, Code 61A Earth Sciences Division NASA Goddard
voice: 301-614-6464 fax: 301-614-6522
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Space Flight Center Building 34, Room S280 Greenbelt, MD 20771
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email: [email protected]
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Abstract
Collisions with dust particles in retrograde orbits cause space erosion on stony
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meteoroids in addition to the particle drag which causes drift toward resonances. The spacing between resonances determines the maximum drift time and sets upper limits on
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the neon-21 cosmic ray exposure (CRE) ages for meteoroids less than ~1 m in radius,
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while space erosion controls the limit for radii greater than ~1 m; the limits accord well
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with the measured CRE ages of stony meteorites.
1. Introduction
Currently the Yarkovsky effects (Öpik, 1951; Rubincam, 1987, 1995) are thought to be the main mechanism for delivering meteoroids to the resonances and explaining
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their cosmic ray exposure (CRE) ages (Peterson, 1976, p. 106; Rubincam, 1995, p. 1592; Afonso et al., 1998; Farinella et al., 1998; Bottke et al., 2000, 2002, 2006). However, Rubincam (2015, p. 117-118) has proposed that space erosion and Wiegert (2015) that drag, both resulting from the collisions with small particles, may also be important for
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CRE ages and resonance delivery.
Wiegert (2015) investigated drag from cometary dust in retrograde orbits. He found that high energy retrograde impacts on meteoroids in prograde orbits produce drag that could be competitive with the Yarkovsky effects for delivering meteoroids in the
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asteroid belt to the resonances.
Rubincam (2015) investigated the effect on meteoroid CRE ages from space erosion caused by dust impacts. However, only impacts from Divine’s (1993) core and
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asteroidal dust populations were considered; these populations have no retrograde orbits. Retrograde dust impacts have a much more profound effect on meteoroid CRE
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ages than Divine’s populations. Accordingly, Wiegert’s (2015) retrograde collision scenario is investigated here for its effect on CRE ages.
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The present paper puts upper limits on
21
Ne CRE ages, as determined by space
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erosion and as well as drift time to resonances. The upper limit on neon-21 CRE ages is determined not just by the maximum drift time, which depends on the spacing between
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resonances, but also depends on how quickly meteoroids are eroded. It is found that drift time controls the upper limit on CRE age for stony meteoroids with Rres ~1 m, where Rres is a meteoroid’s radius when it reaches a resonance, while space erosion controls the limit when Rres ~1 m. The limits agree well with observed CRE ages of stony meteoroids, which are ~150 106 y (e.g., McSween, 1999; Eugster et al., 2006).
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Only stony meteoroids originating in the asteroid belt are investigated. Meteoroid sources closer to the Earth, such as Near-Earth Objects (NEOs) (Morbidelli et al., 2006) are not dealt with here. Iron meteoroids are much less susceptible to space erosion and are only briefly mentioned. No Yarkovsky effects are considered here; only drag from
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collisions with small particles.
2. Terminology
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Wiegert (2015) uses the term “meteoroids” for the small objects that hit a much larger object. He calls the larger object an “asteroid”. The terminology here instead follows Rubincam (2015), using “dust” or “particles” to describe the small objects, while
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“meteoroid” describes the large object. The “dust” can be larger than what is normally considered to be dust-sized (Shirley, 1997), perhaps ranging up to a few grams in mass,
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while a “meteoroid” might range in size from several centimeters up to tens of meters.
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“Asteroid” is reserved for bodies larger than meteoroids.
3. Assumptions
Consider a meteoroid in a circular prograde ecliptic orbit about the Sun in the plane of the Solar System, with orbital semimajor axis a and mean motion
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n = (GMS/a3)1/2
(1)
where G is the universal constant of gravitation and MS is the Sun’s mass. The meteoroid
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is taken to be a sphere, with radius R and density , so that its mass M is
M = 4R3/3 .
(2)
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The meteoroid is assumed to suffer only head-on collisions (see Fig. 1), meaning that the dust is in a circular orbit with the same semimajor axis as the meteoroid, but in the retrograde orbit, and is presumably cometary dust spiraling in toward the Sun from
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Poynting-Robertson drag (Wiegert, 2015). Also, the meteoroid is assumed to be regolith-
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free.
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4. Single impact on a meteoroid
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Two things happen when a dust particle of mass mD impacts the meteoroid. One is that the impact creates a microcrater, so that fragments with a total mass Mej are ejected
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from the crater and leave the meteoroid. Thus the meteoroid undergoes space erosion and becomes less massive and smaller in size. The other is that momentum is transferred to the meteoroid, altering its orbit. The momentum transfer comes from two sources: the momentum of the impacting particle and the momenta of the ejecta. By action-reaction,
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the departing ejecta kick the meteoroid in the opposite direction; see Housen and Holsapple (2011) and Wiegert (2015). The total momentum from the ejecta is estimated as follows. Nakamura et al.
dN ej dm
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(1994) find that an impact on basalt ejects fragments according to a power law
= N ej0 m-a
(3)
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where dNej is the number of fragments ejected in the range m + dm, where m is the mass of a fragment, and Nej0 and are constants. Also, they find that the fragments are ejected with speeds
(4)
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M
Vej = Vej0 m-g
where Vej0 and are also constants.
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The total amount of mass Mej ejected from a single collision will then be by (3)
ò
m2
m1
æ N0 ö mej (dN ej / dmej )dmej = çç ej ÷÷ ( m22-a - m12-a ) è 2 -a ø
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M ej =
(5)
where m2 and m1 are the maximum and minimum masses of the ejecta and 2 0. The total amount of kinetic energy carried away by the ejecta by (3) and (4) is
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ò
1 Eej = 2
m2 m1
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mej (Vej ) (dN ej / dmej )dmej = 2
N ej0 (Vej0 )
2
(m 2(2 - a - 2g )
2-a -2g 2
- m12-a -2g )
(6)
N ej0 =
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where 2 2 0, so that
2 -a M ej m - m12-a
(7)
2-a 2
(V )
=
2(2 - a - 2g ) æ m22-a - m12-a öæ Eej ö ÷ . ç 2-a -2g ÷ç 2 -a - m12-a -2g øçè M ej ÷ø è m2
(8)
M
0 2 ej
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and
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The fragments will be assumed to be ejected uniformly over a hemisphere centered on the microcrater, no matter how grazing the impact. The component pej of the
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ejecta’s momentum normal to the meteoroid’s surface will then be
1 2
ò
m2
m1
mejVej (dN ej / dmej )dmej =
N ej0 Vej0 2(2 - a - g )
(m
2-a -g 2
- m12-a -g )
(9)
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pej =
from all fragments, where the factor of ½ comes from integrating over the hemisphere and 2 0. Thus by (5)-(9)
pej = P0 ( M ej Eej )
1/2
,
(10)
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where
2 ù 1/2 é m22-a -g - m12-a -g ) é (2 - a )(2 - a - 2g ) ù ê ( ú P0 = ê ú ê 2-a 2 2-a 2-a -2g 2-a -2g ú 2(2 a g ) m m m m ë û ( 2 )( 2 )û 1 1 ë
1/2
2 ù 1/2 é 1- f 2-a -g ) é (2 - a )(2 - a - 2g ) ù ê ( ú P0 = ê ú ê 2 2-a 2-a -2g ú 2(2 a g ) ë û (1- f ) (1- f )û ë
(11)
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Writing m1 = fm2 in the above equation, where f < 1, yields
.
1/2
,
(12)
M
so that P0 depends only on the ratio of the largest and smallest masses, and not the actual
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values of the masses.
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Gault et al. (1972) find for polycrystalline rocks that
1+2z
cos2 Q ,
(13)
CE
M ej » K éëmD (Dv)2 / 2ùû
AC
where is colatitude on the meteoroid as shown in Fig. 1, K is a constant, and mD(v2)/2 is the kinetic energy of the impacting dust particle, with v = 2na being the relative speed between the meteoroid and the dust particle. Gault et al. find = 0.065. Also, K = Kstone = 7 10-6 for stony meteoroids, where Mej is measured in kilograms and the kinetic energy mD(v2)/2 is measured in joules (Rubincam, 2015, p. 114).
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9
Not all of the kinetic energy of the dust particle will be converted into the kinetic energy of the ejecta (Housen and Holsapple, 2011; Wiegert, 2015, p. 24). Thus the total kinetic energy of the ejecta will be
Eej = e éëmD (Dv)2 / 2ùû
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(14)
(M
1+z
ej
AN US
where is an efficiency factor such that 1. By (13) and (14)
1/2 1/2 Eej ) = (e K ) éëmD (Dv)2 / 2ùû cosQ .
(15)
The dust will pepper the meteoroid’s leading hemisphere. It is reasonable to assume that
M
the impacts will be spread uniformly over the R2 cross-sectional area seen by the dust
ED
particles (Fig. 1). The impacts on an annulus of radius Rsin and width Rcos d and averaged over the leading hemisphere give a factor of ½, so that by (13) the average
CE
PT
amount of ejected mass M ej is (Rubincam, 2015, p. 114)
1+2z
AC
M ej = R2 éëmD (Dv)2 / 2ùû
1+2z æK ö . = ç ÷éëmD (Dv)2 / 2ùû è2ø
2p
p /2
0
0
ò ò
cos3Q sinQ dQ dF / p R2
(16)
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Similarly, by (10) and (16) the average amount of momentum carried off in the meteoroid’s along-track direction will be
=
2p
p /2
0
0
ò ò
cos3 Q sinQ dQdF / p R2
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1+z
1/2 pej = R2 P0 (e K ) éëmD (Dv)2 / 2ùû
1+z P0 (e K)1/2 é 2 ù m (Dv) / 2 D ë û 2
(17)
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where one factor of cos in the integral comes from (15), another from the annulus, and another from the component of momentum in the along-track direction.
M
5. Time to reach a resonance
ED
Section 4 considered the average effect of a single dust particle impact on the meteoroid. It is now time to consider the evolution of the meteoroid’s orbit from multiple
PT
impacts. A meteoroid shrinks in size and mass from space erosion (Whipple and Fireman,
CE
1959; Fisher, 1961; Whipple, 1962; Schaeffer et al., 1981; Hughes, 1982; Wieler and Graf, 2001, p. 227; Welten et al., 2001; Rubincam, 2015), altering the magnitude of the
AC
forces on it from the size change. The shrinkage affects the drift time to the resonances (Wiegert, 2015), as shown below. Let ND be the number density of the dust particles in the mass range mD to mD +
dmD. Here ND is assumed to be high enough that it is as though the meteoroid is
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11
immersed in a gas and has a constant drag force on it. The total density retro of the dust particles by adding up all the masses mD is
rretro = ò N D mD dmD .
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(18)
Let Pej be the sum of all the ejecta momentum in the meteoroid’s along-track
dPej dt
= - ò (mD Dv + pej )(N D Dv)p R 2 dmD
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direction. The force on the meteoroid in the along-track direction will then be by (17)
(19)
M
where t is time. In (19) the mDv comes from the inelastic collision with the dust particle, while the pej comes from the ejecta momenta.
ED
If a meteoroid is in the asteroid belt and the resonances are a distance a apart,
PT
then how long does it take the meteoroid to reach a resonance? The rate of change of the meteoroid’s orbital semimajor axis a with time for a circular orbit will be given by (e.g.,
CE
Blanco and McCuskey, 1960, p. 178)
AC
0 0 3Falong 3Falong da 2S 2Falong = = ==dt n nM 2nr R 2nr (R0 - R& t)
(20)
where S is the along-track acceleration, n is given by (1), M is given by (2), and MS = 0 0 Falong = R2 Falong , with Falong being the along-track force and Falong is positive and
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independent of R. Also in (20), the meteoroid’s radius shrinks from abrasive space erosion (Gault et al., 1972) with time according to
R = R0 - R& t ,
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(21)
where R0 is the meteoroid’s radius at time t = 0 and R& is the rate at which the radius changes with time. It turns out that R& is constant so long as the particle environment
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does not change (Rubincam, 2015). If it is assumed that a is a short enough distance so that every quantity on the right-hand side of (20) can be considered constant except t,
(22)
ED
0 æ ö R& 3Falong ç Da = ln 1- Tdrift ÷ . ÷ 2nr R& çè R0 ø
M
then (20) can be integrated to give
R0 1- e- L ) , ( & R
(23)
AC
CE
Tdrift =
PT
Solving for Tdrift, the time to reach a resonance, yields
where
L=
2nr Da R& 0 3Falong
.
(24)
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13
Using (21) to eliminate R0 in (23) yields
Rres L (e -1) , R&
(25)
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Tdrift =
where Rres is the meteoroid’s radius when it reaches the resonance.
Equations (20)-(25) are general for a meteoroid in a circular orbit; they hold
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regardless of which directions the impacting particles bombard the meteoroid. For the particular case considered here of dust particles in the retrograde circular orbit
1+z 1 2N D mD (Dv)2 + P0 (e K)1/2 (N D Dv)éëmD (Dv)2 / 2ùû dmD ò 2
(26)
M
0 Falong,retro =
PT
ED
The rate of change in radius R& retro can be calculated via (2) and (16):
(27)
CE
dM dR = 4pr R 2 = - ò M ej (N D Dv)p R 2 dmD dt dt retro
AC
so that
dR K R& retro = =dt 8r
ò (N
1+2z
D
Dv)éëmD (Dv)2 / 2ùû
dmD
(28)
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14
and
ò (N
1+2z
D
Dv)éëmD (Dv)2 / 2ùû
dmD 1+z
ò 12N D mD (Dv)2 + 6P0 (e K ) (N DDv)éëmD (Dv)2 / 2ùû dmD 1/2
.
(29)
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Lretro =
nK Da
6. Stony meteoroids
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For simplicity it will be assumed for stony meteoroids that = 0 in (16), so that mass loss is proportional to energy. However, K will be replaced with K1, such that
1+2z
M
K1 éëmD (Dv)2 / 2ùû = K éëmD (Dv)2 / 2ùû
ED
for mD = 1.5 10-8 kg, for which ND peaks at the Earth (Love and Brownlee, 1993), and v = 34.470 103 m, as is appropriate for the asteroid belt at 3 AU. For K = 7 10-6,
CE
PT
these assumptions give K1 = 9.33 10-6 s2 m-2 and (26) becomes
{
}
1 2 rretro (Dv)2 + P0 (e K1 )1/2 Dv éërretro (Dv)2 / 2ùû 2
(30)
AC
0 Falong,retro »
where retro is given by (18), while (28) becomes
K R& retro » 1 Dv éërretro (Dv)2 / 2ùû . 8r
(31)
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15
Using (30) and (31) in (25) yields
Rres Lretro (e -1) , R& retro
(32)
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Tdrift,retro »
Lretro »
nK1 Da Dv 24 + 6P0 (e K1 ) Dv 1/2
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where now
,
(33)
PT
2 æ 3AU ö æ Da ö 184.78 ç ÷ ÷ ç è a ø è 0.1AU ø » é P ùæ e ö1/2 æ 3AU ö1/2 24 + 297.82 ê 0 úç ÷ ç ÷ ë (2 / 3) ûè 0.5 ø è a ø
(34)
CE
Lretro
ED
ND. This can be written
M
so that retro is independent of retro and thus does not depend on the functional form of
AC
where a is now in AU. Writing retro for stony meteoroids in the form of (34) makes the quantities in
parentheses and square brackets equal to 1 for the chosen nominal values. The resonances in the middle of the main asteroid belt at 3 AU are ~0.1 AU apart (Nesvorný et al., 2002).
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The value for is taken to be 0.5 as assumed by Wiegert (2015), for the reasons he gives on his page 25. Finding a nominal value for P0 is harder work. The experimental data of Nakamura and Fujiwara (1991) find that 5/3; a value of = 5/3 exactly is adopted
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here. Giblin (1998) finds that ~0 ~1/6, with an average value of 1/13. It is assumed here that the range is exactly 0 1/6. When < 1/6, (12) can be used to find P0. However, if = 1/6 exactly, then 2 2 = 0, and P0 can be obtained via a limit
2 ù 1/2 é é (2 - a )x ù ê (1- f 2-a -g ) ú P0 = ê 2ú ë 2(2 - a - g ) û ê (1- f 2-a ) (1- f x ) ú ë û
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process as follows. Set x = 2 2 in (12) so that
1/2
M
.
ED
Using
(x ln f )2 (x ln f )3 f =1+ x ln f + + +... 2! 3!
CE
PT
x
AC
(Selby, 1974, p. 472) in the equation and letting x 0 yields
2 ù 1/2 é é (2 - a ) ù ê (1- f 2-a -g ) ú P0 = ê 2ú ë 2(2 - a - g ) û ê (1- f 2-a ) ( -ln f ) ú ë û
for 2 2 = 0.
1/2
(35)
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The range 10-3 f 10-5 is assumed here, as seems appropriate from Nakamura and Fujiwara (1991) and Nakamura et al. (1994), where (3) appears to start failing at the smaller masses. Using the adopted ranges and values of , , and f in (12) and (35) yields the tight range of values 0.623 P0 0.699 (Fig. 2). Values of < 5/3 lead to P0 0.704.
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Hence a mid-range value of P0 = 2/3 appears to be quite reasonable.
R& retro as given by (31) can also be put into a form similar to (34). Love and Brownlee (1998) find that the Earth accumulates dust at the rate E = ~4 107 kg y-1 =
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1.27 kg s-1. This is the total rate at the Earth for dust in all orbits. Also, some of the observed rate E is due to gravitational focusing by the Earth. Hence to find the rate in free space for dust in circular retrograde orbits, E must be adjusted downward to sE, where s 1 (Wiegert (2015).
M
The total retrograde dust density at the Earth retro,Earth can be found from
ED
RE2retro,EarthvE = sE, where vE = 60000 m s-1 and RE = 6.371 106 m is the Earth’s
smE æ 3AU ö -3 -20 æ 3AU ö ç ÷ = 5.53´10 s ç ÷ kg m . 2 è a ø 3p RE DvE è a ø
(36)
AC
CE
rretro =
PT
radius. The value for retro extrapolated to the asteroid belt is then (Wiegert, 2015, p. 25):
Here the nominal value of s is taken to be 0.5 as assumed by Wiegert (2015). A density of
= 2800 kg m-3 is typical for stones. Thus (31) with the aid of (36) can be written:
æ s öæ 2800 kg m -3 R& retro = 0.0074 ç ÷ç è 0.5 øè r
öæ 3AU ö5/2 ÷ ÷ç øè a ø
m My-1 .
(37)
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Eq. (37) can also be written
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æ 1m ö R& retro » çç ÷÷ è Tmetre,retro ø
(Rubincam, 2015, p. 115), so that
öæ a ö æ 0.5 öæ r Tmetre,retro =134 ç ÷ç ÷ç ÷ è s øè 2800 kg m -3 øè 3AU ø
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5/2
My .
(38)
M
Here Tmetre,retro is the time it takes to wear down a meteoroid’s radius by 1 m from the dust in the retrograde orbit.
ED
For the nominal values of a = 3 AU, a = 0.1 AU, P0 = 2/3, = 0.5, s = 0.5, and
PT
= 2800 kg m-3, (34) works out to be
retro 0.574 ,
CE
(39)
AC
while (38) is
Tmetre,retro (40)
=
134
106
y
.
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Hence for the chosen parameters
Tdrift,retro » ( eLretro -1) Tmetre,retro (Rres /1m) = (104 ´10 6 y)(Rres /1m)
(41)
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where Rres is in meters. The diamonds in Fig. 3 show Tdrift,retro/(Rres/1 m) as a function of a.
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7. Stony cosmic ray exposure ages
For meteoroids traveling to the resonances from the Yarkovsky effects or dust impacts, but with no space erosion, Tdrift would tend to roughly coincide with the CRE
M
age. In the Yarkovsky scenario, the material that becomes a meteoroid is most likely deep inside an asteroid and shielded from cosmic rays. It is then liberated by a catastrophic
ED
impact. Thereafter the meteoroid spends most of its time journeying to the resonance
PT
exposed to the cosmic rays. Once the meteoroid reaches the resonance, relatively little time is spent within the resonance before hitting the Earth (Gladman et al., 1997). Hence
CE
the CRE age is essentially the drift time to the resonance. In the collision scenario, dust impacts cause space erosion, which affects the CRE
AC
age. Though the surface of any object is exposed to cosmic rays, the surface gets worn away, taking its cosmic ray products with it. Thus a meteoroid has an erosion CRE age , which is younger than the length of time the meteoroid has existed as an independent body (Rubincam, 2015).
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Fig. 4 (solid curve) shows the
21
Ne erosion age Ne,retro as a function of Rres for
meteoroids at a = 3 AU for Tmetre,retro = 134 106 y as given by (40). Here the erosion age is computed for a meteoroid with an initial radius of 5 m that shrinks steadily down by space erosion from retrograde particle impacts, all the while being bombarded by cosmic
The computation of the
21
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rays; the drift time Tdrift,retro does not enter into the calculation.
Ne erosion age Ne,retro follows Rubincam (2015). The
neon production at various depths in a meteoroid of radius R is parameterized according
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to equations (11) – (13) of that paper. The amount of neon at depth is computed as the meteoroid’s radius shrinks from space erosion until the meteoroid reaches its final radius. The erosion age is then chosen to fit the neon concentration inside the meteoroid at half its final radius as though the meteoroid always had its final radius and never eroded. It
M
was found that
(42)
PT
ED
Tmetre .
to a high degree of approximation.
CE
Note in Fig. 4 that Ne,retro 0.41Tmetre for Rres ~1 m, regardless of how long it
takes the meteoroid to reach a resonance. (How 0.41Tmetre varies with a is shown as the
AC
open circles in Fig. 3.) But once the radius shrinks below ~1 m, Ne,retro rises sharply. Of interest in Fig. 4 is the shaded region under the curves. For Rres ~0.7 m, the
CRE age of a meteoroid journeying to a resonance via collisions is less than or equal to the time Tdrift,retro to travel between resonances spaced a distance a= 0.1 AU apart. But for Rres ~0.7 m, it is space erosion which controls the upper limit on the CRE age,
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so that ages have to be less than or equal to the erosion age Ne,retro. Therefore a meteoroid must have an age somewhere in the shaded region under the dashed line or solid curve for the assumed travel distance. The maximum CRE age at 3 AU is ~75 106 y for Rres 0.7 m, which is where Tdrift,retro = Ne,retro.
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Fig. 5 is similar to Fig. 4, but with Divine’s (1993) core and asteroidal populations added to the retrograde population. Divine’s populations, which are prograde, do not overlap with the retrograde population. Divine’s populations are
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assumed to cause space erosion but not drag. At 3 AU, Tmetre,Divine 666 106 y (Rubincam, 2015, Fig. 8), so that Tmetre, retro+Divine 112 106 y. The maximum CRE age then lowers to ~69 106 y at 3 AU when Rres = ~0.6 m, and the erosion age for Rres > ~1 m drops below 50 106 y. (Because (42) holds, the erosion age curve in Fig. 5 suffers
M
only a change in the vertical scale. However, Tdrift, retro+Divine suffers more than a change in
ED
vertical scale because the increased R& in (24) changes , so that the dashed line meets the curve at Rres = ~0.6 m instead of ~0.7 m.)
PT
Eugster et al. (2006) show the CRE ages of many stony meteorites in their Fig.s 4, 5, and 6 compiled from various papers. The CRE ages are ~100 106 y for chondrites
AC
CE
and achondrites except for the aubrites, which range up to ~150 106 y.
8. Iron meteoroids
For iron meteoroids, it was assumed in Rubincam (2015, p. 116) that K = Kiron 7.8 10-8, a value estimated from the experimental results of Matsui and Schultz (1984), who chilled iron targets down to 107 K. However, the value of Kiron may be higher.
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Assume that Mej/m = Kiron(v)2/2, so that the ejecta mass is proportional to energy. According to Table 1 of Matsui and Schultz, the Gibeon meteorite yielded Mej/m = 2.55 when hit by a basaltic projectile with speed 5000 m s-1, giving Kiron = 2 10-7. Using this
2 æ 3AU ö æ Da ö 3.96 ç ÷ ÷ ç è a ø è 0.1AU ø » é P ùæ e ö1/2 æ 3AU ö1/2 24 + 43.60 ê 0 úç ÷ ç ÷ ë (2 / 3) ûè 0.5 ø è a ø
(43)
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Liron,retro
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value in (33) and (31) results in
and
öæ a ö æ 0.5 öæ riron Tmetre,retro =17618ç ÷ç ÷ -3 ÷ ç è s øè 7870 kg m øè 3AU ø
My .
(44)
ED
M
5/2
where the nominal density of iron is taken to be 7870 kg m-3. This and the other nominal
CE
PT
values at 3 AU yield iron = 0.0585 and
(
Liron,retro
)
-1 Tmetre,retro (Rres /1m) =1057´10 6 (Rres /1m)y
(45)
AC
Tdrift,retro » e
for an iron meteoroid traveling 0.1 AU. On the other hand, the Toluca meteorite yielded Mej/m = 23.3 when struck by a steel projectile traveling at 3380 m s-1 because the meteorite had oxide shales that spalled on impact. This case gives Kiron = 4 10-6, iron = 0.362, and Tmetre,retro = 384 106 y. These numbers are in rough accord with the CRE
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ages of irons: iron meteorites tend to have CRE ages in the range 108 - 109 y (e.g., McSween, 1999; Eugster et al. 2006). So it may be that drift time controls the CRE ages for irons. However, the parameters are uncertain, and no attempt is made here to estimate
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erosion ages of irons meteoroids.
9. Discussion
In the collision scenario investigated here, da/dt is negative and meteoroids drift
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sunward toward a resonance. Drag from collisions with dust particles inevitably implies space erosion. If the drag on a stony meteoroid is mainly due to the ejecta, as in the case of particles in the retrograde orbit as investigated by Wiegert (2015), then space erosion
M
will also be substantial. Space erosion can profoundly affect CRE ages of meteoroids. Space erosion, rather than drift time Tdrift to a resonance, controls the upper limit of CRE Ne ages for meteoroids with R ~1 m, as shown in Fig.s 4 and 5.
ED
21
The upper limit on allowed ages in Fig.s 4-5 fall in the range ~70 106 y for
PT
stony meteoroids at 3 AU. This accords well with the observed CRE ages, most of which
CE
range from ~106 y to ~100 106 y (e.g., McSween, 1999; Eugster et al., 2006). The exception is the aubrites. The aubrites are physically weak, but yet have ages
AC
which range up to ~150 106 y. They may come from asteroids in high-inclination orbits (McSween, 1999, p. 247; Eugster et al., 2006, p. 841; Ćuk et al., 2014). If so, they may have suffered fewer collisions than they would have in ecliptic orbits and thus have longer CRE ages.
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The results depend upon the chosen values of parameters such as , f, and P0, which are not certain. If s or are larger or smaller than the nominal values, then the drift times and the erosion ages will go up or down. The main dependence of P0 appears to be on f (Fig. 2), and f is not well determined. Also, the approximation = 0 was used get
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(34) and (38), avoiding needing to know ND as a function of mD. More exact calculations would use = 0.065, but necessitate knowing the functional form of ND in order to perform the required numerical integrations.
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The results obtained here assume that the meteoroids are regolith-free. A regolith would complicate calculating drift time and erosion age, and is beyond the scope of this paper. The effect of multiple large collisions on CRE ages is not considered here (Vokrouhlický and Farinella, 2000), nor is the catastrophic destruction of meter-sized
M
objects (Bottke et al., 2005).
All the calculations assume that dust particle concentrations have remained steady
Shaeffer et al., 1981).
ED
for hundreds of millions of years. This may not be the case (e.g., Farley, 2001, p. 1194;
PT
It was implicitly assumed here that Rres, the meteoroid’s radius when it reaches a
CE
resonance, is the same radius it had just before entering the Earth’s atmosphere. In other words, the time between injection into the resonance and falling on Earth is so short that
AC
not much erosion takes place. However, the case of meteoroids pumped up into highly elliptical Earth-crossing orbits and hit by retrograde-orbit impactors should be investigated to see how high or low the erosion might be. The Yarkovsky effects give CRE ages similar to those found here (e.g., Bottke et al., 2006; Ćuk et al., 2014). Hence a question arises: what percentage of CRE ages is due
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to Yarkovsky and what percentage is due to collisional drift/space erosion? Could something interfere with the operation of the Yarkovsky effects, such as the YarkovskyO’Keefe-Radzievskii-Paddack (YORP) effect (Paddack, 1969; Paddack and Rhee, 1975; Rubincam, 2000; Bottke et al., 2006; Vokrouhlický et al., 2007) or collisions (Wiegert,
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2015)? It may be that the Yarkovsky effects play a smaller role in determining CRE ages than currently thought.
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Acknowledgments
I thank Susan Fricke and Susan Poulose for excellent programming support. I thank Richard Durisen and an anonymous referee, both of whom corrected errors and
M
made excellent suggestions which greatly improved the paper. This paper was inspired by
ED
Wiegert (2015). I thank Herbert V. Frey for being supportive of my research.
PT
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CE
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Earth’s History. Kluwer Academic/Plenum Publishers, New York, pp. 221-240.
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Fig. 1 Geometry of dust particles impacting on a spherical meteoroid. The along track direction is shown by the thick arrow. Dust particles (black dots) traveling in the
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retrograde direction collide head-on with the meteoroid. The particles are spread evenly
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over the cross-sectional area R2 seen by the particles, where R is the meteoroid’s radius.
is the longitude on the meteoroid, while is colatitude. A particle impacting on the
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annulus of radius Rsin and width Rcos d ejects mass which is proportional to cos2
. The meteoroid is assumed to randomly orient itself in space, so that it always remains an eroding sphere.
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Fig. 2 P0 computed for = 5/3 and = 0, 1/12, and 1/6 when f = 10-3, 10-4, or 10-5. The symbols are joined with straight lines for clarity. A nominal value of P0 = 2/3 is assumed
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in this paper.
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Tdrift,retro/(Rres/1 m) (diamonds) and = 0.41Tmetre (open circles) for stony
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Fig. 3
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meteoroids as a function of semimajor axis a for the nominal values a = 0.1 AU, P0 =
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2/3, = 0.5, s = 0.5, and = 2800 kg m-3. Rres is the radius when the meteoroid reaches the resonance. Erosion age = 0.41Tmetre is the age for meteoroids when Rres > ~1 m. The
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Fig. 4 Drift time Tdrift, retro (dashed line) to a resonance and 21Ne erosion age Ne,retro (solid curve) plotted as a function of Rres, which is the meteoroid’s radius when it reaches the
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resonance, for the nominal values of the parameters. The meteoroid is at a = 3 AU in a prograde circular orbit in the ecliptic and drifts inward to a resonance. The distance
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between resonances is 0.1 AU. The shaded region gives the allowable CRE ages in the collision scenario for meteoroids between the resonances. Here Tmetre = 134 106 y.
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Fig. 5. Plot similar to Fig. 4, but for the combined retrograde and Divine populations. Divine’s populations, which are prograde, are assumed to cause space erosion but not
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drag. The addition of Divine’s populations shortens Tmetre to 112 106 y.