1
Terrestrial crater counts: Evidence of a two to four-fold
increase in bolide flux at 125 Ma.
Steven N. Ward and Simon Day
Institute of Geophysics and Planetary Physics
University of California, Santa Cruz
Bounded before and after by long periods
of stability, terrestrial crater counts document a
rapid (25Ma duration) increase in crater production less crater destruction near 100Ma. Investigating these features quantitatively using
crater population models, we propose that crater
production accelerated near 100Ma driven by a
two to four-fold increase in impactor flux spanning all bolide sizes. Current best estimates of
power law asteroid flux based on astronomical
methods understate the post-125Ma counts of
large (dc>10km) terrestrial craters by as much
as a factor of four and overstate counts of small
(dc<4km) terrestrial craters by as much as a
factor of ten. We suspect that stronger than generally supposed atmospheric shielding or a "dip"
in true bolide flux relative to a power law rate
probably generate the misfit at the small end. A
possible explanation for the increased post125Ma bombardment are encounters with members of the Baptistina family of asteroids conceived in a collision of large planetoids at
160Ma. Regardless of the cause, rapid and dramatic swings in bolide fluxes and large uncertainties in atmospheric effects bode trouble for
hazard evaluations that hinge on wellconstrained and current surface impact rates.
(v2.4) Submitted to Earth,Planets and Space. .October, 2007
1) Introduction
Two techniques exist for estimating terrestrial bolide flux. The astronomical method
maps the number and brightness of NEAs,
translates brightness into bolide diameter, then
computes the fraction of these that might intersect the Earth’s orbit and position in a given period of time. The terrestrial method tabulates the
number, size, and age of impact structures within
a target area; then, together with a relation between crater size and impactor diameter, returns
bolide flux. The great advantage of the terrestrial
method is that it better represents ground truth -it is hard to argue with the number and age of
existing craters.
In this article we focus on the Earthbound method for fixing bolide flux. We place
particular emphasis on potential geological biases in the terrestrial counts.
2) Earth Impact Database
Figure 1 maps the distribution, size and
age of all 174 known terrestrial impact craters
listed in PASSC Earth Impact data set compiled
by the University of New Brunswick
http://www.unb.ca/passc/ImpactDatabase/ind
ex.html The craters listed span diameters dc, of
10m to 300km. The crater ages run from just 59
years to 2,400Ma. The most striking aspect of
Figure 1 is the highly irregular distribution of
2
Figure 1. Earth Impact Database from http://www.unb.ca/passc/ImpactDatabase/index.html (top) Colors
indicate crater diameter. Note the very non-uniform crater coverage on land. (bottom) Colors indicate crater
age. Note the lack of craters older than 100Ma in Asia.
craters on land. Great expanses of South America, Africa and Asia romp craterless, whereas
North America, Northern Europe and Australia
are densely pocked. Asteroids strike the Earth
with generally uniform spatial density, so some
geological processes must account for the spotty
crater coverage. To greater or lesser extent, four
geological factors hold responsibility for the
coverage variations:
(1) Preservation: Many craters may have
existed previously but have been totally eradicated by erosion. Erosion entails not just loss of
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topographical expression but also the elimination
of the rock unit containing traces of the impact
structure and any impact metamorphism or
melts. The largest impact structures extend
downward tens of km (Grieve and Thierrault,
2000) and should survive anything save a complete reworking of the crust in a continental collision. The smallest impact structures however,
may vanish easily. A transition in survivability
occurs for craters of 3 to 5 km diameter. Structures of this size persist to 1 or 2 km depth --beyond erosion's reach outside of orogenic belts.
(2) Discovery: Many craters may still
exist, but have not yet been recognized. Lack of
discovery is especially probable where craters
have been buried under sediments or extensive
units of volcanic rocks. Buried craters often go
unrecognized unless they have gravity or geomagnetic signatures, or they locate in areas intensively searched by seismic surveys.
(3) Youthfulness: Many bits of crust are
simply too young to record older impacts. Conversely, the presence of burial-sensitive deposits
such as oil shales testify that some crustal bits
have been stable for the entire Phanerozoic.
(4) Isolation: Many pieces of crust may
have been insulated against cratering for periods
of varying length by shallow seas or ice cover.
Water of depth H shields the seafloor from attack
under bolides of diameter dI<~H/10, so shallow
seas (H<1000m) on the continents hinder the
formation of craters of diameter <3km. Isolation
by inland seas is less of an issue for larger impactors (dI>100m) as evidenced by several big
craters (Chesapeake Bay, Montagnais, Mjolnir)
known to have formed in shallow water.
3) Dog Leg bend at 125Ma
Figures 2a and 2b plot cumulative and
differential numbers of craters listed in PASSC
Figure 2a. Cumulative counts of terrestrial craters versus age and diameter for all regions. Dashed lines are linear
fits to the data between 0-80Ma and 150-500Ma. The former is required to run through the origin. The right hand
box quotes the ratio of slopes before and after the 125Ma transition.
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data set versus age 0 to 500Ma; and size >0.5km
to >40km diameter. (Because the smallest craters
are most prone to erosion and non-discovery, we
exclude members dc<500 m from analyses.) Cumulative plots count craters less than a given age
and greater than a given size. Differential plots
count craters less than a given age within distinct
size brackets. The figures include all 131 craters
within the stated size and age limits regardless of
location. [Appendix A discusses crater age uncertainties.] In crater count plots like Figure 2a
or 2b, the slope of the lines correspond to the net
crater production rate -- the rate of crater production less crater destruction. In lumping the
entire global data set into a single group, one
might expect that the geological complications
listed above would be working to confuse the
picture. Surprisingly however, for all of the
500Ma interval save the most recent 25Ma, the
straight line form of the counts indicates net
crater production seems fairly constant for long
periods of time. The most prominent feature of
Figure 2a or 2b is the sharp contrast in slopes for
the period <80Ma relative to the period >150Ma.
Slope changes in Figure 2b, average about a
factor of five and show in every range of crater
size. Although net crater production trended constant before and after, 125 million years ago, the
terrestrial crater record suffered a dramatic increase in production less destruction.
A critical question lurks, Does the dog
leg bend at 125Ma represent an increase in crater
production, a fall off in the rate of crater erasures, or some bias in discovery, youthfulness or
isolation? Using similar presentation as Figure 2
but with smaller data sets, researchers at least as
Figure 2b. Differential counts of terrestrial craters versus age and diameter for all regions. To avoid overlap,
each crater size group has been shifted upward. Dashed lines are linear fits to the data between 0-80Ma and 150500Ma. The former is required to run through the origin. We interpret the change in slope at 125Ma being due to
a change in crater production. The right hand box quotes the ratio of slopes before and after 125Ma.
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far back as 25 years ago [Grieve 1984, 1987;
Grieve and Shoemaker 1994; Hughes 2000] also
noted a tail off in counts of older and smaller
craters. They generally ascribed the reduction to
various erosive processes or vulgarities in sampling. The crater data set has grown steadily in
the past 25 years; still, the notion persists that the
terrestrial crater record is too complex, uncertain,
and incomplete to draw meaningful conclusions.
In this article, we swim against the current by
maintaining that the existing data set, combined
with numerical models of crater production and
loss, are sufficient to test certain hypotheses. In
particular, we argue that the transition at 125Ma
reflects a true change in crater production rate.
3.2) Lines of Reasoning. Our lines of reasoning backing a change in crater production at
125Ma versus a change in crater destruction are
three fold:
(A) The crater counts in Figures 2a and
2b are remarkably linear over all crater diameters
for 350Ma prior to the transition and for 60Ma
after the transition. Whatever mechanisms control crater populations, they must be able to deliver long intervals of stability in net production.
(B) From Figure 2b, you can see that
relative to the long stable intervals before and
after, the transition in net production happens
almost instantly (~25Ma) and nearly contemporaneously over the entire spectrum of crater diameters. Whatever mechanisms control crater
populations, they must be able to jump between
stable states quickly. If the transition related to
changes in climate, tectonic environment or
weathering affecting crater erosion and burial
(Section 2), it would not affect all crater sizes at
the same time and over the same duration.
(C) The ratio of slopes before 80Ma and
after 150Ma computed by least squares fall
nearly in the same ratio (~1/3-1/5, median 1/4.2
right hand box Figure 2b) for all crater sizes.
Whatever mechanisms control crater populations, they must be able to change the rate of
crater production less destruction in the same
proportion over all crater sizes.
4) Crater Population Model.
Short of dismissing the terrestrial crater
record as being too flawed to consider seriously,
the observations above demand a quantitative
explanation -- an explanation best built on a
Crater Population Model (CPM). Everyone
agrees that the time rate of change of a population N(dc,t) of craters of diameter dc equals the
rate at which they are created minus the rate at
which they are destroyed. As an analog to crater
generation and destruction, we take clue from
laws of radioactive decay and propose
∂N ( dc , t )
= β ( dc , t ) − α ( dc , t ) N ( dc , t ) (1)
∂t
Equation (1) assumes that the rate of crater production β(dc,t), being tied to the flux of asteroids
striking the Earth, is independent of the number
of craters that exist at any time. On the other
hand, like a radioactive decay process, (1) also
assumes that the number of craters of any size
lost in any time interval tracks in proportion to
the number of craters of that size that existed at
that time. In words, α(dc,t) is the fractional rate
of crater loss per unit time. Crater half-life equals
0.69/α(dc,t). (For shorthand, sometimes we call
α(dc,t) 'the erosion rate'.) Like radioactivity, (1)
allows for craters to be lost at anytime after their
creation and for rates of production and fractional destruction to vary with crater size. Unlike
radioactivity however, (1) allows crater half-life
to vary with time. Given β(dc,t) and α(dc,t), integrations of (1) over time or crater diameter return
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CPMs for cumulative or differential crater counts
like Figure 2a or 2b or variations on the theme.
For instance, a CPM for the number of craters
less than age t, is
t
 tˆ


N ( dc , < t ) = ∫ β ( dc , tˆ )exp − ∫ α ( dc , t )dt dtˆ (2)
0
 0

The first term in the integrand (2) fixes the number of craters created in a short interval tˆ years
ago. The second term expresses the fraction of
those craters that survive to the present day. We
intend to make quantitative inferences about
crater production and loss given a CPM and the
terrestrial crater count data. The strength of our
inferences should be judged against the quantity
of the data and adequacy of the model.
4.1) Constant Erosion/Constant Production
(CE/CP) CPM. As a start, suppose production
and fractional loss rates were constant in time
back to age t, then CPM (2) follows
β ( dc )
1 − e − α ( dc ) t
α ( dc )
For small and large values of α(dc)t,
N ( dc , < t ) =
[
]
(3)
N ( dc , < t ) ~ β ( dc )t : α ( dc )t << 1 (3a)
and
N ( dc , < t ) ~
Figure 3. Observed differential crater counts for versus age and predictions of the best fitting CE/CP
model (3) (lines) which has a fractional loss rate
(α=1/225Ma). It is not possible to match both the
strong change in slope and the low count curvature
given a constant rate of crater production and erosion.
β ( dc )
: α ( dc )t >> 1 , (3b)
α ( dc )
so even for constant production and fractional
loss rates, counts would trace curved paths (3)
backward in time and approach a saturation
value where production equals destruction (3b).
Could the dog leg in crater counts observed at
125Ma be a misinterpreted manifestation of this
curvature? Figure 3 tests the idea. It plots differential crater counts for dc>0.5km from Figure 2b
and several CE/CP populations (3) using a constant production βi(dc) for each crater size group
and a single-valued fractional loss rate α. The six
βi(dc) and α were determined by least squares fit
to the 131 data points.
Superficially, the best CE/CP model
(Figure 3) reproduces the trends of data, but the
fit is poor because the free parameter α acts in
cross-purposes. Large α values dog leg the cumulative counts, but result in a strong curvature
inconsistent with the data. Small α values keep
the counts fairly straight but do not produce sufficient change in slope. Faced with a constant
crater production rate, it is statistically unlikely
that any constant or simple increase in fractional
erosion rate near 100Ma can account for the observed crater counts (See Appendix B for more
details).
4.2) Constant Erosion with Step in Production (CE/SP) CPM. Figure 4 plots the differential crater counts of Figures 2b again, overlain
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by best-fitting CPM predictions (2) assuming a
constant erosion rate α(dc,t)=α, but now including a step increase in crater production β(dc,t) at
near 100Ma. These CE/SP models have the same
seven parameters as the CE/CP models (6 βi(dc)
and α) plus the time of the step in production, TS
and the relative step size ∆ [i.e. β i (dc,t<TS)=
∆βi(dc,t>TS)]. The fractional step in production
∆, is the same for all crater sizes and trades off
with α. Nevertheless, all of the existing crater
observations can be well-explained by a CPM
with a constant and size-independent crater half
life, coupled with a size-independent post100Ma increase in crater production of between
1.5 to 4 times pre-100Ma levels (lines Figure 4).
Figure 4. (lines) Predictions from best fitting CE/SP
model (2). Differential crater count data spanning all
ages (<500Ma) and sizes can be explained by a constant fractional crater loss rate coupled with a step
increase in crater production of 1.5 to 4 times near
100Ma.
4.3) Selection of Sampling Areas. We are
unable to explain the dog leg in crater counts
near 100Ma with any constant production CPM
having a fixed or simply changing erosion rate;
however, other geological factors might be at
play -- in particular, youthfulness, discovery and
isolation. To try to minimize these effects, we
restrict attention to selected areas of crust that
are generally older than 500Ma, well explored,
and not long blanketed by water or ice. Three
areas fit the bill -- cratonic North America,
Northern Europe and West Australia. Truth to
say, in whittling areas to those with ideal geological histories, it is easy to end up with provinces so small that crater counts fall below testable numbers. Here, we simply bound regions by
latitude/longitude boxes (Table 1 and Figure 5).
Combined, the selected regions cover
14.6x106km2, or 2.8% of the Earth’s surface
(5.1x108km2). In any piece of the Earth this
large, there certainly will be variation in youthfulness, discovery and isolation. Likely however,
geological variations in the selected areas should
be less than those in the Earth's surface taken as
a whole.
Figure 6 plots cumulative counts verses
age and size for all 64 craters (0 to 500Ma age,
>0.5km diameter) within the selected regions.
With fewer points, the counts are a bit noisier
than Figure 2a, but the critical aspects of cratering record found for all regions remain: (1) A
factor of 3 to 5 abrupt change in net crater production at 125Ma and (2) Stable trends for 60Ma
after the transition and 350Ma years prior. The
selected regions have been far-distanced from
each other for 200 to 500 Ma. Regional changes
in net crater production through changes in erosion or burial could not affect three widely dispersed sites simultaneously with the same magntiude. Also, none of the selected regions experi-
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process (Section 5). Although one could test
many sample areas based on other geological
criteria, it is unlikely that these features of the
crater record are flukes of youthfulness, discovery or isolation.
5) Quantification of Bolide Flux for the latest
125Ma
Apparently, the Earth has endured a two
to four-fold increase in relative impactor flux
near 100Ma. Can we employ a CPM and the data
from the selected regions to estimate an absolute
post-125Ma bolide flux rate, and compare it with
astronomical estimates?
5.1) Observed Crater Production rates.
Figure 7 shows cumulative crater counts versus
crater diameter including all 31 impact structures
within the selected areas that have ages less than
125Ma and diameters >100m. A log-log format
draws out any power-law dependencies. For the
largest craters, cumulative counts fall in propor-
Figure 5. Regions selected for flux quantification.
North America (top row), Northern Europe (middle
row), Australia (bottom row). Columns show crater
diameters (left) and crater ages (right).
enced glaciation, drastic uplift followed by erosion, or flood basalt volcanism around 125 Ma.
The observed change in net crater production
rate cannot be attributed to a clean sweep type of
Region:
Latitude
Extent:
30 – 60.5 N
North
America
Northern 48 – 64 N
Europe
Australia 17 – 28 S
Longitude
Extent:
85 – 118 W
Area
106 km3
8.66
10 – 48 E
4.19
124 – 136 E
1.76
Table 1. Selected Regions for flux quantification.
Figure 6. Cumulative counts of craters versus age and
diameter for the selected regions of Figure 5. Dashed
lines are linear fits to the data between 25-80Ma and
150-500Ma. The change in slope near 100Ma and the
linear trend in counts before and afterward are not
likely a bias due to geological factors of youthfulness,
discovery or isolation.
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tion to dc-λ with λ~3 as might be expected in the
absence of contaminating effects on unsaturated
surfaces (see equation 9). Two kinks mark reductions in net crater production rates at
dc=20km and at dc=2km.
5.2) Bolide Flux rates from Crater Production rates. Linking crater production rate
with a bolide flux rate requires both a bolide flux
distribution and a crater scaling law. For first cut,
consider a power law form of bolide flux versus
impactor radius RI
n>(RI)= aRI-b
(4)
n>(RI) represents the annual rate of impacts of
asteroids of radius greater than RI per square
meter of planet surface. For the current, nearEarth environment, Ward and Asphaug (2000)
established constants a and b
a= 3.89 x10-14 m1/3/y; b = 7/3
(5)
Later, following the more recent astronomical
data of Brown et al. (2002), Chesley and Ward
(2006) reduced the a-value above by a factor of
five
a=0.778 x10-14 m1/3/y; b = 7/3 (6)
Schmidt and Holsapple (1982) developed a
crater diameter dc - bolide radius RI scaling law
d
S-H
c
 V 2  β  ρ  1 / 3 2C  1-β
T
(R I ) =  I   I 
 RI
 3.22g   ρT  1.24 
(7)
where VI and ρ I label the velocity and density of
the impactor, and g is the acceleration of gravity.
Parameters β and CT depend on target properties
and derive from laboratory impact experiments.
Schmidt and Holsapple (1982) provide four cases
β=[0.22, 0.16, 0.17, 0.22] and CT=[1.88, 1.40,
Figure 7. Observed 125Ma cumulative crater counts
(triangles) for the selected regions of Figure 5 versus
crater diameter (bottom scale) or bolide radius (top
scale). Flattening of counts to the left reflects nearly
total atmospheric shielding for bolides <30m radius.
Steeply sloping counts reflect uncontaminated rates
for bolides of radius >800m. The gently sloping
counts 30m<RI<800m at the center may reflect enhanced atmospheric shielding.
1.68, 1.60] for targets of water, quartz sand, ottawa sand, and competent rock/saturated soil respectively. For competent rock, and assuming
VI=20km/s and ρI=ρT, scaling law (7) simplifies
roughly to
dc(RI) = Q R 3/4
I ;
Q= 113.593 m1/4 (8)
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For RI=500m, (7) and (8) are equal. Relation (8)
is plotted along the top of Figure 7. Although
idealized, a power law bolide flux (4) combined
with a power law crater scaling (8) predict a
power law production rate of craters of diameter
greater than dc per unit area as
np>(dc)=a (dc/Q)-4b/3 = apdc-λ
(9)
Crater production parameters ap=aQλ and λ=4b/3
are analogous to the a and b values governing
asteroid flux. For b=7/3, λ equals 28/9≈3.1.
Readers should not confuse the crater production
rate dc-λ with the ~dc-2 crater distribution found on
saturated impact surfaces like most of the moon
(Neukum and Wise, 1975). Early in the history of
a fresh surface, crater populations grow in proportion with the crater production (9). With time,
the rate at which smaller craters become obliterated by larger impacts equals the rate at which
they are produced, and the small crater population reaches an equilibrium at dc-2 (Ward, 2002).
The surface of the Earth surely is not saturated
The solid lines in Figure 7 plot crater
production (9) using flux parameters (5) and (6)
scaled by the total area of the selected regions
(14.6x106km2) and the 125Ma exposure duration.
The power law slope of observed crater production for craters dc>20km is close to the –3.1
value predicted in (9). Agreement of slopes suggests that we are looking at uncontaminated production flux rates for the largest craters. On the
high end of crater size, latest astronomical estimate (6) begins to understate rates for craters
dc>10km (bolide radius RI>400m) and falls a full
factor of four short for craters dc>20km
(RI>900m). On the low end, the latest astronomical estimate begins to overstate rates for craters
dc<10km (RI<400m) and falls a full order of
magnitude high for craters dc<4km (RI<100m).
5.3) Nature of the inconsistency- High
End. Best estimates of crater formation based on
astronomical impactor flux are inconsistent with
crater counts on Earth both on the high end
(dc>20km) and low end (dc<7km) of crater sizes.
While several processes might limit low end
cratering, few methods exist to produce extra
high end ones, so the discrepancy dc>20km
looms serious.
Possibly the inconsistency might be due
to sampling – that is, the selected areas suffered
eight impacts dc>20km in the past 125Ma rather
than the two impacts expected by (7), just by
chance. If impacts are Poissonian in time, then
this can be tested. For a Poissonian process, the
probability of N or more occurrences of an impact-related event in interval T is
N −1
P( T, N) = 1 -
(NT)k e -NT
∑
k!
k =0
(10)
where N measures the mean rate of the event
(Ward, 2002). If T=125Ma, N=2/125MA per (6),
then the likelihood of getting eight or more craters is just 0.00109.
Possibly the inconsistency might be due
to a bias low in the crater diameter–bolide radius
relation (8). From (9), cumulative crater production rate depends on Q3.1 . A 25% increase in
crater diameter at fixed bolide size (the Q value)
would shift the predicted crater production
curves in Figure 7 upwards by a factor of 2 and
reduce the high end discrepancy. If β and CT in
(7) were fixed, a 25% increase in crater diameter
would require 50% higher impact velocities or a
factor of two increase in the ratio of target to impactor density. (Arguing systemic differences in
impact velocities and target properties between
the Moon and Mars, Neukum and Wise (1975)
shifted crater production curves in a similar way
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attempting to establish a universal impact flux
curve.)
Possibly the inconsistency might be real
undercounts of NEAs by astronomers or biases
in the translation of NEA brightness into bolide
size, i.e. asteroids are darker, hence bigger, than
commonly thought. Alternatively, astronomical
counts are correct but somehow they just do not
pertain to the current era. Possibly the high end
inconsistency reflects the contribution of comet
impacts not included in asteroid flux (6). Shoemaker (1998) championed the idea that a large
fraction of terrestrial craters dc>20km were born
from comet showers with variable or cyclical
intensity.
5.4) Nature of the inconsistency – Low
End. The atmosphere almost completely shields
the Earth from stony bolides less than 100m diameter. Smaller rocks burn or breakup in transit.
[PASSC lists crater diameters down to a few
dozen meters, a RI equivalent of a few meters.
Small things do fall from the sky. Witness the
"Poison Meteorite" that punctured the Peruvian
highlands in September, 2007. Perhaps these
smallest holes parent from fragments of larger
asteroids or from impactors of iron composition.]
Both Ward and Asphaug (2000) and Chesley and
Ward (2006) attempted to account for standard
atmospheric shielding by “rolling off” impacts in
the range 30<RI<100m (dashed curves in Figure
7). If the Earth had no atmosphere, a 100m diameter asteroid might dig a 2.4km diameter cavity at impact. Logically, the counts in Figure 7
do flatten almost completely at that size. Few
smaller bolides get through. For the largest asteroids however, the atmosphere offers faint
protection. This seems to be mirrored in the predicted ~-3 power law slope dependence on crater
size for dc>20km.
Figure 8. Observed 125Ma cumulative crater counts
(triangles) versus size for the selected regions of Figure 4. The dark line is CPM (11) using crater loss rate
(12), production rate (5), and standard atmospheric
shielding (dashed lines Figure 7).
The fact remains that the difference in
predicted and observed terrestrial craters counts
on the small end is huge -- a factor of four for
RI=250m (dc= 7km) and ten for RI=100m (dc=
4km). From the beginning of research on terrestrial crater counts (e.g. Grieve, 1984, 1988), the
gap between the observed low end counts and a
power law production rate has been ascribed to
erosion. If all one knew was Figure 7, the increasing spread in predicted versus observed
counts with decreasing crater size might be easily written off that way. For example in a CE/CP
model, equations (3) and (9) predict that number
of craters greater than diameter dc with age less
than t in the selected area A would be
∞
 dn>p (d c )   1 − e −α ( d c )t 
N> ( d c , < t ) = A ∫ 

 dd c (11)
dd c   α ( d c ) 
dc 
If we use the Ward and Asphaug (2000) cratering
rates (5) including standard atmospheric shield-
12
ing (thin dashed line, Figure 7) and conjure a
strongly size-dependent fractional crater loss rate
of
1  21866 m 
α (d c ) =


125 Ma  d c 
3.75
(12)
it is possible to reproduce the observed 125Ma
crater counts with CPM (11) (Figure 8). If there
was no other information, the low-end discrepancy might indeed be dismissed as erosion, but
for the story to be closed, (11) and (12) must fit
crater counts of all other ages too. Figure 9 plots
(11) and (12) versus crater size from the data
from Figure 6 at several ages. The erosion explanation shows its failure here. Although one can
tune a CPM to fit a group of craters of one age
(125Ma here) huge misfits appear at others.
Clearly erosion alone can not explain all
of the low-end discrepancy, but can erosion explain any of it? To answer this, we must estimate
fractional loss rate α(dc,t) from the data. Figure
10 plots the entire crater data set of Figure 1 versus age (excluding the very youngest ones
T<10Ma) and grouped by size. The number of
craters in each group has been normalized to
unity to eliminate differences in production rates
at different sizes. If production and erosion were
constant over time, the curves should follow the
straight lines with slopes proportional to loss rate
α(dc). Although one might be inclined to think
that smaller craters erode more quickly than
larger ones, for the last 500Ma at least, Figure 10
suggests that erosion loss rates trend nearly constant with α=1/400Ma for craters 1km<dc<30km
regardless of size (equivalent half-life of about
300Ma). If crater production increased by a factor of two to four at 125Ma as we infer, α would
be smaller still. Per equation (11), referenced to
the latest 125Ma in Figure 6, an erosion rate of
α=1/400Ma could eliminate about 15% of the
dc>1km craters produced in that interval. With
low-end discrepancies exceeding 90%, we estimate that erosion accounts for perhaps 1/5 to 1/4
of the gap.
From a geological perspective, the constancy of crater loss rate α in the range
1km<dc<30km T<500Ma is curious -- whatever
eradicates these impact structures must remove
them in equal proportion regardless of size. Conventional erosion does not operate this way, but
certain "clean sweep" processes might (e.g. glaciation, cover by flood basalt plateau). In clean
sweeps, occasionally some random fraction of
the Earth gets re-surfaced, obliterating or burying
all of the craters in that region. If terrestrial craters in a selected area have a half life of 300Ma,
then 1/2 the area needs to be swept every 300Ma,
Figure 9. Cumulative counts of craters (squares) versus size and age for the selected areas given in Figure
5. Solid curves are predictions from CPM (11) using
(12). Although the erosion hypothesis can be made to
fit craters of a certain age (<125Ma in this case) it
seriously misfits counts of craters of other ages.
13
or 1/3 the area every 100Ma, or 1/10 the area
every 30Ma, etc. This is not to say conventional
size-dependent erosion does not operate on craters dc<1km. These may not live long enough to
experience a sweep.
Regardless of specifics, we conclude that
loss by erosion can not account for the short fall
in number of low end craters on Earth. If losses
can not explain the short fall, then smaller craters
must not be being produced as abundantly as (9)
predicts, possibly because:
1) Actual flux rate for RI<500m bolides
deviates considerably from the assumed power
law distribution. Indeed, the latest information
(Harris, 2007) hints at a "dip" in true flux relative to the power law for bolides 10m<RI <500m.
The dip bottoms near RI~50m falling to 1/5 of
the power law rate. A factor of five fewer impacts in this size range would go a long way, but
not completely, toward covering the low end discrepancy.
2) Atmospheric shielding is far stronger
Figure 10. Fractional existence of craters of various
sizes versus age. Craters T>10Ma in all regions are
included. Straight lines correspond to constant rates
of production and erosion. Up to at least 500Ma, an
erosion loss rate of α(d c)=1/400Ma seems to fit all
craters dc>1km
than accounted for by Chesley and Ward (2006).
Their standard shielding rolls off surface impact
rates for bolides less than 100m radius. Enhanced shielding begins the roll off at much
larger sizes, perhaps 800m radius. Enhanced
shielding cuts crater production at a given bolide
flux, so the observed trend in crater counts between 2 and 20 km diameter (Figure 7) might be
interpreted in this way. Air friction may not stop
all bolides 30m<RI<800m but it might slow
stronger objects and disrupt or airburst weaker
projectiles so that smaller crater(s) are made than
expected otherwise. Recent computer simulations by Bland and Artemiva (2003) do argue
that atmospheric shielding may be more effective
than had been thought. Too, evidence seems to
be mounting that smaller asteroids are quite
"fragile" bodies. Disparate bits held together
largely by self-gravitation may have less chance
to penetrate the atmosphere than a cohesive rock
mass.
Whether by a "dip" in flux or by enhanced atmospheric shielding, a factor of five or
ten drop in smaller asteroids impacting earth
would drastically cut impact tsunami hazard
forecasts because fully 50% of the currently estimated hazard accrues to bolides RI<300m.
One stray notion is that the change in
crater production at 125Ma sourced from a rapid
shift in atmospheric shielding (or asteroid
strength) at that time rather than by a transition
in bolide flux. Admittedly we do not know what
modification in atmospheric structure could
spawn a pronounced step in asteroid shielding or
even if this is plausible. Like erosion changes
however, we tend to discount the notion because
whatever shielding variation is proposed, it must
affect all crater sizes in the same proportion at
the same time.
14
6) Conclusions
Although bounded before and after by
long periods of stability, terrestrial crater counts
document a two to four-fold increase in crater
production less crater destruction over a 25Ma
interval near 100Ma. Investigating these features
quantitatively using crater population models, we
conclude that crater production must have increased at that time, most likely driven by a rapid
two to four-fold increase in impactor flux spanning all bolide sizes.
Current best estimates of a power law
asteroid flux based on astronomical methods understate the post-125Ma counts of large
(dc>10km) terrestrial craters by as much as a
factor of four and overstate counts of small
(dc<4km) terrestrial craters by as much as a factor of ten. Stronger than generally supposed atmospheric shielding or a "dip" in actual bolide
flux relative to the power law rate probably generate the misfit at the small end.
We can only speculate what could cause
a rapid and dramatic increase in bombardment at
125Ma. Hills (1981) estimated that every 100
million years or so another star system passes
within 3000 AU of the Sun - well within the Oort
Cloud. Paine (2001) suggested that these close
approaches might disturb bodies within the
Cloud and subsequently increase the rate of
bombardment of the Earth by comets. Perhaps a
smaller scale event within the asteroid belt could
sweep several rocky bodies out of their normal
orbits and increase bombardment of the Earth by
asteroids. Figure 2b hints that the smallest craters
took the transition about 20Ma later than the
largest craters. This behavior might be consistent
with the collision and break up of a few large
bodies. Initially, NEA population would be overweighted in larger fragments. It takes some period (20 Ma?) of follow-on collisions of these
larger fragments to fully populate the small bolide ranks.
Just recently, Bottke et. al. (2007) analyzed the orbits of the Baptistina family of asteroids and concluded that the group conceived
from a collision of 60 km and 160 km diameter
planetoids at 160Ma. They calculated that the
slow diffusion of the daughter rocks into Earth
crossing orbits doubled the rate of terrestrial impacts over a period of 100 million years peaking
about 40 million years after the collision. Bottke
et. al. (2007) findings eerily echo our contentions of a post 125Ma increase in bombardment
based on the Earth's cratering record.
Regardless of the mechanism, a drastic
change in bolide flux at 125Ma calls for reinterpretations of crater count studies on the
moon and elsewhere where crater populations
spring from a mix of old and new impact fluxes.
Rapid and dramatic fluctuations in bolide fluxes
and large uncertainties in atmospheric shielding
also bode trouble for asteroid hazard evaluations
that hinge on well-constrained and current surface impact rates.
Appendix A. Treatment of Crater Age
Uncertainty.
Of the 174 craters in the PASSC data base, 59
have 'greater than' or 'less than' ages. Generally,
'less than' ages date the rock unit which houses
the crater, and 'greater than' ages date material
accumulated within the crater. How should
'greater than' and less than' ages be handled?
Most of the crater count plots in this article
are cumulative in age. Cumulative plots count all
craters less than a certain age, for instance
<125Ma. In some cases then, a 'less than' crater
age (say <80Ma) makes no difference in a
<125Ma count. Still, other craters with 'less than'
ages (say <1000Ma) may be younger than
15
125Ma or perhaps not. Too, some 'greater than'
ages (e.g. >10Ma) might be less than 125Ma or
not. One approach to this problem just takes the
stated crater age limit as the actual age. That is, a
<350Ma age means exactly 350Ma and a >50Ma
age means exactly 50Ma. We follow this 'Age
Limit as Age' approach. An alternative might be
to exclude every crater with a 'less than' or
'greater than' age. Would this approach invalidate
the main features of the crater record discussed
here?
The top row in Figure A reproduces the
cumulative and differential counts of Figures 2a
and 2b using the 131 craters (d>0.5 km
age<500Ma) where age limit is age. The bottom
row in Figure A plots the cumulative and differ-
ential crater counts using the 91 craters (d>0.5
km age<500Ma) with all 'less than' and 'greater
than' ages excluded. Either way, the driving features of the data are not greatly affected: (1)
Crater counts show largely linear growth rates
prior to and after 125Ma, and (2) About a factor
of four change in slope occurs near 125Ma over
all crater sizes.
Presumably as time goes on, crater age information will continue to improve and age uncertainty issues will ease. We suspect that uncertainties will narrow closer to the stated age
than farther from it. Given the similarities in the
comparison in Figure A, we prefer not to exclude
such a large fraction of the crater data strictly on
this account.
Figure A. Cumulative and differential crater counts using "Age Limit as Age" approach (top row) versus an
approach (bottom row) that drops all craters with 'less than' or 'greater than' ages. The sharp increase in net
cratering rate at 125Ma and the long intervals of stability before and after is evident regardless.
16
Constant Erosion/Constant Production (CE/CP)
N ( dc < 0.5km, t2 < t < t1 ) =
β ( dc < 0.5km) −αt2
e − e −αt1
α
[
]
(B1)
and
Constant Erosion/Step in Production (CE/SP)
N ( dc < 0.5km, t2 < t < t1 ) =
β ( dc < 0.5km) −αt2
e − e −αt1 ; t1 < TS
α
[
]
(B2)
N ( dc < 0.5km, t2 < t < t1 ) =
β ( dc < 0.5km) −αt2
e − e −αt1 ; t2 > TS
∆α
[
Figure B. All crater data dc>0.5 km, grouped by age
in 25Ma bins (red squares). The curves are best fitting CE/CP (black) and CE/SP (colors) CPMs. The
white boxes list β, α and SSE for each model. With
>90% confidence, the terrestrial crater data demand a
step in crater production near 100Ma of between 1.5
and 4 times pre-100Ma rates.
Appendix B. Statistical case for a step increase in crater production near 100Ma.
Although a rapid change in slope of crater
counts near 100Ma seems obvious in Figures 2a
and 2b, consider a statistical test to determine
whether a step in crater production is demanded
by the data. To make the test, we group the data
in Figure 2, (dc>0.5km) by age in twenty, nonoverlapping 25Ma bins (squares, Figure B). The
number of craters in each bin can be deemed independent random samples on which standard
statistics apply.
The two CPMs to test are:
]
In (B1) and (B2), ages t2 and t1 mark the lower
and upper bin boundaries, (e.g. t2=125Ma,
t1=150Ma). (B1) has two free parameters,
β(dc>0.5km) and α. (B2) has these two parameters plus two others -- the time of and step in
production, TS and ∆.
Figure B, plots the best fitting CE/CP
model (black) and five best fitting CE/SP models
with Ts=fixed at 75Ma and ∆=1.5, 2, 2.5, 3 and
4. Even by eye, the step near 100Ma and relative
stability in net production pre-100Ma are poorly
matched by the CE/CP model versus CE/SP.
Goodness of fit of two models is often
judged by the F-statistic
F(ν1 − ν 2 , ν1 ) =
( SSE1 − SSE2 ) /(ν1 − ν 2 )
(B3)
SSE1 / ν1
Where SSE is summed square error and ν is the
degrees of freedom νi=N-pi-1; N and pi being the
numbers of data (20 here) and free parameters
(either 2 or 4). If F(2,17) from (B3) exceeds 2.64
(3.59), we can be 90% (95%) certain that CE/SP
17
provides a better explanation of the crater data
than does the CE/CP hypothesis.
Given the CE/CP SSE 1 =191.490, any
CE/SP model with Ts=75Ma and ∆ between 1.5
and 4 better explains the crater data than does
CE/CP at the 90% confidence level. Moreover,
any CE/SP model with Ts=75Ma and ∆ between
2 1/4 and 2 3/4 better explains the crater data
than does CE/CP at the 95% confidence level.
At greater than 90% probability, the dog
leg bend in crater counts near 100Ma demands
interpretation as a step in crater production versus an interpretation with constant crater production and constant or simply variable erosion.
18
Acknowledgements: We thank Michael Paine, Steve
Chesley and Clark Chapman for helpful comments.
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Steven N. Ward
Simon Day
Institute of Geophysics and Planetary Physics
Earth and Marine Sciences Building
University of California
Santa Cruz, CA 95064 USA
ward@es.ucsc.edu
http://es.ucsc.edu/~ward/