A model of possible variations of the galactic cosmic ray intensity over the recent billion years

Abstract

Based on the analysis of published data on exposure ages of iron meteorites determined with the ⁴⁰K/K method (TK) and ages calculated using short-lived cosmogenic radionuclides (with the half-life T1/2 < 1 Myr) in combination with stable cosmogenic isotopes of noble gases (TRS), the following results have been obtained. (1) The distribution of TRS ages (106 values) has an exponential shape, similar to that for ordinary chondrites, but different from the distribution of TK ages (80 values). The difference is most likely due to small amounts of data for meteorites with low TK ages (less than ~200–300 Myr). The latter can be ascribed to the difficulty of measurement of small concentrations of cosmogenic potassium isotopes. This circumstance makes the selection of meteorites with ⁴⁰K/K ages nonrepresentative and casts doubt on the correctness of conclusions about the variations of the intensity of galactic cosmic rays (GCR) based on the analysis of distribution of these ages. (2) The magnitude of the known effect (systematic overestimation of TK ages in comparison with TRS ages) has been refined. The value k = TK/TRS = 1.51 ± 0.03 is acquired for the whole population of data. We have shown the inefficiency of the explanation of this effect on account of an exponential change in the GCR intensity (IT) with time (T) according to the relation IT = I0exp(–γT) over the whole range of ages of iron meteorites. (3) In order to explain the overestimation of TK ages in comparison with TRS ages, a model has been proposed, according to which the GCR intensity has exponentially increased in the interval of 0–1500 Myr governed by the relation: IT = IT = 1500 (1 + αexp(–βT)). For one of the variants of this model, the GCR intensity has exponentially increased by a factor of two only over the recent ~300 Myr, remaining approximately constant for the rest of the time. The data acquired with the use of this model indicate that the measured TK ages are close to the actual time that the meteorites existed in space; the data are in agreement with the observed exponential distribution of TRS ages.

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196
ISSN 0038-0946, Solar System Research, 2017, Vol. 51, No. 3, pp. 196–203. © Pleiades Publishing, Inc., 2017.
Original Russian Text © V.A. Alexeev, 2017, published in Astronomicheskii Vestnik, 2017, Vol. 51, No. 3, pp. 216–224.
A Model of Possible Variations of the Galactic Cosmic Ray Intensity
over the Recent Billion Years
V. A . A lexeev
Vernadsky Institute of Geochemistry and Analytical Chemistry, Russian Academy of Sciences, Moscow, 119991 Russia
е-mail: AVAL37@chgnet.ru
Received September 29, 2016
Abstract⎯Based on the analysis of published data on exposure ages of iron meteorites determined with the
40K/K method (TK) and ages calculated using short-lived cosmogenic radionuclides (with the half-life
T1/2 < 1 Myr) in combination with stable cosmogenic isotopes of noble gases (TRS), the following results have
been obtained. (1) The distribution of TRS ages (106 values) has an exponential shape, similar to that for ordi-
nary chondrites, but different from the distribution of TK ages (80 values). The difference is most likely due
to small amounts of data for meteorites with low TK ages (less than ~200–300 Myr). The latter can be ascribed
to the difficulty of measurement of small concentrations of cosmogenic potassium isotopes. This circum-
stance makes the selection of meteorites with 40K/K ages nonrepresentative and casts doubt on the correct-
ness of conclusions about the variations of the intensity of galactic cosmic rays (GCR) based on the analysis
of distribution of these ages. (2) The magnitude of the known effect (systematic overestimation of TK ages in
comparison with TRS ages) has been refined. The value k = TK/TRS = 1.51 ± 0. 03 i s acq uir ed f or t he wh ole
population of data. We have shown the inefficiency of the explanation of this effect on account of an expo-
nential change in the GCR intensity (IT) with time (T) according to the relation IT = I0exp(–γT) over the
whole range of ages of iron meteorites. (3) In order to explain the overestimation of TK ages in comparison
with TRS ages, a model has been proposed, according to which the GCR intensity has exponentially increased
in the interval of 0–1500 Myr governed by the relation: IT = IT = 1500 (1 + αexp(–βT)). For one of the variants
of this model, the GCR intensity has exponentially increased by a factor of two only over the recent ~300 Myr,
remaining approximately constant for the rest of the time. The data acquired with the use of this model indi-
cate that the measured TK ages are close to the actual time that the meteorites existed in space; the data are
in agreement with the observed exponential distribution of TRS ages.
Keywords: iron meteorites, exposure ages, variations of the galactic cosmic ray intensity
DOI: 10.1134/S0038094617030017
INTRODUCTION
Nuclear reactions with galactic cosmic ray (GCR)
particles lead to the formation of cosmogenic stable
and radioactive nuclides in meteorites. The data on
the concentration of these nuclides can be used to
obtain information on temporal variations of the GCR
intensity. The time scale available for studies is defined
by the exposure age of the meteorites, i.e., the time for
which a meter-sized or smaller meteoroid exists in
space after its separation from the parent body before
falling to the Earth. The exposure age of stone meteor-
ites usually does not exceed 100 Myr (see the overview
(Alexeev, 2005)). Studies of the concentrations of cos-
mogenic isotopes in stone meteorites made it possible
to conclude that for the recent ~10 million years the
average GCR flux remained constant within approxi-
mately ±10% on time scales from hundreds of thou-
sands to millions of years (see, for example, Wieler et
al., 2013 and the references quoted there).
Studying the variations of the GCR intensity over a
longer period (the last billion years) is possible using
the data on the concentration of cosmogenic potas-
sium isotopes—stable 39,41K and long-lived radionu-
clide 40K (T1/2 = 1.277 Gyr)—as well as the data on the
concentration of cosmogenic radionuclides 36Cl (T1/2 =
0.301 Myr), 26Al (0.72 Myr), 10Be (1.39 Myr), and oth-
ers in combination with isotopes of noble gases
(Lavrukhina, 1969; Kolesnikov et al., 1972; Lavru-
khina and Ustinova, 1990; Lavielle et al., 1999; Wieler
et al., 2013). Due to higher mechanical strength of iron
meteorites in comparison with stone ones and, possi-
bly, because of different orbits, the exposure ages of
iron meteorites are approximately an order of magni-
tude higher, up to one billion years. Thus, the system

SOLAR SYSTEM RESEARCH Vol. 51 No. 3 2017
A MODEL OF POSSIBLE VARIATIONS 197
of 40K/K radiation dating provides a potential oppor-
tunity to study the cosmic radiation flux over the last
~109 years.
Voshage and others (Voshage and Feldmann, 1979;
Voshage et al., 1983; Voshage, 1984) determined 81
values of exposure ages up to 2275 Myr using the
40K/K method. The analysis of the distribution of this
set led to contradictory conclusions regarding the vari-
ations of the GCR intensity. For example, Shaviv
(2002; 2003) concluded that this distribution of ages is
the evidence for variations of the GCR intensity with
the period of 143 ± 10 Myr (or, according to more
recent data (Scherer et al., 2006), with the period of
143 ± 6 Myr), which are caused by presumed periodic
passages of the Solar System through the spiral arms of
the Galaxy. However, the very fact of variations with
such a period, as well as the authors’ conclusion about
the supposed correlation between periodic changes in
the GCR intensity and climatic changes on the Earth,
are disputed by many researchers, as both the proce-
dure of sampling the ages for analysis and the interpre-
tation of the data are considered poorly grounded
(Rahmstorf et al., 2004; Jahnke, 2005; Bailer-Jones,
2009; Overholt et al., 2009; Wieler et al., 2013). We
have analyzed the distribution of the sampling of ages
of iron meteorites obtained from the available dataset
by a more rigorous selection (Alexeev, 2016). Based on
this analysis, it has been concluded that the variations of
the GCR intensity with the period of ~400–500 Myr
over the recent billion years are possible. However, as
it is shown further, this result of the analysis cannot be
considered reliable either.
Many researchers paid attention to the systematic
difference between the 40K/K ages (TK ages) and the
ages acquired using relatively short-lived cosmogenic
radionuclides (T1/2 < 1 Myr) in combination with the
data on the concentration of stable cosmogenic iso-
topes of noble gases (TRS ages) (Schaeffer and Hey-
mann, 1965; Hampel and Schaeffer, 1979; Schaeffer et
al., 1981; Aylmer et al., 1988; Lavielle et al., 1999). The
difference between TK and TRS ages is usually inter-
preted as evidence of an increased (by ~30–50%) flux
of cosmic rays over the last several million years com-
pared with the average flux over a longer time period;
other hypotheses, for example, erosion in outer space,
fail to satisfactorily explain these differences.
In this paper, we present the results of the analysis
of distributions of exposure ages of iron meteorites
determined with different methods and consider the
possibility of explaining the characteristic features in
the distribution of these ages.
DISTRIBUTIONS OF EXPOSURE AGES
The distribution of exposure ages for iron meteor-
ites noticeably differs from that for ordinary chon-
drites. The distribution of ages of H-, L-, and LL-
chondrites can be generally characterized by the expo-
nential decrease in the number of meteorites with the
increase in the exposure age according to the relation
NT = N0exp(–T/τ), (1)
where NT is the number of meteorites with age T. The
value τ determines the average “lifetime” of the mete-
orite in space after its separation from the parent body.
For ordinary chondrites, τ is ~35 Myr (Alexeev, 1993;
2005). The upper value of the range of ages for most
iron meteorites (~1000 Myr) exceeds the respective
value for chondrites (~50 Myr) approximately by a
factor of 20. In this connection, one can expect an
exponential distribution of the ages of iron meteorites
with τ = 700 Myr in equation (1). However, the data
for TK ages reveal a completely different picture
(Fig. 1a). The distribution of ages (80 values in the
interval 0–1500 Myr) can be approximated with a
Gaussian curve with the maximum at ~600 Myr. The
Fig. 1. Distributions of exposure ages of iron meteorites.
(a) Ages determined with 40K/K method; 1 is all meteor-
ites (N = 80, from the data (Voshage and Feldmann, 1979;
Voshage et al., 1983; Voshage, 1984); 2 is the same, after
excluding “paired” meteorites and meteorites with com-
plex exposure history (N = 29). (b) Ages determined from
short-lived radionuclides; 3 is all meteorites (N = 106,
from the data (Lipschutz et al., 1965; Chang and Waenke,
1969; Hampel and Schaeffer, 1979; Lavielle et al., 1999 et al.);
4 is the same, after excluding “paired” meteorites and
meteorites with complex radiation history (N = 45). The
distributions are approximated with the “best” Gaussian
(a) and exponential (b) curves according to equation (1).
10
20
30
500
4
3
01000
(b)
5
10
15
N
2
1
0
(а)
1500
Т, Myr

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SOLAR SYSTEM RESEARCH Vol. 51 No. 3 2017
ALEXEEV
distribution is probably bimodal, but statistics are too
few for a reliable conclusion.
Along with 40K/K ages, many values of the ages of
iron meteorites determined with another method have
been published using the abovementioned cosmo-
genic radionuclides with a short half-life period. The
systematics of these data allowed us to compile a set of
more than 200 values of ages for 106 meteorites. For
each meteorite with several results of age determina-
tion, the average value was calculated (considering the
errors of individual measurements). The distribution
of the population of ages thus obtained is shown in
Fig. 1b. This distribution significantly differs from the
distribution of 40K/K ages. Generally, it can be
approximated, as for ordinary chondrites, with an
exponential curve of the form (1).
For a further analysis of the distributions, it is nec-
essary to introduce the correction for the “paired” val-
ues of ages.
Identification of clusters. Gatherings of close values
(clusters) are often observed in the distributions of
exposure ages, when “paired” meteoroids are split
from the surface of parent bodies in one collision.
Such an event explains, for example, a well-known
peak at 7 Myr in the distribution of exposure ages of
H-chondrites. These peaks are of a separate interest
for establishing the collision history of meteorites.
However, the study of the cosmic ray variations
requires the repeated ages of “paired” meteorites to be
excluded. In studying the distribution of exposure ages
of iron meteorites, Shaviv (2003) and Wieler et al.
(2013) considered all meteorites from the same chem-
ical group with identical exposure ages (within 100
Myr or within the error of age determination) to be
formed in one collision, and the cluster of exposure
age values for such meteorites was replaced with a sin-
gle averaged value. Such an approach allowed the
authors to isolate a group of 50 (Shaviv, 2003) and 38
(Wieler et al., 2013) ages from ~80 initial values; these
groups correspond to presumably independent events
of formation of these meteorites in the asteroid belt.
Considering the importance of identifying the
amount and time of independent meteorite-forming
events, we used a more reliable information criterion
by Akaike (Akaike, 1974; Strashnov et al., 2013) for
distinguishing clusters in the distribution of exposure
ages. The procedure of applying this criterion is con-
sidered in detail in (Alexeev, 2016). This procedure
allowed us to obtain a set of 29 ages (from 80 initial val-
ues of 40K/K ages) corresponding to the most probable
times of individual meteorite-forming collisions in
space (histogram 2 in Fig. 1a). A similar correction for
106 values of TRS ages allowed us to obtain a set of
45 ages, the distribution of which is shown in Fig. 1b,
histogram 4. This distribution is approximated with a
curve of the form (1).
COMPARISON OF AGES CALCULATED
WITH DIFFERENT METHODS
As mentioned above, comparison between the ages
obtained with the 40K/K method (TK) and the expo-
sure ages of the same meteorites determined from the
data on the concentration of short-lived nuclides (TRS)
revealed a characteristic feature. The ages obtained with
40K/K method appeared to be 35–50% higher. At the
same time, the ages obtained using short-lived nuclides
agree with each other within experimental error.
Among the compiled sets of meteorites with TK and
TRS exposure ages (80 and 106 respectively), the age of
37 meteorites was determined using both methods. To
reduce the uncertainties caused by large errors of some
measurements, we studied the distribution of TK ages
against TRS ages in the selected group of 20 pairs of val-
ues, whose error of determination did not exceed 20%.
The obtained dependence is shown in Fig. 2. Coefficient k
Fig. 2. Comparison of exposure ages determined with
40K/K method (TK) and from the data on the concentra-
tions of short-lived radionuclides (TRS). The solid line
marks the regression line (TK = kT RS). The dashed line
corresponds to the value k = 1.
200
400
600
800
1000
1200
2000 400 600 800
TRS, Myr
TК, Myr
Table 1 . Values of coefficient k in the regression line equa-
tion TK = kT i
n is the number of meteorites; 1) is this compilation; 2) from the
results of (Lavielle et al., 1999).
Age (Ti)kn
TRS 1.51 ± 0. 03 201)
36Cl/36Ar 1.52 ± 0 .0 4 152)
26Al/21Ne 1.41 ± 0. 0 4 152)
10Be/21Ne 1.45 ± 0.0 4 152)

SOLAR SYSTEM RESEARCH Vol. 51 No. 3 2017
A MODEL OF POSSIBLE VARIATIONS 199
in the equation of the regression line (TK = kT RS), which
is drawn taking into account the “weight” of experimen-
tal points, is found to be 1.51 ± 0.03 (Table 1).
A clear connection between TK and TRS values is
characterized by a high correlation coefficient: R =
0.99. Similar overestimations of ages determined with
40K/K method in comparison with the ages deter-
mined with 36Cl/36Ar, 10Be/21Ne, and 26Al/21Ne meth-
ods are found in the analysis of the data acquired by
Lavielle et al. (1999) (Table 1). All values of coefficient k
within ~(1–1.5)σ are in agreement with each other.
DISCUSSION OF RESULTS
Comparison of distributions. In the case of chon-
drites, the exponential shape of the distribution of
exposure ages (Alexeev, 2005) is conditioned by a
decrease in the percentage of meteorites with an
increase in their age among those that fell on the
Earth. The longer the meteorite stays in outer space,
the higher the probability of its disappearance from
the population of meteoroids (potential meteorites) on
account of being “scooped up” by the planets or
destroyed in collisions and other processes. A similar
picture can be seen for iron meteorites with exposure
ages (TRS) determined using the data on the concen-
trations of short-lived radionuclides (Fig. 1b).
The difference between the distribution character
of TK and TRS ages, which is shown in Fig. 1, is most
likely caused by different capabilities of dating meth-
ods. To determine the age of the meteorite with the use
of 40K/K method is often not possible due to the rare
occurrence of 40K. For example, for this reason in
(Voshage et al., 1983), from 31 of the studied meteor-
ites it was possible to determine the exposure age only
for 10 meteorites; when the relative 40K occurrence in
the isolated potassium was less than ~4%, the age of
the meteorite was not determined due to high error of
the result.
Analysis of the dependence of the age determina-
tion error on the value of this age for all the population
of TK ages has shown that for 40K/K systematics, all
meteorites with TK < 200 Myr have a high error of age
determination, exceeding 40% and reaching values σ >
100% for TK ~ 100 Myr. The difficulty of measuring
small concentrations of cosmogenic potassium iso-
topes in meteorites with the age TK ~ 200 Myr and
lower is the most likely cause for the “deficit” of mete-
orites with low TK ages. The number of meteorites with
the age TK < 200 Myr was only 9%, and only 1 mete-
orite (less than 1%) with the age TK < 100 Myr was
identified.
The method of determining the age with the use of
short-lived nuclides has a significantly higher sensitiv-
ity, making it possible to determine ages down to
TRS ~100 Myr with the accuracy of ~5% and better
(Lavielle et al., 1999). This, apparently, is the defining
reason for a completely different picture of distribu-
tion of ages calculated by this method (Fig. 1b). In this
case, compared to 40K/K systematics, low-aged mete-
orites appeared significantly more often: 43% with the
age TRS < 200 Myr and 29% with TRS <100 Myr.
Such an “artificial” selection makes the sampling
of meteorites with determined 40K/K ages nonrepre-
sentative and casts a doubt on both the possibility of
using such a selection of ages for studying the varia-
tions of the GCR intensity and correctness of conclu-
sions.
Difference between ages calculated with different
methods. As noted earlier, a suggested explanation for
the observed difference by a factor of ~1.5 between TK
and TRS ages was that the intensity of cosmic rays had
increased over the recent several million years. Let us
consider, for example, the determination of age with
the 36Cl/36Ar method (Lavielle et al., 1999). The age is
determined from the concentration of stable 36Ar and
from its calculated formation rate, assuming that the
GCR intensity is constant. The rate of 36Ar formation is
calculated from the concentration of radioactive 36Cl,
whose equilibrium concentration, the same as for
other short-lived radionuclides (10Be, 26Al), is defined
by the GCR intensity over the recent several million
years. The GCR intensity that increases with time
(i.e., is currently higher) will condition the seemingly
higher rate of formation of 36Ar, which is accumulated
over the whole time that a meteoroid existed in space,
and thus distort (reduce) the actual exposure age of
the meteorite.
It was suggested in (Voshage, 1962; Voshage and
Hintenberger, 1963) that the intensity of cosmic rays
(and thus the rate of formation of cosmogenic iso-
topes) changed according to the following relation:
IT = I0exp(–γT), (2)
where I0 and IT are the current GCR intensity and the
GCR intensity at time T respectively. From the data
for several meteorites, the value of constant γ was
found to be within 1.1 × 10–9 > γ > 0.6 × 10–9 year–1.
We have applied this relation to bring into agree-
ment the TK and TRS ages of the more representative
population of meteorites which we compiled. For this
purpose, a program has been developed that allows
calculating TK and TRS ages, given that the GCR
intensity changes according to equation (2) for various
values of parameter γ. Using this program, TK and TRS
values were calculated for the model population of

200
SOLAR SYSTEM RESEARCH Vol. 51 No. 3 2017
ALEXEEV
200 “true” (i.e., matching the actual time that the
meteorite existed in outer space) values of ages, ran-
domly distributed over the range from 0 to 1500 Myr.
It was considered, as noted before, that the number of
meteorites decrease as their age increases according to
equation (1) with parameter τ = 700 Myr.
For each value of γ, we calculated 200 pairs of TK
and TRS values, determined mean values of coefficient
k = TK/TRS, and calculated the parameters of curve
y= axb that approximates the distribution of ages in
the dependence of TK(y) versus TRS(x). The acquired
data are presented in Table 2.
It can be seen that value k =1.51 ± 0.04 at γ =
(1.8 ± 0.2) × 10–9 year–1 corresponds the most closely
to the obtained experimental value k = 1.51 ± 0.03.
This value of γ significantly differs from the values
1.1 × 10–9 > γ > 0.6 × 10–9 year–1 in (Voshage, 1962;
Voshage and Hintenberger, 1963), for which value k is
significantly lower than 1.5 (Table 2).
For comparison of the calculated and experimental
data, the curve approximating the population of 200
calculated TK and TRS values for γ = 1.8 × 10–9 year–1
is shown in Fig. 3.
The same figure illustrates similar curves acquired
for γ = 1.2 × 10–9 and 2.4 × 10–9 year–1; experimental
dots are shown for 20 meteorites. It can be seen that
most the data lies within the range of values limited by
the curves for 2.4 × 10–9 > γ > 1.2 × 10–9 year–1. How-
ever, as seen in the inset in Fig. 3, the distribution of
calculated TRS values does not match the exponential
shape of the distribution of measured TRS values for
106 meteorites shown in Fig. 1b. This difference indi-
cates the inefficiency of the hypothesis of the exponen-
tial rise of the GCR intensity according to relation (2),
which corresponds to the whole range of ages of iron
meteorites.
Agreement between shorter exposure ages deter-
mined from pairs of radioactive and stable nuclides
and 40K/K ages can be achieved under the assumption
that the GCR intensity has increased only over the
recent ~100 million years, remaining constant for the
rest of time (Schaeffer and Heymann, 1965; Hampel
and Schaeffer, 1979; Lavielle et al., 1999). In this con-
nection, we have considered a model, according to
which the GCR intensity (IT) in the time period from
0 to 1500 Myr has exponentially increased according to
the relation:
IT = IT = 1500(1 + αexp(–βT)). (3)
For the given values of parameter α from 0.6 to 2,
we have considered the dependence of coefficient k on
the value of parameter β. The acquired dependences
are shown in Fig. 4.
Table 2. Values k = TK/TRS and parameters of the equation y = axb of the curve, approximating the distribution of TK ages
in dependence of TRS, for different values γ (in units 10–9 year–1) in equation (2)
Parameter γ = 0.6 1.1 1.2 1.8 2.4
k1.21 ± 0.02 1.35 ± 0.03 1.38 ± 0.03 1.51 ± 0.04 1.62 ± 0.05
a0.40 ± 0.04 0.21 ± 0.04 0.19 ± 0.04 0.10 ± 0.03 0.06 ± 0.02
b1.16 ± 0.01 1.29 ± 0.03 1.31 ± 0.03 1.45 ± 0.05 1.56 ± 0.07
Fig. 3. Comparison of exposure ages determined with
40K/K method (TK) and from the data on the concentra-
tions of short-lived radionuclides (TRS). The dots show
experimental data (N = 20). Solid lines show the expected
ratio of TK and TRS values under the assumption of the
change of the GCR intensity according to the relation IT =
I0exp(–γT), where I0 is the contemporary GCR intensity.
The dashes correspond to the age ratio TK/TRS = 1. The
values of γ in the f igure are given in 10–9 year–1 units. Inset:
Distribution of the calculated values (N = 200) of (1) TRS
and (2) TK ages for γ = 1.8 × 10–9 year–1.
200
400
600
800
1000
1200
200
50
40
30
20
0200 400 600 800
N
1
2
0 400 600
γ = 2.4 1.8 1.2
800
TRS, Myr
T, Myr
TК, Myr

SOLAR SYSTEM RESEARCH Vol. 51 No. 3 2017
A MODEL OF POSSIBLE VARIATIONS 201
From these data, it can be seen that for all values
α≥ 0.8 there are two values of β that allow k = 1.5.
However, in selecting α and β values, it is also neces-
sary to consider that, as noted above, the GCR inten-
sity over the recent ~10 million years was, on average,
constant within ±10%. This condition is fulfilled by
choosing the lesser of two possible values of β for the
chosen value of α. Fig. 5 shows several options which
satisfy both requirements of the considered model.
Further, we performed the calculations for one of
the variants of changes in the GCR intensity, specifi-
cally, the one described by curve 4. According to this
variant, a significant rise of the GCR intensity began
from the point T ~ 300 Myr and by the point T = 0
increased by a factor of two. The distribution of “mea-
sured” TK ages almost coincides with the distribution
of “true” ages in the model set (Figs. 6a, 6b). The dis-
tribution of “measured” TRS ages in the model popu-
lation (Fig. 6c) was close to the distribution of TRS ages
of actual meteorites (Fig. 1b).
Thus, the proposed model agrees with the experi-
mental data, both with the difference between TK and
Fig. 4. Dependence of coeff icient k = TK/TRS on parameter β in equation IT = IT = 1500(1 + αexp(–βT)) for different values of
α. Symbols 1, 2, 3, 4, 5, and 6 correspond to the values α = 2, 1.5, 1.2, 1, 0.8, and 0.6 respectively.
0.01
1
2
3
4
5
6
0.1
β, Myr
1
1.0
k
1.5
2.0
2.5
1E–3
Fig. 5. Relative change in the GCR intensity with time T according to the dependence IT = IT = 1500(1 + αexp(–βT)). Values β
(in units Myr–1) and α for curves 1–5 are the following: (1) 0.008 and 2; (2) 0.01 and 1.5; (3) 0.012 and 1.2; (4) 0.016 and 1;
(5) 0.03 and 0.8 respectively.
0.5
1.0
1.5
2.0
2.5
3.0
IT/IT = 1500
0100
5
4
3
2
1
200 300 400 500
T, Myr
1000 1500

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TRS ages by a factor of ~1.5 and the exponential shape
of the distribution of TRS ages. The data acquired
according to this model indicate that measured TK
ages are close to the actual time that meteorites existed
in space.
CONCLUSIONS
(1) We have analyzed the published data on the
exposure ages of iron meteorites determined with the
40K/K method (TK ages) and ages calculated using
short-lived cosmogenic radionuclides (with the half-
life T1/2 < 1 Myr) combined with stable cosmogenic
isotopes of noble gases (TRS ages).
(2) The distribution of TRS ages (106 values) has an
exponential shape, similar to that for ordinary chon-
drites, but different from the distribution of TK ages
(80 values), which can be approximated with a Gauss-
ian. The difference is most likely due to a small num-
ber of meteorites with low TK ages (less than ~200–
300 Myr). The latter may be associated with complex-
ity of measurements of small concentrations of cos-
mogenic potassium isotopes in meteorites with low
exposure ages. Such an artificial selection by ages
makes the sampling of meteorites with determined
40K/K ages nonrepresentative and casts doubt on the
correctness of conclusions about the variations of the
GCR intensity which were based on the analysis of the
distribution of TK ages.
(3) The analysis of data on meteorites for which
both TK and TRS ages are determined with an accuracy
higher than 20% made it possible to refine the magni-
tude of the known effect, i.e., systematic overestima-
tion of TK ages in comparison with TRS ages. The fol-
lowing value is acquired: k = TK/TRS = 1.51 ± 0.03.
The explanation of this effect on account of an expo-
nential change in the GCR intensity according to depen-
dence IT = I0exp(–γT) over the whole interval of ages of
iron meteorites has been shown to be inefficient.
(4) In order to explain the difference between TK
and TRS ages, a model has been proposed, according to
which the GCR intensity (IT) in the considered time
interval (0–1500 Myr) exponentially increases accord-
ing to the relation: IT = IT = 1500 (1 + αexp(–βT)). This
model agrees with the experimental data: the differ-
ence between TK and TRS ages by a factor of ~1.5 and
the exponential shape of the distribution of TRS ages.
For one of the variants of this model, the GCR inten-
sity has exponentially increased by a factor of 2 over
the recent ~300 million years, remaining approxi-
mately constant for the rest of the time. The data
acquired from this model indicate that the measured
TK ages are close to the actual time that the meteorites
existed in space.
ACKNOWLEDGMENTS
The author would like to thank A.V. Fisenko for
valuable remarks in discussion of the results and the
reviewer for constructive recommendations.
The work was supported in part by the Program 7
for Basic Research of the Presidium of the Russian
Academy of Sciences.
Fig. 6. Distributions of exposure ages of iron meteorites.
(a) Model set of “true” values of ages (N = 200), randomly
distributed in the interval of 0–1500 Myr; the number of
meteorites decreases according to equation (1) for the
average lifetime of iron meteorites in space τ = 700 Myr.
(b) “Measured” values of TK ages for the model set under
the assumption that the GCR intensity changed according
to the variant shown in Fig. 5 (curve 4). (c) Solid curve
marks the same for the “measured” values of TRS ages.
The dashes are the normalized exponential curve, which
characterizes the actual distribution of TRS ages of iron
meteorites (Fig. 1b, curve 4). The distributions are approx-
imated with the “best” exponential curves according to
equation (1).
10
20
30
40
50
5000
0
0
1000
T, Myr
1500
(c)
10
20
30
40
(b)
10
20
30
N
(а)

SOLAR SYSTEM RESEARCH Vol. 51 No. 3 2017
A MODEL OF POSSIBLE VARIATIONS 203
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Translated by M. Chubarova