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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

198

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

202

SOLAR SYSTEM RESEARCH Vol. 51 No. 3 2017

ALEXEEV

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