Changing of the Modulation Region Structure With Particle
Rigidities According to Data of Hysteresis Phenomenon
(Cycle 22)
Lev I. Dorman
Israel Cosmic Ray Center and Emilio Segre’ Observatory, affiliated to Tel Aviv University, Technion and Israel Space
Agency, Israel; IZMIRAN, Russian Academy of Science, Troitsk, Russia;
Abstract. Our studies of neutron monitor observatory data with different cut-off rigidities during solar cycle 22 have made it
possible to investigate the hysteresis properties of the relationship between the variations in solar activity and galactic cosmic
ray intensity. Hysteresis arises due to the delay of interplanetary processes (responsible for cosmic ray modulation) with
respect to the initiating solar processes, which correspond to some effective solar wind and shock wave propagation velocity.
It allows determination of the effective dimension of the modulation region as a function of the effective energy of galactic
cosmic rays. We extend previous investigations made in the framework of the convection-diffusion model by taking into
account drifts that change sign in periods of solar magnetic field reversal. From comparisons with experimental data on longterm cosmic ray variation in cycle 22, we determine the role of convection-diffusion and drifts in global modulation, and the
effective dimension of the modulation region in dependence of particle rigidities.
INTRODUCTION
approximation X o max and Adr are about the same in
odd and even solar cycles. In the present paper we try to
solve the problem of determining Adr and X o max only
on the basis of data in solar cycle 22. We will therefore
correct the observed cosmic ray long-term variation in
cycle 22 for drift effects with different values of the
amplitude Adr ; for each Adr we determine the
correlation coefficient R ( X o , Adr ) of corrected cosmic
ray long-term variation according to a convectiondiffusion model for different values of the time-lag X o
(from 0 to 60 av. months with monthly steps). Then we
determine the value of X o max ( Adr ) when R( X o , Adr )
reaches the maximum value Rmax ( X o max , Adr ) . For
each Adr we will determine Rmax and X o max . It is
A short historical introduction to the research of the
lag between long-term variations of cosmic ray (CR)
intensity observed at Earth and solar activity (SA) is
given in [1]. Analysis made in [1] leads to the conclusion
that observed long-term CR modulation is caused by two
processes: a convection-diffusion mechanism that does
not depend on the sign of the solar magnetic field
(SMF), and a drift mechanism (e.g. [2-7]) which gives
opposite effects depending on the sign of the SMF. In [1]
the relative role of convection-diffusion and drifts in the
long-term CR modulation on the basis of a comparison
of observations in odd and even cycles of SA was
considered: it was shown that the time–lag X o max
between CR and SA in the odd cycles 19, 21 decreases
with increasing of the amplitude of the drift effect Adr ,
but in the even cycles 20, 22, X o max increases with
increasing
natural to assume that the most reliable value of Adr
will correspond to the biggest Rmax ( X o max , Adr )
Adr . To determine
X o max and Adr
separately, in [1] was assumed that for a first
value, i.e. when the correction for drift effects is the best
(in the frame of the model used for drift effects for longterm CR variations). This way will also determine the
CP679, Solar Wind Ten: Proceedings of the Tenth International Solar Wind Conference,
edited by M. Velli, R. Bruno, and F. Malara
© 2003 American Institute of Physics 0-7354-0148-9/03/$20.00
164
of the SMF polarity reversal. We used data on tilt-angles
for the period May 1976-September 1993. On the basis of
these data we determined the correlation between T and
W for 11 month-smoothed data as
most reliable value for X o max characterizing the
dimension of the cosmic ray modulation region in the
Heliosphere.
T = 0.349W + 13.52°
(3)
with correlation coefficient 0.955±0.013. Important for
the Cycle 22 reversal periods were: March 1981±5
months, and June 1991±7 months. As example, the drift
effect according to this model for the period January
1985-December 1996 is shown in Figure 1 for the 11month-smoothed data of W according to (3) and
Adr =1% at W=75.
COSMIC RAY LONG-TERM
VARIATION CAUSED BY
CONVECTION-DIFFUSION
Because the basic convection-diffusion quasistationary model of CR-SA hysteresis phenomenon was
described in details in [1], we will give only the final
equations used in this paper. According to this model the
expected contribution to cosmic ray long-term variations
caused by the convection-diffusion mechanism at the
Earth’s orbit is:
E
)
obs
o
1
2
o
1
(
2
Xo
o
XE
2
)
1.5
DRIFT EFFECT, %
(
ln n(R,r ,t) = A(X , β,t ,t )− B(X , β,t ,t )× F t,X , β,W(t − X)
2.5
, (1)
where
(
F t, X , β ,W(t − X )
o
Xo
XE
 W (t − X )

= ∫ 

 W

)
Xo
XE
(
1 2
+ 1− W ( t− X ) Wmax
3 3
,(2)
)
−β
X dX
max
(
(
F t, X ,W t − X
o
)
XE
0
-0.5
-1.5
-2
1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997
2.628 × 10 6 s). As it was mentioned in [1], the regression
coefficient A(R, X o , β , t1 , t 2 ) determined the CR
intensity out of the Heliosphere, and B(R, X o , β , t1 , t 2 )
characterized the effective diffusion coefficient in the
interplanetary space; these coefficients can be determined
from correlation of observed values ln (n(R, rE , t ))obs
Xo
0.5
-1
with X = r u , X E = 1 AU u , X o = ro u ( X E and X o
are in units of average months = (365.25/12) days =
with values of
1
)
FIGURE 1. An example of assumed drift modulation in Cycle
22 for Adr =1% at W=75.
NEUTRON MONITOR DATA
Results for Climax NM
calculated according
According to the procedure described above we
correct the 11-month-smoothed data on the drift effect
for different values of Adr from 0% (no drift effect) up
to 4% at W=75. The dependence of the correlation
coefficient on the value of the expected time-lags is
shown in Figure 2. For each value of Adr in Figure 2,
X o max ( Adr ) can be easy determined when the
to (2) for different values of X o in the time interval
t1 ≤ t ≤ t2 . Here r is the distance from the Sun, ro is the
assumed radius of modulation region, rE = 1 AU , W is
the monthly sunspot number, and Wmax is the sunspot
number at the maximum of SA.
correlation coefficient reaches a maximum value Rmax .
COSMIC RAY LONG-TERM VARIATION
CAUSED BY DRIFTS
According to the main idea of the drift mechanism
(see [2-7]), we assume that the drifts are proportional to
the value of tilt angle T and changed sign during periods
165
amplitude Adr
Xo, av.month
5
10
15
20
25
30
35
-0.97
according to (5) and the function
-0.96
Xo, av. month
-0.95
10
-0.94
-0.93
-0.92
-0.91
-0.9
-0.89
-0.88
-0.87
Adr=0.5%
Adr=2.5%
Adr=1%
Adr=3%
Adr=1.5%
Adr=4%
FIGURE 2. Correlation coefficient R ( X o , Adr ) according to
11-month-smoothed data of Climax NM (N39,W106; height
3400 m, 2.99 GV) in Cycle 22 for different Adr from 0% up
to 4% at W=75.
12
13
14
15
16
25
-0.94
15
-0.955
13
Rmax(Adr)
-0.95
-0.96
11
-0.965
9
7
1
1.5
Xomax(Adr)
2
2.5
Rmax(Adr)
3
3.5
4
Adr, %
FIGURE 3. Functions Rmax ( Adr ) and X o max ( Adr ) for
Climax NM data in Cycle 22.
Results for Kiel NM Data
The function Rmax ( Adr ) for Kiel NM (sea level;
2.32 GV) data can be approximated with a correlation
coefficient 0.9992±0.0004 by (4) with regression
coefficients a=0.0095±0.0001, b=-0.0250±0.0004, c=0.960±0.014, what gives, according to (5), Adr max =
The function Rmax ( Adr ) can be approximated with
correlation coefficient 0.9985±0.0007 by parabola
2 + bA + c ,
Rmax ( Adr ) = aAdr
dr
-0.961
connection between expected and observed CR intensity
is
characterized
by
correlation
coefficient
Rmax ( X o max , Adr max ) = 0.9652 (see Figure 4).
-0.97
0.5
-0.962
R ( X o , Adr max ) = dX o2 + eX o + f ,
(6)
where d=0.000377±0.000002, e=-0.00942±0.00004, and
f = -0.906±0.004. By (6) we can determine the most
reliable value of X o max corresponding to Adr max :
X o max = − e 2d = 12.5 ± 0.1 av. month. (7)
At obtained values of Adr max and X o max the
-0.945
17
-0.963
From Figure 4 can be seen that the function
R ( X o , Adr max ) can be approximated with a correlation
coefficient 0.99994±0.00003 by a parabola:
21
19
-0.964
FIGURE 4. The function R ( X o , Adr max ) for Climax NM
data in Cycle 22
-0.935
23
-0.965
-0.96
The functions Rmax ( Adr ) and X o max ( Adr ) are
shown in Figure 3.
0
11
-0.966
Adr=0%
Adr=2%
Xomax, av. months
max
R ( X o , Adr max ) is shown in Figure 4.
40
CORRELATION COEFFICIENT R(Xo,Adrmax)
CORRELATION COEFFICIENT R(Xo, Adr)
0
(4)
where a = 0.004065±0.000079, b = -0.01253±0.00024,
and c = -0.9551±0.0185. From (4) we can determine
Adr max when Rmax reaches the biggest value:
Adr max = − b 2a = 1.54 ± 0.04% .
(5)
With this information, we can now correct the Climax
NM data of Cycle 22 for drifts, with the most reliable
1.32±0.04%. Next, we determine R ( X o , Adr max ) that
can be approximated with a correlation coefficient
0.99988±0.00006 by (6) with a regression coefficients
d=0.000466±0.000003,
e=-0.01191±0.00007,
f=0.897±0.005, that gives, according to (7), X o max =
13.4±0.2 av. months. The obtained values for Adr max
166
and for X o max are about the same as for the Climax
NM. In this case the correlation between the predicted
and observed CR intensity is characterized by a
coefficient of Rmax ( X o max , Adr max ) = 0.977.
Adr max (at W=75) and the time-lag X o max (the
effective time of the solar wind moving with frozen
magnetic fields from the Sun to the boundary of the
modulation region on the distance ro ≈ uXo max ). We
found that with an increasing effective CR primary
particle rigidity from 10−15 GV (Climax NM and Kiel
NM) up to 35−40 GV (Huancayo/Haleakala NM) are
decreased both the amplitude of drift effect Adr max
(from about 1.5% to about 0.15%) and time-lag X o max
(from about 13 av. months to about 10 av. months). It
means that in Cycle 22, for the total long term
modulation of CR with rigidity 10-15 GV, the relative
role of the drift mechanism was ≈1/4 and the convectiondiffusion mechanism about 3/4; for rigidity 35-40 GV
these values were ≈ 1/10 for the drift mechanism, and
about 9/10 for the convection-diffusion mechanism. If
we assume that the average velocity of the solar wind in
the modulation region was about the same as the
observed average velocity near the Earth’s orbit in 19651990: u=4.41x107=7.73 AU/av.month, the predicted
dimension of modulation region in Cycle 22 will be 100
AU for CR with rigidity of 10-15 GV and about 80 AU
for CR with rigidity of 35-40 GV. It means that at
distances more than 80 AU the magnetic field in solar
wind is too weak to influence intensity of 35-40 GV
particles.
Results for Tyan-Shan NM Data
The Tyan-Shan NM (43N, 77E, near Alma-Ata; 3.34
km, 6.72 GV) is sensitive to more energetic particles than
the Climax NM and the Kiel NM. For the Alma-Ata NM
the function Rmax ( Adr ) was approximated with
correlation coefficient of 0.9996±0.0002 by (4), with
regression
coefficients
a=0.0149±0.0015,
b=0.019±0.002, c=-0.957±0.009, that gives Adr max =
0.634±0.012%. Next, we determined R ( X o , Adr max )
that can be approximated with a correlation coefficient of
0.9997 by (6) with a regression coefficients d=0.000388,
e=-0.00845±0.00005, f=-0.917, that gives, according to
(7), X o max = 10.9±0.2 av. months. In this case the
correlation between the predicted and observed CR
intensity is characterized by a coefficient of
Rmax ( X o max , Adr max ) = 0.963.
Results for Huancayo/Haleakala NM Data
The Huancayo NM (12S, 75W; 3.4 km, 12.92 GV)/
Haleakala NM (20N, 156W; 3.03 km, 12.91 GV) is
sensitive to primary CR particles of 35−40 GV which is
about 2-3 times larger than for the Climax and Kiel NM.
For Huancayo/ Haleakala NM the function Rmax ( Adr )
was approximated with a correlation coefficient of 0.9998
by (4) with regression coefficients a=0.0621, b=-0.0165,
c =-0.978, which gives Adr max =0.133±0.002%. Next,
we determined R( X o , Adr max ) that can be approximated
with a correlation coefficient 0.99998±0.00001 by (6)
with regression coefficients d=0.000406, e=-0.00842, f=0.935±0.002, that gives X o max =10.38±0.05 av. months
according to (7). In this case the correlation between the
predicted and observed CR intensity is characterized by
Rmax ( X o max , Adr max ) = 0.979.
ACKNOWLEDGEMENTS
This research was partly supported by Israel Cosmic Ray
Center and Emilio Segre’ Observatory (Tel Aviv University),
and by INTAS grant 0810.
REFERENCES
1. Dorman, L. I., “Using galactic cosmic ray observations for
determination of the Heliosphere structure during different
solar cycles” This issue, Paper SI-36.
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(1981).
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1981.
4. Lee, M. A., and Fisk, L. A. , Ap. J., 248, 836-844, 1981.
DISCUSSION AND CONCLUSIONS
5. Kota, J., and Jokipii, J. R., Proc. 26th ICRC, 7, 9-12, 1999.
6. Burger, R. A., and Potgieter, M. S., Proc. 26th ICRC, 7, 1316 (1999).
The taking into account drift effects (see Figure 1)
gives an important possibility, using data only for solar
cycle 22, to determine the most reliable amplitude
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26th ICRC, 7, 77-80 (1999).
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