On the possibility to check the magnetosphere’s model by CR: The strong geomagnetic storm in November 2003

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Advances in Space Research 42 (2008) 1556–1563 www.elsevier.com/locate/asr

On the possibility to check the magnetosphere’s model by CR: The strong geomagnetic storm in November 2003 M.I. Tyasto a, O.A. Danilova a, L.I. Dorman b

b,c,*

, V.M. Dvornikov d, V.E. Sdobnov

d

a St. Petersburg Filial of IZMIRAN, Russia Israel Cosmic Ray & Space Weather Center Affiliated to Tel Aviv University, Technion, and Israel Space Agency, Israel c IZMIRAN, Russian Academy of Science, Troitsk, Moscow Region, Russia d Institute of Solar-Terrestrial Physics SO RAN, Irkutsk, Russia

Received 15 December 2006; received in revised form 9 April 2007; accepted 26 June 2007

Abstract We report on a study of cosmic ray cutoff rigidity variations during the strong geomagnetic storm of 18–24 November 2003. We employed the most recent Tsyganenko magnetospheric model to represent the very strong disturbed Magnetosphere. We used this magnetic field for the cosmic ray trajectory calculations to determine the geomagnetic cutoff rigidity throughout this period of severe geomagnetic disturbance. We determine the cutoff rigidity changes during this period by two methods, by trajectory calculations and by the spectrographic global survey method. The values of the change of cutoff rigidities obtained by two different methods are correlated with the Dst and interplanetary magnetic field and plasma parameters and result in correlation coefficients in the range 0.63– 0.84 for the various cosmic ray stations. The result of this study indicates that the most significant contributions to the cutoff rigidity changes are due to Dst variation although the influence of solar wind density and Bz and By components of IMF variations is significant.  2008 Published by Elsevier Ltd on behalf of COSPAR. Keywords: Cosmic rays; Magnetosphere; Cutoff rigidity; Magnetic storms

1. The matter of problem Magnetic fields in the Earth’s magnetosphere change in response to solar wind disturbances. Dynamic processes in the magnetosphere lead to variations in cosmic ray (CR) cutoff rigidity and CR asymptotic directions that results in changes CR fluxes into the magnetosphere and on the Earth’s surface. Thus magnetospheric CR effects reflect exciting, developing and decaying of current systems in the magnetosphere and they can be used as an independent

*

Corresponding author. Address: Israel Cosmic Ray & Space Weather Center Affiliated to Tel Aviv University, Technion, and Israel Space Agency, POB 2217, Qazrin 12900, Israel. Tel.: +972 4 696 4932; fax: +972 4 696 4952. E-mail address: [email protected] (L.I. Dorman). 0273-1177/$34.00  2008 Published by Elsevier Ltd on behalf of COSPAR. doi:10.1016/j.asr.2007.06.057

information source for additional testing of magnetosphere models. Magnetospheric CR effects are mainly due to cutoff rigidity variations that are most intense during great geomagnetic storms. CR cutoff rigidities can be obtained mainly on base of particle trajectory calculations in the magnetic field of any magnetosphere model (McCracken et al., 1962, 1965; Dorman et al., 1971, 1972; Shea and Smart, 1975; Shea et al., 1976; Smart et al., 2000; see in detail Chapter 3 in Dorman, 2007). Empirical magnetosphere’s models are used widely for this purpose. The accuracy in determining of effective geomagnetic cutoff rigidities substantially depends on a magnetospheric model used in calculations. Empirical magnetosphere models take into account theoretical representations of dynamical processes in the magnetosphere from one side and direct magnetic field measurements in space from another side. Strong

M.I. Tyasto et al. / Advances in Space Research 42 (2008) 1556–1563

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geomagnetic storms are relatively rare events therefore the data of very disturbed periods represents a small part of the data used in the derivation of empirical geomagnetic field models. This circumstance explains why the magnetospheric magnetic field model for strong magnetospheric disturbances did not exist before recent time. For checking magnetosphere’s models for great magnetic storms we used the period of 18–24 November 2003. To calculate geomagnetic cutoff rigidities during disturbed period of 18–24 November 2003, we used the Tsyganenko magnetosphere’s model Ts03 which was derived on base of measurements during 37 geomagnetic storms with jDst j P 65 nT (Tsyganenko, 2002a,b; Tsyganenko et al., 2003). This model describes a strong disturbed configuration of the magnetospheric magnetic field and its evolution during the storm. Geomagnetic cutoff rigidities calculated by the trajectory tracing method in the model magnetospheric field can be named theoretical. Another possibility to determine the geomagnetic cutoff rigidities is to use the spectrographic global survey (SGS) method (see review in Dorman, 1974 and Chapter 3 in Dorman, 2004) which is based on the assumption that the anisotropy in the CR distribution along the directions of arrival is attributed to dependence of their intensity on pitch-angle in the interplanetary magnetic field and to a density gradient (Dvornikov and Sdobnov, 2002). Variation amplitudes of an integral flux of secondary particles DNc/Nco (with respect to a certain background level Nco) observed at a geographical site with cutoff rigidity Rco at level ho in the Earth’s atmosphere may be represented as follows   DDðRco Þ DN c =N co ¼ DRc W c ðRco ; Rco ; ho Þ  1 þ Do ðRco Þ Z 1 Z 2p Z p=2 DD da sin b db þ 0 0 Rco Do  ðR; WðR; a; bÞ; KðR; a; bÞÞW c ðRco ; R; b; ho Þ dR; ð1Þ

M0 Mn N X X X DD ðR; W; KÞ ¼ a0k Rk þ ½ðcnk Rk ÞP n ðcos HÞ Do k¼0 n¼1 k¼1

where a and b are the azimuth and zenith angles of arrival of primary CR particles at the atmospheric boundary, W c ðRco ; R; b; ho Þ is the coupling function between primary and secondary CR variations; W(R, a, b) and K(R, a, b) are asymptotic angles of arrival CR particles; DRc ðRco Þ is a probable variation in geomagnetic cutoff rigidity Rco which was approximated as  pffiffiffiffiffiffiffi   exp  Rco 2 DRc ðRco Þ ¼ b1 Rco þ b2 Rco ð2Þ 1 þ aR1 co þ Ra22 þ Ra33

bmn ðRÞ

co

co

and 0 DDðRco Þ X ¼ a0k Rk co Do ðRco Þ k¼1

M

ð3Þ

are the amplitudes of CR global intensity variations at the cutoff rigidity Rco. According (Dvornikov and Sdobnov, 2002) flux density variations of primary CR particles was represented by

þ ðd nk Rk ÞP n ðcos UÞ;

ð4Þ

where P n ðcos HÞ and P n ðcos UÞ are the Legendre polynomials, H is angle between the IMF vector B and the particle velocity vector (pitch angle), U is the angle between the particle velocity V and the vector B · $n^, where $n^ is the CR density gradient component transverse with respect to B. The pitch angle is defined by expression cos H ¼ sin k sin ko þ cos k cos ko cosðw  wo Þ

ð5Þ

and the angle between V and B · $n^ is represented by cos U ¼ sin k sin no þ cos k cos no cosðw  /o Þ

ð6Þ

If to take into account Eqs. (3)–(6) it is possible to obtain the following system of algebraic equations: Z 2p Z p=2 DN c ¼ DRc W c ðRco ; ho Þ þ da sin b db Nc 0 0 Z 1 (X M0 N X n X  m  a0k Rk þ an ðRÞ cos mWc ðR; a; bÞ Rco

k¼1

n¼1 m¼0

þbmn ðRÞ sin mWc ðR; a; bÞ

 m P n ðsin kðR; a; bÞÞ

 W c ðRco ; R; b; ho Þ dR;

ð7Þ

where a0n ðRÞ; amn ðRÞ; bmn ðRÞ have the form ! ! Mn Mn X X 0 k k an ðRÞ ¼ cnk R d nk R P n ðsin ko Þ þ P n ðsin no Þ; k¼1

amn ðRÞ ¼

k¼1

2ðn  mÞ! ðn þ mÞ! þ

Mn X

2ðn  mÞ! ¼ ðn þ mÞ! þ

Mn X

ð8Þ

! cnk Rk P mn ðsin ko Þ cos mWo

k¼1

!

d nk R

k¼1

Mn X

"

"

d nk R

k

#

P mn ðsin no Þ cos m/o

Mn X

ð9Þ

! cnk R

k

P mn ðsin ko Þ sin mWo

k¼1

!

k

;

P mn ðsin no Þ sin m/o

# ;

ð10Þ

k¼1

and the angles Wo and ko characterize the direction of IMF, /o and no stand for the orientation of the vector B · $n^. So the first member of Eq. (7) characterizes CR variations due to geomagnetic cutoff rigidity changes and the second and the third ones+ characterize variations of the isotropic and anisotropic components of the CR angular distribution. The system of Eq. (7) gave possibility to search for as the unknown parameters b1, b2, a1, a2, a3 needed to calculate the change of geomagnetic cutoff rigidities as unknown parameters a0k, cnk, dnk which characterize cosmic ray variations out of the magnetosphere. The system of equations obtained for two spherical harmonics N = 2 and M0 = 3, M1 = M2 = 2 was used to calculate, in particular, geomag-

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ΔR (GV)

0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2

a

ΔR (GV)

0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4

b

ΔR (GV)

c

ΔR (GV)

d

0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4 -1.6 -1.8 -2

ΔR (GV)

e

0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4 -1.6

ΔR (GV)

f

0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4 -1.6 -1.8

20 0 -20 -40 -60

BZ

Dst (nT)

V (km/s)

N (sm-3)

800 700 600 500 400

0 -100 -200 -300 -400 -500 24 16 8 0

6

12

0

18

18

0

6

12

18

6

12

6

12

0

0

19

18

20

21

18

0

6

12

22

18

0

6

12

0 23

18

6

12

24

18

hours DAYS

Fig. 1. CR cutoff rigidity variation during the geomagnetic storm of 18–24 November, 2003 (DRef – circles, DRsgs – crosses) for different stations (a – Tokyo, b – Alma-Ata, c – Rome, d – Irkutsk, e – Moscow, f – Hobart).

M.I. Tyasto et al. / Advances in Space Research 42 (2008) 1556–1563

netic cutoff rigidities on base of data of neutron monitors corrected for atmosphere effects from the worldwide network. In this paper, we restrict our research only geomagnetic cutoff rigidity variations. Geomagnetic cutoff rigidities obtained by the SGS method can be named experimental because for their determination we used only the experimental cosmic ray data and do not used any theoretical or experimental representations of the magnetospheric magnetic field. 2. Comparison experimental DRsgs derived from CR data and theoretical DRef obtained by trajectory tracing in the frame of Ts03 Tsyganenko model Using the SGS method we obtain experimental CR cutoff rigidity variations (with respect to the quiet level of 12 October 2003) DRsgs for every hour of the geomagnetic storm of 18–24 November 2003. From other side, the theoretical effective cutoff rigidities DRef were calculated in the magnetic field of the Ts03 model by the trajectory tracing method for the stations Tokyo, Alma-Ata, Roma, Irkutsk, Moscow, and Hobart. Quiet cutoff rigidities of the chosen stations occupy the main part of cutoffs influenced on CR by the geomagnetic field. Theoretical cutoff rigidity changes DRef were determined also with respect to the quiet level of 12 October 2003. The daily averaged effective cutoffs at 12 October, 2003 are 11.02 GV (Tokyo), 6.19 GV (Alma-Ata), 6.08 GV (Rome), 3.25 GV (Irkutsk), 2.10 GV (Moscow) and 1.75 GV (Hobart). It is necessary to notice these cutoff rigidities are lower by 0.2–0.4 GV than cutoffs in the main geomagnetic field. Fig. 1 displays the time variations in the calculated cutoff rigidities obtained by two named above different methods theoretical DRef (open circles) and experimental DRsgs (crosses) during 18–24 November 2003. The curves in Fig. 1 correspond to the stations a – Tokyo, b – AlmaAta, c – Rome, d – Irkutsk, e – Moscow, f – Hobart. The low part of Fig. 1 shows the Dst-variation (filled circles), IMF Bz (open circles), the velocity Vsw (filled stars) and the density Nsw(open stars) of solar wind. It is seen in Fig. 1 that curves DRef and DRsgs are in general consistent with each other and with the Dst-variation. Some differences between DRef and DRsgs are noticeable at the Dst-minimum for Moscow and Hobart. Maximum decreases of the geomagnetic cutoffs are observed 20 November during the main phase of the geomagnetic storm but the hours of maximum geomagnetic cutoff decreases does not coincide always with Dst-minimum ones. 3. Comparison of absolute and relative maximum decreases of CR cutoff rigidities The comparison of absolute and relative maximum decreases of the theoretical and the experimental CR cutoff rigidities at each of the six chosen stations are shown in Table 1.

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Table 1 Decreases of the CR cutoff rigidities during the Dst-minimum Station

Tokyo Alma-Ata Rome Irkutsk Moscow Hobart

DRsgs

DRef [GV]

[%]

[GV]

[%]

0.8 1.18 1.33 1.67 1.81 1.51

7.2 19.1 21.9 51.4 86.2 86.3

1.06 1.30 1.32 1.63 1.62 0.96

9.6 21.0 21.7 50.2 77.1 54.9

As it can be seen from Table 1, the theoretical cutoff rigidity DRef decrease with the latitude decreasing (with the geomagnetic cutoff increasing at quiet time) by values in a rather wide limits from 86% before 7%. The experimental geomagnetic cutoffs DRsgs change in a more narrow limits from 77% at Moscow before 9.6% at Tokyo. We see that geomagnetic cutoff decreases DRef and DRsgs at low latitude stations are similar. The geomagnetic cutoffs DRef and DRsgs decrease by 50–85%. at middle latitude stations Irkutsk, Moscow, Hobart. It means, for example, that the theoretical cutoff rigidity of Moscow at the Dst-minimum time is 0.3 GV because the cutoff rigidity of Moscow at quiet time is 2.10 GV and it decreases by 1.81. So the geomagnetic cutoff became less than the cutoff of the auroral zone station Apatity during quiet periods. Such considerable decreasing of cutoff rigidities during this strong geomagnetic storm can explain the aurora appearance at the middle latitudes which was observed at several stations (Ermolaev, 2005). 4. The behavior of the difference DRc ¼ DRsgs  DRef Fig. 2 shows the difference dRc ¼ DRsgs  DRef for all chosen stations (symbols are the same). Positive values dRc predominate mainly at times before the Dst-minimum for all stations except Hobart. Irregularities of the curves dRc are more noticeable during the main and recovery phase of the magnetic storm. Values dRc for Irkutsk and Moscow are positive at given times and at the Dst-minimum. Values dRc for Hobart is rather big during the Dstminimum. Sometimes geomagnetic cutoffs Ref are systematically slightly lower than Rsgs. Differences between DRsgs and DRef are in limits of ±0.7 GV with 90% of differences in limits of ±0.3 GV for all station except Moscow where 83% of differences are in the ±0.3 GV limits. 5. On the correlations of the experimental DRsgs and the theoretical DRef with parameters Dst, Bz, By, Nsw and Vsw It is very interesting to see how geomagnetic and interplanetary parameters are reflected in cutoff rigidity variations of DRsgs and DRef. The geomagnetic activity index Dst, the IMF Bz and By, the solar wind density Nsw, the solar wind velocity Vsw are geomagnetic and interplanetary input parameters of the Ts03 model.

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M.I. Tyasto et al. / Advances in Space Research 42 (2008) 1556–1563

0.8

δR (GV)

a

δR (GV)

b

0.4 0 -0.4 -0.8 0.8 0.4 0 -0.4 -0.8

δR (GV)

0.8

c

0.4 0 -0.4 -0.8

δR (GV)

0.8

d

0.4 0 -0.4 -0.8

δR (GV)

0.8

e

0.4 0 -0.4 -0.8

δR (GV)

0.8

f

0.4 0 -0.4 -0.8 20 0 -20 -40 -60 800 700 600 500 400

BZ

Dst (nT)

V (km/s)

N (sm-3)

6

0

12 18

18

0

6

12 18

19

0

6

12 18

20

0

6

12 18

21

6

0

12 18

22

0

6

12 18

23

0

6

12 18

24

0 -100 -200 -300 -400 -500 24 16 8 0 hours DAYS

Fig. 2. Differences dRc ¼ DRsgs  DRef for different stations (Tokyo, Alma-Ata, Rome, Irkutsk, Moscow and Hobart).

Coefficients of correlation between these parameters with the theoretical cutoff DRef and with the experimental DRsgs are shown in Tables 2 and 3, respectively, and correlation coefficients between DRsgs and DRef is shown also in Table 3.

From Table 2 can be seen that correlation coefficients of the theoretical DRef with the Dst-variation, Bz-component of the IMF and the solar wind density Nsw are rather high with highest for correlations with Dst for every station. IMF By and the solar wind velocity Vsw are weakly

M.I. Tyasto et al. / Advances in Space Research 42 (2008) 1556–1563 Table 2 Correlation coefficients of the DRef with geomagnetic and interplanetary parameters Station

Dst

Bz

By

Density Nsw

Velocity Vsw

Tokyo Alma-Ata Rome Irkutsk Moscow Hobart

0.98 0.98 0.98 0.98 0.98 0.96

0.66 0.68 0.71 0.72 0.69 0.71

0.18 0.18 0.18 0.16 0.14 0.08

0.68 0.68 0.67 0.69 0.69 0.68

0.06 0.05 0.05 0.04 0.04 0.05

Table 3 Correlation coefficients of the DRsgs with geomagnetic and interplanetary parameters and correlation coefficient K between DRsgs and DRef Station

Dst

Bz

By

Density Nsw

Velocity Vsw

K

Tokyo Alma-Ata Rome Irkutsk Moscow Hobart

0.66 0.81 0.82 0.87 0.86 0.78

0.26 0.35 0.37 0.49 0.55 0.48

0.43 0.48 0.47 0.38 0.29 0.16

0.37 0.46 0.47 0.56 0.56 0.55

0.08 0.15 0.16 0.14 0.12 0.17

0.63 0.77 0.78 0.84 0.85 0.76

reflected or nearly not reflected in DRef if to take into account correlation coefficient values. Correlation coefficients between experimental DRsgs and Dst, Bz and Nsw are less than the coefficients in previous case but sizeable enough (see Table 3). As before, the correlation of the experimental DRsgs with Dst is highest and with Vsw is lowest respectively. Component IMF By has more close relation with the experimental DRsgs than with the theoretical DRef. The Birkeland currents of the Ts03 model are represented to be symmetric with respect to the noon-midnight plane. This restriction did not allow the author (Tsyganenko, 2002a,b; Tsyganenko et al., 2003) to take into account the observed longitudinal rotation of the Region 1 Birkeland currents during periods with strong IMF By. More high value of correlation IMF By with the experimental D Rsgs than with the theoretical DRef can indicate that a down-dusk asymmetry of the Ts03 model magnetosphere is approximated not enough exactly. 6. On the relations between the experimental DRsgs and the theoretical DRef for different CR stations It is interesting to see also how DRef and DRsgs correspond each other. Fig. 3 displays scatter plots of the theoretical DRef against the experimental DRsgs for different CR stations and the approximating regression lines. In Fig. 3 correlation coefficients are rather high and equal 0.63, 0.77, 0.78, 0.84, 0.84 and 0.76 for Tokyo, Alma-Ata, Rome, Irkutsk, Moscow and Hobart correspondingly. It is clearly seen that the slopes of the best linear fit to the scatter plots depend on the station latitude to be minimum for the low latitude station Tokyo and maximum for the middle latitude station Hobart. We see that

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the theoretical cutoffs at all station except Hobart are less ‘‘sensitive’’ to interplanetary parameter variations than the experimental ones. One of the reasons of this effect may be the following. Cosmic ray particles go throughout the whole magnetosphere and are influenced by entire magnetospheric magnetic field before reaching Earth’s surface. So the experimental cutoffs reflect the influence of magnetosphere magnetic field. However, the T03 model (Tsyganenko, 2002a,b; Tsyganenko et al., 2003) describes the magnetic field of the inner magnetosphere in region R < 15RE (Earth’s radius) and, consequently, the magnetic field influence of the far magnetosphere tail is not reflected in the theoretical cutoffs DRef. 7. Main results and conclusion The analysis shows that time variations of the theoretical CR cutoff rigidity DRef and the experimental one DRsgs determined by two quite different methods are similar in general during the considered geomagnetic storm. Sufficiently high coefficients of correlation between the experimental DRsgs and the theoretical cutoff rigidities (0.63–0.84) indicate that the T03 model reflects at first approximation main features of the very distorted real magnetosphere and CR data give the possibility to check the magnetosphere’s model. CR cutoff rigidities obtained by both methods decrease substantially during the Dst-minimum so that, for example, the geomagnetic cutoff at Moscow became less than the cutoff rigidity at the auroral zone station Apatity during quiet periods. It explains the appearance of the aurora at middle latitudes during this geomagnetic storm. Main part of cutoff rigidity variations are connected with the Dst-variation. The influence of the solar wind density Nsw and IMF Bz-component in DRsgs is less noticeable than in DRef but the influence of By-component of IMF on the contrary is seen in DRsgs more clear. The Birkeland currents of the Ts03 model are represented to be symmetric with respect to the noon-midnight plane and this restriction did not allow the author (Tsyganenko, 2002a,b; Tsyganenko et al., 2003) to take into account the observed longitudinal rotation of the Region 1 Birkeland currents during periods with strong IMF By. More high value of correlation IMF By with the experimental DRsgs than with the theoretical DRef can indicate that a down-dusk asymmetry of the Ts03 model magnetosphere is approximated not enough exactly. The dependence of DRef on DRsgs demonstrates that the slopes of the best linear fit to the scatter plots increase with decreasing the station latitude to be maximum at Hobart. So the theoretical cutoffs at all station except Hobart are less ‘‘sensitive’’ to interplanetary parameter variations than the experimental ones. It can be related with that the experimental cutoff rigidities DRsgs reflect the influence of whole magnetospheric magnetic field because cosmic ray particles run throughout the magnetosphere. However, the T03 model describes the magnetic field of the inner

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M.I. Tyasto et al. / Advances in Space Research 42 (2008) 1556–1563 0.2

0.2

0.1

0

-0.1

-0.2

-0.2

-0.4

Δ RE F_AA

Δ RE F_T ok

0

-0.3 -0.4 -0.5 -0.6

-0.6 -0.8 -1

-0.7 -1.2

y = 0,4464x - 0,0559

-0.8 -0.9

y = 0,6694x - 0,0957

-1.4

-1.5

-1

-0.5

0

0.5

-1.5

-1

Δ RS GS _To k

0.2

0.2

0

0

0.5

-0.4 Δ RE F_Ir

-0.4 Δ RE F_R

0

-0.2

-0.2

-0.6 -0.8 -1

-0.6 -0.8 -1 -1.2

-1.2

-1.4

-1.4

y = 0,7937x - 0,1231

-1.6

y = 0,7541x - 0,0875

-1.6 -1.5

-1.8 -1

-0.5

0

-2

0.5

-1.5

Δ RS GS _R

-1

-0.5

0

0.5

Δ RS GS _Ir

0.5

0.5

0

0

-0.5

-0.5

Δ RE F_H

Δ RE F_M

-0.5 Δ RS GS _AA

-1

-1.5

-1

-1.5 y = 1,0989x + 0,0247

y = 0,8594x - 0,134

-2

-2 -2

-1.5

-1 ΔR

-0.5 S GS

0

0.5

_M

-1.5

-1

-0.5

0

0.5

Δ RS GS _H

Fig. 3. Scatter plots of the DRef against the DRsgs with regression lines for different CR stations: Tokyo, Alma-Ata, Rome, Irkutsk, Moscow and Hobart.

magnetosphere in region R < 15 RE and, consequently, the magnetic field influence of the far magnetosphere tail is not reflected in the theoretical cutoffs DRef. References Dorman, L.I. Cosmic Rays: Variations and Space Exploration. NorthHolland Publ. Co., Amsterdam, 1974. Dorman, L.I. Cosmic Rays in the Earth’s Atmosphere and Underground. Kluwer Academic Publishers, Dordrecht/Boston/London, 2004.

Dorman L.I., Cosmic Rays in Magnetospheres of the Earth and Other Planets. Springer, Netherlands, 2007, in press. Dorman L.I., Smirnov, V.S., Tyasto, M.I. Cosmic Rays in the Earth’s Magnetic Field. Physmatgiz, Moscow, 1971 (in Russian), translation in English: Cosmic Rays in the Earth’s Magnetic Field (L.I. Dorman, V.S. Smirnov, M.I. Tyasto, Trans.) [Washington] National Aeronautics and Space Administration. For sale by the National Technical Information Service. Springfield, VA, 1973. Dorman, L.I., Gushchina, R.T., Smart, D.F., Shea, M.A. Effective CutOff Rigidities of Cosmic Rays. Nauka, Moscow, in Russian and in English, 1972.

M.I. Tyasto et al. / Advances in Space Research 42 (2008) 1556–1563 Dvornikov, V., Sdobnov, V. Variation in the rigidity spectrum and anisotropy of cosmic rays at the period of Forbush effect on 12–25 July 2000. Intern. J. Geomagn. Aeron. 3 (1), 1, 2002. Ermolaev, Yu.I. et al. Solar and heliospheric disturbances leading to the strong magnetic storm in 20 November, 2003. Geomagn. Aeron. 45 (1), 23, 2005. McCracken, K.G., Rao, U.R., Shea, M.A. The trajectories of cosmic rays in a high degree simulation of the geomagnetic field, M.I.T. Tech. Rep. 77, Lab. for Nucl. Sci. Eng. Mass. Inst. Technol. Cambridge, 1962. McCracken, K.G., Rao, V.R., Fowler, B.C., Shea, M.A., Smart, D.F. Cosmic ray tables (asymptotic directions, variational coefficients and cut-off rigidities), IQSY Instruction Manual. No. 10, London, 1965. Shea, M.A., Smart, D.F. Asymptotic directions and vertical cutoff rigidities for selected cosmic ray stations as calculated using the Finch and Leaton geomagnetic field model, Report AFCRL-TR-75-0177,

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Environ. Res. Pap. No. 502, Air Force Cambridge Res. Lab. Hanscom AFB, Massachusetts, 100 p., 1975. Shea, M.A., Smart, D.F., Carmichael, H., Summary of Cutoff Rigidities Calculated with the International Geomagnetic Reference Field for Various Epochs, Rep. AFGL-TR-76-0115, Environ. Res. Pap. 561, Air Force Geophys. Lab., Bedford, Mass., 1976. Smart, D.F., Shea, M.A., Fluckiger, E. Magnetospheric models and trajectory computations. Space Sci. Rev. 93, 271–298, 2000. Tsyganenko, N.A. A model of the near magnetosphere with a dawn-dusk asymmetry 1. Mathematical structure. J. Geophys. Res. 107 (A8), 1179, doi:10.1029/2001JA00021, SMP 12-1, 2002a. Tsyganenko, N.A. A model of the near magnetosphere with a down-dusk asymmetry: 2. Parameterization and fitting to observations. J. Geophys. Res. 107 (A8), 1176, doi:10.1029/2001JA000220, 2002b. Tsyganenko, N.A., Singer, H.J., Kasper, J.C. Storm-time distortion of the inner magnetosphere: how severe can it get? J. Geophys. Res. 108 (A5), 1209, doi:10.1029/2002JA009808, 2003.