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S... . • -J 4 .6;"a.•.p%...inZj._.s.,. -
AFCRL-TR-7S-0185ENVIROKMEHTAL RtE$EARCH PAPERS, 1O. 503
C'q Tables of Asymptotic Directions and Vertical- Cutoff Rigidities for a Five Degree by Fifteen
C Degree World Grid as Calculated Using the"International Geomagnetic Re terence Field forEpoch 1975.0
M. A. SHEA0. F. iMART
3 April 1975 JUL 2IM
B
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TABLES OF ASYMPTOTIC DIRECTIONS AND VERTICALCUTOFF RIGIDITIES FOR A FIVE DEGREE BY 1-IrTEEN DEGRE ___
WORLD GRID AS CALCULATED USING TIHE I1NtERNATIONAL 6 PERAORMING ONG. RE,-ORT I•juSERGEOMAGNETIC REFERENCE FIELD FOR EP&H 1975.0, ERP No. 503
S.. .. O *......... -- *-~*-.. •6 CONTRACT OR GR4ANT NUM9•ER()
/ M. A.:Shea -"" "
. F. /Smart A, ,N AND•
Z. - a5OlM4U A~il T,01 4AMCAN0 CL 0 P R 'i_-
Air Force Cambridge Research Laboratories (PHE) . " foRK UNT NUMBERSHanscorm AFB ,, 860007W 681311Massachusetts 01731 . 6l102FI I 'ONTROLLING OFFICE NAME AND ADORE-S
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13 KEY WORDS Con•linue ar tgovers. side it n.e..,.•y and Identify by block nub.er)
Cosmic radiationVertical cutoff rigiditiesGeo agnetic field modelsAsy ptotic directions of approach
20 A( RACT oCnre•erIu s, on e side if nece.sary and Identify by block nu•ber)
Tables of asymptotic directions and vertical cutoff rigidities for a fivedegree by fifteen degree world grid are presented. These values were ob-tained by the integration of vertically incident cosniic-ray trajectoriesthrough the International Geomagnetic Reference Field with time derivativesapplied so that the coefficients form a quiescent model o', the geomagneticfield for Epoch 1975.0. The table of asymptotic directions contains theasymptotic latitudes and longitudes which specify the direction of approach.,
DDJ 1473 -101 1 OP01 IS IS OSSOLETE UnclassifiedSECURITY CLASSIFICATION OF THIS PAGE (WhW D.I.•"v.d)'
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20. Abstract (Cont)
from interplanetary space. of a cosmic-ray particle. These asymptoticdirections are given for each grid point for specific rigidities above themain cone cutoff rigidity. The table of vertical cutoff rigidities containsthe geographic coordinates and L value of each location together with themain cone cutoff rigidity, the Stirmer cutoff rigidity, and the effectivecutoff rigidity. ,,
Unclas!s:ifiedSECURITY CLASSIFICATION OF THIS PAGE(Whýn Doee Estertod)
Preface
This is the fifth in a set of reports presenting, in tabular form, the iesults
of utilizing high speed digital computers to calculate, by the trajectory-tracing
method, the motion of charged partic'es in a mathematical simulation of the
internal 2eornagnetic fieli. The pr ;pal results of these ertensive calculations,
which have been made over the past 15 years. have beer the determination of the
geomagnetic cutoff rigidity for soecific locations an the earth and the deternina-
tion of the asymptotic directions of the cosmic rays arriving at these locations
frcm specific directions,
It is the intent of the authors to assemble these values in a group of reports
so that they will be available to the scientific community both for analysis of
scientific data and for comparison with other similar values that may become
available by more sophisticated methods. The first four reports in this set
presented various tables of asymptotic directions and cutoff rigidities for Epochs
1955 and 1960. This report contains *-bles of trajectory-derived asymptotic
directions and vertical cutoff rigidities for a world grid as calculated employing
a geomagnetic field model formed by the international Ceomagnetic Reference
Field coefficients with time derim atives appropriate for Epoch 1975. 0.
3
AlI FOICD (I) JILM 1), 1975--475
Contents
1. :NTRODUCTION 7
2. COSMIC-RAY ASYMPTO'IIC DIRECTIONS 8
3. VERTICAL CUTOFF RIGIDITIES 14
REFERENCES 17
APPENDMX A: Table of Asymptotic Dir-ctions for the World Gridas Calculated Utilizing the International Geo-magnetic Reference Field far Epoch 1975.0 21
APPENDIX B: Table of Vex tical Cutoff Rigidities for a World GridCalculated Utilizing the International GeomnagneticReference Field for Epoch 1975.0 155
Illustrations
1. Illustration of the Definition of the AsymptoticDirection of Approach 12
5
BLn-o FTU
Tables of Asymptotic Directions and Vertical Cutoff
Rigidities for a Five Degree by Fifteen DegreeWorld Grid as Calculated Using the International
Geomagnetic Reference Field for Epoch 1975.0
I. INTR(ŽI)CTIO%
The knowledge of the asymptotic direction of a cosinic-ray particle and bhe
cutoff rigidity of an observation point on the earth is often npcessarv and some-
times essential for a meaningful analysis of various problems in cosmic-ray
physics. Tie asymptotic direction of a pfrticle having a specific rigidity is the
direction in interplanetary space from which the particle comes prior to its inter-
action with the earth's magnetic fields, and is a primary means of relating cosmic-
ray intensity variations at the ea-"th with the direction of these particles in inter-
planetary space. The cutoff rigidity of a specific location on the earth is the
rigidity below -which cosmic rays are inac2cssible to that location.2' 3
Since both the asymptotic directior.s and the cutoff rigidities are a function of
geographic location, zenith and azimuthal angles. and the calculation of these
values involves considerable computational time, it is impractical to determine
(Received for publication 2 April 1975)
I. McCracken, K.G. (1962) The cosmic-ray flare effect. 1. Soene new methods3f analysis J. Geophxc, lies. 67:423.
2. Stbrmer, C. (1930) Perlodische Elektronenbahen im Fi-'ie eines Elexnentar-magnetedi und Lire Anwendung auf Briiehes Modellversucne und auf Eschen-hagens Elementarwellen des Erdxnagnetismus, Z.Astrophys. 1:237 -274.
3. Rossi, B. (1940) System of units for nuclear and cosmic-ray phenomena,Phys.n N t. 5T:G6G.
PRECDII PACE BWLAN-MW FUMED
lit, n-! }c-o) it 0Th- atnf lirectrons on the earth.- . v, -, - " .f Ai " <'f, (' : - i n :u ,t rs has allow4'edw a
I -1 1 A. r 1 t' ,- A I al -,Il it i n• to¶ h - r ,a,1e , \ ith ' 'st of thuse calculatior.n
ix.:'. : i r fir l s:i Ž¶:Ivs at- f''•Ži f-- rrax- satirns.
-- r !n' .2. !r ' mute tioe as,-,'n.ptotic,
-',- ." . 1 .... ,ral ,-an2
*.. .. V .* ¼-*.'h-, " t,'l•' the re, ac to r
" r t . i j '. , l" , n 1! 11 eot :):-i-, ar'il% '1' a r7 ie , "' 4s -5 ) analysI 's. 13-! 20
! .. '" I' .',A" " ork a have :: 711arlre I a c "or oisite listinn of all locations for
, 'a "i .t - :T .- , :7r;'r- e r)f this set of nublications
"iistzn-s of as tptotic directions and cutoff rigid-
"If, . in.. 1 a-rn t tr e "Icrv utra -inc- :cethod. The first four publications
-af'-i -e resultl on ,he extensive cutoff rii-dit2t calculations
-- te . an I .E'attn F- o(- h 1155. 0) intermnal eoot>acnet ic field
Z- rel, , ;.ell a, a s-.-atler Set of results obtained usinr- the Jensen and Cain
* 1 6).01 e, ,,",ef..cients. 1 fT ii rep-rt ontains the asv;>- ptotic directions.
ý, n ienio e r-', 0. .a ria cci " ith .:,rd Dn ints 5 ie1 -,rees in latitude and 15
- 'to .- as al-ulat, I ulili in•2 the International ('eontacnetic Refer-
e'::,, '•i'' ii , *'flt t-:,.: - ier;vat',-'e. aipl i ( so that the coefficients for the
fe' i -o'nel are ainro.-riate for a V-75.0 1-hooch. These asymptotic directions
are jive.- in Appenixx A. with a su:x.rnarv of the vertical cutoff rizidities for each
of the wor",t '"ri- hlcatiors j'",en in A.ppendix B.
2. COSMI(R %I XS)Mr•TOTT: l)IRECTIONS
T-n the "ariq,,- t.ujies of COsmic radiation, it is often necessary to know the
-lirection (! oartcle nmotion in the interplanetary magnetic field near the earth
in order I.. relate the intensity observations at a specific location, azimuth, and
zenith: a.. ile on the earth to an interplanetary flux. With this purpose innmind, the
concept cf asymptotic directions of approach has been developed over the years by
researcoers such as Nlalmofors, 28 Brunbery. 29,30 Brunbera and Dattoer3 and
"McCracken. I The asym-totic direction of approach defines the direction of a
velocitv vcctor in Leocenicic coordinates such that the direction of this vector is
narallel to a radial fromn the earth's center. A ,ctarate numerical calculations of
the as'mnt-tiric direction: of approach have long been advocated, 2&.32,33 however,
the earlier work was hninted to dipolar field models because of the complexity
tDue to the larae number of references in the above text, refer to the list ofreferences at the end of this report for spe•ific "if in..tin on reference-% 4 thru33.
8
of the Prchlern , he a 'Ixe't of hii h-snered diatital con-.puters mari it
nosshble t,) consitr;ct relatively accurte athe-atirai -7.odels of the cenn..•'netlc
field an i to lrteirate the equation of chanced particle rotion in these magnetic
field r..odels. The standard rr-etho I for representin - the r: agnetic field is to
-jefi..e a .a., .- te ntia f!ury, tion t' in tern. -, f -a s •oner, cal harr:.onic exnansion.
In sn•,crr ,,. ',Sri P:a,.ue t:rim tri:;:n,::: it:
n-1I0re (:0 cos . - n sin rn.o I (cos
n7O trio n n0n, R
where j" Sr 7 are the Gauss efcnt,1-'.. (ens O) are- the paitially nor-n nr;alize'I lje e' r functions, and a is the a-,'erage radius of the earth. Thisrmezhod i -,rie- a-e'uately by Chapm..an and Bartles 136 is the method
c.--comronl. used fn-r constructin• models of the ceonmaznetic field. The n_.ainetic
fieldi verTr, I, -:.a" bhe ,7nonp'red t any r."oint where the potential function is valid
by obtainrinic !he derivati.-es of -he potential f ,unction:
Sg = "=I"•r~e o)6r s r
'¾ -I 'U (r,8,c)
£3 -I •U frO8.6)B6 - rsn '•r
The equation for charged particle motion in a magnetic field, in vector form,
is
In spherical coordinates this equation expands to the following differential
equations:
34. 1ory. F-.F. Vi,56'5 S'elected cosmic-ray orbits in the earth's magnetic field.
Phvs. HRev. l03:l1OW.
35. Lust, R. I 1957) impac' zones for solar cosmic-ray particles. Phys. Rev. 105:1827.
36. Chapman, S., and BIrtels, 1. (19511 Geomragnetisrn Vol. 2, Oxford Univ.Press, London.
• l l ll l . . .... • • lll9
"r e "9~ 6
. fl 6n - ve l~ • . .. 7
2:" M 0 - r r tran
"r-- n-•-T - -9 -- - - r -c
Wj ET ½6 Ar unw^here.
is Ve
In these equations we use the followinoc notatinn-
B -.-osition vector of a cosnmic-ray particle
r radial distance froin the center of the earth
B c o-l at itude
6longitude .-easured eastward from the Greenwich mneridian
v ntagnitude of particle velocity
Vr. vesv velocity conmponents in r, 6, and 6 directions
c velocity of l ight
n" inertial mass of cosmic-ray particle, mc z c -
w-here m is the rest mass o
B vector magnetic induction in Gauss
e cosmic-ray charge in esu.
Since the general equation of particle motion in a magnetic field does not
have a solution in closed form, it is necessary to emnploy numerical methods to
detcr-mine th'P trajectory of a charged particle in the geomagnetic field. Withthe advent of high-speed digital computers, it becamee)hsihI7le toý obtain an
10
a T Iratr . . i. , -'i r a s ,pecifie, 'i fiel, n-tiel. riqidity, and
.; ,. . . -;rljfjaton of the TR'in" ., Kutta iterationaiii. , -ecarne the standard method, of tracing
S -: . : ". ,- .a ieti- .field. In this prncess• a know-
!e t.'e-ihn t:c ;,:t<-. o i . -or-inrates of one nf)int on the traiectorw is
u i,,:.'. ¼, , , it . erential equations, to give the coordinates of a
.; .r:•r•t. , r ', .t,,r , lReneate-I an.-li-cation ..-C. suifficient n'ints
r: ,i i "' .- 4-:,. we -;t:1ize **e fas t tat the orbit of a ne-a-
t v tl : t l l l cutwarj fIro:-. the earth frItor- a snecific location
-to 1.re:, " I .e -itr-hi of a .,sitt'c narti!-'e of equal rigidit:;
ý', 1.- .* j t. atelv arriving at the same location in the sam-e
e"et' i that use the lifferential equatior. )f mrotion to
, -- a.t:l'me : the IeIon agne! c field bL using the
"-, .. u lave been nuhishe.1. ý-e'.eral times and will not
,-., !o -4ese '
a- p tie ;* w• ,o ":regtrate the trajectorv utiliz.ng the rr. iinim num ber
o -a'cul irn This iv.onlves a co-mpromise between choosing t'"e 1aro•e--t step
-ze t,-, ':.'im-ize co" puter t
ir e anti thpe necess;.tv of using a small step size in
*t}e iv e-n:-t,'nress to keen the error within acceptable lit.its. For these
- aic.-',a1;n r,, the 1/75 world --ril, A we replace-i the library of step sizes Provid-38
ed in the original McCracken progran: with a computed step size that is between
1 :25 anti I ,50 of the distance traveled during one -,yration. In a uniform, field
the time tequired for one gvi' ation is
33.33 P 2-
3T. il•i. - 1951 .'i Arocess for the step-by-step integration of differentialeont.ionsg in. an autom-atic digit-A computing machine. Proc. CambridgePhil. Soc. 47.O6.
38. acCr"racken. K. G. . Rao, 1. R.. , .nd Shea. L .A. (1962" The trajectories ofcosmic rays in a high degree simulation of the geomagnetic field. M. I. T.Tech. Rept No. 77, NYO-2670.
39. S;hea, M. \., `-mart. I). F. . and McCracken. K. G. !19651 A study of verti-calyv incident cosmic-ray traiectories using sixth-degree simulations ofthe geomagnetic field. AFCRI. Environmental Research Papers No. 141.AFC'RL-65-705.
40. Shea, M,. A. , Smart, D. F. . lcCracken, K. G. . and Rao. U. R. (1968) Sup-plernent to IQSY Instruction Manual No. 10. cosmic ray tables - asymp-totic directions, variational coefficients and cutoff rigidities, AFCRL.pSec. Reerts No. 71. AFCRL-68-0030.
11
A nere P' is; in (xts fA- i iý in nimt , 'n r' sec, an-H i~ sý tha. ratio between the
na -tli le an-I th le e ,)f i hi. if th.- list ,ance rave 'led tir ~j': a uvra -
hon is :tvidelt :,to 50 steps, trier, the stetp size, II, m urits of tit: , ;, a xi-• ::matelv
-•e t.he ear-tL; sna 1'. t' fie! s : n nt ' , w re ,):- nute'i the step lenrtlI
.or each tlunge Kutta iteration. sten. At each sae-n the ofelct-: the )article
,was ,hecke and if thef qurr-nt value n! p iiere,i Cr, the initial value by nore
tha;n I • 10 the interatxn,i was. ieclare. u-.a,-, eatable an:, the traiector-v re-
c.xv ateI witn -nPt C nretinus ite: aize - II ,i. in haT., th~e ,i !work9,befre the existence of the internlanetarv rT-aznetic
:i tl was the orýt' a r, nar-et pte was traced thr'ouah the eeo-
n. a ýnetic ftelI to a listan.e s':-xentv' ar fr,. the earth (selected as 25 earth
ra 1-i by M-(ra-ckeni that t'., effects of the Ceormaznetic fieli on the orbit became
essenti llx inziznn.ficant. At this noint the direction of !notion of the particle was
calculate I in tern-s of tht 4euocentric -oordmnate svstem-. as illustrated in Figare 1.
The aisLvn totic lattude , A, * s v ven by
Ssin v - t cos B
vn (ve CosS v sin C)2]} r
VectorAA
Stationi Meriiofl
%_ S'oTro I ccto't
Stttatoon LLtituo
SStafion Longitude
Figure 1. fllustratictn of the Definition of the AsymptoticDirection of Approach
12
and the "asy-nrptotic longitude'. 4, bv
L'e - arc tan ve cos e'v; sine
In. i2 Vt ',fice c:urrent know..le.dge o. the magnetosphere and the rrcanetosnheric
tail, the original definition of the asvmptotic direction of anuproach has been some-
what modified to mean the direction (in geocentric coordinates) of the velocity
vector of the cosmic ray when it penetrates the mragnetopause or at some arbitrary
distance down the n-a•netospheric tail, For cosmic-ray trajectories with rigid-
ities far above the cutoff rigidity, the computed asymptotic directions of approach
at the nosition of the mlagnetopause obtained using, a nmagr'etospheric field model
andI those calculated using oniy an internal field are verN innilar. For cosmic-
ray traiectories having rigidities 2 to 3 CA: above the cutoff rigidity (or - 5 GV for
the very high polar latitudes), significant differences occur between the asymptotic
directions com:puted using only internal fields and those computed using magneto-
spheric m,:odels containing both the magnetic fields of internal origin and those due41
to currents on the magnetopause and the magnetospheric tail. Near the cosmic-
ray cutoff riLidity, the asymnptotic directions of approach of cosmic ravs with
rigidities of only a few GTV computed with an internal field, have relatively little
correspor~dnnce to those calculated using rnagnetospheric models.
The assmptotic directions contained in Appendix A of this report are intended
as a reference standard for Epoch 1975.0. These calculations were made using
the international ,eoniagnnetic Reference Field t model of the quiescent internal
geomagZnetic field with the time derivatives applied such that the coefficients were
appropriate for describing the geomagzentic field for Epoch 1975.0. The calcula-
tions were initiated at 20 kin above the surface of an oblate earth where the radius
of the earth r(initiaD is guiven by
r II bK -1 1cosA]k 1/ - 20.0~G(nitial) -ao s
In this equation b is the minimum (polar) radius of the earth (6356. 9 kin). E is the
eccentricity (E 2 6.725 X 10-3) and a is the average radius of the earth (6371. 2 kin).
Calculations were continued to a distance of 25 earth radii, at which time the
asymptotic directions of approach were computed. For a number of cosmic-ray
41. Gall, Ruth, Iimenez, .1., and ()rozco, A. (1969) Directions of approach ofc.mi. rn. fnr high .t e stations, J. G .E.,-s. lRes. 74:352; .
13
stutiles these asym ptotic directitos are quite useful ,n as-ertainirn2 the amount
of 'ecn:at_$netic hetielint_ -5m ravys un'lerito be fore arricvinn at a specific location
on the earth. ' hesse tables are li•nite'l to the vertical ,'trection with 20 ;V as the
hii'hest rigiditv contained, Except for locations near the equator. hidher rigi -
litto:; A',hav- 5 aym-toti, t(reci,,r s for verticallv incident particles that approach
tIw x- t'c r;-l:al,
Ili ,ur research otilizung : .asvc:.t,)tit dlirectitnw ,f aiitoa-., for cos,:.ic-rav
studies, we fiuiI that tli' asvn:,ptoit i<. -i etions f v'. rtitallv inicijtcýnt particles
are a reasonable approxir:.antiun t, the entire aivsn!ptotic -one of aceptance ex-
.e!,t f't.' i itlitdsC a > 0 (V above the coScic-ra's cutoff rig idity of a specific
location. At rigidities where the :yvro radius of the particle is small compared
to the cad.ius of the earth, the trajectories of all other azimuths and zeniths
anctes are siu-:Itar to the vertical tralectory with a phase angle displacement
which accounts for the initial pitch angle. TI is is sometimes referred to as the,nc11s ~rteffect of the geor-magnetic field..
The asvniptotic directions o)f approach presented in Appendix A are at stand-
ardized intervals of 1 (A' from a magnetic riglidity of either 10 or 20 GV (depen-
dent upon location) down to near the mrain cone cutoff rigidity. Approxim-.ately
0. 5 6% above the r-ain :one cutoff rigidity, the ,eon-aznctic bending is more
extensive and asyvt;ntotic directions are given in 0. 1 (A intervals. Very near
the main cone cutoff rigidity, asymptotic d:rections are given at 0.01 CV intervals.
This report does not include the asynmptotic directions of the allowed trajectories
in the penumbra; however, the results cf the trajectory calculations at 0.01 CV
intervals through the penumbra, including the asymptotic directions for the
allowed rigidities, are contained on magnetic tape available through the World
Data Centers.
3. NERTI'%L CI.ITOFF RCGII)I'UI.S
Cosmic ray cutoff rigidities are obtained by determining the "fate" (whether
an individual rigidity has a trajectory accessible from infinity or is forbidden)
of cosmic-r', trajectories over the entire rigidity spectrum. Starting at a
rigid.ity hig' sbove the highest possible cutoff, cosmic-ray trajectories are cal-
culated at discrete intervals, decreasing in rigidity until we are satisfied that
the cutoff has been reachtd. As the calculations progress down through the
rigidity spectrum-., the results change from the easily allowed orbits to a complex
structure of allowed, forbidden, and quasi-trapped orbits (loosely called penumbra).and finally to a set of rigidities where the trajectories all intersect the solid
ea-th, As a res,..t of thes- tvne of traiectory calculations, we can define three
14
distinct rigidities: the main cone cutoff. P IM), above which all rigidities are
allowed, the Stbrmer cutoff, P(S), below which all rigidities are forbidden, and
an effective cutoff rigridity, PC, which we have defined as[P (MbDPC = P (M) iP
IP fIs) ]
thus allowing, for the opacity of the -)enumbra. A letaile-1 definition of these
effective cutoffs has been pmblished, and the effect of different spectral slopes
and coupiingc functions has been investigated by Dorman et al.42 In the penumbral
reýion of allowed and forbidden orbits., calculations were mnade at discrete 0.01
CV intervals in an effort to define the structure within this region.
There is an inherent hazard in using the trajectorv-tracing method to deter-
mine the structure within the penumbra in that some of the allowed rigidities
may not be found due to a systematic lin.itation resultine from performing the
trajectory calculations at discrete intervals. In our work -we have made the
specific assumption that if a trajectory is accessible (or forbidden) at a rigidity
Pi, then this result is anplicablt over the intervals between P.i AP/2 where AP
is the size of the rigidity interval. However, we recog, ize that the penumbra
consists of a very complex structure of allowed and forbidden bands, and this
structure is most certainly finer than the 0.01 GV intervals employed. Consequently,
calculations down the rigidity spectrum at finite discrete intervals might miss
some of the allowed rigidities for a specific location and direction.
It is.difficult to be assured that the lowest possible rigidity which may be
allowed for a given loction in a specific direction (that is, the Starmer cutoff
value) has been preciselv determined. We have found that for complex penumbra,
extending the calculations to lower rigidities at smaller rigidity intervals, often
results in an additional allowed rigidity considerably below the previous last
allowed rigidity. This problem and the degree of confidence to which we feel
we have located the lowest possible allowed rigidity is discussed in detail by
Shea and Smart,
42. Dorman, I._ I., Gushchina, H. T. , Shea, M.A., and Smart, ), F. 11972)Cosmic Rays Fffective Cutoff Rigidities, Publishing House "Nauka",Moscow, 1.SS1R.
43. Shea, M. A., and Smart, 1) F. (1974) Tables of asymptotic directions, cut-off rigidities, and reentrant albedo calculations for Palestine. Dallas,and Midland. Texas, AFCRL Special Reports No. j75, AFCRL-TR-74-0159.
S~15
Tie effect of the currents in the ::,agnetosphere and the magnetospheric tail
were not considlered in these calculations. The work of Gall etal, 44Smart et al,45
46ari Žc.art and >.itiea have shown that the inclusion of these currents and the
magnetospheric tail results in a sienificant lowering of the cutoff values at loca-
tinrs ffir which the internal fielI vertical cutoff rigidity would be < 0.5 CV', and
the-ef ;,-, :are -4houldi he taken when using these tables for these locations. The
effe:t- f these ei-e:nal -,)urice- would result in slight decreases (-0.1 GV) for
tI';-ationsw hOlere the vertica! cutoff rioidity calculated with the internal veornaz -
netic field ,nodet is between 0,5 and 2.0 GV. Little if any effect is found for
locations where the vertical cutoff rigidity is higher than 2.0 G-V.47
Vertical c, .ff risiltities for each of the world grid locations are given in
Appendix L ,';ach grid pocnt is identified by the geographic coordinates, together
with the L vI-ue calculated using the same field model utilized in the trajectory -
tracing proor an; ithat is, the IGRF 2 model). All L values were calculated for
an altitude cf 20 kin, consistent with the initializin, of the trajectory-tracing
technique. The various cutoff rigidity data for each location are in order of main
cone cutoff rigidity, P((MI, the St6rmer cutoff rigidity, P(S), the width of the
penumbra, and the effective cutoff rigidity, PC. All rigidities are in GV.
44. 1,11. Rt.. .- menez, ,nd (.,rncho, L. 196.) l Arrival of low-energy cos-mnic revs via the m-agnetospheric tail, .1, Geophvs. lIes. 73: 1953.
45. Smart, D. F., Shea, M. A.. and Gall, Ruth 119691 The daily variation oftraiectorv-derived high-latitude cutoff rigidities in a model magneto-sphere, .. Geophvs. Res. 74:473l.
46. Smart. IJ. F. . and Shea, M. A. (1972) Doily variation of electron and protongeomagnetic cutoffs calculated for Fort Churchill, Canada. J.Geophys.lRes. 77:4595.
47. Shea, M.A. and Smart, D. F. (1971) Calculation of the magnitude of thedaily variation of vertical cutoff rigidities and associated changes in theneutron monitor response for selected North American neutron monitors.12th International Conf. on Cosmic Rays, Hobart, Conference Papers-niversit- of Tasmania) 3:854.
*The abbreviations for the main cone cutoff rigidity, the Stbrmer cutoff rigidity,and the effective cutoff rigidity in this paper have been changed slightly to permitcomputerized headings in the tables. Earlier worn used the abbreviations Pm,Ps, and Pc for the main cone, Stbrmer, and effective cutoff rigidities respectively.
16
References
1. McCracken, K.G. t19621 The cosmic-ray flare effect. 1. Some new methodsof analysis, J.Geophvs. Hes. 67:423.
2. Stbrnrer, C. (1930) Periodische Elektronenbahen im Felde eines Elementar-mnaeneten und ihre Anwendungý auf Brdches Modellversuche und auf Eschen-hagens Elementarwellen des Erdrnagnetismus, Z. Astrophbs. 1:237 -274.
3. ikossi, B. (1940) System of units for nuclear and cosmic-ray phenomena,Phvs. Rev. 57:660.
4. Freon, A., and McCracken, K.G. (1962) A note on the vertical cutoff rigid-ities of cosmic rays in the geomagnetic field, J. Geophvs. Res. 67:888.
5. Kodama, NI. (1965) Trajectory-computed vertical threshold rigidities for85 neutron monitor stations, Can. J. Phys. 43:836.
6. Kondo, I., and Kodama, %I. (1965) Geographic distribution of vertical cos-mic ray threshold rigidities. Proc. International Conf. Cosmic flays 1:558.
7. McCracken, K.G., Rao, U.TR.. Fowler, B.C.. Shea, NIA., and Smart. D.F.(1965) Cosmic ray tables (asymptotic directions. variational coefficientsand cutoff rigidities), IQSY Instruction Manual No. 10.
8. Shea, M.A. , Smart, D. F., and McCracken. K.G. (1965) A study of verticalcutoff rigidities using sixth degree simulations of the geomagnetic field,J. Geophys. Res. 70:4 117.
9. Shea, I. A., Smart, D. F., and McCall, John H. (1968) A five degree byfifteen degree world grid of trajectorv-determined vertical cutoff rigid-ities, Can..i. Phlys. 46:SI098.
10. Shea, M.A.. and Smart, D.F. (1966) Vertical cutoff rigidities in the southAtlantic. Space Research VI, ed. R. L. Smith-Rose. Spartan Books,Washington, n. 7 :177
11. Shea. M. A., and Smart, D. F. (1967) Worldwide trajectory-derived cutoffrigidities and their application to experimental measurements, J. Geophys.Res. 72:2021.
17
References
12. Shea, M. A. 1 19469) A crumpir IOn s o1 0 heoretIci.I ;,ad exper`:,rent :1 ('osmý.i- r;- .'.equ tors. T. Geopbvs. Ties. 74:2407.
t1. Kennev, 1. - u g( er.. . Shea, x .. z. , rid n.1 rt. I). I-, E x',pe.. I p nremen-hi evilu•tion of virio rigidit.: molll I.Geophvs. e. 71ties
14. (a:rr:ich el, Ht I .ra " "h. '.A. SV ,- !. \ .. :La ý n. .. and lPeterson.11. ', . I!tit :\ttfi!iuu( : a!f ne::trao :: umtar . , , tn i; 'he ,' .osphere.(,n. I. P'hxis 4, :'100 ,.
15. (-,rrnichiel, If. Iierrovitch, N.. -eliea, - N I " id' n (1ri9II, Cosmip -r:iv lzilitude s 'r-ev' in Norff? :\rerir:v in, s;ilnrner, I j9 Can..* Phvs. 4.:2037.
.1(. Carmichael, t.. S;he; M, N.A., and I'eterson, .. V N,. f IIl I'I. ("osr ic-raz,"lititude survey in W\estern I ý A nd Ifa:.w:,ii in suninier, 19i66, (an. I.Phvs. 47:2057.
,17. Coetzee. W, V. , Konig, P. 11. , r . . , <toker. 11. I. . v:n der WValt.AX. .1. , and Vrekani. N. .1. (10701 (Cutoff rigidities and lat itude dependenceof muons at 307 gIcm2 in inclined directions. A.ta Phvsica. AcadentiaeScientiaru~i " Iungaricae 29. 1 Proc. I 1th International Con!. on<osmic Rays. Budapest 196
18. Sporre. B. , and P'omerantz. WA. (1970' Time dependence of the positionof the cosmic ray equator, Acta Phvsica Acaderniae Scientiarum Tlungar-icae 29. Suppl. 2 (Proc. 11Ith international Con!, on cosmic Rays, BudapestT' 9-, ):.545.
19. van der Walt, A. 1. , Stoker, P.!t. , ?!iiree, .. P. , and Coetzee, W. F-'. (1970)Latitude distributions of cosmic ray components at sea level and at air-plane al~titudes in the South Atlaintic magnetic anomnaiv, Acta P-hvsica.Academiae Scientiarum itungaricae _% Suppl. (Proc. 11th InternationalConf. on Cosmic FRays, . udaqpest 191591553.-
20. Kent, D. V. , and Pomerantz. 'I.A. 119711 Cosmic ray intensity variationsin the lower atmosphere, J.Geophvs. Res. 76:1652.
21. Shea, '1.A., Smart, D. 1.F, 'McCall, 1. R., and Gumm. B.S. (19741 Tablesof vertical cutoff rigidities for Epochs 1955 and 1960, AFCRL FEnviron-mental Research Papers No.493, AFCRl.-TR-74-C550.
22. Shea, M. A. . Srr.mart, 1, V.. McCall, T. H., and Gumm, B. S. (1974) Tablesof non-vertical cutoff rigidities for 76 various locations, AFCRL Environ-mental Riesearch Papers No.497, AICRL-TR-75-0008.
23. Shea, M. A.. Smart. D. F., 'McCall, .1. R., and GCmm, B. S. (1975) Tablesof Asymptotic Directions and Vertical Cutoff Rigidities for a Five Degreeby Fifteen Degree World Grid Using the Finch and Leaton GeomagneticField Model. AFCRL Environmental Research Papers No. 498, AFCRI-TR-75-0042.
"24. Shea, NI. A., and Smart, D. F. (1975) Asymptotic directions and vertical cut-off rigidities for selected cosmic-ray stations as calculated using theFinch and Leaton geomagnetic field model, AFCRL Environmental Re-search Papers No.502, AFCRL-TH-75-0177.
25. Finch. i. P. , and Leaton, B. R. (1957 ) The earth s main magnetic field-epoch 1955.0 . Monthly Notices Roy. Astron. Soc.. Geophys. Suppl. 7:314.
26. lensen, D.C., and Cain, .. C. (1962) An interim geomagnetic field, (abstract),.1. Geophvs. Ries. 67:3568.
%1
References
27. fAGA Commission 2, Workrng Group 4 t 1C69) International GeomagneticziReference F-ield 1965.0, J.Geophys. Res. 74:4407-4408.
128, NlAimfors, Karl Gustav 11945) D)etermnination of orbits in the field of a mag-netic dipole w.th apphcations to the theory of the diurnal variation of cos-mic ridiation. Arkiv for 1:±tematik AFtronomie och Evsik 32A, No. 8.
2:. ltrunhierg. tKr'9st-Ake E1953) lxperimental determination of electron orbitsin the field of a magnetic dipole, VPart 1. Tellus V:135.
30. Brunherg, V. -A. (1956) Cosmic ravy in the terrestrial magnetic dipole field,'I ellus VIII, -2.
.II. pA runberg. Erist-Ake and )attner, Adam (1953) Experimental determinationof electron orbits in th,, field of ai magnetic dipole. Part II, Tellus V:269.
32. lern.itre, G., and " allart:,, M. .19-5!j) On the allowed cone of cosmicradiation, Phvs. Rev. 50:493.
33. 'VAlart:i, M. S. (l19i81 An outline of the Lheorv of the allowed cone of cosmicr-dition. TUniv. of Toronto Studies. . lath. Series, No. 3 .
34, Jory, F.S. (1956) Selected cosmic-ray orbits in the earth's magnetic field.P~hvs. Rev. 103: 106',-.
35. ILust, R. 11957) Impact zones !or solar cosm.ic-ray particles, Phys. Rev. 105:1827.
U6 "napian. " a, nd Bartels, .. (1951' Geoimagnetism Vol. 2. Oxford Univ.Press, London.
37. Gill, S. (195l) A process fur the step-by-step integration of differentialequations in an s--itomrlitic dtgital cimputing machine, Proc. CambridgePhil. ioc. 47 : P6.
38. McCracken, K. G., Rao, U. R.. and Shea. M.A. (1962) The trajectories ofcosmic rays in a high degree simulation of the geomagnetic field, 'Al. I. T.Tech. Rept No. 7?, NYO-2670,
39. Shea, M. A.. Smart, D. F.., and McN- racken, K.G. (1965) A study of verti-cally incident cosmic-raŽ- trajector es i.Sing sixth-degree simulations ofthe geomagnetic field. AFCRL Environmental Research Papers No. 141,AECRL-65-705.
40. Shea, M. A,. Smart. D. F. , McCracken, K. G., . and Rao, U. R. (1968) Sup-
plement to IQSY tnstructicn Manual No. 10. cosmic ray tables - asymp-totic directions, variational coefficients and cutoff rigidities, AFCRLSpec. Reports No.'1. AFCRL-63-0130.
41. Gall. Ruth, Jimenez, J., and Orozco. A. (1969) Directions of approach ofcosmic rays fo:' high 'atitude stations, J. Geophys. Res. 74:3529.
42. Dorman, L. 1., (.iUsl:china. F. T., Shea. M. A., and Smart. D. F, (1972)Cosmic Rays Effe ýtive Cutoff. Rigidities, Publishing House "Nauka",
43. Shea, M. A., and Smart, D. F. 11974) Tables of asymptotic directions, cut-off rigidities, and reentrant albedo calculati-ns for Palestine, Dallas.and Midland. Texas, AFCRL Special Reportc No. 175. AFCHL-TR-74-0159.
44. Gall. R., Jimenez. J. , and Camacho, L. (1968) Arrival of iow-energy cos-mic rays via the magneto.spheric tail. J. Geophys. Res. 73:1593.
19
.----- --------. -"-- ------- t----....... ----- - - - ,- .._•... .. .- "- -- • . .. """. .
Referances
i,. Srmal., D. F. Shea, M.A,, and Gall, He'th ('969) The daily variation of"I tra•i-torv-derived hgh-lathtude cutoff rigidities in a mooel magneto-
sphee,. ,(eaophvs. ies. 74:4731.
4,i. Sm; rt. 1). '.. ,und .hea, M.A. (I'1 92' Dailv v:.riation of electron and protonj•eomagneti- cutoffs calculated for F.ort Churchill. (C'naida..1. (eoph.
47. Shea. M .A.. : Smart, ). D. ( 1971,0 Calcuilation of the -nagnitude of thedaly v,,riatxon of vertical cutoff rigidities and associated changes in thene'utron monitor response for selected North Amerinan neutron monitors,12th International Conf. on Cosmic H .vs, Hobart, Conference Papersft-iiverstT Tasmania) 3:5•.•
20
Appendix A
Table of Asymptotic Directions for the World Grid as CalculatedUtiizuig the International Geomagnetic Reference Field for Epoch 1975.0
"Table Al lists the asvmiptotic directions for world Lrid locations with grid
points each 5 deerees in latitude and 15 degrees in longitude. The listings are
arranged by latitude with six longitudes to a page, starting with the northern
polar regions and decreasing in latitude to the southern polar regions.
The geographic coordinates are given across the top of each page with the
appropriate rigidity and asymptotic directions given underneath the cocrdinates.
Eacl listing continues with decreasing rigidity values until the first forbidden
orbit is reached, Theso forbidden orbits are desiirnated byv either an "R" or an
"F", representing reeantrant or failed-to-reach-a-solution respectively. The
rigidities are in GV, and the asymptotic directions given in geocentric coordi-
nates to the nearest degree with positive latitude in the northern hemisphere,
negative latitude in the southern hemisphere, and longitudes measured east from
the Greenwich meridian. Negative asymptotic longitudes indicate that the grid
point is in the Western Hemisphere and the cosmic-ray trajectory did not cross
the Greenwich meridian during its orbit from the interplanetary medium to its
detection point. Asymptotic longitudes greater than 360' are indicative of the
number of times the cosmic-ray particle crossed the Greenwich meridian during
its trajectory as it circumscribed the earth.
21
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Appendix B
Table of /ertical Cutoff Rigidities for a World Grid as CalculatedUtilizing the International Geomagnetic Reference Field for Epoch 1975.0
Tne table in this appendix summrarizes the vertical cutoff rigidity values for
a world grid with locations 5 degrees in latitude and 15 degrees in longitude.
Each of these values has been de.termined by the trajectory-tracing technique
utilizing the International Geomagnetic Reference Field 27 with time derivatives
applied so that the coefficients of the model are appropriate for a 1975.0 Epoch.
The format of this table is:
Geographic Location: The geographic coordinates of each gridpoint are given with the latitude as posi-
tive for the northern hemisphere andnegative for the southern hemisphere,and the longitude in degrees East ofGreenwich.
L value: The L value, in Earth radii, calculatedusing the International GeomagneticReference Field 27 (Epoch 1975.0).
P (1%): The main cone cutoff rigidity (in GV)for this location.
P (S): The Stbrmer cutoff rigidity (in GV) forthis location.
Penumbral width: The difference between the main conecutoff rigidity and the Storrner cutoffrigidity (in GV).
PC: The effective vertical cutoff vi.gidity(in GV) for this location.
155
"P�.ACDIDG PAGE BLAIENO 7• F 4WD
T e iLfI 31
5u!q.q"Iq 07671E6TICAL CUTOFF RI;IDIOIES FOD OE otRL0 ;RID
AS ICALCU.A0E j IG TONE OGRE (EPOCh 1975.4) GEDRAG.NE1IC FIELD RODEL
GEDGR6PNIC L 0 s Pt MUMOM1AL
IMNHTflC6TIo LA,. 1.00.. vALuE PLP P (Sb MIDTH PC
WOI•s 0 G1Ro 50.38 05,00 2.2 00. 57 a.04 0.00 0.z
WORLD GR00 56.00 15.00 160-0 01. k 0.4 08.00 3.36
WORLDO 300 00.00 100.00 15 -5 -" 0.3. a .04 0..on .04
WORLD G;R1 $50.0 6.S.0 0 " 1.141 3.09 . Q 0.00 00
WORLD GID 50,0 ) 35.00 13.0? 0,. .
WORLD GRID 5,00] 75.0,0 02.-100 o0,00., .
WORLD GRO ID5 .00 9I Q1 a 0.10 0.0 o 0.o
wOlI- GRID 60 1.01 0.11 0. a- 0.11
wof0 G 1400 00.00 1 .00 i .66 ? 0.11 0.10 0.8o 0.11
W ew RD 10412 10,3 1.05.0 023 01 0.01 0. 00.1
WOBe G.ID q50.0 1S.00 .51. 0.11 1,t0 0.80 0.10
WORLD GID 0.5 1I•5.0 0.0 3. .5 3.00 0.06
WO RO GRID sc oS 105.00 7,39 0.4 0.30 0, 0.86
WORLO GRIo Sila 1 q5.00 .'. 03 0.- 4 0 .0 0.00 0, s6
wORLD CR1 5•0.10 -70. 00 .4.,07
W 0R.D GR 160g.010 625. 00 07 20. an0
WORLD GRID 0.I•10 240.00 b6 . .6..00. 1 0
.041.0O GR I0 00.00 255.090 6a .351l 0., 00
WORLD GRIDO 50.0n . )9.0 1166.2 8D0.o0
,OILD GRID 5,GA. 105.00 2 Ž71.02 0.00WORLDGR TO0 . 00
WORL 000 .3 0.0. no2.175
WORLD GI5.1 3,0 5 1 .mG .6 10 0.00
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wORLD SoD. a0 1 0 9.6 0.1s 0.Do 7.wORLD ?RID ?i.30 0.0 aO.2t5 0.0 0.O3 0.80 20.WORLD GRID 1.)15.00 1.092 0.1 0.14 0.00 0.0'
sORLO GRID '1,,5O 30.00 9.0?'I 0.18 .10~k 8.80 0.10
wORLDOHIO 71.00 b3.5a, 58.97 0.05 0.00 0.0 a• 0.0
wORLo GRID 76,1 1. 0.175 1, ?5 1.72 3 00 0,20
WORLD GRI• •75.00 75.00 O,0. 0.25 0. 0.20
WORLD GRID 700105.00 7.502 0.. 0, 0 9..
1,5.'
wO7t. GRID 75.3z 120.00 7.53Z0 0.77 0.22 8.6o 0.26
WORLD GRID 75.00 1350 1.090 0.207 0.27 0.0o 0.78WORLDý GRID0 75.0 i50.0at 1.50 .02.6 6.82
WORLD GRI0D 75.6 165.00 7.•93 0.24 2a , 0.6 0 0.2
WORLD AR D10 70.00 156.0o 0.040 0.01 0.28 0.00 0.70
WORLD GRID ?.0 o I0.. 00,8.1 0.14 0.80 0.16ORILD rR 10 75 .906 l10.66n 13.1a 6.09 66a 8.60o 8.09
NORLD GRID 15.00 075.80 10.966 0. 85 0.84 0.88 0.16
WORLD GRto 75.08 "•o.so 30. 0. 0.0 0
WORLD GRID 75.00 203?.0 51 603 1.86
WOLDGRD75 .00 ?'16.00 127.499 0*0
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WORLD 6600 70.06 315. 00 35.608 5.g
wORL D 6RID 75.00 000.00n 201.64 8.. 02 0.02 0. 00 0.07
WORLD GRID 75.00 065.0 00 5.571 0.0OF 0.87 6.00o 0.07
WORLD aIzo DD.uO 80.0 I.S'.6 0.26 0.20 0.89 0.o6
WORLD 6110 76.00 15.80o 6.626 2.04 o0'.3 0.00o 0.34.
WORLD GRID 78.s 30.90 6.07 06 06 00 04
wOMILD GRID 765:0 415.00 5.061 oa 00 .406
v0MLO GRID 76.30 60.08 0.541 0.9 4 0..19 8.1' 0.1.9
WORLD GR 1 75.00 10.00 5 .6016 0.06b 0.60 0.0 0.51
WOmmý R ID Or 7.00 90.68 5.293 A.9'I 6.52 8.68e 0. F?
401. 0*10 GRID.08 165.88 0.170 0. 55 6.55 6.00o 0.50
WORLD Gsam 70.5 no L . 00 5.860 .0 0. 6 5tI. 609 0.00
WORLD Ga10 7 0.08 13S..18 4.555 0. 60 0.60 0.0 0o .60
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WORLD GRID ?a6.08 1tO.5q 0.21 6.56 6.56 0.60o C.55
WORLD GRID 76.06 066.68 5,053 0.4.? 0.47 0.06 0.47
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WORLD GRID 76.06 255.66 z5.009
%OILD GRID 76.00 079.40O 37.786 0.* 6
WORLD (AID 76.00 285.66 38.496620
WORLD GRID ?6.66 30665 0. 071ib6T
WORLD GRID 76.6 31.0 7.4 68 ' 01 10
WORLD OtlOD 76.06 303.66 12.654.1 .1 8.081
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WORLD aGRI 65.30 15.00 4.597 0.76 0.71 0,07 0.772
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wORLD GRID 65.20 120. 00 3.E03 1.26 6.99 0.27 1.12
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wORL GR 10 65.)0 1600-0 3.156 1.07 0,511 10.6 0.95
WORLD GRID 65.10 055.00 ...%.6G 0.76 0.73 1.03 0.76,
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v ORL0 rID 65.30 330.60 7.655 0.70 0.26 6.00 0.26
btlOWL GRID 65.05 065.90 S..174. 0.42 0.62 0.00 0.62
W0O00m D (A10 60.00 0.80 3.f90 1.22 1 .08 01.l 1.14
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WORLD GR I 60.10 30.30 3.220 1.6. 0.16 3.26 1.46
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WORLD GRI0 60.33 90.00 Z.560 1.09 1.60 0.29 1.73
WORLD GRID0 60.20 105.60 2.062 1.-46 i.?7 0.26 1.82
WORLD G.010 63.30 17Z.00 2.816 2.37 1.01 0.2Z6 1.9s
WORLD GR ID 60.10 135.00 2.75 2.71 1.51 0.00 2.35
iOta GRID 60.'5 156.60 2.731 .76P 1.86 0.42 2.05
WORLD GRI0 60.)o 16b500 2.790 2.15 1.7Z 0.1"3 1.*"
WORLD CA00 63.36 156.68 7.967 1.5y7 1.63 0.Z 1.75
W ORLD0 rID 60.13 15.00 3.296 1.5? 0.27 0.25 1.60WORLD GRI•D 63.00 z13, go 3.866 1. to 0.09 6.29 1L.
WORLD GiD0 60.10 ?ZS. 00 6.750 0.6 b .o60 0.1? 0.65
WORLD GRID 60.1• Z29.00 6.163 6.%3 0.39 0.06 0.40
WORLD 0GR ID 60.00 55.00 6.01 3. ?z O.2Z 0.00 1.27WORLD GRID 60.06 7t7.06 9.979 0.17 6.15 9.02 0.16
WORLD GoID 60.30 205.00 16.158 0.01 0.16 0.00 0.14WOlRLD GRID 60.5 938.00. 6.6.37 0.Z1 i.21 0.00 0.z1WORLD GRID 60.00 315.10 6.669 0.60 3.37 0.03 0.38
WORLD GRID 60.30 330.00 5.060 09.65 .56 8.65 0.59Wo* O CRLO GRID 66.10 36S.00 6.277 0.94 .6.0 0.12 0.06
WORLD GRID 0 55.03 0. Be Z.620 2.11 1.08 0.31 19,.
WORLD GR1ID 55 .10 15.00 Z.643 7.96 1.99 0.37 2.20WORLD Goto 55.30 30.61 2.533 2.6. 2.36 8.30 7.47
WORLD 910 S 55.11 6,5.09 2..65 2.3 2.30 U.50 2.61
WO RLD GRID 55.50 61.00 2.613 2.t0 Z.37 0.63 2.b4
WORLD GRID 55.06 75.00 Z.367 3.06 2.16 0.88 2.70WORLD GRID 55.90 91.01 2.3Z5 3.16 2.36 3.60 2.85
WOR.D GRID 55.10 105.00 2.786 3.29 2.65 0.6'. Z.92
WORL D GRID 56.00 178.80 2.230 3.63 2.76 6.67 3.12
iOtLa G1110 55.10 135.00 2.093 3.55 7.91 O.b4 3.31WORLD GRID 55.61 153.03 2.176 3.5) 3.03 0.54 3.35
WORLD GRID 55.20 165.00 2,223 3.65 2.77 0.66 3.15
W04L.O GR1ID $i.0 166.00 2.010 3.05 2.70 0.45 2.66
WORLO Jk100 55.30 195.30 Z.579 2.69 1.16 0.53 '.22
WORLD GRID 55.09 213.00 2.0s5 0. 1.7 0.6 0.26 1.75
WORLD 0*00 55.08 2253.3 3.505 1.31 10.1 0.10 1.23WORLD GRID 55.00 263.06 4.33? 0. 56 0.76 0.39 0.70
WORLO GRID 55.00 255.11 5.624 0.5o 0.56 8.00 09.5
WORLD GRID 54.00 Z7178. 6..36 0.36 0.36 6.00 0.36
WORLD GRID 55.00 Z75.40 6.605 0.30 0.95 0.03 0.14
WORLD GRIO 55.050 36.01 5..696, 5.0 0.46 0.$0 a.4,
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wOR GR 4D10 55.00 365.00 3.16h? 1.7 1.60 0.31 1.55
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WORLD GRI q 35.30o 30.100 1.Z93 10.u,? 8.31 2. 36 10.10o
wORL "RI0 35.0 .500 ,29S 11,. 8.S0 2.51 L0.53
WORLD GrID 35.00 60.00 1.270 11.49 9.30 2. 11 11.15WORLD GRI•D 5.30 75.00 1.23 12.00 9.S5 2.45 11. 66
w ORLD, Gq ID 5 .30 90. 00 1 .204 12. 34 9.41 2.43 11.5?WORLD GRI1D .00 LOS. Go 1.06 12.4- 10.12 Z.37 11.71
WORLD GRID !,.10 120.05 1.211 12.88 10.49 •.09 11.93
WORLD GRAIT r v.10 135.00 1 .216 12. .9 10.1S 1 .5`6 12e.3.WOOLD GRID 35.10 180.00 1.229 1z.I2 10.56 1.1S6 11.55
WOOLD GRI: 3,.'a 165.0 I .Z64 11.44 9.51 1.93 10.60
WORLD GRID 35i.00 1o8.0 1.32'. 10.57 7.33 2.6CA4 9WORLD &10 35.00 195.300 1.403 4.45 7.72 1.73 3.97
WORLD GRID 35.30 210.? 0 1.34 7.90 6.40 1.50 7.65
WORLD GRI 35.00 2S.08 1.602 6.26 5.-1 0.65 6.12
WORLD GRIo 35.03 21.0.00 1.1740 5.35 6. .80 0.55 5.21
WRL GRID 35.830 ?SS.00 1.921 6. 3 I.e 3 0.5-1 ..2S
WORLD GR In 5.00 278.00 2.123 3..4. 2.27 3.67 3.19eofDD G40 35.33 385.13 2.23. 3. 29 2-0o 0.6q 2.89WORLD GI 5s.05 100.03 2.122 4.01 3.2. 0.77 3.58
WOqLD GRID 15.00 515.00 1.851 5.76 6.49 1.75 4.98
WORLD GRID 15s.5 338.88o 1I583 7.37 5.05 1.22 6.80
.DRLD. GRID 55.030 3.5.05 1.'137 9.95 7.53 0.55 J.10
wORL 0 G(TO 30.00 0.00 1.186 11. ? 110. 7 1..0 11.30
WORLD GRoID 3000 18.09 1.166 12.30 10.64 I.*9 11.71
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WORLD GRI0 30.50 68.80 1.146 13.56 12.05 1.51 13.34
wORLD Gýr1 30.10 75.80 1.113 16.97 14.a7 0.03 14.87WORLD GR IO 30.00 90.03 1.993 16.57 146.3' 8.80 go 1.3
WORLD GRID n.O 1 9is.0g 1.396 14..'0 16.40 0.80 1 -. 40
WORLD GRID 13.10 10z.0o 1.185 14.26b 16.2 0.00 1'..2b
WORLD GRID 30.00 135.00 1.115 13.95 13.05 0.00 13.95WORLD GAID 10.10 150.00 1.131 03.44 13..• . 0.00 13.*fwORLDL GRID 3.90 &6.800 1.163 1Z.76 11.83 0.03 12.7?WOL•lO GD0 38.08 108.88 1.215 11. #9 18.70 1.29 1.65WORLO GRID 33 .0 195.03 1.270 11.1 9 9.42 1.77 10.48
WOIL0 GR OG 30.3 210.08 1.34. 10.32 7.38 z.94 9.63
wORLD GRAID 10.10 225.88 1.416 9.38 7.35 2.03 $.78ORLD0 GRD0 38.30 2.0.10 1.508 7.08 6.19 0,79 .88
WORLD GRID 1.03 .o 25.0 1.631 S.78 S.35 0.43 5.60WORLD GRID 3.130 270.00 1.781 6.133 .14 0.76 4.66
WORLD GRIO 30.33 205.03 1o.8s 4.76 3.61 1.15 .. a?
1WORLD ;RID 30 368.00 1.0z 5.?5 4.z0 1.05 O.-6
WORLD GRID 18.20 315.88 1.618 7.06 5.75 1.t1 6. 8WORLD GRIO 30.30 338.38 1.378 10.06 0.68 1.38 1.67
WORLD GRID ,30.0 365.03 1.243 11. 2 9.60 1.6Z 10.46
1ORLLD GRID 25.38 8.88 1.092 13.21 12.29 0.92 11.18WORL 0 GRI o 25.10 15i.8 1.073 13.60 12.69 0.99 13.66WORLD GRID 25.50 30.08 1.067 14.13 16.10 8.80 1*.18WORLD GRID 7S5.30 k6.30 1.070 16.53 16.53 8.00 14.53WORLD GRID 2S.30 68.88 1.09s6 15. O0 iS.ib 0. .o 15.80WORLD GRID 25.30 75.30 1.828 15.5 15.58 0.00 15.58WORLD GRID 25 .8 90.00 1.011 15.85 15.05 0.00 15.85
ORLD GRID0 Z5.28 105.02 1.016 15.79 15.7q 0.30 15.15WORLD RIO 25.30 1028.0 1.028 15. 69 15.-9 8.0 11,49WORLD GRID 25.28 135.80 1m041 15.03 15.03 0.00 15.83
WORLD GRID 25.10 15.0.0 1.059 146.6 14.66.• t u. 14.66WORLD GRID 25.30 165.80 1.888 13.6? 11.76 8.00 13.76WORL0D GR I 25.10 188.08 1.132 13.07 13.91 0.go 13.87mOnLD GRID 2i.00 195.88 1.182 12.-3 I12.63 0.00 12.643WORLO GRI0 25.08 2108.6 1.229 11.82 11.76 0a.8 11.61W ORLD GRID 25.30 225.88 1.278 11.07 10.10 0.97 10.94
WORLD GRI 25.00 246.80 1.368 9.78 8.66 1.12 .76.WORLD0 GRIDIn 29.38 ZS.00 1.6zb 8.15 7.47 0.28 7.119WORLD GR1I Z5.08 Z278 .8 1.53A b6.4z 5.66 0.96 6.80WORLD GR1D ZS.38 Z85.o 0.0e36 S.92 6.6z 1.30 5.46WORLD GRID0 25.0 $88.88 1.609 7.06 5.Z 1.80 A.56Wo RLD GRI 2405.10 315.80 1 .636 10a. 13 7.87 2.20 .6WORLD GRqD 2.80 338.88 0.238 11.71 10.5'% 1,1? 11.47WORftLD GRID1 25.08 365.08 1*13 12'.61 11.70 8.91 12.66
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WORLD GRID 30. 3 90 225.0.?5 12.37 i.12 5 .07 12.33"WORLD GRID0 -1.0 2~00 1.1 1-? 1.-?.0 1WOtO GCIA 39.30 25.00 1.7?7 12.12 9.21 0.91 ".69
WORLD 100 20.3 270.00 1.35 A.11 7. E a.076wOW3L0 RID -0.30 205.00 0..54 9.34 5.71 2.53 ).9G.
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wR 1.-3 15.33 60.00 .a55 .9 16.1, It 0g.01 15.76MORL 10117 1,0 15.20 .501 17.C5 17.1 -2.00
wORL G0R9 1T.20 90.10 .036 17.4 iT..? 0.00 17.II
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