GEOPHYSICAL COORDINATE TRANSFORMATIONS CHRISTOPHER T. RUSSELL NInstitute of Geophysics and Planetary Physics, University of Calififornia, Los Angeles, CA 90095, U.S.A. (Received l2 January, 1971; in revised form 26 March, 1971) Absract. The use of a vector-matrix formalism to describe the transformation from one cartesiancoordinate system to anothcr results in simple-to-use and easy-to-understand relationships.Furthermore, the required matrix transformations may be derived directly. The commoncoordinate systems in use in solar-terrestrial relationships are described and then thetransformation matrices required to convert vectors in one system to another are derived. Introduction Many differcnt coordinate systems are used in experimental and theoretical work onsolar-terrestrial relationships. These coordinate systems are used to display satellite trajectories,boundary locations, and vector field measurements. The need for more than one coordinatesystem arises from the fact that often various physical processes are more understood,experimental data more ordered, or calculations more easily preformed in one or another of thevarious systems. Frequently, it is necessary to transform from one to another of these systems.It is possible to derive 1he transformation from one coordinate system to another in terms oftrigonometric relations between angles measured in each system by means of the formulas ofspherical trigonometry (Smart, 1944). However, the use of this technique can be very tricky and can result in rather complex relationships. However, this method is at times used. A recentexample of the use of this technique to transform from geographic to geomagnetic coordinatescan be found in Mead (1970). Another technique is to find the required Euler rotation angles and construct the associatedrotation matrices. Then these rotation matrices can be multiplied to give a single transformationmatrix (Goldstein, 1950). The vector-matrix formalism is attractive not only because it permitsa shorthand representation of the transformation, but also because it permits multipletransformations to be preformed readily by matrix multiplication and the inverse trsnsformationto be derived readily. The matrices required for coordinate transformations need not be derived from Euler rotationangles, however, It is the purpose of this note to explain how to derive these coordinate transformations without deriving the rcquired Euler rotation angles as well as to describe the mostcommon coordinate systems in use in the field of solar terrestrial relationships. Discussions of the transformations for some of the coordinate systems to be treated in this reportmay aIso be found in papers by Olson (1970), and by the Magnetic and Electric Fields FieldsBranch (1970) of the Goddard Space Flight Center. The former paper differs from the present(0*0*0*work primarily in notation and the number of systems treated. Anothcr difference is that the Earth's orbit is considlered to be TABLE I Names and abbreviations given to the coordinate systems common to this paper, to the reportof Olson (1970), and to the report of the Magnetic Electric Field Branch (1970). _____________________________________________________________________________ Present Work Olson (1970) MEFB (1970) ______________________________________________________________________________ Coordinate System Abbr. Coordinate System-Abbr. Coordinate SystemxxAAbbr. _______________________________________________________________________________ ` hp x (#%'0*,.8135@8: O_ K_ w s o k!g!c0!_2![;!WE!SR!OV!KV!`!wo!s!o!k!g!c"_<"[>"Wv"S"O"K""w"s#o#k<#g@#cI#_S#[U#W#S#O%K%%w%sb&oe&k&g&c&_'['W(S;(O=(K=())w,)sP)o\)k`)gg)cx)_|)[7*W<*Sa*Oc*Kc**w*s*o*k*g*c*_*[*W5,S9,OH,KH,%-w*-s.o.k_/gd/ck/_}/[~/W/S0O0K00w1s1o 1k#1g'1cB1_G1[_1Wd1Sv1O1K11w1s1o1k1g1c2_2[2W3S 4O4K44w4s4o4kA6gE6ce6_i6[p6W6S6O]7K]7e7w~8s8o8k8g8c8_9[9W9S9O9K`99w9s!:o%:k:g:c:_:[;W;S;O;K;;w;sw>so?os?k?gTAcZA_A[AW0CS7COzEKzE~EwEsEoEkEgEc-F_1F[FWFSGOHKHHwHsHoHkHgHcI_I[IWIS.IO0IK 0I;Iw?IsJoJj1Kf5KbVK^ZKZjKVnKRvLNyLJyL"Mw(MsMoMkMgMcM_N[UNWXNSNONKN~OwOsOoOkOgOcP_P[PWPSPOPKPPwPsPo3Qk\QguQcwQ_Q[QWQSQOQKQQwQsQoQkQgQclS_S[SWSSSOVKVVwVsVocWkeWgyWcW_W[WWWS=XO@XK@XRXw`XsoXosXkzXgXcX_Y[YWYSYOYK YYwYsYoYkYgYcY_[[8[W[S[O[K[[w\s\o^k^g1_c4___[_W`S`O`K``w7`s9`oN`kP`gS`c^`_c`[`W`S`O`K``wZas]aoaakagacc_c[bdWdSdOdK dew es;eo>ekegecg_g[yiWiSiOiKivmw~msmomkngncn_o[goWjoSoOoKopwpsqoqkqgqcq_q[qWrSrOrKrswsssosksgscs_s[sWtStOtKt'twtsto(uk*ugucu_u[uW4vS6vObvKbvfvwpvsvovkwgwcx_x[xWxSxOyKyyw:ysLyo_ykS{g^{c{_{[|W|S|}O}K} ~w~s~o2~k~g5cr_[WSOKwsGoIkgc/_c[eWSOKw-/vxDFTV]f /kkmvx)+VRXRRRRRR(< ``RR#RIR-\$< ``(< ``ZZZZZZZZ4Z6ZwZyZZZ <``6Z8ZL9}999999 < @ <`` ZZRZTZkZmZZZZZ_ZaZnZpZ < @pZZZZZZ[Z]ZkZmZRZTZoZqZ < @qZZZ Z Z =!=!=2!=T!=< < @ T!V!^!^!^>"^"^"^#^#^>#^@#^U#^%^%^%^%^<%=(^+)^^)^`)^z)^|)^9*^;*^c*9\$<< c**V*V*V*V*V*V*V7,V9,VF,VH,V.Va/Vc/V\$yZyZyZzZizZzZ {AI{A^{A< < @ ^{}{b{b{=| !| <`\$<` !< !|W|Z|Z|Z}Z;}Z\}Z|}Z}Z}Z~Z ~~~~ < `~4~6~7`bprIe=/2'=/20!=/2'=/20!=/2'=/2P=/2'=/2=/2'=/20=/2'=/2=/2'=/2 =/2=/2'=/2=/2'=/2@=/2'=/2=/2 =/2p#=/20!=/2'=/2P"=/2'=/2 =/2'"'B !)4*".3/D0U0f1PEQ__x4z0{N{{"{3{D}U}f]^