Calculation attitude parameters:

1. Theory.

Axes position of the GSE system  relative to the s/c coordinate system is determinated by 3 angles Alpha, Betha and GammaAlpha and Betha determinate coordinates of a new XGSE axis, while Gamma determinates YGSE or ZGSE.

Designating constructive axes of s/c by X, Y and Z with X axis going along nominal spin axis let us introduce Alpha and Beta angles as  the angles between X axis and Sun direction projection onto XY and XZ planes respectively.

Alpha and Betha angles can be approximated by the trigonometric  functions of time t :

Alpha = A1 + A2sinW1t + A3cosW1t + A4sinW2t + A5cosW2t,
Betha  = B1 + B2sinW2t + B3cosW1t + B4sinW2t + B5cosW2t,

where W1 is a mean spin rate of s/c, W2 is a mean angular velocity of the angular momentum projection on YZ plane.

As a third attitude parameter Gamma angle has been taken which is the angle between Y axis of s/c and projection of the North Pole of ecliptic direction on YZ plane.   This angle is approximated by the linear function of time

Gamma = c1 + c2*t

Let us calculate the coordinates new X,Y and Z axes in GSE system:

X' = [XXG YXG ZXG]
Z' = [XZG YZG ZZG]

X GSE coordinates are :

XXG = 1/SQRT(1 + tg^2(Alpha) + tg^2(Betha));
YXG = tg(Betha)/SQRT(1 + tg^2(Alpha) + tg^2(Betha));
ZXG = tg(Alpha)/SQRT(1 + tg^2(Alpha) + tg^2(Betha));

Z GSE coordinates are :

XZG = -A/SQRT(A^2 + XXG^2);
YZG = cos(Gamma) * SQRT(1-XXG^2);
ZZG = sin(Gamma) * SQRT(1-XXG^2);

where A = YXG * cos(Gamma) + ZXG * sin(Gamma);

Y GSE coordinates may be calculated as as vector product of vectors X and Z:

Y' = [XYG YYG XYG] = X' o Y'

Matrix M can be contructed from X' Y' Z' columns: M = [X' Y' Z']. Any vector in s/c coordinate system can be transformed to GSE system :
____       ___
Vgse = M * Vsc

2.Example.

String in the attitude file look like this:

1 1998 3 7 33.377  .890  .205  .232 -1.719  .105  .061  .298  1.716  .253 .063 -.110  52.5669 39.0572 -2.6696 -52.5669

It means that for 1998 March, 7, for time of 33.377 thousand seconds from 0.00 hour, for interval length of 0.890 thousand seconds, the attitude coefficients are:

A(5) = 0.205  0.232  -1.719  0.105   0.061
B(5) = 0.298  1.716  0.253   0.063  -0.110
W1 = 52.5669     W2 = 39.0572
c1 = -2.6696     c2 = -52.5669

Application of these coefficients to corresponding experimental data yield following results:

Data table:

 Input vector Time            Bx   By   Bz     980307 09 16 21 604 -665.00 233.00 264.00  980307 09 16 24 604 -666.00 275.00 218.00  980307 09 16 27 604 -668.00 310.00 168.00  980307 09 16 30 604 -669.00 336.00 113.00  980307 09 16 33 604 -669.00 355.00 56.00  980307 09 16 36 604 -670.00 366.00 -2.00  980307 09 16 39 604 -671.00 366.00 -59.00  980307 09 16 42 604 -672.00 359.00 -114.00  980307 09 16 45 604 -672.00 342.00 -166.00  980307 09 16 48 604 -672.00 317.00 -214.00  980307 09 16 51 604 -673.00 285.00 -255.00  980307 09 16 54 604 -673.00 245.00 -290.00  980307 09 16 57 604 -672.00 199.00 -318.00  980307 09 17 00 604 -672.00 149.00 -337.00  980307 09 17 03 604 -672.00 95.00 -348.00  980307 09 17 06 604 -671.00 39.00 -350.00  980307 09 17 09 604 -671.00 -19.00 -343.00  980307 09 17 12 604 -670.00 -75.00 -327.00  980307 09 17 15 604 -670.00 -129.00 -303.00  980307 09 17 18 604 -669.00 -180.00 -271.00  980307 09 17 21 604 -669.00 -226.00 -232.00  980307 09 17 24 604 -669.00 -267.00 -188.00  980307 09 17 27 604 -668.00 -300.00 -138.00  980307 09 17 30 604 -668.00 -327.00 -85.00  980307 09 17 33 604 -669.00 -345.00 -29.00  980307 09 17 36 604 -669.00 -355.00 28.00  980307 09 17 39 604 -669.00 -356.00 84.00  980307 09 17 42 604 -669.00 -348.00 139.00  980307 09 17 45 604 -669.00 -331.00 190.00  980307 09 17 48 604 -669.00 -306.00 237.00 Output vector BxGSE     ByGSE    BzGSE   -666.1537  217.2564 -274.3152  -667.1761  211.6487 -277.1026  -669.2543  208.6975 -281.2331  -670.4191  206.1493 -285.0732  -670.6675  206.2733 -290.4797  -671.9767  208.9620 -296.0442  -673.2971  212.8188 -298.4244  -674.6415  219.2784 -300.3945  -675.0048  225.2637 -299.5475  -675.3804  231.1883 -297.1232  -676.6790  237.3895 -291.4288  -676.9938  240.6227 -284.3134  -676.2987  241.6170 -276.6772  -676.5246  240.8757 -267.6702  -676.7437  237.2624 -259.6874  -675.9178  231.4441 -252.6475  -676.0935  222.7400 -248.0635  -675.1897  213.5963 -244.8460  -675.2586  204.0340 -244.4298  -674.3103  194.2222 -246.9522  -674.3244  185.4542 -251.7030  -674.3270  179.2297 -259.5260  -673.2791  174.1737 -267.6318  -673.2338  172.5618 -278.1308  -674.1508  173.0241 -288.1151  -674.0475  176.5205 -298.1115  -673.8899  182.9417 -306.1989  -673.7102  190.6397 -312.6614  -673.4671  199.8289 -315.8044  -673.1952  209.3942 -316.7464 Angles Alpha  Beta  Gamma  (rad)  0.0151 -0.9736 -0.2279  0.0196 -0.9972 -0.0722  0.0237 -0.9961 0.0853  0.0274 -0.9702 0.2407  0.0306 -0.9203 0.3901  0.0331 -0.8475 0.5298  0.0349 -0.7536 0.6564  0.0360 -0.6411 0.7666  0.0363 -0.5127 0.8578  0.0359 -0.3715 0.9277  0.0347 -0.2211 0.9746  0.0327 -0.0653 0.9973  0.0301 0.0922 0.9953  0.0269 0.2473 0.9686  0.0231 0.3964 0.9178  0.0188 0.5356 0.8443  0.0143 0.6615 0.7498  0.0095 0.7710 0.6367  0.0046 0.8614 0.5078  -0.0003 0.9305 0.3663  -0.0050 0.9764 0.2158  -0.0095 0.9982 0.0598  -0.0136 0.9951 -0.0977  -0.0173 0.9674 -0.2527  -0.0204 0.9156 -0.4015  -0.0229 0.8412 -0.5403  -0.0246 0.7458 -0.6657  -0.0257 0.6320 -0.7746  -0.0259 0.5024 -0.8642  -0.0254 0.3604 -0.9325