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 Gamma.  Alpha 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
 

 

 Irina Belova charlota@iki.rssi.ru
Natan Eismont