Alpha and Betha angles can be approximated by the trigonometric functions of time t :
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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
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.
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:
Application of these coefficients to corresponding
experimental data yield following results:
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Time
Bx By Bz
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BxGSE ByGSE BzGSE -666.1537 217.2564 -274.3152
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0.0151 -0.9736 -0.2279
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